Predicting sintering deformation of ceramic film constrained by rigid substrate using anisotropic constitutive law

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Acta Materialia 58 (2010) 5980–5988 www.elsevier.com/locate/actamat

Predicting sintering deformation of ceramic film constrained by rigid substrate using anisotropic constitutive law Fan Li a, Jingzhe Pan a,*, Olivier Guillon b, Alan Cocks c b

a Department of Engineering, University of Leicester, Leicester, LE1 7RH, UK Institute of Materials Science, Technische Universita¨t Darmstadt, D-64287 Darmstadt, Germany c Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK

Received 14 January 2010; received in revised form 14 May 2010; accepted 8 July 2010 Available online 9 August 2010

Abstract Sintering of ceramic films on a solid substrate is an important technology for fabricating a range of products, including solid oxide fuel cells, micro-electronic PZT films and protective coatings. There is clear evidence that the constrained sintering process is anisotropic in nature. This paper presents a study of the constrained sintering deformation using an anisotropic constitutive law. The state of the material is described using the sintering strains rather than the relative density. In the limiting case of free sintering, the constitutive law reduces to a conventional isotropic constitutive law. The anisotropic constitutive law is used to calculate sintering deformation of a constrained film bonded to a rigid substrate and the compressive stress required in a sinter-forging experiment to achieve zero lateral shrinkage. The results are compared with experimental data in the literature. It is shown that the anisotropic constitutive law can capture the behaviour of the materials observed in the sintering experiments. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Constrained sintering; Anisotropic constitutive law; Coatings

1. Introduction Constrained sintering of ceramic films has become a popular technology to fabricate various products such as solid oxide fuel cells, micro-electronic PZT films and protective coatings. A single layer, or several layers, of powder material is applied to a substrate using deposition techniques such as inkjet printing, tape casting or dip coating. The system is then exposed to elevated temperatures to cause it to consolidate. During heating the porous layers attempt to shrink, first due to drying and then due to sintering. The shrinkage is, however, constrained by the substrate and adjacent layers, giving rise to stresses which may lead to cracking [1]. Many researchers have carried out fundamental studies to understand the process of constrained sintering. Bordia et al. [2] compared the sintering *

Corresponding author. Tel.: +44 116 223 1092. E-mail address: [email protected] (J. Pan).

behaviour of alumina powder in constrained and free sintering. They observed significant differences in the densification behaviour between constrained and free sintering, which cannot be explained using an isotropic constitutive law [2]. Guillon et al. [3] carried out similar experiments over a range of temperatures and came to a similar conclusion. Lin and Jean [4] conducted constrained sintering experiments on silver circuit paste and observed similar densification curves as those obtained by Guillon et al. [3]. Lin and Jean also measured the in-plane stress and found that the stress increases rapidly at the beginning of sintering, then decreases to an intermediate level, after which it remains constant. Another finding of Lin and Jean [4] is that there is little difference in the grain-growth behaviour between constrained and free sintering, which was later confirmed by Guillon et al. [5]. However, microstructural anisotropy clearly develops with density, especially above 90% of relative density, as shown by the orientation of elongated pores [5] or the difference in pore separation

1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.07.015

F. Li et al. / Acta Materialia 58 (2010) 5980–5988

[6]. Direct measurement of the viscosities for an anisotropic constitutive law is difficult. Bordia et al. [2] suggested that three additional viscous parameters are required to describe a transverse isotropic body but only two have to be measured for constrained sintering compared to the isotropic situation. A number of attempts have been made to model the constrained sintering of thin films, which have been reviewed recently by Green et al. [7]. Micromechanical approaches [8] and the discrete element method [9] have been employed to model anisotropic sintering. In particular, Carroll and Rahaman [10] developed an anisotropic model for constrained sintering. Their model provides a reasonable fit to the early stage of the experiment data. However, the model is based on a simple cubic array of particles rather than a realistic random packing. Wakai and his co-workers [11,12] numerically calculated the anisotropic sintering stress and viscosity tensor for a range of particle packing patterns and different degrees of contact anisotropy, which provide a useful guide to the development of analytical constitutive laws. In isotropic constitutive laws for sintering deformation, the relative density and average grain size have been widely used as the state variables to describe a sintering material. If anisotropy exists, or develops, during the sintering process then the relative density is no longer sufficient to define the state of the material. Jagota et al. [13] used the orientation distribution of contacts, contact area and length of the link connecting particle centroids, and the coordination number and volume fraction of the particles to define the state of a powder compact in their anisotropic constitutive law for sintering and compaction. Cocks et al. [14] used a simpler analytical approach in the development of their anisotropic model, by replacing the relative density by the sintering strains. The model of Jagota et al. [13] can fit their experimental data fairly well, while the model developed by Cocks et al. [14] has not yet been validated by any experimental data. The purpose of the current paper is to show that the anisotropic constitutive law developed by Cocks et al. [14], with some modifications and extensions, is able to capture the anisotropic sintering behaviour observed by Bordia et al. [2] and Guillon et al. [3]. Firstly, the constitutive model due to Cocks et al. [14] is briefly introduced. Then modifications and extensions to this model are proposed. Finally predictions of the modified model are compared with the experimental data obtained by Bordia et al. [2] and Guillon et al. [3]. 2. An anisotropic constitutive law for sintering deformation 2.1. The original constitutive law due to Cocks et al. [14]

Fig. 1. Micromechanical model of a powder compact.

sintering” in the current paper. Sintering beyond this stage is referred to as the “later stage” sintering. Solid state diffusion in the particle contacts (also referred to as inter-particle boundaries or grain-boundaries) is taken as the dominant mechanism for matter transportation. As matter is removed from the inter-particle contacts, the particles approach each other, resulting in the densification of the particle system. The particles are assumed to be spherical. However the contact size between the particles is orientation-dependant, which is the origin of the anisotropy in this micromechanical model. Cocks et al. [14] assumed that the * contact and its orientation n are relative to the principal directions of the macroscopic strains. The macroscopic stresses Rij, sintering stress Rs , and strain rates E_ ij are related to each other through a strain rate potential X such that: Rij þ ðRs Þij ¼ where XðE_ ij Þ ¼

@X @ E_ ij

Z 3ZDR _ _ Eij Ekl Emn Eop ni nj nk nl nm nn no np dS ~b 8pD S Z 3ZDc _ þ Eij ni nj dS 4pR3 S

ð1Þ

ð2Þ

in which Z is the number of contacts per particle, D the rel~ b the effective coeffiative density, R the particle radius, D cient for grain-boundary diffusion, i, j, k, l, m, n, o, p the direction indices (i, j, k, l, m, n, o, p = 1, 2, 3), ni the ith * direction component of the outward normal n of a contact as shown in Fig. 1, and c the specific surface energy. The integration is over the surface S of a representative particle. Cocks et al. [14] were concerned with the response at small strains and assumed that the particle coordination number, Z, is constant. In the extension of the model presented here we adopt the expression for Z suggested by Helle et al. [16]: Z ¼ 12D

Fig. 1 illustrates the micromechanical model of powder compacts that was firstly used by McMeeking and Kuhn [15] for isotropic sintering and then by Cocks et al. [14] for anisotropic sintering. The assumption in this model is that the contacts between particles are circular and do not interact with each other, which defines the “early stage

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ð3Þ

The viscosity tensor Cijkl and the sintering stress (Rs)ij can be obtained by comparing Eq. (2) with a standard expression for X for a linear material: 1 XðE_ ij Þ ¼ E_ ij C ijkl E_ kl þ ðRs Þij E_ ij 2

ð4Þ

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F. Li et al. / Acta Materialia 58 (2010) 5980–5988 Table 1 Viscosity tensors calculated from Eq. (2). 3

C ijkl ¼ 3ZDR ~ cijkl 4pD b

c1111 c2222 c3333 c1122 = c2211 c1133 = c3311 c2233 = c3322

1:396E211 þ 0:398E11 ðE22 þ E33 Þ þ 0:12ðE222 þ E233 Þ þ 0:08E22 E33 1:396E222 þ 0:398E22 ðE11 þ E33 Þ þ 0:12ðE211 þ E233 Þ þ 0:08E11 E33 1:396E233 þ 0:398E33 ðE11 þ E22 Þ þ 0:12ðE211 þ E222 Þ þ 0:08E11 E22 0:199ðE211 þ E222 Þ þ 0:24E11 E22 þ 0:04ð2E11 þ 2E22 þ E33 ÞE33 0:199ðE211 þ E233 Þ þ 0:24E11 E33 þ 0:04ð2E11 þ 2E33 þ E22 ÞE22 0:199ðE222 þ E233 Þ þ 0:24E22 E33 þ 0:04ð2E22 þ 2E33 þ E11 ÞE11

Table 1 provides a set of expressions for the viscosity tensor in terms of the principal strains Eij. Some calculation errors in Ref. [14] have been corrected. It is worthwhile to point out that the strains Eij and strain rates E_ ij when presenting the constitutive law refer to the nominal strains and nominal strain rates. Because sintering deformation is usually large, true strains and strain rates have to be used when performing the calculation (using the finite element method for example). The nominal strains and true strains can be converted between each other through   Enominal ¼ exp Etrue 1 ð5Þ ij ij while the corresponding strain rates can be converted through   E_ true E_ nominal ¼ exp Etrue ð6Þ ij ij ij where in Eq. (6), summation is not implied by repeating an index. Note that since the strain rates are described with respect to the principal axes, the shear components of the true and nominal strain rates are equal to each other. In the present paper we do not distinguish Enominal from Eij, ij and use Etrue where the true strains have to be used. ij 2.2. Modified anisotropic law for early stage sintering The constitutive law of Eq. (2) predicts a constant viscous Poisson’s ratio of m = 0.25 when being reduced to isotropic sintering (by equating E11 = E22 = E33). Casagranda et al. [17] and Ma [18] observed that the viscous Poisson’s ratio is close to zero in the early stage of sintering. Zuo et al. [19] on the other hand found that the viscous Poisson’s ratio increases with the relative density from 0.2 at D = 0.65 to 0.4 at D = 0.95, which correlates to the calculation by Riedel et al. [20]. Eq. (2) therefore does not capture the observed variation in the viscous Poisson’s ratio. The problem can be resolved by taking into account the energy dissipation rate due to the relative sliding between the particles [21]. McMeeking and Kuhn [15] suggested that this energy dissipation rate, ws, for a single grainboundary takes the following form: 1 * 2 ws ¼ gc px2 ðD v t Þ ð7Þ 2 in which gc represents a sliding viscosity of the grain* boundary, x is the radius of the grain-boundary and D v t is the sliding velocity as shown in Fig. 1. Raj and Ashby

[22] pointed out that grain-boundary sliding is a diffusion-controlled process with its rate depending on the grain-boundary roughness and diffusion coefficient. Kim et al. [23] proposed an explicit expression for the sliding viscosity gc ¼ g

R2 ~b D

ð8Þ

where g is a dimensionless parameter depending only on the roughness and thickness of the grain-boundary. The grain-boundary radius is given by [14] xðni Þ ¼ 2RðEij ni nj Þ1=2

ð9Þ

*

The sliding velocity D v t can be calculated as *

* D v t ¼ 2RE_ ij t i nj

ð10Þ

* ti

where is the projection of ni in the plane of the grainboundary. The total energy dissipation rate due to grainboundary sliding is then given by   Z D 1 Z s W ¼ ws dS 4pR3 =3 S 2 4pR2 Z ** 3g ZD ¼  c E_ ij E_ kl Emn t i t k nj nl nm nn dS ð11Þ 4pR S which is an additional term in XðE_ ij Þ of Eq. (2). Eq. (4) can be now rewritten as 1 XðE_ ij Þ ¼ E_ ij C ijkl E_ kl þ ðRs Þij E_ ij 2 1 ¼ E_ ij ðC nijkl þ C sijkl ÞE_ kl þ ðRs Þij E_ ij ð12Þ 2 in which the viscosity tensor Cijkl has been split into two parts: a normal part C nijkl and a shear part C sijkl . The expressions of C nijkl and C sijkl are provided in Table 2. It can be observed from the table that the sliding term does not contribute to the bulk viscosity (for free sintering) of a powder compact. This is because only the normal motion between particles results in the increase in the relative density. However the sliding term significantly affects the viscous Poisson’s ratio. When the anisotropic law is reduced to an isotropic one by setting E11 = E22 = E33 = E, where E represents the average strain, the corresponding viscous Poisson’s ratio can be calculated as m ¼ 0:25 

1:0425g jEj 3:352E2 þ 0:838g jEj

ð13Þ

F. Li et al. / Acta Materialia 58 (2010) 5980–5988

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Table 2 Viscosity tensors calculated from Eqs. (11) and (12). 3

3

3gc ZDR s s 3ZDR n n  s n  s C ijkl ¼ C nijkl þ C sijkl , C nijkl ¼ 3ZDR ~ cijkl ; C ijkl ¼  4p cijkl , C ijkl ¼ 4pD ~ ðcijkl  g cijkl Þ, cijkl ¼ cijkl  g cijkl 4pD b

cs1111 cs2222 cs3333 cs1122 cs1133 cs2233 cn1111 cn2222 cn3333 cn1122 cn1133 cn2233

¼ cs2211 ¼ cs3311 ¼ cs3322

¼ cn2211 ¼ cn3311 ¼ cn3322

b

0.718E11 + 0.479E22 + 0.479E33 0.479E11 + 0.718E22 + 0.479E33 0.479E11 + 0.479E22 + 0.718E33 (0.359E11 + 0.359E22 + 0.12E33) (0.359E11 + 0.12E22 + 0.359E33) (0.12E11 + 0.359E22 + 0.359E33) 1:396E211 þ 0:398E11 ðE22 þ E33 Þ þ 0:12ðE222 þ E233 Þ þ 0:08E22 E33 1:396E222 þ 0:398E22 ðE11 þ E33 Þ þ 0:12ðE211 þ E233 Þ þ 0:08E11 E33 1:396E233 þ 0:398E33 ðE11 þ E22 Þ þ 0:12ðE211 þ E222 Þ þ 0:08E11 E22 0:199ðE211 þ E222 Þ þ 0:24E11 E22 þ 0:04ð2E11 þ 2E22 þ E33 ÞE33 0:199ðE211 þ E233 Þ þ 0:24E11 E33 þ 0:04ð2E11 þ 2E33 þ E22 ÞE22 0:199ðE222 þ E233 Þ þ 0:24E22 E33 þ 0:04ð2E22 þ 2E33 þ E11 ÞE11

By tuning the grain-boundary viscosity constant, g , Eq. (13) can reproduce the evolution of viscous Poisson’s ratio observed in the various sintering experiments. 2.3. Empirical extension of the anisotropic law to later stage sintering

(c) An isotropic and fully dense solid has a viscous Poisson’s ratio of 0.5, i.e. we have h iiso C II ijkl i¼j–k¼l m¼h iiso h iiso II C ijkl þ C II ijkl i¼j¼k¼l

The constitutive law based on the model shown in Fig. 1 is valid until the inter-particle contacts start to interact with each other. If the relative density D exceeds a certain value, Ds, then sintering enters the later stage and a different constitutive law is required. Several isotropic constitutive laws have been developed for later stage sintering, some of which are based on micromechanical models (e.g. Refs. [24,25]) while others are based on empirical fitting to experimental data (e.g. Refs. [26,27]). For later stage sintering, it is difficult to establish a general and anisotropic micromechanical model similar to that shown in Fig. 1. Here an empirical approach is adopted following Hsueh et al. [26]. It is assumed that the anisotropic viscosity tensor for the later stage sintering, C II ijkl , can be scaled from the early stage one such that C II ijkl ¼ f ðDÞC ijkl

if ði ¼ j ¼ k ¼ lÞ

ð14Þ

C II ijkl ¼ gðDÞC ijkl

if ði ¼ j–k ¼ lÞ

ð15Þ

The later stage model must satisfy the following conditions: (a) It reduces to the early stage model at D = Ds, i.e. f ðDs Þ ¼ gðDs Þ ¼ 1

ð16Þ

i¼j–k¼l

 iso gð1ÞC ijkl i¼j–k¼l ¼ ¼ 0:5 iso  iso f ð1ÞC ijkl i¼j¼k¼l þ gð1ÞC ijkl i¼j–k¼l

ð18Þ

A possible choice of f(D) and g(D) satisfying Eqs. (16)–(18) is  1D f ðDÞ ¼ 1 þ C f exp  ð19Þ aðD  Ds Þ  1D ð20Þ gðDÞ ¼ 1 þ C g f ðDÞ exp  bðD  Ds Þ in which Cf ¼

2:01 2

ðlnðD0 Þ=3Þ  0:222g lnðD0 Þ

1 Cg ¼ 1 þ Cf

1

6:04 2

ðlnðD0 Þ=3Þ þ g lnðD0 Þ=3

ð21Þ ! ð22Þ

are two material constants depending on the sliding viscosity constant g and the initial relative density D0; a and b are two parameters depending on the “degree of anisotropy”, e, defined as " # 3ðjE11 E22 E33 jÞ1=3 e ¼ exp ð23Þ jE11 þ E22 þ E33 j such that

(b) An isotropic and fully dense solid deforms by Coble ~ b [25,26], i.e. creep with a shear viscosity of 2R3 =27D we have h iiso h iiso C II  C II ijkl ijkl 

i¼j¼k¼l

¼ f ð1ÞC ijkl

iso

~b ¼ 4R3 =27D

i¼j¼k¼l

i¼j–k¼l

 iso  gð1ÞC ijkl i¼j–k¼l ð17Þ

a ¼ a1 þ a2 e;

b ¼ b1 þ b2 e

ð24Þ

in which the numerical coefficients, ai and bi (i = 1, 2) are adjustable parameters in the constitutive law that have to be obtained by fitting the experimental data. For an isotropic constitutive law, Du and Cocks [27] used Ds = 0.95 while McMeeking and Kuhn [15] used Ds = 0.90. For constrained sintering Ma [18] used a much lower value of Ds = 0.85. The lower switching density can be justified

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F. Li et al. / Acta Materialia 58 (2010) 5980–5988

for constrained sintering because densification is retarded by the constraint and the inter-particle contacts parallel to the substrate start to interact at a lower relative density. In the current study we use

0:85; constrained films Ds ¼ ð25Þ 0:95; free sintering In more general situations, further study is required to establish how the switching density depends on the degree of anisotropy. The case studies in the current paper suggests that an empirical relation of Ds ¼ 0:792 þ 0:058e would be able to fit the data. 2.4. The sintering stress The sintering stress Rs in Eq. (1) is anisotropic in general [11–13,28]. Considering two spherical particles in contact with each other, a simple relation between the contact size, x, the particle approaching distance, DR, and particles size, R, exists [29,30]:  x 2 DR ð26Þ ¼4 R R the validity of which is independent of whether the particle compact is isotropic or not. Eq. (26) leads to Eq. (9), which is a fundamental starting point of the anisotropic model. Cocks et al. [14] showed that Eq. (9) leads to a conclusion that the rate of free energy change is independent of the macroscopic strain state, Eij, i.e. the sintering stress is isotropic even if the viscosity tensor, Cijkl, is anisotropic. This is consistent with the detailed numerical calculation by Wakai and Sinoda [11], which shows that the difference between the different components of the sintering stress tensor is not very large. We then have ðRs Þij ¼ Rs dij , where dij is the Kronecker delta. However, this study revealed that the expression for the sintering stress derived in Ref. [14] is not as good as that suggested by Ashby [31] when capturing the trend in the experimental data:   3c 2 2D  D0 Rs ¼ D ð27Þ R 1  D0 which is used in the current paper for early stage sintering. For later stage sintering we once again take the empirical approach and use the following expression following Ashby [31]: Rs ¼

2c 1=3

R½ð1  DÞ=6

R_ ¼

  3 e Qg R0 A exp  ð1  DÞ3=2 ; e 4R0 T R RT

for D 6 0:9 ð29Þ

  3 e Qg R0 A 4=3 R_ ¼ exp  ð1  DÞ ; e 5:4R0 T R RT for 0:9 < D 6 0:972    e Qg 4A R0 exp  R_ ¼ ½1  5:46ð1  DÞ2=3 ; e R0 T R RT

ð31Þ

for 0:972 < D 6 1

in which R0 is the initial value of particle radius R, T the ~ the universal gas consintering temperature in Kelvin, R ~ a stant, Qg the activation energy for grain-growth and A material constant. The grain-growth rate calculated from this model is used for both free and constrained sintering in the current paper. 3. Comparison with sintering experiments by Guillon et al. [3] Guillon et al. [3] carried out a series of free and constrained sintering experiments using alumina TM-DAR. The constrained sintering condition is illustrated in Fig. 2. The three orthogonal directions are noted as 1, 2, and 3. The thickness of the film is much smaller than its other dimensions and it is assumed that the film is perfectly bonded to the rigid substrate. Shrinkage only takes place in the thickness direction, i.e. we have E11 ¼ E22 ¼ E_ 11 ¼ E_ 22 ¼ 0. It is further assumed that the green film is deposited onto the substrate homogenously in the film plane. We then have R11 = R22 – 0 and R33 = 0. Using these simplifications in the anisotropic constitutive law, we obtain E_ 33 ¼ Rs =C 3333

ð32Þ

and R11 ¼ R22 ¼ C 1133 E_ 33 þ Rs ¼ C 2233 E_ 33 þ Rs ¼ Rs ð1  C 1133 =C 3333 Þ ¼ Rs ð1  C 2233 =C 3333 Þ

dD ¼ DE_ true 33 dt

ð34Þ

3 1

Densification is always accompanied by grain-growth. Guillon et al. [5] observed that there is no significant difference in the grain-growth behaviour (grain size as a function of the relative density) for free and constrained sintering. Du and Cocks [27] and He and Ma [32] proposed the following grain-growth law:

ð33Þ

i.e. the sintering response of the constrained film is uniquely determined by, for example, C1133, C3333, and Rs. The relative density of the film is calculated by integrating

ð28Þ

2.5. Grain-growth law

ð30Þ

2 film substrate Fig. 2. Sintering of a film constrained by a solid substrate.

F. Li et al. / Acta Materialia 58 (2010) 5980–5988

Fig. 3. Fitting the grain-growth data (discrete symbols) obtained by Zuo and Ro¨del [33] for alumina powder TM-DAR using the grain-growth law (dashed lines) by Du and Cocks [27]. The grain-growth law is then used to predict the grain size-density relation (solid lines) in the sintering experiment Guillon et al. [3] to validate the constitutive law.

Zuo and Ro¨del [33] have studied the grain-growth behaviour of the alumina TM-DAR powder. Fig. 3 shows their data of grain size versus relative density at two different sintering temperatures. By fitting these data with the grain-growth law given by Eqs. (32)–(34), we obtained the activation energy and the material constant as ~ ¼ 90 m2 K1 s1, which are consistent Qg = 475 kJ and A with those obtained by other studies in the literature [9,16,32]. The fitting of the grain-growth model with the experimental data is shown in Fig. 3. Fig. 3 also shows the grain size versus relative density predicted by the grain-growth model for the sintering temperatures and the initial green density that were used in the experiments conducted by Guillon et al. [3]. According to the observation of Guillon et al. [5], the relation between grain size and relative density remains the same for free and constrained sintering. The grain size–density relations shown in Fig. 3 are used together with the constitutive law for both free and constrained sintering. Firstly, we study the sintering experiment conducted by Guillon et al. [3] at a holding temperature of 1150 °C in both free and constrained conditions. Their densification curves are reproduced in Fig. 4. The initial particle radius was R0 = 75 nm. The specific surface energy is taken as c ¼ 1:1 J m2 according to Wonisch et al. [9]. The effective ~ b is obtained by fitgrain-boundary diffusion coefficient D ting the constitutive law with the free sintering data in ~ b ¼ 1:5  1036 m5 s N1. The grain-boundary Fig. 4 as D sliding constant is set as g ¼ 0:005, which can be compared to g = 0.007–0.02 used by Wonisch et al. [9] in their discrete element modelling. The fitting for free sintering is shown in Fig. 4. The constitutive law with this set of data (together with the grain size-density relation shown in

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Fig. 4. The fitting of the densification curve for free sintering and subsequent prediction of the densification curve for constrained sintering using the anisotropic constitutive law. The experimental data is for alumina powder TM-DAR with a holding temperature of 1150 °C, reproduced from Guillon et al. [3].

Fig. 3) is then integrated for the case of constrained sintering. The comparison between the model prediction and the experimental data is shown in Fig. 4. Only the early stage model is used for constrained sintering, as the relative density stays below the switching value Ds. It can be observed from the figure that the anisotropic constitutive law performs reasonably well in predicting the constrained sintering behaviour. Next we study the data by Guillon et al. [3] at the holding temperature of 1300 °C as reproduced in Fig. 5. Both the early stage and later stage models have to be used, as the relative density goes above the switching value Ds. Again, the initial particle radius is R0 = 75 nm and the specific surface energy is taken as c = 1.1 J m2. By fitting the constitutive law with the free sintering data, the effective grain-boundary diffusion coefficient is obtained as ~ b ¼ 1:0  1034 m5 s N1. The effective coefficient of D

Fig. 5. The fitting for free sintering and subsequent comparison between the model prediction and experimental data for constrained sintering. The experimental data is for alumina powder TM-DAR with a holding temperature of 1300 °C, reproduced from Guillon et al. [3].

F. Li et al. / Acta Materialia 58 (2010) 5980–5988

grain-boundary diffusion depends on the temperature through   ~ bD ~ b0 Qb Xd ~ Db ¼ exp  ð35Þ ~kT ~ RT ~ is the atomic volume, ~k Boltzmann constant, db in which X ~ b0 a pre-exponential constant, grain-boundary thickness, D and Qb the activation energy for grain-boundary diffusion. ~ b ¼ 1:5  1036 m5 s N1 at T = 1150 °C and 1:0 D 1034 m5 s N1 at T = 1300 °C lead to an activation energy of Qb = 530 kJ mol1 for grain-boundary diffusion, which is a little larger than values obtained by Zuo and Ro¨del [33] (480 ± 15 kJ mol1) for the same powder. These parameters are then used in the constitutive law (together with the relation between the grain size and relative density shown in Fig. 3) to fit the constrained sintering data as shown in Fig. 5. The numerical parameters in Eq. (24) are determined as a1 = 0.903, a2 =  0.303, and b1 =  0.371, b2 = 0.431. Fig. 5 compares the prediction of the anisotropic constitutive law with the experimental data. Once again, the anisotropic constitutive law is able to capture the sintering behaviour for both free and constrained sintering. Table 3 summarizes all the parameters required by the constitutive law. Using the parameters shown in Table 2, the three viscous Poisson’s ratios for constrained sintering can be calculated as mconstr: ¼ mconstr: ¼ 12 21

constr: constr: constr: C constr: 1122 C 3333  C 1133 C 2233 constr: constr: constr: C 3333 C 2222  C constr: 2233 C 2233

ð36Þ

¼ mconstr: ¼ mconstr: 13 23

constr: constr: constr: C constr: 2222 C 1133  C 1122 C 2233 constr: constr: constr: C 3333 C 2222  C constr: 2233 C 2233

ð37Þ

constr: constr: constr: C constr: 2222 C 1133  C 1122 C 2233 constr: constr: constr: C 1111 C 2222  C constr: 1122 C 1122

ð38Þ

and ¼ mconstr: ¼ mconstr: 31 32

Fig. 6 shows the viscous Poisson’s ratios as functions of the relative density calculated using the constitutive law for free sintering (E11 = E22 = E33) and constrained sintering (E11 = E22 = 0) respectively. For free sintering in the early stage, Casagranda et al. [17] and Ma [18] concluded from

1.2

Viscous Poisson’s Ratio

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1 ν 12constr. = ν 21constr.

ν13constr. = ν 23constr.

0.8

ν 31constr. = ν 32constr.

0.6

ν free

0.4 0.2 0 0.7

0.75

0.8

0.85

0.9

0.95

1

Relative Density Fig. 6. Viscous Poisson’s ratios as functions of the relative density calculated from the anisotropic constitutive law for free sintering (solid line, E11 = E22 = E33) and constrained sintering (discrete lines, E11 = E22 = 0) respectively.

their experiments that the viscous Poisson’s ratio is close to zero while Zuo et al. [19] concluded that the viscous Poisson’s ratio increases from 0.2 at D = 0.65 to 0.43 at D = 0.97. The variation of the viscous Poisson’s ratio for free sintering shown in Fig. 6 is within the range of these somewhat conflicting experimental data. For constrained is much sintering, it can be observed from Fig. 6 that mconstr: 31 larger than mconstr: and mconstr: , that mconstr: is the smallest 12 13 12 among the three and that mconstr: and mconstr: are considerably 12 13 smaller than the viscous Poisson’s ratio for free sintering. Similar trends were also observed in the discrete element simulations by Wonisch et al. [9]. It is worth noting that mconstr: becomes larger than 1 at higher density, which is 31 thermodynamically allowed for anisotropic materials. 4. Relating densification curve of constrained sintering with that of free sintering In the analysis shown in Section 3, one needs to know the specific surface energy, grain-boundary diffusion coefficient and grain-growth law in order to calculate the densification curve for constrained sintering. In fact, the use of

Table 3 List of parameters used in present model to predict the data by Guillon et al. [3]. Particle radius (initial) R0 Initial green relative density D0 Surface energy c ~ Atomic volume X ~ b0 Grain-boundary thickness  pre-exponent diffusion coefficient db D Activation energy for grain-boundary diffusion Qb Activation energy for grain-growth Qg ~ Grain-growth constant A Relative density Ds to switch from early and intermediate stages to final stage Normalized grain-boundary viscosity g* Final stage model fitting parameters a1 and a2 Final stage model fitting parameters b1 and b2

75 nm 0.685 1.1 J/m2 8.47  1030 m3 1.02  107 m3 s1 530 kJ/mol 475 kJ/mol 90 m2 K1 s1 Ds = 0.792 + 0.058e g* = 0.005 a1 ¼ 0:903; a2 ¼ 0:303 b1 ¼ 0:371; b2 ¼ 0:431

F. Li et al. / Acta Materialia 58 (2010) 5980–5988

these data can be avoided in the special case of the constrained film as shown in Fig. 2. By manipulating the anisotropic constitutive law, it is possible to calculate the densification curve for constrained sintering from the densification curve for free sintering. The free sintering data must be obtained using the same powder compact and identical heating schedule as used for constrained sintering. Using the anisotropic constitutive law, at the same relative density (rather than the same sintering time), the ratio between the strain rate in the thickness direction of the constrained film and the free sintering strain rate can be calculated as: E_ constr: 33 E_ free 33

¼

ðc3333 Þ

free

free

þ ðc1133 Þ þ ðc2233 Þ ðc3333 Þconstr:

free

;

D < 0:85; ð39Þ

E_ constr: 33 E_ free 33

¼

2ð1  D0 Þ½ðc3333 Þ 2

free

þ ðc1133 Þ

3D ð2D  D0 Þ½ð1  D=6Þ

1=3

free

þ ðc2233 Þ

f ðDÞðc3333 Þ

free



constr:

ð40Þ

0:85 < D < 0:95 E_ constr: f ðDÞðc3333 Þ 33 ¼ E_ free 33

free

;

free

þ gðDÞðc1133 Þ þ gðDÞðc2233 Þ constr: f ðDÞðc3333 Þ

D > 0:95

free

ð41Þ

Despite their apparent complexity, Eqs. (39)–(41) are simply functions of the relative density D and the principal strains, with the parameters cijkl expressed as a function of these strains according to the relationships given in Table 2. Knowledge of the surface energy, grain size and grain-boundary diffusion coefficient is not required. For the constrained film we have Econstr: ¼ ðD0  DÞ=D 33

ð42Þ

For free sintering at the same relative density, we have free free constr: Efree þ 1Þ=3Þ  1 11 ¼ E 22 ¼ E 33 ¼ expðlnðE 33

ð43Þ

5987

using the same alumina powder but different green density. The densification curve for free sintering obtained by Bordia et al. [2] is projected to constrained sintering using Eqs. (39)–(43), which is then compared to the experimental curve for constrained sintering. It can be observed from the figure that the anisotropic constitutive law provides an excellent description of the material data. 5. Comparison with sinter-forging experiment by Bordia et al. [2] Bordia et al. [2] measured the uniaxial compressive stress that is required to achieve zero radial shrinkage in their sinter-forging experiment using cylindrical samples. They concluded that the required compressive stress cannot be calculated using an isotropic constitutive law. Their experimental data and calculation using an isotropic constitutive law are reproduced in Fig. 8. Using 1 and 2 to represent the radial direction and 3 the loading direction, for the zero radial shrinkage experiment we have R11 = R22 = 0, E11 ¼ E22 ¼ E_ 11 ¼ E_ 22 ¼ 0 and   C 3333 R33 ¼ Rs 1  ð44Þ C 1133 The various data required by Eq. (44) to calculate the compressive stress are not available from Ref. [2]. However it is possible to translate their isotropic calculation to our anisotropic calculation using the constitutive law developed in this paper. At the same relative density D, the ratio between the compressive stress predicted by the anisotropic constitutive law and that by reducing the anisotropic constitutive law to an isotropic law can be calculated as: aniso:

aniso:

Raniso: 1  ðc3333 Þ =ðc1133 Þ 33 ¼ ; iso: iso: iso: R33 1  ðc3333 Þ =ðc1133 Þ

D < 0:85

ð45Þ

Eqs. (39)–(43) are applied to the experimental data as shown in Fig. 7, which were obtained by Bordia et al. [2]

Fig. 7. Projecting densification curve of the constrained film from the densification curve for free sintering data using the anisotropic constitutive law. The experimental data are reproduced from Bordia et al. [2].

Fig. 8. Compressive stress required to achieve zero lateral shrinkage in sinter-forging calculated using isotropic and anisotropic constitutive laws respectively in comparison with the experimental data. The isotropic calculation and experimental data are reproduced from Bordia et al. [2].

5988

F. Li et al. / Acta Materialia 58 (2010) 5980–5988

h i ðDÞðc3333 Þaniso: 2ð1  D0 Þ 1  fgðDÞðc aniso: Raniso: 1133 Þ 33 h i; ¼ iso: 1=3 2 3333 Þ Riso: 3D ð2D  D0 Þ½ð1  D=6Þ 1  ðc 33 iso: ðc Þ

Acknowledgement

1133

Raniso: 33 Riso: 33

0:85 < D < 0:95 h i h i aniso: aniso: 1  f ðDÞðc3333 Þ = gðDÞðc1133 Þ h i h i ; ¼ iso: iso: 1  f ðDÞðc3333 Þ = gðDÞðc1133 Þ D > 0:95

ð46Þ

This work is funded by the EPSRC (EP/F037430), which is gratefully acknowledged. References

ð47Þ

Calculating the terms on the right hand side of Eqs. (45)– (47) and taking the values of Riso: 33 from the calculation by Bordia et al. [2], we can obtain the values of Raniso: at var33 ious densities using Eqs. (45)–(47). The results are compared to the data obtained by Bordia et al. [2] in Fig. 8. The anisotropic model reproduces the major features of the experimental data and provides a significantly better prediction than the isotropic model. 6. Conclusions The deformation of a constrained film during sintering cannot be calculated using isotropic constitutive laws. The anisotropic microstructure developed during constrained sintering has to be taken into account. In the early stage sintering when the particle contacts do not interact, the anisotropy arises from the fact that the contact size is orientation-dependent during constrained sintering. The anisotropic constitutive law used in this study is based on a micromechanical model that takes this effect into account. For the later stage of sintering it is difficult to characterize the anisotropic microstructure and develop a similar micromechanical model. However, the form of the early stage law can be empirically extended into the later stage to obtain a complete constitutive law for constrained sintering. It is shown that this constitutive law can capture the trend observed in constrained sintering and sinter-forging experiments. The advantage of the present constitutive law is that it has an analytically explicit form, which can be used directly in finite element analysis to calculate sintering deformation of more sophisticated systems (a multi-layered system of complicated geometry for example) than the cases studied in the current paper.

[1] Bordia R, Jagota A. J Am Ceram Soc 1998;76:2475–85. [2] Bordia R, Zuo R, Guillon O, Salamone S, Ro¨del J. Acta Mater 2006;54:111–8. [3] Guillon O, Aulbach E, Ro¨del J, Bordia R. J Am Ceram Soc 2007;90:1733–7. [4] Lin Y, Jean J. J Am Ceram Soc 2004;87:1454–8. [5] Guillon O, Weiler L, Ro¨del J. J Am Ceram Soc 2007;90:1394–400. [6] Guillon O, Nettleship I. J Am Ceram Soc 2010;93:627–9. [7] Green D, Guillon O, Ro¨del J. J Eur Ceram Soc 2008;28:1451–66. [8] Ch’ng HN, Pan J. J Comput Phys 2005;204:430–61. [9] Wonisch A, Guillon O, Kraft T, Moseler M, Riedel H, Ro¨del J. Acta Mater 2007;55:5187–99. [10] Carroll D, Rahaman M. J Eur Ceram Soc 1994;15:473–9. [11] Wakai F, Sinoda Y. Acta Mater 2009;57:3955–64. [12] Wakai F, Akatsu T. Acta Mater 2010;58:1921–9. [13] Jagota A, Dawson P, Jenkins J. Mech Mater 1998;7:255–69. [14] Cocks ACF, Gill SPA, Pan J. Adv Appl Mech 1999;36:81–162. [15] McMeeking RM, Kuhn L. Acta Metall Mater 1992;40:961–9. [16] Helle AS, Easterling KE, Ashby MF. Acta Metall 1985;33: 2163–74. [17] Casagranda A, Xu J, Evans AG, McMeeking RM. J Am Ceram Soc 1995;79:1265–72. [18] Ma J. PhD thesis. University of Cambridge, Department of Engineering; 1997. [19] Zuo R, Aulbach E, Ro¨del J. J Mater Res 2003;18:2170–6. [20] Riedel H, Zipse H, Svoboda J. Acta Metall Mater 1994;42:445–52. [21] Ashby MF. Surf Sci 1972;31:498–542. [22] Raj R, Ashby MF. Metall Mater Trans 1971;2:1113–27. [23] Kim BN, Hiraga K, Morita K. Acta Mater 2005;53:1791–8. [24] Pan J, Cocks ACF. Acta Metall Mater 1994;42:1215–22. [25] Pan J, Cocks ACF. Acta Metall Mater 1994;42:1223–30. [26] Hsueh C, Evans A, Cannon R, Brook R. Acta Metall 1986;34: 927–36. [27] Du ZZ, Cocks ACF. Acta Metall Mater 1992;40:1969–79. [28] Olevsky EA, Kushnarev B, Maximenko A, Tikare V, Braginsky M. Philos Mag 2005;85:2123–46. [29] German RM. Sintering theory and practice. New York: WileyInterscience; 1996. [30] Exner HE. Rev Powder Metall Phys Ceram 1979;1:7–251. [31] Ashby MF. HIP 6.0 Background reading, the modelling of hotisostatic pressing. Cambridge University Engineering Department Report, Cambridge; 1990. [32] He Z, Ma J. Mater Sci Eng A 2003;361:130–5. [33] Zuo R, Ro¨del J. Acta Mater 2004;52:3059–67.