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Acta mater. 48 (2000) 1153±1166 www.elsevier.com/locate/actamat

EFFECT OF GRAVITY ON DIMENSIONAL CHANGE DURING SINTERINGÐI. SHRINKAGE ANISOTROPY E. A. OLEVSKY 1{ and R. M. GERMAN 2 1

Department of Mechanical Engineering, College of Engineering, California State University, San Diego, 5500 Campanile Drive, San Diego, CA 92182-1323, USA and 2P/M Lab, Engineering Science and Mechanics Department, The Pennsylvania State University, 118 Research West, University Park, PA 16802-6809, USA (Received 25 November 1998; accepted 23 September 1999) AbstractÐThe e€ects of gravity on sintering shrinkage and dimensional uniformity are analyzed using a continuum theory of sintering. Shape change caused by gravity during sintering is described both analytically and numerically. For a cylindrical sample shape, analytical approximations to characterize gravityinduced nonuniformities in shrinkage and compact aspect ratio are shown. As an illustration of the concept, analysis is performed for the solid-phase sintering of a copper powder cylindrical specimen to replicate earlier experiments reported by Lenel et al. (Trans. Am. Inst. Min. Engrs, 1963, 227, 640) and Exner (Rev. Powder. Metall. Phys. Ceram. Soc., 1968, 51, 604). The intensity of shrinkage anisotropy is compared for viscous and di€usional mechanisms of sintering. An algorithm is introduced for minimization of gravity-induced shrinkage anisotropy, suggesting an asymptotic approach to the peak temperature for best dimensional uniformity. 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Sintering; Gravity

1. INTRODUCTION

Sintering typically relies on surface energy as a primary driving force for particle bonding and densi®cation. Surface tension contributes a weak sintering stress that couples with atomic motion to form interparticle bonds. Because the sintering stress is low, most materials require a high temperature to induce signi®cant ¯ow to obtain measurable shrinkage. However, gravity provides another fundamental stress that acts on the powder compact during sintering. Unfortunately, gravity is not isotropic, so it induces anisotropic deformation. Curiously, gravity is not often incorporated in sintering models, although it contributes to nonuniform shrinkage, microstructure gradients, and compact slumping. This paper treats the sintering with the simultaneous action of surface tension and gravity. 1.1. Gravity e€ect on solid-phase sintering A limited number of articles have been dedicated to this research topic. Lenel et al. [1, 2] reported on nonuniform shrinkages in the sintering of loose

{ To whom all correspondence should be addressed.

copper powder and compacts with variations in orientation and support. They showed di€erences in the axial and radial shrinkages for loose powder aggregates [1], attributing this di€erence to gravity. Cutler and Henrichsen [3] provided an analogous result for sintering soda-lime glass for powder compacts from 40 mm particles. They observed a larger shrinkage in the vertical direction as compared with the horizontal direction. Following the qualitative experimental investigations, Lenel et al. [4] carried out a quantitative study directed to the analysis of shape distortions for cylindrical copper powder specimens sintered with di€erent supports. The arrangements included full bottom support [Fig. 1(a)], partial support [Fig. 1(b)], and top support [Fig. 1(c)] to measure the di€erences in radial and axial shrinkage with position and support as rationalized to gravity. Besides this work, only the experimental study of Exner [5] has provided information on the variation in sintering shrinkage with position for loose powder arrays. He measured the variation in radial shrinkage with position and used his results to estimate the stresses induced by gravity. From the experimental shrinkage di€erences and his calculations, Exner suggested that the ®ndings from Lenel et al. were biased by anisotropic pores associ-

1359-6454/00/$20.00 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 9 9 ) 0 0 3 6 8 - 7

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OLEVSKY and GERMAN: DIMENSIONAL CHANGE DURING SINTERINGÐI

ated with the nonspherical powders. However, his assessment of the stress contribution from gravity and its relative e€ect on shrinkage leads to smaller estimates of the gravitational e€ect than observed experimentally. Although the magnitude of the

gravity e€ect on solid-state sintering is uneven between experiments, these few observations all verify anisotropic behavior. 1.2. Gravity e€ect on liquid-phase sintering Studies on microstructural and con®guration changes in liquid phase sintering have provided insight into the sintering and composition factors that in¯uence distortion [6±14]. Most of the research in this area analyzed microstructure gradients as in¯uenced by gravity. The in¯uence of gravity-induced grain settling was examined by Kohara and Tatsuzawa [6], Niemi and Courtney [7], and German and co-workers [8±14]. The last group also studied gravity e€ects on the grain structure under liquid-phase sintering, with emphasis on grain packing [11] and grain coarsening [13]. Experimental evidence of gravity-induced shape distortions is given in several studies [8, 12]. A geometry described as an ``elephant foot'' shape was frequently observed for distorted W±Ni±Fe powder compacts. A publication by Raman and German [12] provided a ®rst mathematical model for gravityinduced shape distortion during liquid-phase sintering. In this study, the evolution of the shape of a ``top hat'' (cylinder with ¯ange) was modeled and experimentally observed. One of the model assumptions is constant density during the distortion process. This is based upon the hypothesis that bulk densi®cation occurs during heating and is essentially complete prior to liquid formation [15]. In light of this idea, shape change is calculated for an incompressible viscous material in a gravity ®eld [12]. Although qualitatively successful for liquid phase sintered tungsten heavy alloys with excessive liquid quantities, such an assumption substantially restricts the model basis and excludes shrinkage and its interplay with gravity forces. 1.3. Rationale and organization of the paper

Fig. 1. Experimental arrangements of Lenel et al. [4] for measuring the di€erences induced by gravity in radial and axial shrinkage. Position and support are rationalized to gravity: (a) full bottom support; (b) partial support; (c) top support.

Until recently there has been little analysis of the macroscopic behavior of powder compacts during sintering. Most of the e€ort has been devoted to consideration of theoretical links between mass transport mechanisms and descriptions of shrinkage at the microscopic level [14]. No relevant theory was available to bridge between sintering phenomena and macroscopic component shape retention or distortion. The continuum theory of sintering has been under intensive development since the late 1980s [16±28]. This approach is a basis for the analysis of gravitational e€ects on the distortion during sintering. It originates from continuum mechanics and exhibits the required traits for modeling macroscopic deformation of a porous structure to predict ®nal density and dimensions.

OLEVSKY and GERMAN: DIMENSIONAL CHANGE DURING SINTERINGÐI

The present work elaborates a mathematical model of sintering to incorporate both phenomenaÐshrinkage and gravitational e€ects. Based on this model, we describe the sintering behavior of a cylindrical powder specimen. The qualitative and quantitative experimental data of Lenel et al. [4] for solid-phase sintering, as well as of Exner [5] are used for veri®cation of the model. The paper is organized as follows: Section 2 includes the derivation of approximate evolution equations for shape changes of a cylindrical porous specimen sintered in a gravitational ®eld. The shape change factors are related to the bottom and top specimen radii, specimen height, and aspect ratio. The model predictions are compared with experimental data of Lenel et al. [4] and of Exner [5]. The modeling results of the ``sintering under gravity'' are presented in Section 3 in the framework of di€erent sintering mechanisms (viscous and di€usional). Section 3 also includes a comparison of the corresponding intensities of the shrinkage anisotropy. Section 4 describes an algorithm for optimization of the temperature regime in order to minimize gravity-induced shrinkage anisotropy with relevant calculations for copper powder. This paper encompasses an analytical approximation for the estimation of the dimension changes. Numerical (more realistic) ®nite-element solutions describing gravity-induced shape distortions under both solid and liquid phase sintering are presented in a companion paper [29].

in Fig. 2 to obtain an analytical solution for the gravity-induced shape distortion during sintering. At the onset of sintering, the specimen is assumed to have an initial radius R0 and initial height h0. Because gravity imposes an axial load which increases from the specimen's top to the specimen's bottom, during sintering the specimen assumes the shape of a truncated cone (Fig. 2). Two shrinkage anisotropy parameters can be suggested for analysis of the shape change. One parameter to describe shape change is the ratio between the bottom Rb and the top Rt radii, which evolve from their initial values during sintering. The second parameter is the ratio between the height and average radius normalized with respect to its initial value (h0/R0). Assume that the axial velocity Vz and porosity y depend only on the axial coordinate: Vz 6ˆ var…r†, y 6ˆ var…r†: Then the condition of momentum conservation with respect to the axial direction in a cylindrical coordinate system (taking into consideration the axisymmetric character of the problem) has the form:   @ Vr @ Vr @ sz r …1† ‡ Vz ˆÿ ‡ Fz @t @z @z where Vr is the radial velocity component, r is the volumetric mass of the porous material, sz is the axial stress, t is time, and Fz is the volumetric mass force (in this case induced by gravity) in the axial direction: Fz ˆ ÿrg:

2. AN APPROXIMATE ANALYTICAL ASSESSMENT OF THE GRAVITY-INDUCED SHAPE DISTORTION UNDER SINTERING

We consider a specimen con®guration as sketched

1155

…2†

Here g is gravitational acceleration. Neglecting the inertia forces, the condition of momentum conservation in equation (1) can be rewritten as

Fig. 2. Initial and current con®gurations of a porous specimen. During sintering the specimen assumes the shape of a truncated cone. The gravity forces induce a ``distributed up-setting'' load.

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OLEVSKY and GERMAN: DIMENSIONAL CHANGE DURING SINTERINGÐI

@ sz ˆ ÿrg: @z

In light of a continuum theory of sintering [18, 23±25, 27, 28] (see Appendix A), the axial sz and the radial sr stresses can be represented in the form (in axisymmetric formulation):     s…W † 1 sz ˆ j_e z ‡ c ÿ j …2_e r ‡ e_ z † ‡ PL W 3     s…W † 1 j_e r ‡ c ÿ j …2_e r ‡ e_ z † ‡ PL …4† sr ˆ W 3 where s(W ) and W are equivalent e€ective stress and strain rate, respectively (see Appendix A); j and c are the normalized shear and bulk response moduli, and PL is the e€ective sintering stress. Here we assume a linear dependence between equivalent e€ective stress and strain rate without a yield function: …5†

where Z0 is the shear viscosity of the porous body. The parameters e_ r and e_ z are the radial and axial strain rates, respectively: @ Vr e_ r ˆ , @r

@ Vz e_ z ˆ : @z

j ˆ …1 ÿ y† ,

2 …1 ÿ y†3 cˆ : 3 y

…7†

For the e€ective sintering stress, the following simple expression will be used [31]: PL ˆ PL0 …1 ÿ y†

…8†

where PL0 is a local sintering stress which can be represented as PL0 ˆ

2a r0

8    > @ 2 @ Vz > > 2Z j c ‡ > 0 > @ z 3 @z > > >     > > > 1 @ V r > > ‡2 c ÿ j ‡ PL ˆ rT g…1 ÿ y† < 3 @r      > > 1 @ V 1 @ Vr z > > > > 2Z0 c ÿ 3 j @ z ‡ 2c ‡ 3 j @ r ˆ ÿPL > > > > > @ y=@ t > @ Vz @ Vr > : ˆ ‡2 1ÿy @z @r …11† Here we have taken into account the relationship between volumetric mass r, porosity y, and theoretical density rT for the sample: r ˆ rT …1 ÿ y†:

…9†

with a being the surface tension and r0 being the average particle radius. We shall solve the problem of sintering for cylindrical green compacts assuming that variation of radial stress with respect to the radius is negligible. Also, unlike gravity e€ects on the axial direction, there are no mass forces applied in the radial direction. Hence, due to the boundary condition at the lateral surface: sr jexternal surface ˆ 0, radial stress should be equal to zero everywhere in the porous volume:

…12†

Expressions (11) represent a set of nonlinear di€erential equations related to the unknown coordinate- and time-dependent functions Vr, Vz, and y. An exact analytical solution of this set is impossible. To obtain an approximate analytical assessment let us average the right-hand part of the ®rst equation in (3) with respect to the axial coordinate z: @ fsz g ˆ rT g…1 ÿ y † @z

…6†

The normalized parameters j and c depend on porosity y and can be approximated as follows [30]: 2

…10†

From equations (3)±(6) and (10) it follows that:

2.1. Derivation of the kinetic equations for porosity and geometrical sizes of a cylindrical specimen

s…W † ˆ 2Z0 W

sr ˆ 0:

…3†

…13†

where 1 y ˆ h

…h 0

y dz

h is the current height of the porous specimen. Approximation (13) is equivalent to the assumption of the uniform spatial porosity distribution (which agrees well with experimental observations of Exner [5]). Integration of equation (13) with respect to the axial coordinate z gives sz ˆ rT g…1 ÿ y †…z ÿ h†:

…14†

In deriving equation (14), a boundary condition of a free top surface …sz jzˆh ˆ 0† is taken into consideration. It follows from equation (14) that the gravity forces induce a ``distributed up-setting'' loading mode when the load increases from the top end face of the specimen, where it equals zero, up to the maximum magnitude of rgh at the bottom of the specimen. Schematically, this is shown in Fig. 2. Taking into consideration an average result given by equation (14), the solution of the set of equations in (11) with respect to the radial and axial strain rates gives

OLEVSKY and GERMAN: DIMENSIONAL CHANGE DURING SINTERINGÐI

8 > @ Vz …j ‡ 6c†rT g…z ÿ h†…1 ÿ y † ÿ 3jPL > > > < @z ˆ 18Z0 jc > > @ Vr …j ÿ 3c†rT g…z ÿ h†…1 ÿ y † ÿ 3jPL > > ˆ : @r 18Z0 jc

sintering, we obtain …15†

ln

Rb r g ˆ T 12Z0 Rt

@ Vz @ Vr ‡2 @z @r

can be determined from equation (15) as follows: r g…1 ÿ y †…z ÿ h† ÿ 3PL e_ ˆ T : 6cZ0

…16†

2.2. First shrinkage anisotropy factor: aspect ratio between the bottom and the top radii The term @ Vr =@ r in equation (15) does not depend on the radial coordinate. Therefore @ Vr R_ …z, t† ˆ R…z, t† @r

…t

2 ÿ 3y h dt:  †2 0 …1 ÿ y

…20†

In view of the ®rst equation in (19), the current height of the specimen can be expressed in terms of the average current porosity and its own average (with respect to time) value

For an axisymmetrical case, the shrinkage rate e_ ˆ

1157

…17†

where R(z, t ) is the current radial dimension of the porous specimen, which depends on the axial coordinate, and R_ …z, t† is the corresponding derivative with respect to time. Also, assuming an insigni®cant dependence of the axial strain rate on the axial coordinate (which means that the axial strain rate is associated with the above-mentioned average porosity), we have h_ …t† @ Vz …18† ˆ @ z zˆh h…t† where h(t ) is the current height of the porous specimen, and h_ …t† is the corresponding derivative with respect to time. Using equations (7), (8), (17), and (18), and, averaging expressions in equations (15) and (16) with respect to the axial coordinate, one can reduce equations (15) and (16) as follows: 8 > @ y r gh=2 ‡ 3PL0 > > > ˆ ÿy T > > @t > 4Z0 …1 ÿ y † > > > > < _ R b R_ t R_ …0, t† R_ …h, t† rT g 2 ÿ 3y ˆ 1 ÿ h ÿ Rb > R…h, t† 12Z0 …1 ÿ y †2 Rt R…0, t† > > > > > > > y PL0 @h > > > : @ t ˆ ÿh 4Z …1 ÿ y †2 0 …19† Integrating the second equation in (19) with respect to time, and assuming that Rb ˆ Rt (Rb and Rt are the bottom and top radii, respectivelyÐsee Fig. 2) at the onset of sintering (the specimen has an ideal cylindrical shape at the beginning of the process) and, taking into consideration that Rb > Rt during

1 h ˆ ts

… ts 0

h dt

(ts is the total time of sintering): h1

8y_ y ÿ 1 Z0 :  rT g y 1 ‡ 6PL0 =…h rT g†

…21†

Substituting equation (21) into equation (20) and _ taking into consideration that y dt ˆ dy , we have ln

Rb 2 1 ˆ Rt 3 1 ‡ 6PL0 =…h rT g†

… y y i

2 ÿ 3y  dy y …1 ÿ y †

…22†

where y i is the initial average porosity. Accounting for equation (9), the latter expression is reduced to 2

Rb 4 1 ÿ y i y i ˆ Rt 1 ÿ y y

!2 32=…3…1‡b†† 5

…23†

where b ˆ 12a=…h rT gr0 † is a dimensionless parameter characterizing the relationship between surface tension and gravity sources. Equation (23) determines one of the shape factors of the porous specimen: the ratio of the bottom and the top radii. In Fig. 3, the dependence of this shape factor on porosity is represented. For comparison, the experimentally obtained data points of Lenel et al. [4] and Exner [5] (for the sintering of copper powder) are also given in Fig. 3. The material parameters corresponding to these prior experiments were used in the calculations in accord with equation (23), with y i ˆ 0:4, rT ˆ 8960 kg=m3 , g ˆ 9:81 m=s2 , a ˆ 1:72 N=m, r0 ˆ 4:4  10ÿ5 m, and h set to the arithmetical mean of the initial and ®nal heights or 1:8  10ÿ2 m: As seen in Fig. 3, the experimental data of Lenel et al. [4] (which gives Rb =Rt ˆ 1:0032 for y ˆ 0:24† and Exner [5] (which gives Rb =Rt ˆ 1:0043 for y ˆ 0:26† indicate larger shape distortions in comparison with the predictions. This is attributed to initial density (porosity) nonuniformities caused by die compaction and use of nonspherical powders, as well as substrate friction on the compact bottom during shrinkage. The rate of the shape change increases with density. This can be related to the redistribution of the deformation energy from the volume component (which decreases due to diminished porosity) to the shear component.

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OLEVSKY and GERMAN: DIMENSIONAL CHANGE DURING SINTERINGÐI

2.3. Second shrinkage anisotropy factor: aspect ratio height±radius A second shape change factor is the height± radius aspect ratio. From equations (15), (17), and (18) we have (R is the current radius of the porous specimen which depends on the axial coordinate z ) r g…z ÿ h† h_ R_ : ÿ ˆ T 2Z0 …y ÿ 1† h R

…24†

Accounting for the ®rst condition in equation (19), we obtain from equation (24)    d …h=R†1=2 3PL0 ln ˆ : …25† dt y 4Z0 …1 ÿ y†

The exact solution of equation (25) has the form h h0 y ˆ R R0 y i

!2

"

3 PL0 exp 2 Z0

# dt : 01ÿy

…t

…26†

Using the ®rst equation in (19) and introducing here 1 R ˆ h

…h 0

R dz

we derive an approximate expression for the h=R aspect ratio

Fig. 3. The dependence of the ratio of the bottom and the top radii (®rst shape change factor) on porosity.

OLEVSKY and GERMAN: DIMENSIONAL CHANGE DURING SINTERINGÐI

h h0 y ˆ R0 y i R

!2=…1‡b† :

…27†

The dependence of …h=R †=…h0 =R0 † on porosity (using the same material parameters for copper powder) is presented in Fig. 4. For comparison, the curves which correspond to free sintering …h=R ˆ h0 =R0 † and for the absence of shrinkage, when only the gravity forces are active, are also shown in Fig. 4. In the latter case, the aspect ratio h=R corresponds to that obtained under free-upsetting of a cylindrical powder specimen [32]: h h0 y ˆ R0 y i R

!2 :

…28†

Equations (23) and (27) describe the shape change of a cylindrical specimen. Let us summarize the model assumptions that have been used in the derivation of these expressions: . an idealized specimen shape evolution (assumed uniform porosity and truncated cone); . independence of the axial ¯ow velocity, radial stress and porosity from the radial coordinate;

1159

. neglect inertia forces; . neglect external friction from the substrate; . averaging approximations [equations (19) and (21)]. Due to the above-mentioned model assumptions, the relationships represented by equations (23) and (27) are rough approximations for the evolution of the shape. Nevertheless equation (23) [as well as equation (27)Ðsee above] satis®es a limiting transition to the case when gravity e€ects are negligible: Rb lim ˆ1 …29† Rt b 4 1 which means that there is no shape change under gravity-free sintering of the initially uniform powder specimen. In another limiting case, when sintering shrinkage is negligible and only gravitational forces are active [no shrinkage due to sintering: 12a=…h rT gr0 † 4 0Š the following evolution equation for the shape change is valid: 2

Rb 4 1 ÿ y i y i ˆ Rt 1 ÿ y y

!2 32=3 5 :

…30†

Both h=R and Rb/Rt are shape factors that serve as measures of shrinkage anisotropy. In particular, h=R is the current aspect ratio and characterizes di€erences in the axial and radial shrinkages (for isotropic shrinkage, h=R should be constant and equal to h0/R0 during the entire process). Exner [5] considered the deviations in axial and radial shrinkage from the corresponding shrinkage values, which are not in¯uenced by gravity or other supplementary stress. Using equations (23) and (27), it is possible to obtain the ratio between the deviations in radial and axial directions: DR ˆ Dh

Rb ÿ1 Rt     Rb h h0 2ÿ 1‡ R R0 Rt

ˆ 2ÿ

y y i

1 ÿ y i y i 1 ÿ y y 0 ! 2=…1‡b†

!2=…3…1‡b†† ÿ1

@ 1 ÿ y i y i 1 ÿ y y

!2=…3…1‡b††

1: ‡1 A …31†

The relationship represented by equation (31) is plotted in Fig. 5. Exner showed DR=Dh ˆ ÿ0:5 under the assumption that gravity does not in¯uence overall shrinkage.

Fig. 4. The dependence of the normalized aspect ratio (second shape change factor) on relative density.

3. GRAVITY-INDUCED DISTORTION IN LIGHT OF THE DIFFERENT MECHANISMS OF SINTERING

The

previous

analysis

assumed

a

viscous

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OLEVSKY and GERMAN: DIMENSIONAL CHANGE DURING SINTERINGÐI

Fig. 5. The ratio between the deviations in radial and axial shrinkage from their values, which are not in¯uenced by gravity or any other supplementary stresses.

mechanism of material ¯ow as a dominant mechanism in sintering and is most appropriate to amorphous materials [14, 33]. For the description of sintering of crystalline materials, it can be used only as a rough approximation based upon a corresponding choice of the coecient of viscosity. In this section, we shall consider sintering under gravity with the governing mechanism corresponding to a di€usional creep (a coupled volume and grainboundary di€usion) in accordance with the McMeeking and Kuhn [34] and Du and Cocks [35] models. In the framework of the McMeeking and Kuhn model [34], the bulk c and the shear j

{ It should be noted that j and c are proportional to each other in the framework of this model …c ˆ 5=6j).

moduli are given by{: cˆ

 2=3 1 1 ÿ y0 …1 ÿ y†2 …y0 ÿ y†2 2 1ÿy 36y20 

jˆ

1 ÿ y0 1ÿy

2=3

…1 ÿ y†2 …y0 ÿ y†2 60y20

…32†

where y0 is the porosity of the random packed state …y0 ˆ 0:4). The McMeeking and Kuhn model is relevant for the initial stages of sintering when the porosity y > 0:1: In the capacity of the generalized viscosity coecient (substituting Z0) the ratio r30 =D is used where r0 is the initial average radius of the powder particles, D is an e€ective di€usion coecient and is

OLEVSKY and GERMAN: DIMENSIONAL CHANGE DURING SINTERINGÐI

r g…1 ÿ y † R_ b R_ t ÿ ˆ T 3 h: Rb Rt 12cr0 =D

determined as follows: bDb ‡ r0 …y0 ÿ y†Dv Dˆ O: kT

…33†

Here bDb is the grain boundary di€usion coecient times the boundary thickness, Dv is the volume diffusion coecient, O is the volume of the di€using atom, k is Boltzman's constant and T is the absolute temperature. The e€ective sintering stress determined by the McMeeking and Kuhn model is PL ˆ

 1=3 …1 ÿ y†2 …y0 ÿ y† 1 ÿ y0 PL0 : y0 1ÿy

…34†

Substituting the above-mentioned values in equation (19), one can determine the shrinkage anisotropy factors. In particular, the rate of change of the bottom±top radius ratio is estimated as [compare with the second equation in (19)]

1161

…35†

To compare the intensity of the shrinkage anisotropy for two di€erent sintering mechanismsÐviscous ¯ow and di€usional creep, one cannot relate the second equation in (19) and equation (35) directly. These expressions describe the absolute values of the aspect ratio change rate, which are obviously di€erent because the shrinkage rates for two sintering mechanisms are also unequal. The latter means that values describing shape change should be normalized with respect to the corresponding shrinkage rates. Let us choose porosity change rates both for di€usional and viscous mechanisms under the conditions of free sintering (neglecting gravity in¯uence) in the capacity of normalizing parameters. Thus, for comparison of the intensity of shrinkage anisotropy in the case of viscous or di€usion mechanisms of sintering: " ! # R_ b R_ t _ =y ÿ Rb Rt ! # viscous flow : oˆ " …36† R_ b R_ t _ =y ÿ Rb Rt diffusional creep Using expressions (19) and (35), and substituting porosity evolution rates corresponding to the di€erent mechanisms [for the di€usional creep, sintering rates can be determined from equations (16), (32), and (34)], we obtain oˆ

4…2 ÿ 3y†…1 ÿ y0 †2=3 …y0 ÿ y†…1 ÿ y†1=3 : 3y0 y

…37†

The relationship between o and y (for y0 ˆ 0:4† is presented in Fig. 6. For large porosities, when 0:283 < y < 0:4, the di€usional creep mechanism provides more intensive shape change. For smaller porosities, when 0:1 < y < 0:283, viscous ¯ow causes more intensive anisotropic shrinkage. For the range of small porosities …0 < y < 0:1), the di€usional creep model of Du and Cocks [35] can be used with

Fig. 6. Comparison of the intensity of shrinkage anisotropy in the case of viscous and di€usion mechanisms of sintering: the dependence of criterion o on porosity. Equation (37) is used for y > 0:1 (the McMeeking±Kuhn model) and equation (39) is used for y < 0:1 (the Du±Cocks model). The results for small porosities …y < 0:1† are plotted for three di€erent values of the dihedral angle l.

jˆ

1 …1 ‡ 39:6y† 2

cˆ

0:044…1 ÿ y† p y

PL ˆ PL0 ) (  1=3 1 y ‡4 cos l 2 4p…1 ÿ y†…2 ÿ 3 cos l ‡ cos3 l† …38† where l is the dihedral angle. In this case, the cri-

1162

OLEVSKY and GERMAN: DIMENSIONAL CHANGE DURING SINTERINGÐI

terion o has the form: ( oˆ

y 4p…1 ÿ y†…2 ÿ 3 cos l ‡ cos3 l† 3…1 ÿ y†2 y

In Fig. 6, one can see the plotted dependence o vs porosity y for 0 < y < 0:1 and di€erent dihedral angles l. For smaller dihedral angles, the shape change due to di€usional creep becomes smaller. However, for a dihedral angle equal to 908 the shape change is more intensive due to di€usion if compared with viscous ¯ow when 0:05 < y < 0:1: Physically, in the framework of the di€usional mechanism of sintering, the e€ect of the increase of the shrinkage anisotropy with increasing dihedral angle should be connected with the overall increase of the grain boundary area. The results of this comparison should be important for the analysis of dimension change under sintering with both mechanisms of material ¯ow (viscous and di€usional) being present. This is possible, for example, in the case of transient liquid phase sintering. To minimize the shrinkage anisotropy, it is necessary to enhance viscous ¯ow at the onset of sintering (when porosity is still large) and to provide higher activity of the di€usional creep in the end of sintering (when porosity is smaller). This can be achieved, for example, by optimization of the heating regime during sintering. The initial phase composition that determines the amount of the melted phase in liquid-phase sintering can also be used as a controlling parameter. And, of course, as it follows from Fig. 6, the initial porosity level can be used for an optimization. Another relevant example of an interplay of the di€usional and viscous mechanisms of sintering is the thermoprocessing of glass±metal composites. It should be noted that the higher intensity of the shrinkage anisotropy due to di€usional creep at the initial stage of sintering can be an additional explanation for the deviation of the theoretical and experimental data in Fig. 3. The theoretical results represented in Fig. 3 are based upon the idea of the linear-viscous constitutive properties of the porous body skeleton [in this case it is possible to derive an analytical solution equation (23)]. Therefore, the predicted shrinkage anisotropy should be smaller than that in the case of the di€usional mechanism of sintering.

1=3

) ‡8 cos l …2 ÿ 3y† :

…39†

shrinkage anisotropy are important goals in process optimization. A basis for optimization is to determine a heating cycle that maximizes densi®cation yet minimizes shrinkage anisotropy. Accordingly, during heating the viscosity of the porous body skeleton should be low to provide the highest degree of shrinkage, but high to avoid excessive shrinkage anisotropy. Due to the assumptions, the shrinkage anisotropy factors derived in Section 2 do not explicitly contain any temperature-dependent parameters. However, besides surface tension, which should be slightly dependent on the temperature (this dependence is further neglected), the average values h and R can be taken into account as temperature connected, as they describe shrinkage which depends on the temperature. At the same time, viscosity that should be

4. MINIMIZATION OF SHRINKAGE ANISOTROPY BY OPTIMIZATION OF THE HEATING REGIME

One of the main objectives of modeling of sintering is component dimensions control. In particular, minimizations of the shape distortion and of the

Fig. 7. Optimization strategy enabling high ®nal density with a minimum shape change.

OLEVSKY and GERMAN: DIMENSIONAL CHANGE DURING SINTERINGÐI

considered as the main temperature-dependent material parameter, is not explicitly represented in expressions (23) and (27). As a starting point for the optimization analysis, let us consider expression (25), where viscosity of the porous body skeleton is a function of the temperature: Z0 ˆ Z0 …T †: The left-hand part of equation (25) can be accepted as a measure estimating the relationship between the shrinkage anisotropy rate and the shrinkage rate. It can be assumed that there exists an optimal ratio between the shrinkage anisotropy and the shrinkage rates that can be associated with some value of the left-hand part of equation (25). An optimal sintering regime should provide a constancy of this during the entire sintering process, thereby resulting a linear dependence of the criterion ln‰…h=R†1=2 =yŠ on time t (Fig. 7). In such an optimal case, if we assume some speci®c tolerances prerequested for the change of the aspect ratio hf =R f r…h=R†f min , and for the shrinkage y f Ryf max (subscript f corresponds to the values at the end of sintering), the following condition should be satis®ed for any h, R, and y: 2 !1=2 3  h= R 7 6    7 6 5 d …h=R†1=2 1 4 …h=R †f min ln  ln dt ts y y=yf max ˆ

3PL0 : 4Z0 …1 ÿ y†

…40†

If PL0 does not depend on the temperature, then condition (40) can be satis®ed if d fZ …1 ÿ y†g ˆ 0 dt 0

…41†

y_ Z_ 0 : ˆ 1ÿy Z0

…42†

or

Accounting for equation (19), we obtain Z_ 0 ˆ ÿ

  y h r g ‡ 3P L0 : T 2 4…1 ÿ y †2

…43†

As Z_ 0 ˆ

@ Z0 dT @ T dt

we have   dT 1 h y ˆÿ ‡ 3P r : g L0 dt @ Z0 =@ T 4…1 ÿ y †2 T 2

…44†

The optimal heating regime T ˆ T…t† can be determined from the solution of a set of ordinary di€erential equations [obtained from equations (19) and

(44)]: 8   > y dT 1 h > > ˆÿ r g ‡ 3PL0 > > > dt @ Z0 =@ T 4…1 ÿ y †2 T 2 > > > > < @ y r gh=2 ‡ 3PL0 ˆ ÿy T > @t 4Z0 …1 ÿ y † > > > > > > @h y PL0 > > > : @ t ˆ ÿh 4Z …1 ÿ y †2

1163

…45†

0

The initial value of the temperature, which is necessary for the solution of set (45), can be determined from equation (40), which should be satis®ed for any h, R, and y, and, in particular, for the initial values of these parameters: Z0 …Ti † ˆ

3PL0 ts 3: !1=2 , h =R 0 0 …1 ÿ yi †ln4 …yi =ymax † 5 …h=R †f min 2

…46† From the latter equation we obtain Ti. For the solution of set (45), the temperature dependence of the shear viscosity Z0 ˆ Z0 …T † of the porous body skeleton should be known. Following expression (33), we determine the temperature dependence for the viscosity of copper in the form Z0 ˆ r30 =D ˆ

r30 kT : ‰bDb ‡ r0 …y0 ÿ y†Dv ŠO

Herewith, r0 ˆ 4:4  10ÿ5 m, y0 ˆ 0:4, bDb ˆ 4:4  10ÿ4 exp…ÿ1:07  105 =…8:31T †† m3 =s [14], Dv ˆ 6:0  105 exp…ÿ2:13  105 =…8:31T †† m2 =s [14], O ˆ 1:18  10ÿ29 m3 [36]. The local sintering stress is determined as PL0 ˆ 2a=r0 , with a ˆ 1:72 N=m [14]. The results of the optimization of the heating regime under sintering during ts ˆ 5 h for a copper powder specimen with initial sizes: h0 ˆ 0:04 m, R0 ˆ 0:01 m, initial porosity yi ˆ 0:36, and predetermined tolerances for the change of the aspect ratio   h0 h ˆ 1:05 R0 R min and for the shrinkage yf max ˆ 0:05, are represented in Fig. 8. The solution of set (45) has been carried out by a Runge±Kutta method of the fourth order. It turns out that for the optimization of sintering, it is necessary to gradually decrease the heating rate. Physically, this calculated optimization strategy is understandable. Due to the porosity decrease during sintering, the reduction of the free surface energy contributes more to the shear deformation rather than to the volume one. This causes the acceleration of the shrinkage anisotropy compared with shrinkage. In order to keep the optimal ratio

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OLEVSKY and GERMAN: DIMENSIONAL CHANGE DURING SINTERINGÐI

Fig. 8. Optimal heating regime enabling high ®nal density with a minimum shape change of a copper powder sample with initial sizes: h0 ˆ 0:04 m, R0 ˆ 0:01 m, the initial porosity yi ˆ 0:36, and the predetermined tolerances for the change of the aspect ratio …h0 =R0 †=…h=R †min ˆ 1:05 and for the shrinkage yf max ˆ 0:05:

between the change of the aspect ratio and the change of porosity rates, the temperature rate should be diminished. The same optimization strategy can be used in the determination of the favorable heating regime when the sintering mechanism corresponds to di€usional creep. In that case, criterion (25) can be reformulated in terms of j, c, and PL corresponding to the McMeeking±Kuhn model [34] …0:1 < y < 0:4† or the Du±Cocks model [35] …0 < yR0:1). 5. CONCLUSIONS

1. A mathematical model of sintering e€ected by gravity is elaborated.

2. Analytical expressions for the shrinkage anisotropy factors for a cylindrical specimen: the bottom±top radii ratio and the height±average radius aspect ratio, are obtained. 3. In accord with the McMeeking±Kuhn model, the di€usional creep mechanism provides in the beginning of sintering (when porosity is higher than 28%) greater intensity of anisotropy of shrinkage compared with viscous ¯ow. The viscous ¯ow mechanism causes higher anisotropy for smaller porosities …0:1 < y < 0:28). In the framework of the Du and Cocks model …0 < y < 0:1), for smaller values of the dihedral angle, the di€usional mechanism of sintering provides a smaller degree of the shrinkage aniso-

OLEVSKY and GERMAN: DIMENSIONAL CHANGE DURING SINTERINGÐI

tropy compared with the viscous ¯ow. 4. An algorithm of the optimization of the sintering heating regime, which enables high ®nal density and small shrinkage anisotropy, is developed. The obtained temperature±time dependence for a copper powder specimen indicates the necessity of a gradual decrease of the heating rate in the course of sintering.

AcknowledgementÐThe support of the NSF Institute for Mechanics and Materials, University of California, San Diego is gratefully acknowledged. REFERENCES 1. Lenel, F. V., Hausner, H. H., Hayashi, E. and Ansell, G. S., Powder Metall., 1961, 8, 25. 2. Lenel, F. V., Hausner, H. H., Shanshoury, A. E., Early, J. G. and Ansell, G. S., Powder Metall., 1962, 10, 190. 3. Cutler, I. B. and Henrichsen, R. E., J. Am. Ceram. Soc., 1968, 51, 604. 4. Lenel, F. V., Hausner, H. H., Roman, O. V. and Ansell, G. S., Trans. Am. Inst. Min. Engrs, 1963, 227, 640. 5. Exner, H. E., Rev. Powder Metall. Phys. Ceram., 1979, 1±4, 210. 6. Kohara, S. and Tatsuzawa, K., Bull. Inst. Space Aero. Sci. Univ. Tokyo B, 1981, 17, 493. 7. Niemi, A. N. and Courtney, T. H., Acta metall., 1983, 31, 1393. 8. Kipphut, C. M., Bose, A., Farooq, S. and German, R. M., Metall. Trans. A, 1988, 19, 1905. 9. Yang, S.-C. and German, R. M., Metall. Trans. A, 1990, 22, 786. 10. Heaney, D. F., German, R. M. and Ahn, I. S., J. Mater. Sci., 1995, 30, 5808. 11. Liu, Y., Heaney, D. F. and German, R. M., Acta metall. mater., 1995, 43, 1587. 12. Raman, R. and German, R. M., Metall. Mater. Trans. A, 1995, 26, 653. 13. German, R. M., Liu, Y. and Gri€o, A., Metall. Mater. Trans. A, 1997, 28, 215. 14. German, R. M., Sintering Theory and Practice. WileyInterscience, New York, 1996. 15. German, R. M., Farooq, S. and Kipphut, C. M., Mater. Sci. Engng A, 1988, 105±106, 215. 16. Olevsky, E. A. and Skorohod, V. V., in Technological & Design Plasticity of Porous Materials. IPMS NAS, Ukraine, 1988, pp. 97±103. 17. Jagota, A., Dawson, P. R. and Jenkins, J. T., Mech. Mater., 1988, 7, 255. 18. Skorohod, V. V., Olevsky, E. A. and Shtern, M. B., Proc. IXth Int. Conf. on Powder Metallurgy, Dresden, Vol. 2, 1989, pp. 43±57. 19. Riedel, H., A constitutive model for the ®nite-element simulation of sinteringÐdistortions and stresses, in Ceramic Powder Science III, ed. G. L. Messing. American Ceramic Society, Westerville, OH, 1990, pp. 619±630. 20. Reid, C. R. and Oakberg, R. G., Mech. Mater., 1990, 10, 203. 21. Skorohod, V., Olevsky, E. and Shtern, M., Sci. Sintering, 1991, 23(2), 79. 22. McMeeking, R. M., Mech. Granular Mater. Powder Systems, ASME, 1992, 37, 51. 23. Skorohod, V., Olevsky, E. and Shtern, M., Powder Metall. Metal Ceram., 1993, 362, 16.

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24. Skorohod, V., Olevsky, E. and Shtern, M., Powder Metall. Metal Ceram., 1993, 362, 16. 25. Skorohod, V., Olevsky, E. and Shtern, M., Powder Metall. Metal Ceram., 1993, 363, 208. 26. Cocks, A. C. F., Acta metall., 1994, 42(7), 2191. 27. Olevsky, E., Dudek, H. J. and Kaysser, W. A., Acta metall. mater., 1996, 44, 707. 28. Olevsky, E., Mater. Sci. Engng, R: Reports, 1998, 23, 40. 29. Olevsky, E. A., German, R. M. and Upadhyaya, A., Acta mater., 2000, 48, 1167. 30. Skorohod, V. V., Rheological Basis of Theory of Sintering. Naukova Dumka, Kiev, 1972. 31. Olevsky, E. and Molinari, A., submitted. 32. Olevsky, E., La Salvia, J. and Meyers, M., Adv. Powder Metall. Partic. Mater., 1997, 3(20), 13. 33. Scherer, G. W., J. Am. Ceram. Soc., 1977, 60(5), 236. 34. McMeeking, R. M. and Kuhn, L. T., Acta metall., 1992, 40(5), 961. 35. Du, Z.-Z. and Cocks, A. C. F., Acta metall., 1992, 40, 1969±1980. 36. Ashby, M. F., HIP 6.0, Background Reading. University of Cambridge, Cambridge, 1990. APPENDIX A

A.1. Basic ideas of a continuum theory of sintering A.1.1. Mechanisms of sintering. This appendix provides the reader with the main ideas of the model describing the sintering of nonlinear-viscous porous materials. This model is suitable for treating sintering under mechanisms of viscous, power-law creep, or plastic ¯ow. The same concept can be used to describe sintering controlled by volume or grain boundary di€usion mechanisms. In this case, corresponding expressions for the e€ective equivalent stress s(W ) and strain rate W, as well as for the normalized shear j and bulk c viscosity moduli are used (see, e.g. McMeeking and Kuhn [34] or Cocks [26]). A.1.2. Constitutive behavior of viscous porous bodies. The mechanical response of a porous body with nonlinear-viscous behavior is described by a rheological constitutive relation that connects components of a stress tensor sij and strain rate tensor e_ ij [27]:     s…W † 1 sij ˆ …A1† j_e ij ‡ c ÿ j e_ dij W 3 where j and c are the normalized shear and bulk viscosity moduli, which depend on porosity y [for example, following Ref. [30], j ˆ …1 ÿ y†2 , c ˆ …2=3†……1 ÿ y†3 =y†]; dij is a Kronecker symbol …dij ˆ 1 if i ˆ j and dij ˆ 0 if i 6ˆ j); e_ is the ®rst invariant of the strain rate tensor, i.e. sum of tensor diagonal components: e_ ˆ e_ 11 ‡ e_ 22 ‡ e_ 33 : Physically, e_ represents the volume change rate of a porous body. The porosity y is de®ned as 1 ÿ r=rT , where r and rT are volumetric mass and theoretical density (volumetric mass of a fully dense material), respectively. Alternatively, porosity is the ratio of pore

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OLEVSKY and GERMAN: DIMENSIONAL CHANGE DURING SINTERINGÐI

volume to volume of the porous body. The dependence between the porosity and the shrinkage rates is given by a continuity equation: e_ ˆ y_ =…1 ÿ y†: The e€ective equivalent strain rate W is connected with the current porosity and with the invariants of the strain rate tensor: q 1 W ˆ p j_g 2 ‡ ce_ 2 …A2† 1ÿy where g_ is the second invariant of the strain rate tensor deviator:  g_ ˆ

1 e_ ij ÿ e_ dij 3



1 e_ ij ÿ e_ dij 3

1=2 :

…A3†

Physically, g_ represents the shape change rate of a porous body. This value can be expressed in terms of the main elongation rates e_ 1 , e_ 2 , e_ 3 : q 1 g_ ˆ p …_e 1 ÿ e_ 2 †2 ‡ …_e 2 ÿ e_ 3 †2 ‡ …_e 3 ÿ e_ 1 †2 : …A4† 3 The e€ective equivalent stress s(W ) determines the constitutive behavior of a porous material. If s(W ) is described by a linear relationship: s…W † ˆ 2Z0 W, where Z0 is the shear viscosity of a fully dense material, then we obtain an equation corresponding to the behavior of a linear-viscous porous body:     1 sij ˆ 2Z0 j_e ij ‡ c ÿ j e_ dij : …A5† 3 If s(W ) is a constant ‰s…W † ˆ t0 , t0 is the yield stress for a fully dense material], we obtain an equation corresponding to a rigid-plastic porous body: p     t0 1 ÿ y 1 sij ˆ q j_e ij ‡ c ÿ j e_ dij : …A6† 3 j_g 2 ‡ ce_ 2 In the general case, s(W ) is described by a nonlinear relationship. For example, for hot deformation, a power law is used ‰s…W † ˆ AW m where A and m are the material constants, 0RmR1]. In this case, we have 0 q 1mÿ1     2 2 1 @ j_g ‡ ce_ A p sij ˆ A j_e ij ‡ c ÿ j e_ dij : 3 1ÿy …A7† For the case where m ˆ 1, equation (A5) transforms

into equation (A3) …A ˆ 2Z0 † and, for m ˆ 0, equation (A5) transforms into equation (A4) …A ˆ t0 † Thus, linear-viscous and rigid-plastic behavior are two limiting cases for a nonlinear-viscous rheology. With regard to expressions for j and c, the model can be used for 0 < 2=3: Constitutive relations for sintering of nonlinear-viscous porous materials should incorporate a term corresponding to the in¯uence of capillary (Laplace) stressesÐa sintering factor. A.1.3. Sintering factor. In accord with concepts from irreversible thermodynamics, the following equation is valid: sij e_ ij ÿ

@F ÿ H_ …_e ij , y† ˆ 0 @t

…A8†

where F is the free surface energy of the porous system, H_ …_e ij , y† is the rate of energy dissipation, and t is time. Herewith @F ˆ PL e_ @t

…A9†

where PL is an e€ective Laplace stress (sintering stress), which depends on the local sintering stress PL0 and porosity. (``E€ective'' means that the parameter describes a value in a macroscopic porous volume, while ``local'' designates parameters ascribed to a single pore or particle.) The dissipation of energy is conditioned by the processes taking place in a substance (porous body skeleton). However, at the macro-level, H_ …_e ij , y† has to depend on an averaged parameter of the substance ¯ow. The e€ective equivalent strain rate W de®ned by equation (A2) is such a parameter. Assuming that H_ …_e ij , y† is a quasi-homogeneous function and that @ H_ =@ e_ ij ˆ sij we obtain from equation (A8): ! H_ @ W sij ÿ PL dij ÿ e_ ij ˆ 0: W @ e_ ij

…A10†

Finally, from equations (A1), (A2) and (A10) we have sij ˆ

    s…W † 1 j_e ij ‡ c ÿ j e_ dij ‡ PL dij : …A11† W 3