- Email: [email protected]
Hot-isostatic pressing diagrams: New developments
ACIU metal/.Vol. 33, No. 12. pp. 2163-2174,1985
ooo1-6l60/8553.00+ 0.00
Printed in Great Britain. All rights resuwd
Copyright Q 1985 Pergamon Prcsa L.td
HOT-ISOSTATIC PRESSING DIAGRAMS: NEW DEVELOPMENTS A.S.HELLE,'K.E.EASTERLING'and M. F. ASHBV ‘Department of Engineering Materials, University of LuleA, S-951 87 LuleA, Sweden and %epartment of Engineering, University of Cambridge, Cambridge CB2 IPZ, England (Receiued 9 February 1985; in revisedform
I May
1985)
Abatrati-The equations and procexlurcs for constructing hot-isostatic pressing diagrams are greatly simplified and clarified. In earlier work, three classes of mechanism were modelled: plastic yielding,
power-law creep and diffusional densification. In this paper two further mechanisms are added: the diffusional deformation of the particles themselves when the grain size is much smaller than the particle size. and the separation of pores from boundaries when grain growth occurs. Substantially better agreement between the model and experimental data is now obtained, particularly in cases where grain growth has occuned. Application of HIP diagrams to tool steels, alumina and copper, incorporatiag experimental data, are presented and discussed. R&tar&-Nous avons nettcment simplifiCIts equations et les procedures pour con&ruin les diagrammes de pmssage isostatique i chaud. Dans un article ant&rieur, nous avions mod&a6 trols classes de m&anismes: l’&oukment plastique, le fluage en loi de puissance et la densification par di!Rtsion. Darts cet article, aous ajoutons deux autres m&anismcs: la dCformation par diffusion da particules elks-m&nes quand la taille des grains est beaucoup plus petite que la taille des particules et la s¶t@tdes pores g partir des joints de grains lorsque la croissance des grains SCproduit. Nous obtenoas mam&nant un accord aettemeat meilkur entre k modhle et les r6sultats expcrimentaux, en particulkr daas ks cas oul il s’est produit une croissance des grains. Nous pr4sentons et nous discutons l’applkation der diylnmmes de PIC aux a&m g outils , B l’alumine et au cuivre en y introduisant des r&hats @timentaux. B-Die Gkichungen turd Verfahren xur Konstruktion voa Diagrammen des lwstatischen HeiDpressens we&m stark vereinfacbt uad aulbereitet. In frtiheren Arbeiten wurden dml Klasacn voa Mechanismsn modelliert: plastischer FlieBbeginn, Potenagesetxkriechen und Difhtsionsv&dttung. In dieser Arbeit we&n awei weltere Mechanismen hinxuge!%gt:die Diffusionsverformtmg da Teikben filr den Fall, da0 die Korngri5& vie1 kleiner als die Teilchengriifle ist, und die Abachnflmng der Poren voa den Komgrenaen, wean Kornwachstum vorliegt. Model1 und experimentelle Daten stimmen atm vlel besser fiber&r, insbesoadere im Falle des Komwachstums. Diagramme des isostatischen HeiRpmssens werden auf Werkxeugstilhle, Aluminiumoxid und Kupfer einschlieDlichexperimenteller Daten angewaadt und diskutiert.
NOMJZNCLATtJRF.
R, r
averaae contact ama (m2) slope of radial diibutioa
T
function
relative density initial relative density (0.64 in most of our computations) density at which pores close critical density for pore separation density from yielding densification rate (6) grain boundary diliitsion ccdicimt times boundary thickness (m’s_‘) volume dilfusion coefficient (m’s_‘) average contact force (N) fraction of pores on the boundaries average grain size (m) 1maximum grain sixe (m) Roltxmana’s constant (JK-‘) external messure (Nmm2) effective&sure during initial stage sintering (Nm-')_ internal pressure (lUmm2) limiting pressure causing yielding (Nmw2) outgassing pressure (Nm-‘) oarticle radius (m) mean particle mdius (m)
maximum particle radius (m) pore radius during final stage siateriag absolute temperature (K) melting temperature (K) volume (m3) neck radius (m) average number of contacts per particle initial number of contacts per particle geometric factor in equation (28) creep parameters (s-t, Nm-‘, -) atomic volume (m3) yield stress (Nm-‘) 1. INTRODUCTION
When a child squeezes snow to make a snowball, he is wing the technique of hot-isostatic praaaing (inela gantly, but universally known as “HIPing**). With the increasing interest in a noar net-&ape forming, HIPing has c&god in the last dccade~as an &&ant manufacturing proccas, allowing dense, prcciacly shaped bodies to be formed from pdwdcrs of metal or of ceramics. It is now used CommerciaRy to fabricate fully dense components from dikult
2163
2164
HELLE et 01.: HOT-ISOSTATIC
materials such as tool steels, superalloys, certain titanium alloys and a wide range of ceramics. But the process is an expensive one. The powder, packed into an evacuated sheet metal preform, is heated and simultaneously subjected to a high pressure (usually of argon gas) in a pressure vessel. A single run can cost many hundreds of dollars. Yet there is, still now, no systematic way of selecting the process variables (the pressure, temperature, time, particle slxe, ctc) to give the optimum results: operating conditions are found by trial and error, and may not be optimal. The last few years have seen a major advance in the understanding of the mechanisms which contribute to densificatlon during HIPing [l-8]. There are several such mechanisms some involving diffusional redistribution of ma& others involving plastic flow or power-law creep. Bach depends on the process variables in different ways, and since the overall densillcation rate is a superposition of all these contributions, its dependence on the process variables is very complex. One way of simpllfylng the problem is to construct “HIPing Me&anlsm Diagrams” [S-11] which identify the dominant me&a&n3 of den&ation (so that the dominating characteristics are easily read off) and which also display (as a set of contours) the density reached aRu a given time at a given temperature and pleasure. The diagrams provide a tool for selecting the optimal conditions both for commercial HIP cycles, and for laboratory experiments to study the underlying mechanisms. Previous papers [tll] on HIP-mechanism diagrams used models for three classes of mechanism: plastic yielding, power-law creep and diffusional densification. The equations used to describe these mechanisms were, in some cases, very complicated, with the result that the numerical procedures for computing the diagrams were complex, cumbersome and slow. The 8rst of the three aims of the present paper is to simplify the equations and streamline the computational method. We 8nd that much of the complexity is unnecessary; the densification-rate equations can be greatly simplified without important loss of precision. The second aim is to add an approximate treatment of two further mechanisms: the Nabarro-Herring-Cable deformation of the particles themselves (when the grain size is much smaller than the partide size); and the inhibition of ditfusional densification when grain growth separates pores from grain boundaries. The final aim is to use experimental data to conduct critical tests of the precision of the diagrams. A note on precision. Many material properties (like diffusion coellicients or creep rates) are known, at best, to within a factor of two. There is little point in retaining a precision of (say) one part in one hundred in the modelling when this much larger error is to be superimposed on it. In simplifying the models, we have aimed at a level of simplification which intro-
PRESSING
DIAGRAMS
duced new errors of less than f 10% in densification rate; where the discrepancy between the detailed model-based equation and a possible simplification of it was larger than this we have retained the detailed equation. If a large body of HIPing data were available for a given material, the precision of the diagrams could be increased by inferring material properties from these data (rather than using published material properties obtained by direct measurement). But the HIPing data at present available is not sufficiently extensive or precise to justify this refinement. Even without it, the diagrams have a useful precision, as we demonstrate below.
2. THE MODELLINC OF THE PARTICLE GEOMllTRY
In this and the next section, we develop equations for the rate of den&z&ion b of a powder compact, of initial density Do, by each of several mechanisms. In doing so, we draw heavily on previous work [l-14] in which d&led modela for various mechanisms of HIPlng are developed. However, these rigorous treatments generally lead to equations which are very complicated and diScult to use. One of the principal alms of the present paper is to simplify, as far as possible, the treatment of each me&a&m and the resulting equations for densl&ation-rate. Accordingly, we consider only two stages of sinterlng. During the inirhl sruge (D < 0.9) the individual powder particles are still recognisable, and densiflcation is modelled by the growth of necks where the particles are in contact, and by the increase in the number of such contact points. During thefinal stage (0.9
2165
HELLE et af.: HOT-ISOSTATIC PRESSING DIAGRAMS
And (finally) the growing particles touch more neighbours, causing the number of contacts to increase. The mathematics of all this [12-141is complicated but the results are adequately approximated by two very simple expressions. First, the number of contact neighbours per particle, 2, increases, in a roughly linear way, with density D Z=12D
XV’ -DoJR2 -(l-4)
12 10
5,
8
(1)
Z increases from the value 7.7 at the start of HIPing (assuming dense random packing for which Do =0.64) to a value of 12 when full density is reached. Second, the average area, a, of a contact is close to “3
14
(2)
(where R is the initial particle radius) giving a neck radius
and a total contact area per particle (normal&d by the surface area 4~~) of
0.6
07
0.9
0.6
1.0
DENSITY
RELATIVE
Fig. 1. The approximationfor 2 and u. The figurs shows the total contact area aZ/R’ as a fbnction of density. (A) Exact solution for o from Ref. [14].(R)Our approximation, combining equations (1) and (2). (C) The approximation used in our earlier work [l I].
If an external pressure P is applied to the compact (with a current density D and co-ordination number Z), the average contact force, /, is easily shown [141 to be
f’-&
4nR2
Q
(4) (Note that the normalised contact area correctly goes to unity as D approaches 1.) The total normal&d contact area aZ/R2, calculated from the various equations, are compared in Fig. 1 which shows uZ/R2 as a function of relative density. Although our approximation [from equations (1) and (2)] deviates somewhat from the exact solution given by Fischmeister and Arzt [14]the error is much less than that of an approximation used in our earlier workt [ 111,and is more than adequate for present purposes. The curvature of the neck, p, is given by X2 P*2(R
The contact force produces a contact pressure, Pm, on each particle contact of p&!?Ep a
Using equations (1) and (2) we obtain
4 - 4)
(which correctly goes to P as D goes to 1). The pressure resulting from surface tension is almost always insignificant compared to that from the external pressure during the initial stage of sintering, and we shall ignore it. We ho iguon the effect of gas which is trapped in the pores since compacts are usually outgassed before HIPing. 0.3
(6)
Equations (5) and (6) are compared iu Fig 2. The accuracy of the approximation is adequate: its effect on the dens&&on rate is much less than experimental scatter and uncertainties in material proper-
(9)
p”=D1(D-DO)
This can be approximated by a linear relationship between the neck curvature and the relative density which is particularly convenient for later computation P=R(D-Do)
8)
aZD
I
0.2
-
0.1
-
I
P’A
ties.
me
overlappingvolum6 per particle wao ineom?ctly
in our
uukr
work (111. It should be
V=$Z,,(R’-Ry(ZR’+R)+$(R’-R)‘(SR’+R)] where R’ = (II/&)lRR is the new particle radius.
given
0 0.6
07 RELATIVL
04
03
om!mv
Fig. 2. The approximation for the wck cwvahuc cm uni& of R) as a function of density. Full line, apation (5); broken line. equation (6).
HELLE et ul.:
2166 2.2. Thejind
HOT-ISOSTATIC
PRESSING
stage (0.9 CD < 1)
DIAGRAMS
2
At relative densities above about 0.9 the particle shape is best approximated by a polyhedron. We assume that the porosity is closed and uniformly distributed, such that the pores (spherical and of equal size) are situated at the corners of a tetrakaidecahalron (following Refs [14] and [lq). The pore radius is [IS]
Ignoring again the effects of surface tension and gas trapped in the pores, the effective pressure can simply be considered as equal to the external pressure during the 6nal stage. The internal pressure from the gas trapped in the pores could, however, eventually lead to an upper limit for densiication. The internal pressure is given by [9,11]
5a
3’
0
I
OS
07
I
OS
NELATIW OfNSlTV
Fig. 3. The approximationfor the limiting pressure of yieldingin the initialstags (dividedby the yieldstnxs) as a fbnction of density. Full line, equation (13); broken line. equation (14). cause yielding of the spherical shell surrounding each
p,,p(l-D,)D
‘(1-DID,
(11) pore. In this case the limiting pressure for yielding is
where Pe is the outgassing pressure and D, is the
density at which the pores close. This could he included in the computational scheme if nuxssary simply by subtraction from the effective pressure.
3. THE DENSIFICAI’ION RATE EQUATIONS
given by [1,81
which gives a starting density for the time dependent mech.anisms of .
(17)
3.1. P&uric yieia%lg
When pressure is applied to a powder compact it will first deform by plastic yielding. This causes the average contact arq a, to grow and the effective pressure [equation (811to fall until the yield stress, u,, of the material is no longer exceed& [8-10,121. Yielding will occur during the initial stage provided
P&Z 34,
(12)
The external pressure which willjust cause yielding is thus [using equation (911 Ph=
3D$+, -
0
~ , .3 (D’ - D:) (1 -D,) ar
(13)
(1 - 4)P+ 1.3 aY
D, I” 0 . >
Dedication auk occur by diffusion of material from the contact areas between powder particles, such that the particles move closer together and the pores6llup.Theprocesshasbeulanalysedindetail in Refs [8-lo] and [ll]. The den&cation rate by boundary diffusion during the initial stage is obtained by equating the rate of volume deposition of material on one particle by boundary diffusion to the rate of removal of the overlapping volume from the contact zones of one particle. This gives [8,1 l]
(14)
The approximate equation (14) is convenient for later computation. It is compared with the more precise equation (13) in Fig. 3, which shows that the approximation is satisfactory. Densification by yielding is instantaneous, so that densification by time-dependent mechanisms begins from the density resulting from instantaneous plastic yielding. This starting density is given by inverting equation (14) to give D,P
32 Di&jidon from interparticle boundah
(15)
The compact enters final stage sintering during plastic yeilding only if the pressure is high enough to
where the neck radius p is given by equation (6) and where g(D) is a function of D
(18)
Here SD, is the grain boundary diffusion coefficient times the boundary. thickness, D, is the lattice diffusion coefbcient, n is the volume of the diEusing atom or molecule, k is Roltzmann’s constant and T. is the absolute temperature. Using equation (I) for Z and equation (9) for Pfl, and further simplifying the density-dependent part of equation (18). we can
HELLE et al.: HOT-ISOSTATIC PRESSING DIAGRAMS
2167
3.3. Power-law creep Densification under pmssure can occur by crecpdeformation at the particle contacts. The constitutive equation for a solid exhibiting power-law creep is
(22)
Ob
I
I
01
0.6 RELATIVE
where 4, a0 and n are material properties and Z and c are the equivalent strain rate and stress, respectively. Then the densification rate by power-law creep during the initial stage is [7,8, 1l] B = 5.3(D2DO)le;
DENSITY
Fig. 4. The approximation for the initial stage dcnsitication rate by boundary diffusion. Full tine. equation (18); broken line, equation (19).
$ O()
+
*
(23)
where x and P, are given by equations (3) and (9). During the final stage, the dcnsification rate is 1798, 111
approximate the densification rate by boundary diffusion during the initial stage as *
43(1- Dl# (SD, + PD,) *P ’ = (D -Do)* kTR3
(24)
(1%
This approximate result is compared with the more precise e&ration (18) in Fig. 4. It is obviously adequate. During final-stage sintering the densification rate by diffusion is given by [8-IO]
[l-(l-Dl? 3(1-D)yl-[1+(1-D)YJ]ln(1-D)-3
P
No satisfactory approximations have been found so far for the power-law creep rate-equations, and we thus have to use them in this form. 3.4. Nabarro-Herring and Cable creep We have observed that the grain sixe of copper powder is often significantly smaller than the particle size. Then a new deformation mechanismdiffusional flow (or Nabarro-Hening/Coble creep) contributes to densification. Allowing for both volume (Nabarr+Herring creep) and grain boundary (Cable creep) diffusion the rate-equation for diffusional flow is (161
(20) L where r, the pore radius, is given by equation (10). This is well approximated by (21) Figure 5 compares quations (20) and (21). The approximation is sufbciently accurate.
=gqD*+q
(25)
whereCisthemeangrainsize.ThenifC~<2R,this deformation adds to power-law creep as a way of deforming the contact zone. The same constitutive quation applies as for power-law creep [quation (2011 but in this case n = 1 and duo is given by equating equations (22) and (25). Using equation (23) for the initial stage, we obtain
b =24.9$
(D*D~)‘P; [o.+Flp,
(26)
and using equation (24) for the final stage, we find D = 31.5 Y(l k;G
- D)[4+$+.
(27)
When computing the diagrams shown in Section 4 this mechanism was considered to operate only when the grain size, G, was smaller than the neck sire, 2x. 3.5. Pore separation and grain growth
, a¶6
0.00 MIATIVE
DLNSITV
Fig. 5. The approximation for tonal stage densi!ication rate by boundary difhion. Full line, equation (20); broken line, equation (21).
During HIPing the compact is usually held for a considerable time (an hour or so) at a high temperature (0.5-0.7 T,). This can result in grain growth and pore separation from the grain boundaries, iso-
2168
HELLE et ol.: HOT-ISOSTATIC PRESSING DIAGRAMS
lating the pores in mid-grain. When the pores no longer Iic on grain boundaries, dcnsification by diffusive mechanism is suppressed. Power-law creep and plastic flow are not, of course, affected by pore separation. Three factors inhibit grain boundary motion and pore separation the intrinsic drag on grain boundaries, the drag caused by solute, and the pinning force exerted by particles or pores. As the pores shrink,in sizt the third contribution A, and boundaries start to move. Brook (11 and Yan et al. [181have analysed the conditions for pore separation and have shown that a mini&um intrinsic and solute drag force is required to prevent it. If the drag force lies below the minimum, pore separation will occur when a critical density has been exceeded. In the limiting case of pore drag control it is possible to predict this critical density [l8]: it is
with Ic=l f,= 2.7(1 - D)y’ when this is less than 1 and DzD,.
Here, do is the densification rate when all pores are on boundaries. In computing the diagrams of Section 4 we have considered pore separation in the case of single crystal powders. It should be noted, however, that in polycrystalline powders, grain growth cannot be controlled by a Zener pinning mechanism until the grains have attained the size of the particles. 4. CONSTRUCI’IONOF THE HIP MAPS
We have constructed three types of diagrams. The first type (of which Fig. 9 is an example) is for constant temperature; the axes are the relative density (D) and logarithm of normal&d pressure [log D,- 1 -[;(l -#-)I. (28) (P/u,)]. The second (Fig. 8 is an example) is an enlargement of part of the first one, with relative The mean and the maximum particle radii (R and density (0) and linear pressure (P) as axes; it allows a more detailed picture of densification over a narR_) appear because the the driving force for grain rower range of pressure. The third type (Fig. 7) is for growth is related to the difference (R, - R). The constant pressure; the axes are relative density (0) quantity L is a geometric constant (181which allows for Merent limiting cases in the grain size distribu- and homologous temperature (T/Z’,). The maps are constructed by evaluating the con: tion: 6 - 5/4 is appropriate for a narrow size distributributions to the densification rate from each of the tion (R_ ati), and c - 3/2 is appropriate for a mechanisms described in Section 3 [equations (19). wide distriiution (&>>I). When pore separation occufs, some pores are (21), (23). (24). (26) and (2711,and summing them detached from the boundaries, but not all. Let the (since they are independent) to give the total densification rate; pore separation is included in the fraction of pores on boundaries be&. Then only a way described in Section 3.5. The increment in time fraction /I of all pores continue to shrink by is obtained by dividing a density increment by the ditTusional densifjing processes tier exceeding the critical density, and the densification rate by these densification rate, and the time required to reach the current density is given by summing the increments in “mechanismC is reduced to f#. It is pceaible to estimate the fractionf, for single time. The procedure, summa&d in the flow chart of crystal powders for which grain growth can be as- Fig. 6, is repeated for each value of T or P. Examples of the diagrams are shown in Figs 7 to sumed to be controlled by Zener pinning. The stable 15. They show, first, the field of dominance of each grain siix (Zener [191)is mechanism (that is, the range of P, T and D in which a given mechanism contributes more to the 4r c “3 densification rate than another). The field boundaries are the lines along which two mechanisms contribute where I is the radius of a void during final stage of equally to the total; the boundary of the “YIELD” sir&ring, and f - 1 - D is the volume fraction of field is the density reached by yielding alone [equavoids. The frrction of pores on grain boundarits is tions (15) and (17)]. Superimposed on the fields are obviously proportional to the area of grain boundary contours of constant time; they show the density per unit volume (c/C). We know that oil pores are on reached in a given time (we have used f, f, 1. 2 and grain boundaries when c = 2R. So [using also equa- 4 h). tions (10) and (2911 The material data used to construct maps for tool steel, alumina and copper are listed in Table 1. The /I =g z-27(1 . material properties which enter the equations are, of -D)-" course, those at the temperature of HIPing, so, as far as possible, a temperature coefficient is used to corprovided that & ZG1 and D 2 0,. Now, in computing HIP diagrams, we can allow rect properties like the modulus. Data for the of the high-temperature for pore separation by writing, for the appropriate temperature-dependence yield strength have .proved hard to find, and we have, diffusional densifying mechanisms c
fi =frd,
(31)
instead, used a fixed, average, high-temperature strength. Small errors here do not affect the results
HELLE et al.:
HOT-ISOSTATIC
PRESSING
DIAGRAMS
2169
HIP MAP PROGRAM
I READ MATERIAL DATA atomic volume
n
Burgers vector
b
melting temperature
Tm
shear modulus
uo'
yield strength diffusion creep
TmdP i$q
OY D ov' Q,r n, A
data data
6n,,,’
Q,
SELECT TYPE OF MAP D-LOG (P/oy) DfT;Tm
1 -READ PLOTTING DATA P, T, R. Rmax, d, time contours 1 1 SET STARTING
P
(OR T)j
CALCULATE D FROM YIELDING lq EVALUATE DENSIFICATION RATE-EQUATIONS SUM-UP FOR TOTAL DENSIFICATION RATE INTEGRATE FOR TIME 1 INCREASE DENSITY BY ONE STEP 1 IS FULL DENSITY REACHED? 1
No
Yes
INCREASE P (OR T) BY
ONE
STE
1 HAS WHOLE RANGE BEEN EXAMINED?
No
Fig. 6. Flow chart of the computer program for construction of HIP maps.
much, since underestimating the starting density (by yield) causes more rapid densification by the other mechanisms, largely compensating for the error. How accumte are the diagrams? The biggest uncertainty lies in the material properties: diffusion coefficients (particularly for grain boundary diffusion) and power-law creep constants are, often, very uncertain. We have exercised the maximum
critical judgement in selecting the data, extmcting the material pammeters by the methods used by Frost and Ashby [lq. We then 6nd that the maps give a tolerably good fit to the HIPing data. Further refinement of the material pammekmtohnprovethe fit is possible (by using the HIPing data itself to correct diffusion and power-law creep parameters), but we have not attempted it here.
2170
HELLE et ok: HOT-ISOSTATIC PRESSING DIAGRAMS
I NDuoLDDous
TEMPER*nlaE.
R,
* 5Oum
a=50rm
I
111.
NORLULISEO
Fig. 7. Dedy4emperahtrc diagram for hot isostatic pnssillg of tool 8ted.
Wg.
PWSSLRE.
LCG(P/U,,
10. Density-normaliscdprtswre diagram for hot isostatic-pressingof tool steel (lo~scale). TEbeERATvRE. lc 1200
P-L
of too1 steel (linear scale). PaEssuRE.
Hc44OLOGDUS
lDeEnAllM
TIT,
Fig. 11. Density-temperature diagram for hot isostatic pressing of A&4.
mid
‘c
TEMPERATURE,
loo
10
1500
CREEP
diagram for hot kostatic pressing
Fig. 8. Dady-pressure
8400
1100
lDO0
1200
UDO
1500
BaJNMRv 0.9
-
DIFFUSION BWNMRV
DIFFUSION
P~2WMNlm~ Ral25m &.,s25wn ir = 1DDym 0.6
I -2
I
-1 NORUALISED
Fig.
pM55UUE.
a.25vm
II
1
0
1
. 0.5
LCG(PlU,)
9. Density-nomdiaai pmsmre diagram for hot isostatic pressing of tool steel (log scale).
5. APPLICATION OF THE MAPS
We have constructed HIP diagrams and compared them with experimental data for three different materials: a tool steel, alumina and copper. Hotisostatically pressed tool steel is already a commercial product: it has better uniformitv - and a finer dis-
05
0.7 HOMOLODOUS
Fig.
*SE*
TEWERATURE
1 I 1.
12. Duuity-tanperaturc diagmm for hot isostatic prcsing of Al,o,.
persion of carbide particles than that obtained by conventional methods. Alumina is an example of a ceramic for which HIPing provides a manufacturing route to high densities without the addition of sintering aids such as MgO. For both materials industrial HIPing data exist. Copper was selected as a third material since it is particularly well characterised and
2171
HELLE er al.: HOT-ISOSTATIC PRESSING DIAGRAMS lc
TENPEAATlmE,
c;0999 B e
P= 150MN/m’
f 0.990 l
0.8
1
PQ
1
MEA
I
150
loo
PRESSURE.
200
a*lOvm VIELO
250 0.985
Fig. 13. Density-pressure diagram for hot isostatic pressing of Al@, (linear scale). IO
8 0.45
060
0%
0.10 nOMOLoGWS
EEA
TEMPERATURE.
0.65
0
1 I T.
Fig. 15. Density-tcmpcraturc diagram for hot &static pressing of copper.
PRESSURE. MN I m’ I
.
MN/m*
100
aI-
there is commercial interest in producing high purity copper containers for nuclear waste storage by HIPing. bemuse
5.1. Tool steel A typical composition of a tool steel fabricated by hot-isostatic pressing is given in .Table 2. Hotisostatic pressing data obtained from Helhnan [20] arc listed in Table 3. HIP diagrams based on the c@ations given in the text arc shown in Fig 7-10. Hellman’s data arc plotted on the diagrams. Figure 7, with axes of D and T, was const~cted for a particle diameter of 50 gm and a constant pressure of lOOMPa. It shows that power-law creep is the
I a=lovm
. 0.6 ‘-2 NORM&
PRESSURE.
L&t
PM,)
Fig. 14. Density-nomaliscd pmssun diagram for hot isostatic passing of copper (log scale).
Tabk I. M~terkl data Tool steer”
Materialproperty Atomk vohme, lXm9
1.21 x 10-m
2.58x IO_‘0 1680 2w 8.1 x IO -0.85
1.55 x IOJ -0.35
4.21 x lo’ -0.34
3.7 x lo-’ 280
2.8 x lo-” 477
2 x 10-s 197
2 x lo-” 167
8.6 x IO-lo 419
5 x lo-‘3 104
7.3 ._ I.5 x IO”
::!I
4.8 7.4 x Id
A (Dam conslant)
*)A tanpwanue
A&* 4.25 x IO-” 4.76 x IO-lo 2320 Is00
dependentmodulus I138 been u6cdz
r=l’ol I+ !ig!?!p!z!) I_ VW ‘.u
*)D,= Dti exp( -Q,/RIRT) ma/r. %D,-8D,exp(-Q,/RT)m’/r. “‘he power-hw creep pammeW
1. /_I
in tbc equations (23) and (24) UC given by 4
AbD,
i+=zijF “h
M no cmcp data for tool nedr at HIP tempemWrc9 (around 12WC). In this tanpcmuirc range the steel b mMcnitic and all the alloying ckncnta arc in solid solution. Ilw mccbanicnl
mIbe data am for pure dumirm, taken from Frost and Asbby [la]. @‘fbc data arc for pure copper, taken from Frost and A&y [l6].
2172
HELLE et a/.: HOT-ISOSTATIC PRESSING DIAGRAMS
The diagrams in Figs 11 and 12 are constructed at a constant pressure of 100and 200 MPa, respectively. Both show that boundary diffusion is the dominant mechanism and that grain growth leading to pore separation is anticipated (giving the rather abrupt change in slope of the time contours). The experimental data (solid circles) show good agreement with the model. Initially densification is fast and high densities are reached rapidly when the temperature is above 0.65 T,. By increasing the temperature by only 3O”C,the time required to reach a given density can be halved. But at the critical density for grain growth densiIlcation becomes much slower, and the gain due to increased temperature diminishes. The diagrams show that, to achieve full density, temperatures in the region of 1500°C are required. If grain growth is prevented then full density should be reached in short times at temperatures in the region of 1300°C. Figure 13 shows a HIP diagram at a constant temperature of 1300°C.Again, wcperimental data [21] is in good agreement with it. The effect of increasing pressure on the den&cation rate is rather small: the pressure must be doubled in order to halve the time required to reach a certain density. Grain growth sets a limit to the maximum density obtainable when HIPing at realistic pressures. Since the dominant mechanism is diffusion, the dens&&ion rates could be substantially increased by decreasing the particle sire [see equation (2111,but the maximum density would still ultimately depend on whether or not grain growth occmred.
Table 2. Composition of a typical HlPcd tool steel (wtyJ C
Cr
MO
1.3-2.3
-4.0
S&7.0
w
V
co
-6.5
3.1-6.5
8.5-10.5
Tabk 3. Hot-kostatk pressing data of tool steel f201
1 2
I
Ef 9916
: 1
:: 9918
I
98.9 99.9 full 92.6
: 1 *Not ~tatal.
dominant me&a&n at low densities, switching to boundary dilfusion at high. The data agree fairly well with the predictions of the diagram, which explains the very slow incmase indensityasfulldensityis approached, caused by pore separation. The critical density for separation, equation (28) is D, = 99.30/, Grain growth can, of cow be reduced or prevented by other means-by solute drag or by precipitate pinning for examplo4o our condition gives a lower boundforD,TheFigsSand9(withaxesofDand P) show that full density can be reached by increasing the pressure, an observation conllrmed’by Helhnan’s data. Figure 10 shows what happens if the grain sire of 5.3. Copper an individual particle is smaller than the neck sire. HIP diagrams for copper are shown in Figs 14 and Now a new me&a&m (Nabarr~Herring/Coble 15. The diagram shown in Fig. 14 is for polycreep) becomes dominant at low pressures and accel- crystalline copper powder with a mean particle dierates the rate of densiflcation. ameter of 150pm and a mean 8rain size of 10 pm. It is constructed for a constant temperature of 550°C. 5.2 Allmlina with axes of D and log (P/a,). At this temperature, HIP diagrams for alumina are shown in Figs a pressure of a little above 100MPa is required if full 1l-13, each of them being for single crystal powder density is to be achieved in a reasonable time. Powerwith a mean particle diameter of 2.5pm. Density law creep is the dominant mechanism following an data [21] for high purity alumina powders initial region of plastic yielding, the latter already (A&O,> 99.Yb are listed in Table 4, and plotted on resulting in a considerable dens&ation due to the the diagrams. low yield strength of copper at elevated temperatures. Tabk 4. Hot-ix&&
(rem) 2.5 ;: 2:5 2.5 z 2:5 2.5 2.5 2.5 2.5 0.5
PmIsun (MNm-*)
urmain~ data of ALO, (211
Temperature (‘Cl
Tiie (hour)
Den&y 0
2 2 4 2 4
full 97.5 98. I 99.6 99.5
Es
1350 1200 1200 1200 I200 1300 13tKl 1300
E 9915
t: 200 200 Is0
tz 1400 1400 1350
: 2 2 4 : 2
99:s 99.6 full
150 loo ii 200 100
E
Comments
Green dcmity = 60 to 65% Estimated density at HIP temperature = 70% Cold isostrtic pnuing at lOOMPa used to obtain 6Y/, density Extreme grain growth
HELLE et d.:
HOT-ISOSTATIC
PRESSING DIAGRAMS
2173
Tegman [22] has HIPed copper at a pressure of 150 h&a when exploring the possibility of manufacturing containers for nuclear waste. He used a 60”/,/40”/, mixture of the Alcan UHP and Ecka AK9 1 powders described in Table 5, and obtained a density of 99.7% at 500°C in 5-10 h. Figure 15 shows the densification behaviour of copper powder at 150 MPa. Tegman’s result is in reasonable agreement with the computed time. Since power-law creep is the dominant densification mechanism a&r yielding, the particle size has almost no effect on the dmsification rate. Plastic yielding alone results in a density close to 99% of the theoretical. As can be seen, timedependent densification is very slow below 450°C. An increase in temperature of less than 20°C halves the
time required to reach a certain density, and above 600°C full density can be reached rapidly. Since the dominant mechanism is power-law creep, grain growth with pore separation has little effect on densification. Although the diagrams show a field of Nabarro-Herring/Cable creep, copper cannot easily be used to confirm this mechanism experimentally since it dominates only under conditions which lead to unacceptably long sintering times. 6. CONCLUDING
REMARKS
The simplifications and modifications to the models for hot-isostatic pressing described above, allow the computation of diagrams in a fast, straightforward way. The present equations and procedure are a significant improvement over the earlier methods [!3-1 l] and lead to good agreement between theoretical prediction and experimental data for copper, tool steel and alumina. The simpriscatianr make the computationai program more amenable for use on a microcomputer, it being the ultimate objective of this project to develop a system to provide maps as a direct (on-line) aid in selecting the best combination of process variables for industrial HIPing so that
expensive “trial and error’* runs can be largely avoided: The equations and program described here largely realise that goal. Acknowledgements-We should like to thank SKBS (Swcd-
ish Atomic Fuel Systems) and the SERC (British Science and Engineering Research Council) for partialsupportof this project.
REFERENCES 1. C. Torrc. U. Berg and H. H. Huttcnman, Hoehschule &&en 93, 62 (1948). 2. 1. MacKenzie and R. Shuttleworth. Proc. Phys. Sot., Land. 62, 833 (1949). 3. A. K. Kakar and A C. D. Challandcr, Trans.Am. Ihtt. Min. Izngrs24% 1117 (196%). 4. M. S. Koval’chenko and 0. V. Samsanov. Povoshkhov. hfetull. 1, 3 (1961). R. L. Cable. J. a&. Phys. 41, 4798 (1970). :: R. L. Hewitt. W. Wallace and M. C. de Malhcrbe.. Pow&r Metall. 16, 4798 (1973). 7. D. S. Wilkinson and M. F. Ashby. Aera met&l. 23, 1277 (1975).
2174
HELLE ef al.:
HOT-ISOSTATIC
8. D. S. Wilkinson, Ph.D. thesis, Univ. of Cambridge (1977). 9. D. S. Wilkinson and M. F. Ashby, Proc. 4h hr. &II on Situaud Related Phemmena p. 473. Pknrrm, New York (1975). 10. D. S. Wilkinson and M. F. Ashby, Science of Sfntering, 4th Itu Round Table on Simdtg, Vol. 10, pp. 67-76 (1978). 11. E. Arzt, M. F. Ashby and K. E. Easterling, Mcroll. ~mts. A 14s 211 (1983). 12. E. AIZ& Aclcr metull. 30, 1883 (1982). 13. H. F. Fiscluaistcr, E. Arzt and L. R. Olson, Pow&r Meld/. 21, 179 (1978). 14. H. F. Fiitcr and E. hnt, Pow&r Metull. 26,82 (1983).
PRESSING
DIAGRAMS
15. F. B. Swinkels and M. F. A&by. Acta meroll. 28, 259 (1981). 16. H. J. Frost and .M. F. Ashby, Deformatim-Mechania Maps. Pcrgamon Rat, oxford (1982). 17. R. J. Brook, /. Am. Cerawt. Sot. 52, 56 (1969). 18. M. F. Yan, R. M. Cannon, Jr, H. K. BoWenand V. Chowdluy, Muter. Sri. khgng 60, 27s (1983). 19. C. Zencr, quoted by C. S. Smith, 7hu. Am. hr. Min. &&rJ 175, ls (1948). 20. P. Hellman, Uddcholm AB, Sweden, private communication (1981). 21. H. Larker. ASEA, Rohrtsfors, Swakn, private communicntion (1981). 22. R. Tcgman, ASEA, Robutsfon, Sweden, private cummunication (1983).
ooo1-6l60/8553.00+ 0.00
Printed in Great Britain. All rights resuwd
Copyright Q 1985 Pergamon Prcsa L.td
HOT-ISOSTATIC PRESSING DIAGRAMS: NEW DEVELOPMENTS A.S.HELLE,'K.E.EASTERLING'and M. F. ASHBV ‘Department of Engineering Materials, University of LuleA, S-951 87 LuleA, Sweden and %epartment of Engineering, University of Cambridge, Cambridge CB2 IPZ, England (Receiued 9 February 1985; in revisedform
I May
1985)
Abatrati-The equations and procexlurcs for constructing hot-isostatic pressing diagrams are greatly simplified and clarified. In earlier work, three classes of mechanism were modelled: plastic yielding,
power-law creep and diffusional densification. In this paper two further mechanisms are added: the diffusional deformation of the particles themselves when the grain size is much smaller than the particle size. and the separation of pores from boundaries when grain growth occurs. Substantially better agreement between the model and experimental data is now obtained, particularly in cases where grain growth has occuned. Application of HIP diagrams to tool steels, alumina and copper, incorporatiag experimental data, are presented and discussed. R&tar&-Nous avons nettcment simplifiCIts equations et les procedures pour con&ruin les diagrammes de pmssage isostatique i chaud. Dans un article ant&rieur, nous avions mod&a6 trols classes de m&anismes: l’&oukment plastique, le fluage en loi de puissance et la densification par di!Rtsion. Darts cet article, aous ajoutons deux autres m&anismcs: la dCformation par diffusion da particules elks-m&nes quand la taille des grains est beaucoup plus petite que la taille des particules et la s¶t@tdes pores g partir des joints de grains lorsque la croissance des grains SCproduit. Nous obtenoas mam&nant un accord aettemeat meilkur entre k modhle et les r6sultats expcrimentaux, en particulkr daas ks cas oul il s’est produit une croissance des grains. Nous pr4sentons et nous discutons l’applkation der diylnmmes de PIC aux a&m g outils , B l’alumine et au cuivre en y introduisant des r&hats @timentaux. B-Die Gkichungen turd Verfahren xur Konstruktion voa Diagrammen des lwstatischen HeiDpressens we&m stark vereinfacbt uad aulbereitet. In frtiheren Arbeiten wurden dml Klasacn voa Mechanismsn modelliert: plastischer FlieBbeginn, Potenagesetxkriechen und Difhtsionsv&dttung. In dieser Arbeit we&n awei weltere Mechanismen hinxuge!%gt:die Diffusionsverformtmg da Teikben filr den Fall, da0 die Korngri5& vie1 kleiner als die Teilchengriifle ist, und die Abachnflmng der Poren voa den Komgrenaen, wean Kornwachstum vorliegt. Model1 und experimentelle Daten stimmen atm vlel besser fiber&r, insbesoadere im Falle des Komwachstums. Diagramme des isostatischen HeiRpmssens werden auf Werkxeugstilhle, Aluminiumoxid und Kupfer einschlieDlichexperimenteller Daten angewaadt und diskutiert.
NOMJZNCLATtJRF.
R, r
averaae contact ama (m2) slope of radial diibutioa
T
function
relative density initial relative density (0.64 in most of our computations) density at which pores close critical density for pore separation density from yielding densification rate (6) grain boundary diliitsion ccdicimt times boundary thickness (m’s_‘) volume dilfusion coefficient (m’s_‘) average contact force (N) fraction of pores on the boundaries average grain size (m) 1maximum grain sixe (m) Roltxmana’s constant (JK-‘) external messure (Nmm2) effective&sure during initial stage sintering (Nm-')_ internal pressure (lUmm2) limiting pressure causing yielding (Nmw2) outgassing pressure (Nm-‘) oarticle radius (m) mean particle mdius (m)
maximum particle radius (m) pore radius during final stage siateriag absolute temperature (K) melting temperature (K) volume (m3) neck radius (m) average number of contacts per particle initial number of contacts per particle geometric factor in equation (28) creep parameters (s-t, Nm-‘, -) atomic volume (m3) yield stress (Nm-‘) 1. INTRODUCTION
When a child squeezes snow to make a snowball, he is wing the technique of hot-isostatic praaaing (inela gantly, but universally known as “HIPing**). With the increasing interest in a noar net-&ape forming, HIPing has c&god in the last dccade~as an &&ant manufacturing proccas, allowing dense, prcciacly shaped bodies to be formed from pdwdcrs of metal or of ceramics. It is now used CommerciaRy to fabricate fully dense components from dikult
2163
2164
HELLE et 01.: HOT-ISOSTATIC
materials such as tool steels, superalloys, certain titanium alloys and a wide range of ceramics. But the process is an expensive one. The powder, packed into an evacuated sheet metal preform, is heated and simultaneously subjected to a high pressure (usually of argon gas) in a pressure vessel. A single run can cost many hundreds of dollars. Yet there is, still now, no systematic way of selecting the process variables (the pressure, temperature, time, particle slxe, ctc) to give the optimum results: operating conditions are found by trial and error, and may not be optimal. The last few years have seen a major advance in the understanding of the mechanisms which contribute to densificatlon during HIPing [l-8]. There are several such mechanisms some involving diffusional redistribution of ma& others involving plastic flow or power-law creep. Bach depends on the process variables in different ways, and since the overall densillcation rate is a superposition of all these contributions, its dependence on the process variables is very complex. One way of simpllfylng the problem is to construct “HIPing Me&anlsm Diagrams” [S-11] which identify the dominant me&a&n3 of den&ation (so that the dominating characteristics are easily read off) and which also display (as a set of contours) the density reached aRu a given time at a given temperature and pleasure. The diagrams provide a tool for selecting the optimal conditions both for commercial HIP cycles, and for laboratory experiments to study the underlying mechanisms. Previous papers [tll] on HIP-mechanism diagrams used models for three classes of mechanism: plastic yielding, power-law creep and diffusional densification. The equations used to describe these mechanisms were, in some cases, very complicated, with the result that the numerical procedures for computing the diagrams were complex, cumbersome and slow. The 8rst of the three aims of the present paper is to simplify the equations and streamline the computational method. We 8nd that much of the complexity is unnecessary; the densification-rate equations can be greatly simplified without important loss of precision. The second aim is to add an approximate treatment of two further mechanisms: the Nabarro-Herring-Cable deformation of the particles themselves (when the grain size is much smaller than the partide size); and the inhibition of ditfusional densification when grain growth separates pores from grain boundaries. The final aim is to use experimental data to conduct critical tests of the precision of the diagrams. A note on precision. Many material properties (like diffusion coellicients or creep rates) are known, at best, to within a factor of two. There is little point in retaining a precision of (say) one part in one hundred in the modelling when this much larger error is to be superimposed on it. In simplifying the models, we have aimed at a level of simplification which intro-
PRESSING
DIAGRAMS
duced new errors of less than f 10% in densification rate; where the discrepancy between the detailed model-based equation and a possible simplification of it was larger than this we have retained the detailed equation. If a large body of HIPing data were available for a given material, the precision of the diagrams could be increased by inferring material properties from these data (rather than using published material properties obtained by direct measurement). But the HIPing data at present available is not sufficiently extensive or precise to justify this refinement. Even without it, the diagrams have a useful precision, as we demonstrate below.
2. THE MODELLINC OF THE PARTICLE GEOMllTRY
In this and the next section, we develop equations for the rate of den&z&ion b of a powder compact, of initial density Do, by each of several mechanisms. In doing so, we draw heavily on previous work [l-14] in which d&led modela for various mechanisms of HIPlng are developed. However, these rigorous treatments generally lead to equations which are very complicated and diScult to use. One of the principal alms of the present paper is to simplify, as far as possible, the treatment of each me&a&m and the resulting equations for densl&ation-rate. Accordingly, we consider only two stages of sinterlng. During the inirhl sruge (D < 0.9) the individual powder particles are still recognisable, and densiflcation is modelled by the growth of necks where the particles are in contact, and by the increase in the number of such contact points. During thefinal stage (0.9
2165
HELLE et af.: HOT-ISOSTATIC PRESSING DIAGRAMS
And (finally) the growing particles touch more neighbours, causing the number of contacts to increase. The mathematics of all this [12-141is complicated but the results are adequately approximated by two very simple expressions. First, the number of contact neighbours per particle, 2, increases, in a roughly linear way, with density D Z=12D
XV’ -DoJR2 -(l-4)
12 10
5,
8
(1)
Z increases from the value 7.7 at the start of HIPing (assuming dense random packing for which Do =0.64) to a value of 12 when full density is reached. Second, the average area, a, of a contact is close to “3
14
(2)
(where R is the initial particle radius) giving a neck radius
and a total contact area per particle (normal&d by the surface area 4~~) of
0.6
07
0.9
0.6
1.0
DENSITY
RELATIVE
Fig. 1. The approximationfor 2 and u. The figurs shows the total contact area aZ/R’ as a fbnction of density. (A) Exact solution for o from Ref. [14].(R)Our approximation, combining equations (1) and (2). (C) The approximation used in our earlier work [l I].
If an external pressure P is applied to the compact (with a current density D and co-ordination number Z), the average contact force, /, is easily shown [141 to be
f’-&
4nR2
Q
(4) (Note that the normalised contact area correctly goes to unity as D approaches 1.) The total normal&d contact area aZ/R2, calculated from the various equations, are compared in Fig. 1 which shows uZ/R2 as a function of relative density. Although our approximation [from equations (1) and (2)] deviates somewhat from the exact solution given by Fischmeister and Arzt [14]the error is much less than that of an approximation used in our earlier workt [ 111,and is more than adequate for present purposes. The curvature of the neck, p, is given by X2 P*2(R
The contact force produces a contact pressure, Pm, on each particle contact of p&!?Ep a
Using equations (1) and (2) we obtain
4 - 4)
(which correctly goes to P as D goes to 1). The pressure resulting from surface tension is almost always insignificant compared to that from the external pressure during the initial stage of sintering, and we shall ignore it. We ho iguon the effect of gas which is trapped in the pores since compacts are usually outgassed before HIPing. 0.3
(6)
Equations (5) and (6) are compared iu Fig 2. The accuracy of the approximation is adequate: its effect on the dens&&on rate is much less than experimental scatter and uncertainties in material proper-
(9)
p”=D1(D-DO)
This can be approximated by a linear relationship between the neck curvature and the relative density which is particularly convenient for later computation P=R(D-Do)
8)
aZD
I
0.2
-
0.1
-
I
P’A
ties.
me
overlappingvolum6 per particle wao ineom?ctly
in our
uukr
work (111. It should be
V=$Z,,(R’-Ry(ZR’+R)+$(R’-R)‘(SR’+R)] where R’ = (II/&)lRR is the new particle radius.
given
0 0.6
07 RELATIVL
04
03
om!mv
Fig. 2. The approximation for the wck cwvahuc cm uni& of R) as a function of density. Full line, apation (5); broken line. equation (6).
HELLE et ul.:
2166 2.2. Thejind
HOT-ISOSTATIC
PRESSING
stage (0.9 CD < 1)
DIAGRAMS
2
At relative densities above about 0.9 the particle shape is best approximated by a polyhedron. We assume that the porosity is closed and uniformly distributed, such that the pores (spherical and of equal size) are situated at the corners of a tetrakaidecahalron (following Refs [14] and [lq). The pore radius is [IS]
Ignoring again the effects of surface tension and gas trapped in the pores, the effective pressure can simply be considered as equal to the external pressure during the 6nal stage. The internal pressure from the gas trapped in the pores could, however, eventually lead to an upper limit for densiication. The internal pressure is given by [9,11]
5a
3’
0
I
OS
07
I
OS
NELATIW OfNSlTV
Fig. 3. The approximationfor the limiting pressure of yieldingin the initialstags (dividedby the yieldstnxs) as a fbnction of density. Full line, equation (13); broken line. equation (14). cause yielding of the spherical shell surrounding each
p,,p(l-D,)D
‘(1-DID,
(11) pore. In this case the limiting pressure for yielding is
where Pe is the outgassing pressure and D, is the
density at which the pores close. This could he included in the computational scheme if nuxssary simply by subtraction from the effective pressure.
3. THE DENSIFICAI’ION RATE EQUATIONS
given by [1,81
which gives a starting density for the time dependent mech.anisms of .
(17)
3.1. P&uric yieia%lg
When pressure is applied to a powder compact it will first deform by plastic yielding. This causes the average contact arq a, to grow and the effective pressure [equation (811to fall until the yield stress, u,, of the material is no longer exceed& [8-10,121. Yielding will occur during the initial stage provided
P&Z 34,
(12)
The external pressure which willjust cause yielding is thus [using equation (911 Ph=
3D$+, -
0
~ , .3 (D’ - D:) (1 -D,) ar
(13)
(1 - 4)P+ 1.3 aY
D, I” 0 . >
Dedication auk occur by diffusion of material from the contact areas between powder particles, such that the particles move closer together and the pores6llup.Theprocesshasbeulanalysedindetail in Refs [8-lo] and [ll]. The den&cation rate by boundary diffusion during the initial stage is obtained by equating the rate of volume deposition of material on one particle by boundary diffusion to the rate of removal of the overlapping volume from the contact zones of one particle. This gives [8,1 l]
(14)
The approximate equation (14) is convenient for later computation. It is compared with the more precise equation (13) in Fig. 3, which shows that the approximation is satisfactory. Densification by yielding is instantaneous, so that densification by time-dependent mechanisms begins from the density resulting from instantaneous plastic yielding. This starting density is given by inverting equation (14) to give D,P
32 Di&jidon from interparticle boundah
(15)
The compact enters final stage sintering during plastic yeilding only if the pressure is high enough to
where the neck radius p is given by equation (6) and where g(D) is a function of D
(18)
Here SD, is the grain boundary diffusion coefficient times the boundary. thickness, D, is the lattice diffusion coefbcient, n is the volume of the diEusing atom or molecule, k is Roltzmann’s constant and T. is the absolute temperature. Using equation (I) for Z and equation (9) for Pfl, and further simplifying the density-dependent part of equation (18). we can
HELLE et al.: HOT-ISOSTATIC PRESSING DIAGRAMS
2167
3.3. Power-law creep Densification under pmssure can occur by crecpdeformation at the particle contacts. The constitutive equation for a solid exhibiting power-law creep is
(22)
Ob
I
I
01
0.6 RELATIVE
where 4, a0 and n are material properties and Z and c are the equivalent strain rate and stress, respectively. Then the densification rate by power-law creep during the initial stage is [7,8, 1l] B = 5.3(D2DO)le;
DENSITY
Fig. 4. The approximation for the initial stage dcnsitication rate by boundary diffusion. Full tine. equation (18); broken line, equation (19).
$ O()
+
*
(23)
where x and P, are given by equations (3) and (9). During the final stage, the dcnsification rate is 1798, 111
approximate the densification rate by boundary diffusion during the initial stage as *
43(1- Dl# (SD, + PD,) *P ’ = (D -Do)* kTR3
(24)
(1%
This approximate result is compared with the more precise e&ration (18) in Fig. 4. It is obviously adequate. During final-stage sintering the densification rate by diffusion is given by [8-IO]
[l-(l-Dl? 3(1-D)yl-[1+(1-D)YJ]ln(1-D)-3
P
No satisfactory approximations have been found so far for the power-law creep rate-equations, and we thus have to use them in this form. 3.4. Nabarro-Herring and Cable creep We have observed that the grain sixe of copper powder is often significantly smaller than the particle size. Then a new deformation mechanismdiffusional flow (or Nabarro-Hening/Coble creep) contributes to densification. Allowing for both volume (Nabarr+Herring creep) and grain boundary (Cable creep) diffusion the rate-equation for diffusional flow is (161
(20) L where r, the pore radius, is given by equation (10). This is well approximated by (21) Figure 5 compares quations (20) and (21). The approximation is sufbciently accurate.
=gqD*+q
(25)
whereCisthemeangrainsize.ThenifC~<2R,this deformation adds to power-law creep as a way of deforming the contact zone. The same constitutive quation applies as for power-law creep [quation (2011 but in this case n = 1 and duo is given by equating equations (22) and (25). Using equation (23) for the initial stage, we obtain
b =24.9$
(D*D~)‘P; [o.+Flp,
(26)
and using equation (24) for the final stage, we find D = 31.5 Y(l k;G
- D)[4+$+.
(27)
When computing the diagrams shown in Section 4 this mechanism was considered to operate only when the grain size, G, was smaller than the neck sire, 2x. 3.5. Pore separation and grain growth
, a¶6
0.00 MIATIVE
DLNSITV
Fig. 5. The approximation for tonal stage densi!ication rate by boundary difhion. Full line, equation (20); broken line, equation (21).
During HIPing the compact is usually held for a considerable time (an hour or so) at a high temperature (0.5-0.7 T,). This can result in grain growth and pore separation from the grain boundaries, iso-
2168
HELLE et ol.: HOT-ISOSTATIC PRESSING DIAGRAMS
lating the pores in mid-grain. When the pores no longer Iic on grain boundaries, dcnsification by diffusive mechanism is suppressed. Power-law creep and plastic flow are not, of course, affected by pore separation. Three factors inhibit grain boundary motion and pore separation the intrinsic drag on grain boundaries, the drag caused by solute, and the pinning force exerted by particles or pores. As the pores shrink,in sizt the third contribution A, and boundaries start to move. Brook (11 and Yan et al. [181have analysed the conditions for pore separation and have shown that a mini&um intrinsic and solute drag force is required to prevent it. If the drag force lies below the minimum, pore separation will occur when a critical density has been exceeded. In the limiting case of pore drag control it is possible to predict this critical density [l8]: it is
with Ic=l f,= 2.7(1 - D)y’ when this is less than 1 and DzD,.
Here, do is the densification rate when all pores are on boundaries. In computing the diagrams of Section 4 we have considered pore separation in the case of single crystal powders. It should be noted, however, that in polycrystalline powders, grain growth cannot be controlled by a Zener pinning mechanism until the grains have attained the size of the particles. 4. CONSTRUCI’IONOF THE HIP MAPS
We have constructed three types of diagrams. The first type (of which Fig. 9 is an example) is for constant temperature; the axes are the relative density (D) and logarithm of normal&d pressure [log D,- 1 -[;(l -#-)I. (28) (P/u,)]. The second (Fig. 8 is an example) is an enlargement of part of the first one, with relative The mean and the maximum particle radii (R and density (0) and linear pressure (P) as axes; it allows a more detailed picture of densification over a narR_) appear because the the driving force for grain rower range of pressure. The third type (Fig. 7) is for growth is related to the difference (R, - R). The constant pressure; the axes are relative density (0) quantity L is a geometric constant (181which allows for Merent limiting cases in the grain size distribu- and homologous temperature (T/Z’,). The maps are constructed by evaluating the con: tion: 6 - 5/4 is appropriate for a narrow size distributributions to the densification rate from each of the tion (R_ ati), and c - 3/2 is appropriate for a mechanisms described in Section 3 [equations (19). wide distriiution (&>>I). When pore separation occufs, some pores are (21), (23). (24). (26) and (2711,and summing them detached from the boundaries, but not all. Let the (since they are independent) to give the total densification rate; pore separation is included in the fraction of pores on boundaries be&. Then only a way described in Section 3.5. The increment in time fraction /I of all pores continue to shrink by is obtained by dividing a density increment by the ditTusional densifjing processes tier exceeding the critical density, and the densification rate by these densification rate, and the time required to reach the current density is given by summing the increments in “mechanismC is reduced to f#. It is pceaible to estimate the fractionf, for single time. The procedure, summa&d in the flow chart of crystal powders for which grain growth can be as- Fig. 6, is repeated for each value of T or P. Examples of the diagrams are shown in Figs 7 to sumed to be controlled by Zener pinning. The stable 15. They show, first, the field of dominance of each grain siix (Zener [191)is mechanism (that is, the range of P, T and D in which a given mechanism contributes more to the 4r c “3 densification rate than another). The field boundaries are the lines along which two mechanisms contribute where I is the radius of a void during final stage of equally to the total; the boundary of the “YIELD” sir&ring, and f - 1 - D is the volume fraction of field is the density reached by yielding alone [equavoids. The frrction of pores on grain boundarits is tions (15) and (17)]. Superimposed on the fields are obviously proportional to the area of grain boundary contours of constant time; they show the density per unit volume (c/C). We know that oil pores are on reached in a given time (we have used f, f, 1. 2 and grain boundaries when c = 2R. So [using also equa- 4 h). tions (10) and (2911 The material data used to construct maps for tool steel, alumina and copper are listed in Table 1. The /I =g z-27(1 . material properties which enter the equations are, of -D)-" course, those at the temperature of HIPing, so, as far as possible, a temperature coefficient is used to corprovided that & ZG1 and D 2 0,. Now, in computing HIP diagrams, we can allow rect properties like the modulus. Data for the of the high-temperature for pore separation by writing, for the appropriate temperature-dependence yield strength have .proved hard to find, and we have, diffusional densifying mechanisms c
fi =frd,
(31)
instead, used a fixed, average, high-temperature strength. Small errors here do not affect the results
HELLE et al.:
HOT-ISOSTATIC
PRESSING
DIAGRAMS
2169
HIP MAP PROGRAM
I READ MATERIAL DATA atomic volume
n
Burgers vector
b
melting temperature
Tm
shear modulus
uo'
yield strength diffusion creep
TmdP i$q
OY D ov' Q,r n, A
data data
6n,,,’
Q,
SELECT TYPE OF MAP D-LOG (P/oy) DfT;Tm
1 -READ PLOTTING DATA P, T, R. Rmax, d, time contours 1 1 SET STARTING
P
(OR T)j
CALCULATE D FROM YIELDING lq EVALUATE DENSIFICATION RATE-EQUATIONS SUM-UP FOR TOTAL DENSIFICATION RATE INTEGRATE FOR TIME 1 INCREASE DENSITY BY ONE STEP 1 IS FULL DENSITY REACHED? 1
No
Yes
INCREASE P (OR T) BY
ONE
STE
1 HAS WHOLE RANGE BEEN EXAMINED?
No
Fig. 6. Flow chart of the computer program for construction of HIP maps.
much, since underestimating the starting density (by yield) causes more rapid densification by the other mechanisms, largely compensating for the error. How accumte are the diagrams? The biggest uncertainty lies in the material properties: diffusion coefficients (particularly for grain boundary diffusion) and power-law creep constants are, often, very uncertain. We have exercised the maximum
critical judgement in selecting the data, extmcting the material pammeters by the methods used by Frost and Ashby [lq. We then 6nd that the maps give a tolerably good fit to the HIPing data. Further refinement of the material pammekmtohnprovethe fit is possible (by using the HIPing data itself to correct diffusion and power-law creep parameters), but we have not attempted it here.
2170
HELLE et ok: HOT-ISOSTATIC PRESSING DIAGRAMS
I NDuoLDDous
TEMPER*nlaE.
R,
* 5Oum
a=50rm
I
111.
NORLULISEO
Fig. 7. Dedy4emperahtrc diagram for hot isostatic pnssillg of tool 8ted.
Wg.
PWSSLRE.
LCG(P/U,,
10. Density-normaliscdprtswre diagram for hot isostatic-pressingof tool steel (lo~scale). TEbeERATvRE. lc 1200
P-L
of too1 steel (linear scale). PaEssuRE.
Hc44OLOGDUS
lDeEnAllM
TIT,
Fig. 11. Density-temperature diagram for hot isostatic pressing of A&4.
mid
‘c
TEMPERATURE,
loo
10
1500
CREEP
diagram for hot kostatic pressing
Fig. 8. Dady-pressure
8400
1100
lDO0
1200
UDO
1500
BaJNMRv 0.9
-
DIFFUSION BWNMRV
DIFFUSION
P~2WMNlm~ Ral25m &.,s25wn ir = 1DDym 0.6
I -2
I
-1 NORUALISED
Fig.
pM55UUE.
a.25vm
II
1
0
1
. 0.5
LCG(PlU,)
9. Density-nomdiaai pmsmre diagram for hot isostatic pressing of tool steel (log scale).
5. APPLICATION OF THE MAPS
We have constructed HIP diagrams and compared them with experimental data for three different materials: a tool steel, alumina and copper. Hotisostatically pressed tool steel is already a commercial product: it has better uniformitv - and a finer dis-
05
0.7 HOMOLODOUS
Fig.
*SE*
TEWERATURE
1 I 1.
12. Duuity-tanperaturc diagmm for hot isostatic prcsing of Al,o,.
persion of carbide particles than that obtained by conventional methods. Alumina is an example of a ceramic for which HIPing provides a manufacturing route to high densities without the addition of sintering aids such as MgO. For both materials industrial HIPing data exist. Copper was selected as a third material since it is particularly well characterised and
2171
HELLE er al.: HOT-ISOSTATIC PRESSING DIAGRAMS lc
TENPEAATlmE,
c;0999 B e
P= 150MN/m’
f 0.990 l
0.8
1
PQ
1
MEA
I
150
loo
PRESSURE.
200
a*lOvm VIELO
250 0.985
Fig. 13. Density-pressure diagram for hot isostatic pressing of Al@, (linear scale). IO
8 0.45
060
0%
0.10 nOMOLoGWS
EEA
TEMPERATURE.
0.65
0
1 I T.
Fig. 15. Density-tcmpcraturc diagram for hot &static pressing of copper.
PRESSURE. MN I m’ I
.
MN/m*
100
aI-
there is commercial interest in producing high purity copper containers for nuclear waste storage by HIPing. bemuse
5.1. Tool steel A typical composition of a tool steel fabricated by hot-isostatic pressing is given in .Table 2. Hotisostatic pressing data obtained from Helhnan [20] arc listed in Table 3. HIP diagrams based on the c@ations given in the text arc shown in Fig 7-10. Hellman’s data arc plotted on the diagrams. Figure 7, with axes of D and T, was const~cted for a particle diameter of 50 gm and a constant pressure of lOOMPa. It shows that power-law creep is the
I a=lovm
. 0.6 ‘-2 NORM&
PRESSURE.
L&t
PM,)
Fig. 14. Density-nomaliscd pmssun diagram for hot isostatic passing of copper (log scale).
Tabk I. M~terkl data Tool steer”
Materialproperty Atomk vohme, lXm9
1.21 x 10-m
2.58x IO_‘0 1680 2w 8.1 x IO -0.85
1.55 x IOJ -0.35
4.21 x lo’ -0.34
3.7 x lo-’ 280
2.8 x lo-” 477
2 x 10-s 197
2 x lo-” 167
8.6 x IO-lo 419
5 x lo-‘3 104
7.3 ._ I.5 x IO”
::!I
4.8 7.4 x Id
A (Dam conslant)
*)A tanpwanue
A&* 4.25 x IO-” 4.76 x IO-lo 2320 Is00
dependentmodulus I138 been u6cdz
r=l’ol I+ !ig!?!p!z!) I_ VW ‘.u
*)D,= Dti exp( -Q,/RIRT) ma/r. %D,-8D,exp(-Q,/RT)m’/r. “‘he power-hw creep pammeW
1. /_I
in tbc equations (23) and (24) UC given by 4
AbD,
i+=zijF “h
M no cmcp data for tool nedr at HIP tempemWrc9 (around 12WC). In this tanpcmuirc range the steel b mMcnitic and all the alloying ckncnta arc in solid solution. Ilw mccbanicnl
mIbe data am for pure dumirm, taken from Frost and Asbby [la]. @‘fbc data arc for pure copper, taken from Frost and A&y [l6].
2172
HELLE et a/.: HOT-ISOSTATIC PRESSING DIAGRAMS
The diagrams in Figs 11 and 12 are constructed at a constant pressure of 100and 200 MPa, respectively. Both show that boundary diffusion is the dominant mechanism and that grain growth leading to pore separation is anticipated (giving the rather abrupt change in slope of the time contours). The experimental data (solid circles) show good agreement with the model. Initially densification is fast and high densities are reached rapidly when the temperature is above 0.65 T,. By increasing the temperature by only 3O”C,the time required to reach a given density can be halved. But at the critical density for grain growth densiIlcation becomes much slower, and the gain due to increased temperature diminishes. The diagrams show that, to achieve full density, temperatures in the region of 1500°C are required. If grain growth is prevented then full density should be reached in short times at temperatures in the region of 1300°C. Figure 13 shows a HIP diagram at a constant temperature of 1300°C.Again, wcperimental data [21] is in good agreement with it. The effect of increasing pressure on the den&cation rate is rather small: the pressure must be doubled in order to halve the time required to reach a certain density. Grain growth sets a limit to the maximum density obtainable when HIPing at realistic pressures. Since the dominant mechanism is diffusion, the dens&&ion rates could be substantially increased by decreasing the particle sire [see equation (2111,but the maximum density would still ultimately depend on whether or not grain growth occmred.
Table 2. Composition of a typical HlPcd tool steel (wtyJ C
Cr
MO
1.3-2.3
-4.0
S&7.0
w
V
co
-6.5
3.1-6.5
8.5-10.5
Tabk 3. Hot-kostatk pressing data of tool steel f201
1 2
I
Ef 9916
: 1
:: 9918
I
98.9 99.9 full 92.6
: 1 *Not ~tatal.
dominant me&a&n at low densities, switching to boundary dilfusion at high. The data agree fairly well with the predictions of the diagram, which explains the very slow incmase indensityasfulldensityis approached, caused by pore separation. The critical density for separation, equation (28) is D, = 99.30/, Grain growth can, of cow be reduced or prevented by other means-by solute drag or by precipitate pinning for examplo4o our condition gives a lower boundforD,TheFigsSand9(withaxesofDand P) show that full density can be reached by increasing the pressure, an observation conllrmed’by Helhnan’s data. Figure 10 shows what happens if the grain sire of 5.3. Copper an individual particle is smaller than the neck sire. HIP diagrams for copper are shown in Figs 14 and Now a new me&a&m (Nabarr~Herring/Coble 15. The diagram shown in Fig. 14 is for polycreep) becomes dominant at low pressures and accel- crystalline copper powder with a mean particle dierates the rate of densiflcation. ameter of 150pm and a mean 8rain size of 10 pm. It is constructed for a constant temperature of 550°C. 5.2 Allmlina with axes of D and log (P/a,). At this temperature, HIP diagrams for alumina are shown in Figs a pressure of a little above 100MPa is required if full 1l-13, each of them being for single crystal powder density is to be achieved in a reasonable time. Powerwith a mean particle diameter of 2.5pm. Density law creep is the dominant mechanism following an data [21] for high purity alumina powders initial region of plastic yielding, the latter already (A&O,> 99.Yb are listed in Table 4, and plotted on resulting in a considerable dens&ation due to the the diagrams. low yield strength of copper at elevated temperatures. Tabk 4. Hot-ix&&
(rem) 2.5 ;: 2:5 2.5 z 2:5 2.5 2.5 2.5 2.5 0.5
PmIsun (MNm-*)
urmain~ data of ALO, (211
Temperature (‘Cl
Tiie (hour)
Den&y 0
2 2 4 2 4
full 97.5 98. I 99.6 99.5
Es
1350 1200 1200 1200 I200 1300 13tKl 1300
E 9915
t: 200 200 Is0
tz 1400 1400 1350
: 2 2 4 : 2
99:s 99.6 full
150 loo ii 200 100
E
Comments
Green dcmity = 60 to 65% Estimated density at HIP temperature = 70% Cold isostrtic pnuing at lOOMPa used to obtain 6Y/, density Extreme grain growth
HELLE et d.:
HOT-ISOSTATIC
PRESSING DIAGRAMS
2173
Tegman [22] has HIPed copper at a pressure of 150 h&a when exploring the possibility of manufacturing containers for nuclear waste. He used a 60”/,/40”/, mixture of the Alcan UHP and Ecka AK9 1 powders described in Table 5, and obtained a density of 99.7% at 500°C in 5-10 h. Figure 15 shows the densification behaviour of copper powder at 150 MPa. Tegman’s result is in reasonable agreement with the computed time. Since power-law creep is the dominant densification mechanism a&r yielding, the particle size has almost no effect on the dmsification rate. Plastic yielding alone results in a density close to 99% of the theoretical. As can be seen, timedependent densification is very slow below 450°C. An increase in temperature of less than 20°C halves the
time required to reach a certain density, and above 600°C full density can be reached rapidly. Since the dominant mechanism is power-law creep, grain growth with pore separation has little effect on densification. Although the diagrams show a field of Nabarro-Herring/Cable creep, copper cannot easily be used to confirm this mechanism experimentally since it dominates only under conditions which lead to unacceptably long sintering times. 6. CONCLUDING
REMARKS
The simplifications and modifications to the models for hot-isostatic pressing described above, allow the computation of diagrams in a fast, straightforward way. The present equations and procedure are a significant improvement over the earlier methods [!3-1 l] and lead to good agreement between theoretical prediction and experimental data for copper, tool steel and alumina. The simpriscatianr make the computationai program more amenable for use on a microcomputer, it being the ultimate objective of this project to develop a system to provide maps as a direct (on-line) aid in selecting the best combination of process variables for industrial HIPing so that
expensive “trial and error’* runs can be largely avoided: The equations and program described here largely realise that goal. Acknowledgements-We should like to thank SKBS (Swcd-
ish Atomic Fuel Systems) and the SERC (British Science and Engineering Research Council) for partialsupportof this project.
REFERENCES 1. C. Torrc. U. Berg and H. H. Huttcnman, Hoehschule &&en 93, 62 (1948). 2. 1. MacKenzie and R. Shuttleworth. Proc. Phys. Sot., Land. 62, 833 (1949). 3. A. K. Kakar and A C. D. Challandcr, Trans.Am. Ihtt. Min. Izngrs24% 1117 (196%). 4. M. S. Koval’chenko and 0. V. Samsanov. Povoshkhov. hfetull. 1, 3 (1961). R. L. Cable. J. a&. Phys. 41, 4798 (1970). :: R. L. Hewitt. W. Wallace and M. C. de Malhcrbe.. Pow&r Metall. 16, 4798 (1973). 7. D. S. Wilkinson and M. F. Ashby. Aera met&l. 23, 1277 (1975).
2174
HELLE ef al.:
HOT-ISOSTATIC
8. D. S. Wilkinson, Ph.D. thesis, Univ. of Cambridge (1977). 9. D. S. Wilkinson and M. F. Ashby, Proc. 4h hr. &II on Situaud Related Phemmena p. 473. Pknrrm, New York (1975). 10. D. S. Wilkinson and M. F. Ashby, Science of Sfntering, 4th Itu Round Table on Simdtg, Vol. 10, pp. 67-76 (1978). 11. E. Arzt, M. F. Ashby and K. E. Easterling, Mcroll. ~mts. A 14s 211 (1983). 12. E. AIZ& Aclcr metull. 30, 1883 (1982). 13. H. F. Fiscluaistcr, E. Arzt and L. R. Olson, Pow&r Meld/. 21, 179 (1978). 14. H. F. Fiitcr and E. hnt, Pow&r Metull. 26,82 (1983).
PRESSING
DIAGRAMS
15. F. B. Swinkels and M. F. A&by. Acta meroll. 28, 259 (1981). 16. H. J. Frost and .M. F. Ashby, Deformatim-Mechania Maps. Pcrgamon Rat, oxford (1982). 17. R. J. Brook, /. Am. Cerawt. Sot. 52, 56 (1969). 18. M. F. Yan, R. M. Cannon, Jr, H. K. BoWenand V. Chowdluy, Muter. Sri. khgng 60, 27s (1983). 19. C. Zencr, quoted by C. S. Smith, 7hu. Am. hr. Min. &&rJ 175, ls (1948). 20. P. Hellman, Uddcholm AB, Sweden, private communication (1981). 21. H. Larker. ASEA, Rohrtsfors, Swakn, private communicntion (1981). 22. R. Tcgman, ASEA, Robutsfon, Sweden, private cummunication (1983).