Computer simulation of sintering processes

Computer simulation of sintering processes

SIMULATION PROCESSES P. BROSS Max-Planck-fnstitut fur Metallforschung, OF SINTERING and H. G EXNER Institut fir Werkstoffwissenschaften, Stuttgar...

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SIMULATION PROCESSES P. BROSS Max-Planck-fnstitut

fur Metallforschung,

OF SINTERING

and H. G EXNER

Institut fir Werkstoffwissenschaften,

Stuttgart. Germany

(Rrceicerl 20 June 1975) Abstract-A numerical procedure is described for simulating sintering of model systems. Based on earlier work by Nichols for surface diffusion transport. simultaneously acting mechanisms (surface and grain-boundary diffusion) are treated without simplifying assumptrons concerning the neck geometry developed during sintering. The method is independent of the physical constants specific for the sintering material and normalised with respect to size which makes it rather general. Some results are given for cylinder models including asymmetric arrays. It is thought that the results obtained by this computer simulation greatly enhance our understanding of sintering processes.

R&sum&-Nous presentons un procede numhrique de simulation du frittage dans divers mod&s. En nous appuyant sur un travail ancien de Nichols concernant le transport par diffusion superficielle. nous traitons les mbcanismes agissant simultantment (diffusions superficielle et intergranulaire) sans aucune hypothese simplificatrice concernant la gtometrie des collets form& au tours du frittage. Cette methode est indtpendante des constantes physiques spccifiques du materiau que l’on fritte. et ellr est normaiisee en ce qui concerne la tailie, cc qui la rend g&&ale. Nous presentons quelques risuitats pour des modtles cylindr~ques, comprenant des arrangements asym&ques. Nous pensons que Ies risultats obtenus par nos simulations sur ordinateur augmentent beaucoup notre comprehension du frittage.

Zusammenfassung-Es wird ein numerisches Verfahren fur die Simulation von Sintervorgangen in Modellsystemen beschricben. Auf der Grundlage friiherer Arbeiten von Nichols Wr Oberfl;ichentransport werden gleichzeitig ab~aufende Transportmech~ismen (OberRIchen- und Korn~enzendiffusion~ behandelt, wobei keinerlei vereinfachende Annahmen iiber die sich beim Sintern ausbildende Halsgeometrie gemacht werden. Das Verfahren ist weitgehend unabhangig von den physikalischen Konstanten des sinternden Stoffes und in Bezug auf die TeilchengrGDe normiert, so daB es sehr universe11 einsetzbar ist. Einige Ergebnisse fur Zylindermodelle einschlieDlich unsymmetrischer Anordnungen werden besprochen. Es ist zu erwarten, daD die mit dieser Computersimulation gewonnenen Aussagen das allgemeine Verst;indnis von Sinterprozessen stark erweitern konnen.

1. INTRODUCTION The quantitative description of sintering processes requires simplified models. The geometry of powder stackings is by far too compiicated to allow a precise evafuation of the driving forces and the path of materia) transport. Kuczynski [l] was first to derive relationships describing the kinetics of sintering processes by diffusional mechanisms by means of idealised two particie models (two spheres or two cylinders). These derivations incorporate several approximations. From a physical point of view the substitution of the real neck profile by a surface with circular contours tangential to the original surfaces of the two particles is not satisfying. It requires a sudden change of chemical potential along the surface due to the discontinuous change of surface curvature. Later on, numerous authors (e.g. [Z-6]) have used this approximation. In a later approach [7]. a cartenoid was used instead of the circular contour for describing the neck surface. This, however, does not improve the situation to any considerable degree. No

other attempt to find a better analytical description of the neck contour is known to the present authors. The only successful development for a more accurate evaluation of the real situation is due to Nichols and Mulfins [S-LO] who as early as 1965 used a digital computer to simulate the morphological changes occurring at the contact region between two particles. The only mechanism considered by these authors is surface diffusion. By surface diffusion, hovvever, no center-to-center approach is possible. Therefore, shrinkage of powder during sintering cannot be modelled in this way. Though Nichols and Mullins achieved important progress in the understanding of material transport during sintering, the limited nature of their method caused their results to be almost completely neglected in sintering literature, A mathematical model able to give a deeper insight into the kinetics of sintering processes should consider at least one additional mechanism for material transport allowing for center-to-center approach between the two particles and, in turn, for shrinkage. i.e. volume diffusion or grain boundary diffusion. This 1013

iOl4

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paper describes the computer simulation of material transport due to capillary forces under the simultaneous action of surface and grain boundary diffusion. In this way. the kinetics of contact formation between two particles including center approach and the effects of asymmetric neck formation in irreguiar arrangements during sintering can be quantitatively described. 2. IMODEL GEOMETRY AND THE BASIC DIFFERENTUL EQUATIONS In the following, only prismatic models are considered, the geometry of which can be described by one cross section only. The three-dimensional problem is then reduced to two dimensions. We also assume that the material is monocrystalline at the right and left side of the contact, respectively and that a grain boundary between them is formed during formation of the neck as shown in Fig. 1. The radii of neck curvature at the two intersections between the grain boundary and the neck surface are not necessarily equal. In a previous paper [ll], we have shown that the distribution of stress, the flux along the grain boundary and the rate of material removal or deposition along the grain boundary can be derived rigorously for the symmetric as well as for the asymmetric situation. Disregarding volume diffusion throughout the paper (which would make necessary the definitions of chemical potentials as a two-dimensional vector field rather than as gradients along plane curves), we get the height N of the layer removed or deposited at each point along the grain boundary of length L (see Fig. 1) by grain boundary diffusion as (1 l]

6H(Y) -

dt

5(Kz - K,)Y + (3K, - 2Kz)*L - 2 = C&q’ L’

where K, and K2 are the curvatures at both sides of the neck, respectively. All geometric parameters (H. Y, K,, K1 and L) are normalised with respect to size. i.e. they are given as fractions of the

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cylinder radius r. r is the sintering time. The constant Ccrr is then given by

where L)oe is the grain boundary coefficient, b is the grain boundary width, ‘is is the specific surface energy, a3 is the atomic volume and kT has the usual meaning. The material which flows out from the grain boundary must be distributed by surface diffusion along the neck surface in order to smooth out the profile and the curvature gradient in the vicinity of the intersection of neck surface and grain boundary. The material flow along the surface can be described by the following differential equation: (3) where 6M is the amount of material flowing into or out from a surface element of length ~5.8per time intervaf 6t and unit length. K is the local curvature of the surface element. The rate constant Cs is given by

c

s-

=

DswJ kT

where 0s is the surface diffusion coefficient and the other quantities are defined above. Both types of material transports, i.e. transport along the boundary and transport along the surface, occur simultaneously and influence each other. The only way to treat this problem is the simultaneous numerical evaluation of equations (1) and (3). To do this, the differential equations must be transformed in convenient equations for finite differences, i.e. small time intervals and small surface elements.

3. EQUATIONS FOR THE NUMERICAL SOLUTIOX Following the procedure of Nichols[S-10). the time intervals At are normalised with respect to the physical constants of equation (3) and the cylinder radius r: (5) Then. equation (1) can be rewritten in the following way: AH(Y) = CS

Boundary

Fig. 1. Geometry of the neck formed between two cylindrical particles during sintering.

5(X3 - K,)Y + (3K, - ZKI)L - 2 Ar L3

16) where AH(Y) is the height of the layer removed or deposited at coordinate Y at the grain boundary

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12DGB*ba’ys kTr“

kTr4 .= D5ysa’

12

DGBh D,a’

Ki+r + Ki+* - 2Ki (8)

AS3

whsre AM is the shift of the surface element AS at location i in the direction normal to the surface profile during the normalised time interval Ar. Ki. Ki_l, and Ki+L are the curvatures at location i and the two adjacent intervals i - I and i + I. The curvature of a plane curve is

K,_b” where along angle 6, and

~~ =

(7)

This constant and the constant relating r to real time c [equation (j)] are the only variables depending on the material while all other quantities are independent of the material and the absolute dimensions. The transformation of the differential equation for surface diffusion to differences is not trivial. Since the curvature K cannot be described as an analytical function of length S along the surface, we again followed Nichols’ [S-lo] procedure to substitute the surface profile by a polygon, i.e. by breaking it up into small straight intervals A.S. In contrast to the parameter method used by Nichols, we define the equally spaced comer points of this polygon in Cartesian coordinates (Fig. 2). This makes easier to control the numerical procedures, testing of volume constancy and comprehension of the various steps of the calculations. As a first step we get in analogy with Nichols AIM -= Ar

6.5

dr is the angle change of the surface profile the surface element 6.7. If we take ri as the between the tangent to the curve at location the X-direction (Fig. 2) and average the angle

I

t

2

li-1.

(IO)

(I 1)

and we get ~3 =

Pi-2

+

Pi-1

-

Bi -

Pi-L

(121

4A.S and Ms =

AT.Bi-3

-

Pi-2

-

2(8i-l

-

Bi) +

Pi-1

-Pi-?

4AS3 (13

Repeated numerical evaluations of this equation showed that it is very hard to achieve a stable solution. A steady change of b-values over the neck profile is expected for the real situation of a profile controlled by surface diffusion. Instability (which presents itself as an increasing fluctuation of p-values along the neck profile) occurs unless very small Ar values are chosen. Then, however, computer time gets excessive. After trying out various procedures. vve adopted an additional averaging procedure considering three subsequent intervals: AZ. = ‘iMi + A,Wi-, + A:Mi_ I 2 4

(14

This procedure is based on the fact that. in the real situation, deposition or removal of material in subsequent intervals increases or decreases continuously along the surface profile. By taking the adjacent intervals i - 1 and i + 1 into account, fluctuations in the interval i due to inaccuracy of the numerical method are smoothed and stability is retained for reasonable values of Ar. The final equation for the material transport along the surface resulting in a stepwise dislocation of the surface curvature then is: Pi-4

+

Bi-3

-

3(Bi-2

+

Bi-1

- Pi**) 16AS3

-

Pi Pi-2

-

Bit3

ai

(15)

I

*i yi

Cii-t

Xi-l y i-l

X

-

The tangent angles 2 can be expressed by the angles of slopes of the intervals AS. /? (Fig. 2):

AZi =

AZi .

li-1

2AS

’ yi*t

Y

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change over two intervals. we get

during Ar and C is given by C=

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Fig. 2. Polygon representation of a surface contour (Si, surface elements; xi, tangent angles; pi, slope angles: A&. shift of coordinates Xi and x during AT).

4. NUMERICAL

EVALUATION

The simultaneous action of surface and grain boundary diffusion is considered in the following way: in each time interval Ar, the material flowing out from the grain boundary is determined by equation (6) and distributed evenly over the two surface intervals at the right and the left side of the grain

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boundary-surface intersection. Then, equation (15) is applied to all surface intervals and the new surface profile at the end of time interval AT is obtained by shifting the original corner points by the amount AZi normal to the direction of ASi. The coordinates defining the new polygon are given by Xi(T + AT) =

X,(r) -

AZi sinpi,

(16a)

Yi(r + AT) = K(r) + AZicos/$.

(16b)

At the intervals adjacent to the grain boundary, AZi

= AZ

I

+

E

AS

(where V is the material flowing out from the grain boundary per unit length). The length of these intervals is reduced by the amount AH/Zcos/?, due to material removal from the grain boundary. The intervals of the new profile are no longer of equal length. In order to be able to apply equation (15) again, an equidistant polygon is superimposed by using analytical geometry to find the intersection of a circle of radius AS and the straight sections of the non-equidistant new profile. The computer program first checks if the intersection point falls on that part of the line which is part of the profile, ASi. If not, the next interval ASi+ 1 is considered and the intersection point found. As a result we get the corner points of a polygon with equal edges, and the whole procedure described above is repeated for the next time interval. A flow chart of these steps of the computer program? is shown in Fig. 3.

5. STABILITY AND ACCURACY As mentioned above, the results of the computer simulation is strongly dependent on its stability. Fluctuations which have no physical meaning must be damped in consecutive iterations. Stability is sensitive to the proper choice of the time intervals AT and the profile spacings AS. Also, accuracy and, obviously, computer time are governed by this choice. If the intervals are taken to be small, stability and high accuracy will be achieved but computer time gets excessive. The optimum choice for AT and AS is difficult and proved to be one of the main problems in this work. In the very early stages of sintering, considerable material transport occurs only in a very narrow region near the grain boundary-neck surface intersection as the curvature gradients are zero along the original surface of cylinders or spheres. Only a limited number of intervals need to be considered. t The program is written in FORTRAN obtained from the authors on request.

IV and can be

We adopted the procedure described by Nichols [S-lo] starting with two cylindrical particles making line contact. The initial contact is described by a profile composed of circular contours where the neck width is approximately 3:: of the cylinder radius and the radius of the neck profile is O.Olo/ of the cylinder radius. In this stage, the polygon includes 16 intervals only. Then. the interval width is increased by a factor i., and the number of polygon intervals by 1 in each iteration. This makes the new profile glide up the original particle surface until a symmetry plane of the array is reached. In the following second stage the interval width is again increased but by a much smaller factor L2. This makes the number of contour points decrease slowly in accordance with reduction of surface curvature gradients. By this procedure, computer time gets reduced to a reasonable limit if i, and & are properly chosen. The most economic choice for Ar would be a value which is just below the limit where instability occurs. We tried to establish criteria which would allow us to find this limit by a mathematical procedure. To give an example for the numerous attempts described in detail in [12]: fluctuations of the profile are avoided if piTI 1 /?I for all intervals from the grain boundary to the inflection point and Big, I pi for all the rest of the profile. For this boundary condition, the maximum time interval is given by (pi - fli Arm’x = ZA,CJ, -

AMi_,

1

)AS’ -

A,2Ji,,

(17)

Since AT,,,..~must be evaluated from all points of the profile, the numerical procedure is slowed down and the program cannot be completed in reasonable computer time. With other criteria for stability we encountered the same problem. Therefore, we used the empirical procedure due to Nichols [S-lo]. He takes As = &-AS4

(18)

where i., is an empirical constant. In this way, AT increases automatically if AS increases in the way described above. The stability and accuracy of the numerical evaluation was tested in computer runs with L, values between 0.5 and 1.1. i., = 0.9 was found to be the maximum value to keep fluctuations just below the stability limit. Then, AS depends on the contour factors L, and i2 only. i.e. ATE_, = i3*lfAri for the first stage and Ari+ 1 = i.J*i.:Asi for the second stage where i is the number of iteration. For the first stage, i, was chosen such that the total number of profile points does not exceed 300. For the second stage. i., was found empirically. Figure 4 shows the number of iteration steps for a plane layer of cylinders necessary to reach the equilibrium configuration which is a plane sheet since grain boundary energy is neglected. For 50,000 steps corresponding to a computer time of approx. 15min, j.2

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interval

vidrh

AS and factors

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1017

X

4 Definition of neck region of initial contour equidistant points (interval width 2s)

Calculate

material

transport

by

from the grain

of this volume over the first intervals on the right and the left of grain boundary -1 Determination

of

slope

angles

S from coordinates

Xi and Y. I.

1 Determination

of

time interval

A? = 2SL

1 Increase

time r by AT 1

Calculate

new interval

width

1 Calculate material transport AZi of all steps along the surface by equ. (15)

Calculate

-

No +

Check if

the new coordinates

final

sintering 1

stage

according

is reached

Yes

Fig. 3. Flow chart of the simulation program.

becomes 1.004 (see Fig. 4). This value was used for all our compilations. The accuracy of the simulation is tested by accounting for the fact that the total volume remains constant. Figure 4 also shows the error at the end of the run for the plane cylinder layer. For iz = 1.00004, an error of - l.ZO./,is obtained, i.e. the final volume is lower than the starting volume by this amount. This error is in the same order of magnitude as that obtained by Nichoois [f&10), but of opposite sign. 6. APPLICATION

OF THE METHOD

This numerical procedure was applied to the fol&.W. ?i bc

lowing situations (Fig. 5): (a) a plane row of cylinders, (bf two cylinders, (c) cylinders arranged in a say that the connection planes between the cylinder axes form constant angles. Additionally, equations similar to equation (6) and (15) were found for symmetric arrangements of spheres [t2], and the following situations were considered: (d) straight row of spheres, (e) two spheres. In order to save computer time, the s>-mmetry properties of each of these arrays were considered and only those parts evaluated from which the total array can be built up (see Fig. 5). Three values for C = 12*DGB.b/D,a were chosen for all calculations. two corresponding to the physical constants for copper at 1200 K and 1300K taken

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1 I.0004

I.0002 Factor

r.0000

x2

Fig. 1. Influence of factor i., for increasing surface interval width on the number of iterations for simulating the sintering of a plane array of cylinders by surface diffusion and on the deviation from volume constancy.

from the literature [S, 6, 13, 143, i.e. C = 14 and C = 9, respectively, and one for surface diffusion as the only mechanism acting, i.e. C = 0. The results hold for any other material for conditions where C takes one of these values and other mechanisms do not contribute to sintering. The consequences of the results with regard to sintering theory are far-reaching and will be discussed in detail elsewhere. Here, a few examples are given. In Fig. 6 the results for a plane array of cylinders as obtained by Nichols and in this work are compared. There are deviations mainly due to the fact

that volume constancy is not retained and the deviations are positive in Nichols’ and negative in our

Fig. 5. Model geometries for computer simulations.

X

Fig. 6. Results of computer simulations of sintering a plane array of cylinders according to Nichols (-----) and this paper f----j.

solution. The true profile supposedly is in between these results. Another check of Nichols’ results has been presented recently by German [ 151, however, the results are not yet published to allow quantitative comparison with ours. In all cases, the principal outcome is identical for surface diffusion as the only mechanism acting Figs. 7 and 8: The tangent circle approximation (idea&d model) as used in the derivation of sintering equations since Kuczynski’s early paper Cl] does not yield a good representation of the real situation. Undercutting occurs in the early stages, and a continuous change of curvature occurs along the surface. The most important finding is that the region influenced by material transport extends far beyond the region given by the tangent circle model and, as a consequence, neighboring necks influence each other relatively early in the sintering process. This concise solution for simultaneously acting grain boundary diffusion and surface diffusion allows us to include shrinkage for the first time, without simplifying assumptions. Figure 7 shows that the extension of the neck region is less pronounced than for surface diffusion only for both mechanisms acting, but still reaches far beyond the tangent point of the idealised model. Figure 8 demonstrates the situation in the later stages. Wide extension of the neck region in contrast to the tangent circle approximation is observed, which again is less pronounced in the case of simultaneous surface and gram boundary diffusion. While in cases (a), (b), (d) and (e) the neck region

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I

a

i I

__------_ Surface

---_

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Computer

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result

---_

diffusion

i

i

Surtoce

ond groin

boundary

diffusion

i Fig. 7. Contours of necks between cylinders in the early stages of sinrering for surface diffusion (top) and simultaneous surface and grain boundary diffusion (below).

is symmetric throughout the sintering process. the neck becomes asymmetric in case (c) due to the fact that the neck regions of the contacts overtap on the inside of the ring of cylinders earlier than on the outside. Figure 9 shows the computer simulation of a ring with 110 degree opening angle. This asymmetry becomes obvious at the profile given by curve 5. From then on, the material taken from the grain boundary has a wedge shape. In this case, the angle between the pfanes connecting the cyIinder axes changes. This fact explains rearrangement taking place in solid state sintering of irregularly packed particles [ 16-191.

7. CONCLUSIOSS (I) By means of computer simulation, sintering of geometrically simple systems can be described quantitatively without arbitrary assumptions about neck geometry for simultaneous surface diffusion and grain boundary diffusion. This allows modeliing of the kinetics of neck growth and of center approach between powder particles. (2) Since the surface curvatures at the grain boundary-surface intersections determine the chemical potential gradient in the grain boundary, diffusion along the surface and along the grain boundary are

I .---

Circle oppfoxlmation

/ / / /: ’

/

Computer

result

/’

i’/

Surface

diffusion

Surface

and grain boundary

Fig. 8. As Fig. 7, later sintering stages.

diffusion

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(61 The decisive parameter for aIt geometrical changes in the models considered (since volume diffusion is excluded) is the ratio of grain boundary and surface diffusivity. (T) It is thought that the results of computer simulations presented fiere can give a deeper insight in the sintering mechanisms of models as weft as of real powders. Ack~~vledgemenu-Thanks are due to Prof. Dr. G. Petzow, head of the Powder Metallurgy Laboratory, and his group. as weli as to Prof. Dr. G. C. Kuczynski for valuable discussions during initiation and elaboration of this work, and to Prof. Dr. 3. Predel for assistance in the use of the computer.

REFERENCES

Fig. 9. Computer simulated neck contours between asymmetrically arranged cylinders (ring with 7 = 110’. see Fig 5) for simultaneous grain boundary and surface di& sion. The o&et of the curves from the ordinate corresponds to the amount of material removed from the grain boundary, From curve 4 onwards, the upper and lower profiles differ due to interaction of necks at the symmetry line.

coupled processes. Additivity of neck growth rates for simultaneous acting sintering mechanisms neglecting this interaction and its effects on the geometrical changes is a very crude approximation at best. (3) Undercutting occurs for simuftaneous grain boundary and surface diffusion, however it is less pronounced than for surface diffusion only. (4) In the case of asymmetrically arranged contacts, asymmetric necks form due to the interaction of adjacent necks. This, in turn, causes tilting of particles with respect to each other due to a wedge-shaped removal of material from the grain boundaries. (5) fn irregular arrangements of part&s, this tilting causes rearrangement in satid state sintering which influences shrinkage rates.

1. G. C. Kuaynski, Truns, &fer. Sue. A.I.M.E. 185. 169 (1949)” 2. D, L. Johnson and I. B. Cutler, J. Am. c(?rrtm. Sot. 46, 541 (1963). 3. J. G. R. Rockland. .4cta metal!. 14, 1273 (1966). 4. L. Berrin and D. L. Johnson, in Sin&ring and Readied P~e~~rne~~ (edited by G. C. Kuczynski et ai.X p_ 369_ Gordon and Breach, New York (1967). 5. D. L. Johnson, J. appt. Php. 40, 192 (1969). 6. M. F. Ashby, Acta metall. 22. 275 (1974). 7. R. M. German and Z. A. Munir, bferall. Trans. (A) 6. 2229 11975). 8. F. A. Nichooli and W. W. Mu&as, J. appi. Phyr. 36, 1825 (1965). 9. F. A. Nichols, J. uppi. Pb.w. 37, 2805 (1966). 10. F. A. NichoIs. Acra me&& 16. 103 11968). 11. H. E. Exner and P. Brosa A& met& 2< 1007 (1979). 12. P. Brass, Ph.D. thesis, University of Stuttgart (1976). 13. J. Y. Choi and P. G. Shewmon, Trans. Met. Sot. A.IM.E. 224, 589 (1962). 14. S. R. Srinivasan, Arra met&. 21, 611 (1973). t5. R. M. German, Paper at the 4th Innt. Conf. on Sintering, Dubrovnik, September 1976. Abstract in Sistering Science and Technology (edited by D. P. Wskokovic), p. 6. International Institute for the Science of Sintering, Beograd (1976). 16. H. E. Exner, Rev. Powder Metallurgy Special Ceramics 1 (1979) In press. 17. G. Petrow and H. E. Exner, 2. &&rail& 67, 611 (8976). lg. C. 3. Shumaker and R. M. Fufrarh, in Sintering arrd Refured Phenometra (edited by G. C. Kunnyski), p. 191. Plenum Press, New York (1973). 19. H. E. Exner, G. Petzow and P. Wellner, ibid. 358.