grain growth with boundary pores

310 Journal of Nuclear Materials 152 (1988) 310-322

North-Holland. Amsterdam

GRAIN GROWTH WITH BOUNDARY PORES

P. HARTLAND * and A.G. CROCKER

Department of Physics, Uniuersiry of Surrey, Guildford, Surrey GU2 5XH, United Kingdom

and

M.O. TUCKER

Berkeley Nuclear Laboratories, CEGB, Berkeley, Gloucestershire GL13 9PB, United Kingdom

Received 3 March 1987; accepted 26 November 1987

A quasi-rigid-body model has been developed to describe the motion of grain boundary pores during growth in columnar-type grain structures. Using this model equations have ben derived for the velocity with which pores will move by

surface diffusion while remaining attached to a migrating grain boundary. Two particular pore geometries have been

examined, the lenticular edge pore and a comer pore in the form of a curvilinear isosceles triangle. Where possible the

equations derived in this paper have been applied to UO,. The quasi-rigid-body model has been used to investigate the effects

of pore size, shape, semi-dihedral angle and temperature on grain growth. These effects are demonstrated by comparing

growth in specific ordered arrays of grains in porous and pore-free structures.

1. Introduction

The pattern of grain growth in three-dimensional polycrystalline structures is difficult to analyze due to the random distribution of grain sizes and the convo- luted nature of the grain surfaces. The problem of

isolating effects and interpreting observations becomes yet more difficult in inhomogeneous structures where for example pores are present on grain boundaries. Several authors [l-3] have adopted the approach of simplifying grain geometry by examining growth in essentially two-dimensional or columnar structures. In this way insight may be provided into growth in three- dimensional structures. This technique has been ex-

tended here to analyse the effect of grain boundary porosity in columnar-type grain configurations. Thus the boundaries between grains and between grains and pores are formed by plane curves.

* Present address: Computer Centre, Royal Holloway and Bedford New College, University of London, Egham, Surrey

TW20 OEX, United Kingdom.

In this paper a quasi-rigid-body model for pore motion has been developed. Using this model two pore geometries have been examined, these are a lenticular edge pore and a corner pore in the form of a curvilinear isosceles triangle. Equations have been derived for the velocity with which these pores will move by surface

diffusion while remaining attached to a migrating grain boundary. In common with several previous studies [4,5] surface diffusion is assumed to be the dominant process governing pore motion. The effects produced by pores in polycrystalline materials are demonstrated by comparing ordered arrays of grains with ordered arrays of porous grains where the two grain configurations would be identical if the pores were removed. In this way the retardation produced on grain boundary motion has been examined for factors such as pore size, density and temperature. In addition, the effects of edge and comer pores have been compared.

Since boundary pores are prevalent in UO, nuclear fuel during irradiation and play an important part in fission gas release and swelling, where possible the equations derived in this paper have been applied to that material.

0022-3115/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

P. Hartland et al. / Grain growth with boundary pores 311

2. Pore velocity

2.1. Introductory concepts

In the following model for pore motion the shape of a pore is determined by the boundary conditions im- posed upon the chemical potential which is assumed to be continuous at all points around its surface. In addition at the pore comer the semi-dihedral angle 8 is determined by balancing the surface energy of the grain boundary ys and the pore-grain interface yS to give

e = cos-‘( y$2y,). (1)

The incremental motion of a pore is treated as a two step process. Firstly it is assumed that atoms, diffusing in a thin surface layer, are deposited or removed along the lengths of the pore surface in such a way that as the pore moves forward, it retains its shape and sire. In other words it moves as a rigid body. In the second step the pore readjusts its shape in order to satisfy the boundary conditions in its new position. This two-part process is invoked repeatedly as the pore moves through the material. Throughout, the pore is required to remain attached to the adjoining migrating boundaries which are assumed to have uniform curvature. These structures correspond to the situation where each grain has a uniform associated hydrostatic pressure. In reality, un- less the pore is in equilibrium and has no inclination to migrate, its surface is bound to have curvature which varies from point to point. The diffusion of atoms by which the pore moves is motivated by these local variations in curvature and hence in the chemical potential of the surface [6]. However, in the present treatment to make the problem tractable, these local variations in curvature are replaced by segments of constant curvature giving rise to the same average chemical potential * .

* The effect of this approximation is to assume a slightly

different circumferential distribution in the rate of vacancy

production than would occur in reality, when the total

vacancy migration is minimized and the process occurs in

the most efficient way. The approximate treatment therefore

requires a rather greater overall movement of vacancies than

would occur in practice, giving rise to an apparently larger

drag effect of the pore on grain boundary migration. How- ever the additional movements will be considerably smaller

than those associated with movement of the pore as a whole,

and it is therefore unlikely that this will qualitatively affect

the predictions of the model. (The authors are grateful to the

referee for drawing attention to this point.)

Grain boundary

V

Fig. 1. Geometry of lenticular edge pore.

2.2. General relationships

The shapes of pores considered in this paper are symmetrical about a line passing perpendicularly through the centre of the leading surface, such as those examples in figs. 1 and 2. Along this line of symmetry the motion of the pore is directed with a velocity VP. The grain boundary adjacent to the pore comer will

Fig. 2. Geometry of an isosceles comer pore.

312 P. Hartland et al. / Grain growth with boundary pore.r

migrate towards its centre of curvature with a velocity given by [l]

dR, Msys -- dt R, ’ (2)

where R, and Mg are respectively the radius of curvature and mobility of the boundary.

In order that the pore corner remains in contact with the migrating boundary, VP must satisfy the following

geometric condition:

where 4 is the angle between the grain boundary at the corner and the direction of pore motion.

Atoms diffuse around the surface of a moving pore from regions of high chemical potential on the leading

surface to regions of low potential on the trailing surface to reduce the total free energy of the system . The flux of these diffusing atoms is given by [7]

44 dy -- j’= S2kT ds’

where k is Boltzmann’s constant, T the absolute temperature, D, the coefficient of surface diffusion, ds an element of the pore’s surface, D the atomic volume and 6, the thickness in which surface diffusion takes place (8, = 0’13). Because of conservation of matter the velocity V, of an element along its normal is given by

pf&!!& ” ds ’

Resolving the velocity of the pore at some point on the i th surface of the pore along the direction normal

(outwards into the bulk of the grain) to this surface it is found that

where 9, is the angular co-ordinate of the i th surface. Combining eqs. (3) to (6) one obtains the following

equation [8]:

d2p, - = & sin 9,) de:

where

P, = rt2kTM,yg

D&R, sin 4

and i = t or 1 for the trailing and leading surfaces respectively. In the above equation r, is the radius of

curvature of the ith surface. Integrating eq. (7) with respect to +, produces the expression

dPi - =A,-& cos +,, d+,

where A, is a constant of integration. For a pore situated on the boundary between two

columnar type grains each of which has uniform pressure within its bulk the chemical potential is given by

[61

p, = -QY,/r,. (9)

Matching this value of p, to the mean value of the

chemical potential CL,, obtained by integrating eq. (8) over this surface gives

where $rb and @,, are the limits of the angular aperture

of the i th pore surface. The above equations are used in Appendices 1 and 2

to determine the velocities of edge and corner pores. The pore shape examined in Appendix 1 is the lenticu-

lar edge pore (fig. 1). The symmetry of this pore results in purely translational rigid-body motion. In general, however, the motion of a corner pore will contain both rotational and translational elements. Without reference to a specific pore shape or grain boundary structure

contiguous to the pore the informatiop provided by a general analysis would be severely limited. This is due to the problem of uniquely specifying the boundary conditions to be placed on the flux within the pore surface. Hence to obtain the velocity of a corner pore, a specific pore geometry has been examined in Appendix 2. This is the curvilinear isosceles pore shown in fig. 2. Pores of this shape arise when the curvatures of the boundaries adjacent to the leading surface of the pore are equal. In such a configuration the boundary terminating at the junction of the trailing surface has by

symmetry zero curvature.

2.3. Application to UO,

The instantaneous pore velocities given in Ap- pendices 1 and 2 may be used to examine the motion of pores in specific polycrystalline materials. Because of the frequent observation of such pores in irradiated UO, nuclear fuel these calculations hate been performed for this material. It should be borne in mind of


course that the present models are two-dimensional and that pores in materials such as UO, are necessarily three-dimensional. Nevertheless there is no reason to suppose that effects discussed in the following sections will differ qualitatively from those that would be derived from more realistic calculations.

The coefficient of surface diffusion in UO, is given

by PI 4 = 4 exp( - Q,/RT), where D0=5.6X lo2 m2 s-l and Q,=5.0~ 10’ J mol-‘. The atomic volume B is 4 X 1O-29 m3 and it has been assumed here that the thickness of the surface layer 6, is equal to 01/3. The semi-dihedral angle 8 has the usual magnitude of SO”. By substituting the above values into eqs. (A.8) and (A.9) the velocities of the edge and corner pores may be obtained as functions of size, temperature and the angle #. This angle represents

t \

\ \

--

:

- sl?prn2

-\

\

t

T=2023H - - - - \

T=2523 K

-101 I 1 I

LO0 50' 60' 7o" JI

60'

Fig. 3. Variation in the velocity of a lenticular edge pore of area ap with the angle + at constant values of temperature in

uo,.

Or-----l

-_ -2- - 1-y a =lO.Opm

\ \I \ -5

--

r

- _a,=1.0pm2 ‘\ --

-.

-u

\

. \

\

---

t

_ a,=1D0pm2 \ --

-- \

.\ J

t T=2023 K - - - -

I T=2523 K \I I

-10 I Lao 50'

J, 60'

Fig. 4. Variation in the velocity VP of a symmetric comer pore

of area ap with the angle I) at constant values of temperature

in U02.

the interaction between the pore and its adjoining gram boundary without reference to a particular grain configuration.

Figs. 3 and 4 show the variations in log,,V, again J, for fixed values of pore area and temperature. The temperatures selected in these graphs T = 2023 K and 2523 K are approximately 70% and 90%, respectively, of the melting point of UO,. The three pore sizes used in figs. 3 and 4 correspond to small, medium and large pores in this material.

2.4. Normalised pore velocities

The value of the semi-dihedral angle 0 in UO, is not typical of the semi-dihedral angles in more common polycrystalline materials where it may exceed an angle of 80 O. To examine the effects, which changes in the

314 P. Hartland et al. / Grain growth with boundary pores

il go*

Fig. 5. Variation in the normalized velocity U, of a lenticular edge pore of area 1.0 pm* and semi-dihedral angle B with the angle 4. The curve for B = 60 o (not shown) falls between

thoseof B=50° andB=70°.

angle 8 have on pore motion, a normalised velocity u, is defined by the expression

where VP is defined in eqs. (A.9) and (A.14). The variation in v,, for changes in the angle # are shown in figs. 5 and 6 for the edge and comer pores respectively.

2.5. Comparison of edge and corner pores

When the curvatures of the leading and trailing surfaces are equal, the pores will be stationary. In the case of the edge pore the curvatures are equal at $J = a/2 while for the comer pore this occurs at v/3. These

angles represent upper limits on a range of possible 4 angles for the two configurations. Both these upper

limits are independent of the semi-dihedral angle 8. Below the angle 4 = (m/2) - 0 the curvature of the leading surface of the pore becomes concave. Such configurations do not develop in the ordered arrays of grains examined later in this paper where the curvatures of adjacent grain boundaries are uniform. However, the

crescent-shaped edge pore is an important configura-

tion, as it represents a geometry which develops before the pore detaches from a grain boundary. Increasing the angle tc, from the lower limit for a given pore configura-

tion with a convex leading surface produces an increasingly more symmetric pore. In consequence there is a

decrease in the impetus driving atoms around the surface. As 4 increases its value, so does the area of the leading surface and thus more atoms are required to be transported away from this surface to move the pore forward. This increase in the resistance of the pore to motion together with the diminishing driving force,

1

-i

Fig. 6. Variation in the normalized velocity v, of an isosceles corner pore of area 1.0 pm2 and semi-dihedral angle 13 with

the angle $.


produces a retardation in velocity which falls to zero when the pore becomes symmetric. This effect can be seen in figs. 3 and 4. It is also apparent from these diagrams that increasing the temperature of the material for given values of J/ and 0 increases the pore velocity because of the increased mobility of the surface atoms. The increase in pore velocity with decreasing pore size reflects the smaller distances necessary for surface atoms to migrate and the greater curvatures of surfaces provid- ing higher driving forces for diffusion.

From fig. 6 it can be seen that in the case of the curvilinear isosceles comer pore a decrease in the angle B for fixed values of J, results in an increase in the normalized velocity with decreasing values of 8. This is a result of the fact that, for the same value of J, for smaller dihedral angles, the curvature differences between the leading and trailing surfaces of the pore are greater than for larger values of 0, the difference disap- pearing entirely in the limiting case of 0 becoming 90 ‘. However, a somewhat different situation arises with the lenticular edge pore. In this case, as can be seen from fig. 5, as 0 is reduced from the maximum theoretical value the normalized velocity initially becomes larger in the same way as in fig. 6 for the comer pore. However, for a given value of JI the velocities for 0 = 40 o and 30 o are lower than those for (3 = 50 ‘. The reduction in velocity at these lower values of B is explained by an increase in the resistance of the pore to motion due to the increased length of the leading surface. Because of the geometry of the lenticular pore, as (3 tends towards zero the surface area tends towards infinity. Thus the number of atoms required to be removed from the leading surface to move the edge pore forward becomes progressively larger.

In the case of the comer pore, the increase in the length of the forward surfaces, which occurs as a result of the variation in 0, is smaller than in the case of the edge pore for the same angle. Furthermore, this increase is insufficient to produce a reversal in the trend of increasing values of the normalised velocity u,, with decreasing values of 8.

3. Effects of porosity on selected grain structures

3.1. Grain boundary retardation

The effect of porosity on grain boundary migration is examined by comparing the motion of boundaries containing pores with that of pore free boundaries in identical grain configurations. A measure of the retarda-

tion suffered by a boundary due to porosity is provided by the ratio

r = v**/vs. (II)

In this ratio Vs* is the velocity of a boundary in a system with pores and Vg is the velocity of a boundary in the same grain configuration where pores are absent and the sizes of the grains themselves are unchanged. Since the introduction of pores to a boundary is ex- pected to reduce its velocity of migration, the quantity r will be referred to as the retardation factor.

3.2. Structures examined

To demonstrate the effect grain boundary porosity produces on grain boundary motion, the retardation factor I’ has been determined for two grain-pore geometries.

The first configuration consists of a circular cylindrical grain embedded in a larger single grain. This structure enables the effects of grain edge porosity to be examined in isolation.

The second configuration examined in section 3.4 contains sites for both edge and comer pores and thus enables comparisons to be made between their effects on boundary migration.

3.3. Cylindrical grain with pores

Tucker [8,10] has shown that in irradiated UO, at high temperatures pores on gram boundaries have a tendency to develop a uniform size and distribution. In view of this finding the following pore distribution has been used to represent porosity in irradiated UO,. Around the circumference of a grain of area ag, n equally sized lenticular pores are evenly distributed. From the geometry of this configuration (fig. 7(i) and 7(ii)) the area of the grain is given by

a,=nR:($+ (i - +)77)

+nr,R, cot : cos +r cos # ( 1

+nrjz(~-#I+(cot(~)+sin#1)cos2$l. (12)

In this equation the radii of curvature R, and r, may be expressed as functions of a set of configurational variables a p, a, and $ from equations (A.4) and (A.8) respectively.

An iterative method is used to solve eq. (12) for $,. Once this angle has been determined it is possible to

316 P. Hartland et al. / Grain growth with boundary pores

Fig. 7. A cylindrical grain (if with pores of equal size distributed evenly around its boundary and a detail (ii) of one of

these pores.

calculate from eqs. (2), (3) and (A.9) a value for k’s*, the rate at which the porous boundary migrates.

3.3. I. Influence of pore size on grain boundary retardation Four lenticular pores each of area up are equally

spaced around the boundary of a cylindrical grain. The velocity VP with which the grain boundary moves in the absence of pores is given by

V8 = n”2M,#$a,. (13)

Substituting into eq. (11) this value of Vs together with VP* and taking the loglo value of the resulting equation one obtains

logie(I? = -~log~*(u*/as~ +G

where

(14)

Fig. 8 shows the variation in the retardation factor I with changes in the pore-grain area fraction up/ap. In this figure the three curves represent pores of areas ar, = 0.1 rc.m2, 1.0 pm* and 10.0 pm2 respectively. The parameter cy is fixed at lo3 prnw2. Variations in the area of the pore relative to the size of the grain may be considered as being due to the shrinkage of the grain to which pores of constant area remain attached during the migration of the boundary. It is clear from fig. 8 that for a given area fraction aJag, the larger the pore the greater will be the retardation produced upon the boundary. It is also possible to deduce from the curves shown in this figure that for the same area of grain, the retardation factor r decreases with increasing pore size. Fig. 8 shows that the drag produced on the boundary is not merely a scaling factor. For if this were so, r wouid remain unaltered by changes in the sizes of the pore and grain such that the ratio ap/ng stays constant.

When ap/ap is small r is approximately equal to unity and the mechanism controlling the motion of the combined pore boundary system is the rate at which atoms are able to transfer across the boundary. As the grain shrinks in size the migration velocity of its boundaries increases (eq. (13)). Thus pores attached to these boundaries will also be required to move more rapidly. This results in pore shapes with increasingly lower values of +. In these shapes the length of the pore surface progressively decreases. In addition the shape of the pores becomes less symmetric in the plane of the boundary and hence there is an increase in the potential to drive atoms around the surface. These effects com- bine to increase the rate at which atoms are transferred between grains via the pore surfaces and the pores themselves will move rapidly. Nevertheless as the size OF the pores increases relative to that of the shrinking grain the value of r decreases. This is because the pore progressively takes up more of the grain boundary and in doing so reduces the area of contact between adjacent grains. Surface diffusion thus plays a gradually more important role in transferring atoms from the cylindrical grain to the adjoining one in order to facili- tate growth. It is this effect which is dominant and causes the fall in the value of r with increasing up/as.

The roles which the two processes, transboundary migration and surface diffusion, play in grain growth can be seen by taking the ratio of the number of atoms crossing between adjoining grains through the two routes, that is

c = log,0 2(Qtn2 + Q1)3’2 sin \t 1 _ ,,*r, cos(+-@) n,

na’/2ap($~, + c,)(l - tan 8 tan 4) cos B . n7 R, we sin 4 . (15)


0

-1

L%l,o(r)

-2

-3

-‘ , I I I , 8

-6 -7 -6 -5 -L -2

Fig. 8. Variation in the velocity ratio r for changes in the pore grain area fraction aJag at a constant value of a = lo3 pm-* for a

cylindrical grain having four uniformly sized and distributed boundary pores.

In this equation n, and nr are the number of atoms per unit time passing by surface diffusion and transboundary diffusion respectively and os is the angular aperture of the grain boundary between pores (fig. 7(i)).

From eq. (15) it can be shown that as transport due to surface diffusion increases relative to that due to trans-

boundary diffusion the curve of r (fig. 8) becomes more linear.

3.3.2. Influence of pore density on grain boundary retarda- ti0fl

Consider the situation in which pores of uniform size are evenly distributed around the circumference of a

a9 = lO‘~m , a = lo3 )~m ’

Al : ap=1yn2, A2: aP=5pm2, a,=10~m2

A : Total area of pores 10Oy1~, 9 = 50“

B : Total area of pares 250#, 0 = 50”

Bl : a,,= 1 pm’, B =70°

Cl : aP= lpm2, 0 =89.9”

I I I 100 150 200

N

Fig. 9. Variation in the retardation factor r with pore number N for various distributions of pores and semi-dihedral angles on a

cylindrical grain.

318 P. Harttand et a/. / Grain growth with boundmy pores

cylindrical grain of cross-sectional area lo4 pm’. The variation in the retardation factor r with the number of pores surrounding this grain is shown by the solid lines (Al, A2 and A3) in fig. 9 for the specific pore sizes 1.0, 5.0 and 10 pm2. From these curves it can be s&en that the velocity of migration of a porous boundary decreases with increasing pore density. The number of pores at which the curves in fig. 9 are terminated represents the point where the addition of one more pore would result in the corners of adjacent pores touching causing them to coalesce. An alternative treatment of pore density is to keep the total volume of porosity constant. This porosity is then divided evenly amongst N isolated pores which are again uniformly spaced on the grain edge. Calculations for two specific volumes for the total porosity are considered; these have cross-sectional areas of 100 pm2 and 250 pm2 which represent 1% and 2.5% respectively of the volume of the cylindrical grain. The long broken lines (A and B) in fig. 9 represent the variation in r with changes in pore number for these two volumes. It can be seen from this figure that for the same amount of porosity a few large pores produce more retardation on a grain boundary than do many small pores.

3.3.3. Influence of dihedral angIe on grain boundaT retardation

Consider the range of cases when N lenticular pores, each 1 pm2 in area, are evenly distributed around the edge of a cylindrical grain of area lo4 pm2. In fig. 9 the effect produced upon boundary migration has been examined for three values of 8: these are 50 O, 70 o and 89.9 O. The curve for 90 o is unobtainable. This value of 0 implies that the surface energy of the grain boundary y, is zero and therefore that the boundary exerts no force on the pore. The effects changes in 8 have upon the boundary retardation r for various numbers of pores are shown by curves A,, B, and C, in fig. 9. It follows from these curves that boundary retardation increases with decreasing values of 8. Further it can be seen that grain growth cannot be completely prevented by the presence of pores on the boundary even when the pore density is so great that their comers touch and they are at the point of coalescence.

3.3.4. Influence of temperature in UUz The form of (Y dependency on T can be determined

by substituting expressions for grain boundary mobility [l] and the coefficient of surface diffusion [9] into eq. (AS) to give

a(T) = a0 exp[(Q,- QmPTly (16)

where a0 is a temperature-independent term and Q, is the activation energy for grain growth.

Due to the discrepancy between the basis for the determination of the theoretical and experimental values of Q, [ll] it is not possible to obtain an exact relationship between 01 and T. However, to obtain the approximate form of the variation in a with temperature it is assumed that the theoretical and experimental values of Q, are of the same order of magnitude. Comparing values for Q, [9] and Q, [12] it is found that Q, > Q,. However it is clear from eq. (16) that (Y decreases with increasing temperature. This implies that the mobility of the pore relative to that of the grain boundary is smaller at lower temperatures than at higher ones.

Although a precise value of (Y corresponding to a particular temperature cannot be determined accurately, from eqs. (2) and (AS) the velocity of a boundary which corresponds to a particular value of (Y and temperature can be determined. For example at a = 10’ pm-’ a cylindrical grain with a radius of 10 pm has a boundary velocity of approximately 1 pm/h at 1800°C.

3.4. ~i~~u~ar structure with pores

3.4. I. Introduction In this section a porous bimodular configuration of

grains is examined, fig. 10. Pores of equal size with semi-dihedral angles of 50*

are situated at the centres of grain edges (fig. 10(i)) and corners (fig. lO(ii)) of migrating boundaries. The effect each pore configuration produces upon the motion of these interfaces is examined to ascertain which of these two sites provides the greatest effect on grain boundary motion. In the absence of pores the rate of migration of the four-sided grain boundary is given by

%A vg = - al/2

s 1 1 + 4 - fi] I’?. (17)

Fig. 10. Details of the arrangement of pores on (i) edges and (ii) cmners of a four-sided grain.


\ \ \ A: a,=lpm2 a=103pi2 \ \ \ 6: a,=lpm2 a-5x103pi2\ \ \

C: a,-lop2 a=103pm-’ \ ‘\ .\

D: ap-10pm2 a=5~10~p6~ \ \ \

- Corner Pore \D ‘f \B

se- Edge Pore \ \ \

I I 1 \I 1 \ -8 -7 -6 -5 -L -3 -

40 (2) Fig. 11. Variation in the velocity ratio r with pore grain area

fraction ap/ag for edge and comer pores located on the

boundary of a four-sided curvilinear grain lo4 pm2 in area.

3.4.2. Comparison of edge and corner pores

The retardation factor r was determined in the manner described in section 3.3 using eqs. (2), (3) and (17) together with the equations from Appendices A.1 and A.2. The retardation produced by edge and corner pores is shown in fig. 11 by the broken and solid lines respectively. It can be seen by comparing the curves in this figure that for equal values of or,, ag and CY the retardation produced on boundary migration by comer pores is less than that produced by edge pores. Thus grains containing only comer pores shrink at a greater rate than those with only edge pores present. In addition the difference between the rates of migration increases with increasing pore size and with decreasing temperature (increasing cr). These differences become more marked as the size of the grain decreases for a given sized pore and OL value.

3.4.3. Combined effects of edge and corner pores

It is possible to perform a restricted analysis of vacancy diffusion between edge and comer pores via a conducting grain boundary in a specific configuration using the model developed in previous sections. In the following analysis there is assumed to be no absorption or generation of vacancies by the boundary between the pores. Thus the total volume of porosity within the system remains constant. This situation corresponds to Oswald ripening of the pores. In addition there is assumed to be no net accumulation of matter at the comers of a pore moving by surface diffusion.

The effects which the presence of both edge and isosceles corner pores produces on the motion of a boundary are examined for the configuration shown in fig. 12. All the edge pores are of equal size as are all the corner pores, each has a semi-dihedral angle 0 = 50 O.

By varying the J, angles for edge and comer pores in an iterative manner, configurational variables such as pore and grain curvatures were found which corre- sponded to selected values of 0, OL, grain area, edge and corner pore area. Once the radius of curvature of the grain boundary in fig. 12 was determined the value of r could be calculated.

The gradient in the chemical potential between edge and comer pores was calculated on the assumption that the potential varied linearly along the length of the boundary. This vacancy flux generated by the gradient in the potential is

WBAPB js = QkTw,R, ’ (18)

where os is the angle subtended by the boundary with a radius of curvature R,, Apg the difference in the chemical potential between the corners of the two pores, and D, the coefficient of grain boundary diffusion. The value of Da in UO, is given by Reynolds and Burton

v31.

Fig. 12. Arrangement of both edge and comer pores on a

four-sided grain.

320 P. Hartland et al. / Grain growth with boundaT pores

-1.01 cl 05

0,1/Jm21

Fig. 13. Variation in r for the four-sided grain configuration with both edge and comer pores present, the area of a comer pore being a,. The total area of a pair of edge and comer

pores is constant at 1.0 pm’.

Calculations were performed for various sizes of the four-sided grain with different amounts of porosity. It was found that the vacancy flux was always away from

the edge pore and towards the corner pore. Hence the flow of vacancies occurred in a direction which minimized the interfacial energy of the system. Fig. 13 shows that the retardation on the motion of the boundary is at a maximum when all the porosity is concentrated at the centre of the grain edge. As more of this porosity is transferred to the corner pore the velocity of migration of the grain boundary increases which is reflected in the increased value of I’.

Finally it was found that for a given distribution of porosity the smaller the grain around which this porosity was located the larger was the vacancy flux from the edge to the corner pore. This result implies that Oswald ripening is enhanced by the shrinkage of the four-sided

grain and hence by the growth of the neighbouring grains.

4. Conclusion

Using the quasi-rigid-body model for pore motion the dependence of pore velocity on factors such as size, shape and temperature has been investigated. The following conclusions have been drawn:

(1)

(2)

(3)

(41

(51

(61

(71

(8)

In general two factors govern pore motion. The first is the difference in curvature motivating surface diffusion which increases as the change in orienta-

tion of the boundary across the pore increases. The second factor is the resistance of the pore to motion, a measure of which is the number of atoms required to be transported around the pore to move it for-

ward. This is proportional to the area of the leading surface.

Because of the approximations used, the model is likely to overestimate the effect of pore drag on grain growth, but the magnitude of this overesti-

mate is likely to be small and will not qualitatively affect the predictions. From an examination of a cylindrical grain whose boundary is decorated with lenticular pores, it ap-

pears that the transbounda~ mi~ation of atoms dominates grain growth when smafl pores sparsely

populate the boundary of a large grain. As the fraction of the boundary occupied by pores increases, surface diffusion increases in importance relative to transboundary migration. For a given amount of porosity a few large pores produce more retardation than do many small pores. Boundaries are retarded to a greater extent the smaller the dihedral angle between the pore edges

where they meet the grain boundary. In UO, pores are relatively less mobile than their associated grain boundaries at low temperatures. Using bim~ular grain structures, when all other factors are equal grain edge porosity appears to produce a greater retardation on boundary motion than do corner pores. With increasing pore size and temperature the relative difference between the effects produced by these two pore types becomes more marked. When edge and corner pores are both present and when grain boundary diffusion can occur without the production or absorption of vacancies, corner pores grow at the expense of edge pores thus tend- ing to minimize grain boundary retardation for a given amount of porosity.

P. Hariland et al. / Grain growth with boundary pores 321

Appendices. Dgrivation of pore velocities using the quasi-rigid-body model for pore motion

1. Velocity of a jore on a grain edge

In this Appendix the case of a two-dimensional

lenticular pore htuated on the edge of a columnar grain (fig. 1) is examined.

As a conse&_tence of the symmetry of the grain

boundaries adjacent to the pore there will be no side- ways movement during its quasi-rigid-body motion.

Thus at the centre of the leading and trailing surfaces of

the pore there is no atomic flux. Hence from eq. (4) and the geometry of the pore (fig. 1) it can be seen that

91= Iv, +,-tr for the leading and trailing surfaces. Thus it follows

from eq. (8) that Ai = 0 for both pore surfaces. From

fig. 1 the limits of the angular aperture are found to be

+,,=2rr-+-e; +‘lb=T+#+e

for the leading surface and

9,,=#-0; Gtb=“-q+e

for the trailing surface. When these angles are substituted into eq. (10) the

values of p, and pLt are given by

cm +, pi(+,,r,)= -P, sin+,+- i 1 J-h -~

h-+i r, ’ (A.11

where the negative sign in this equation refers to the trailing surface (i = t) and the positive sign to the

leading surface (i = 1). In the above equation

+1=++e, (A.2)

and

+,=+-e. (A.3)

In order that the flux remains finite at the pore comer the chemical potential must be continuous. Thus at the upper comer of the pore in fig. 1 the values of IL, and pt must satisfy the boundary conditions

~t(~+Gt, rt)=k(k, rl>.

Substituting into this expression the value of p, and IL, given by eq. (A.l) and using expressions following eq. (7) for /?,, the resulting equation may be solved for R,, the radius of curvature of the grain boundary adjacent to the pore, to give

(c,rf + ctrrz) cos e

sin 4 ’ (A.41

where

c,=sin #;+ cm J/L

#, - +n i=lort,

and

ZkTM, (y=-

D,S$ . (A.5)

The parameter (Y reflects the relative mobilities of the

grain boundary and the pore surface. A consequence of

the assumption that the pore remains a rigid body during its incremental movements is that the pore does

not lose or gain atoms as a result of grain boundary diffusion. Thus at the pore comer the atomic fluxes will match, hence from eq. (4)

From eq. (8) the above condition may be written

(-4.6)

(A.71

This form of the ratio of curvatures is also a consequence of the assumption that the pore surfaces have uniform curvature and that surface tension forces bal- ance at the comers.

From the geometry of the lenticular pore an expression may be obtained from the radius of curvature of the leading surface of the pore, namely

r1=( ~2~:Q,)1’z. (‘4.8)

where ap is the area of the pore and

Q, = (Q - 2J/i - sin 2#,)/2, i=lort.

From eqs. (3), (A.4), (A.7) and (A.8) the expression

2 fhs DA VP = -

(q*Q, - QI)~'*

kTa3,/* (n*c,+c,)(l-tan0 tan+)’ (A.9

for the instantaneous velocity of the lenticular edge pore is obtained.

2. Velocity of curvilinear isosceles corner pore

From the symmetry of the two dimensional curvilinear isosceles comer pore (fig. 2) it can be seen that the flux is zero at the centre of the leading surface. It is also clear that there will be no flux at the junction of the trailing surfaces. From the geometry of the pore and

322 P. Hartland et al. / Gram growth with boundacv pores

eq. (4) these conditions may be expressed in the following manner:

for the leading and trailing surfaces respectively. Hence

from eq. (8) the values of the first constant of integra-

tion for the pore surfaces

A,=& cos 8 and A, = 0.

From fig. 2 the limits of the angular aperture are found

to be

+,,=2r-(J,+0); Glb=n+e++

for the leading surface and

$,,=$-6 ‘#‘,b = @

for the trailing surface. Substituting these angles into eq. (10) one obtains

the following equations for the chemical potential on

the leading and trailing surfaces:

QYS -- r,

(A.lO)

and

(All)

where

c, = sin $I, - 4, cos 8 + (I + e2/2 - +:/2)cos 0 - cam +,

26-4

At the junction of the two surfaces the value of p is

assumed to be continuous. Thus

pL,($-O, ‘1)=~,(~+++~* r,).

Substituting into the equation the values of p, and /.I,

given in eqs. (A.lO) and (A.ll) and rearranging, one obtains the following expression for the radius of curva-

ture of the grain boundary adjacent to the pore:

(cd+d) sin #

cos 8, (A.12)

where (Y and c, have the meanings given in Appendix 1. From the geometry of this pore r, may be written in terms of pore area ur, as

r, = Ha’j2 P ’

(A.13)

where

H = [2n2Qt + Q, - 2~) sin( B - &) cos 4, cos+~] -“‘,

Q, = f(2$, - r + sin 24,)

and

Q, = +(20 - 1c, + sin 28 - 4).

Substituting the values of R,, r, and n given by eqs. (A.12), (A.13) and (A.7) into eq. (3) one obtains the following equation for the instantaneous velocity of the

curvilinear isosceles comer pore:

v = 252Y,w3s(1 -I-‘> ’ kTa;“H3( v2c, + c,)

(A.14)

Acknowledgements

The authors are indebted to Visiting Professor P E J Flewitt of CEGB for valuable discussions. The research was supported by an SERC (CASE) studentship awarded to P Hartland. Part of the work was carried out at the Berkely Nuclear Laboratories of the Technol-

ogy, Planning and Research Division and the paper is published with permission of the Central Electricitry Generating Board.

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22.

grain growth with boundary pores

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