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Pore geometry and interfacial energy
Journal of Nuclear Materials 113 (1983) 253-255 North-Holland Publishing Company
253
LETTER TO THE EDITORS PORE
GEOMETRY
AND INTERFACIAL
ENERGY
1. Introduction Pore and grain size and shape are determinant for the development during sintering of ceramic fuel as well as for its irradiation behaviour. The surface energy value ysv is required for any attempt of kinetic calculation and if a more realistic than spherical shape is assumed, the grain boundary energy boa is needed too. From tension equilibrium considerations a simple relation exists between the two energies and the angle + formed at the incidence of grain boudary and the free or pore surface:
k&v
= 2 cos( e/2).
(1)
Kingery and Franqois [l] reported as early as 1965 an important difference between the $I angle values measured on the free surface and those on the internal pores surface, which was attributed to the pressure of the entrapped gas. This explanation was later rejected [2]. For irradiated fuel, fission product contamination was assumed [3], but this does not answer the question of the unirradiated fuel. One also should notice the close values of the dihedral angle between 90 and 100 degrees for pores in such materials as UO,, mixed oxides, carbides both fresh and irradiated while free surface angles cluster roughly around 150 degrees 141. Iron oxide also fits these figures [5]. Our observations
show some dependence of the angle on the number of faces and the sizes of the pore too (see fig. 1). A quantitative description however is difficult to assess due to the uncertainty about the incidence of the fracture surface on the pore. No general explanation has yet been found for this phenomenon.
2. Geometrical model of closed pores In a single crystal isotropic matrix a pore assumes a spherical shape. The pressure exerted by the surface is: $?E,
(2)
r
where r is the radius of the pore. Whether this pressure is compensated or not by the internal gas pressure is immaterial since in the latter case the balance is provided by the matrix stress and we assume the pore contraction is slow enough to allow surface diffusion to maintain equilibrium shape of the pore. In a polycrystalline matrix there will be grain boundaries (GB) intersecting the pore surface. For simplicity we assume a normal incidence of these boundaries. They take over part of the superficial pressure thus modifying the pore shape towards a higher
GB
Fig. 1. Pore in the vicinity of the free surface.
0022-3 115/83/0000-0000/$03.00
0 1983 North-Holland
Fig. 2. Pore surface
geometry.
I. V. Nicolaescu
254
mean curvature
Pore geometry
radius R. Eq. (2) is altered in this case:
2Ysv 2Ysv LYSS
p=R=---
/ F. Glodeanu,
r
4nr2
(3)
’
where r is now an equivalent volumic radius and L is the total length of the surface-boundary intersection of a pore. If eq. (1) is used, eq. (3) becomes: 1 1 _=_R r
L cos( +/2) (4)
’
4or2
We assume the pore surface is shared by n equal-sized grains and focus our attention on a single face and the volume within a sector under it. We consider this sector to be conical rather than pyramidal and an axial crosssection through it is shown in fig. 2 where the left side represents the spherical equivalent pore while at the right the deformed shape is represented. Among the angles defined in fig. 2 an equation exists: rr a ---=__2 2
9 2
P 2’
(5)
which leads to: P siny=-cos
(
2,” 2 2 . 1
(6)
The volume of a sector is:
means:( -) for n < 1 + tan2+/2 and ( +) otherwise. We must admit that in the last term of formula (3) the exact area of the pore should be used rather than the equivalent sphere area in the denominator. But this would complicate to a considerable extent the solution for R/r. Besides, we noticed that the error is negligible for the case of n = 2 where a rigorous solution is easy to compute. For larger n we expect this error to diminish as well as the error of taking circular pore faces instead of polygonal. The relationship between pore curvature and dihedral angle is better expressed by (12) than by the formula proposed in ref. [I].
3. Physical connection of curvature and dihedral angle We mentioned earlier that the matrix is stressed by the pore. This stress is considered isostatic in the immediate vicinity of the surface. Since the grain boundary is formed by a lower atom packing than the matrix, a higher compressive modulus should be expected for it. Since according to (71 the surface energy for UO, is placed around 0.9 J/m2 at elevated temperatures, small pores (R < 10pm) may exert a stress well above 0.1 MN/m’. We assume the thickness b of the grain boundary is increased by Ab which is proportional to the stress p:
The value of L can be computed
as: kss
L=nnRsinz.
11 R
r
(13)
The grain boudary energy may also be assumed proportional to its thickness as a first approximation. This means that the excess energy is:
n-2 n’
and introduced
energ),
p=KAb.
from which results: coscY=2
and interfacial
in eq. (4) gives:
nR.p 4,.2 ‘ln2
’
(14)
= &b/b.
The dihedral unstressed
angle will have a value 9 different
from the
(PO:
+ 2
cos-
We make the notation A = n sin (R/2) cos (e/2) and we notice that by developing (6) and introduction (8) we obtain:
(11) Eq. (10) is solved for R/r:
(12) The maximum value of A is 1 and is reached for: n = 1 + tan2+/2 so that (12) can be used for any combination of n and 9 provided the sign is choosen for a monotonic decrease of curvature for increasing n. This
+ 2
-
cos-
+o= 2b
Ab cos-.+o
-
2
(15)
Using (3) and (13), eq. (15) becomes:
cos$/cos$l=--
~YSV
r
Krb R ’
(16)
Usually b is assumed 5.54 x lo-” m. Then taking for convenience the constant K = 2 X lOI N/m3 we can compute r/R as a function of + and the curves are drawn as descending lines for several equivalent pore radii, in fig. 3. The ascending lines represent the variation of r/R with cp according to eq. (12) for various numbers of faces. Under the above assumption we find dihedral angles varying with the pore size and its number of faces as represented by the intersections in fig. 3.
I. V. Nicolaescu
/ F. Glodeanu, Pore geometry and interfacial
energy
255
done. However we expect the decrease of the angle with increasing curvature will be confirmed. This phenomenon may imply a decrease of the sintering rate of smaller pores compared to the general theory predictions. On the other hand, very small lenticular pores may become unstable on the grain boundary since their dihedral angle tends to zero. Possibly they dissolve in the grain boundary and the vacancies thus generated migrate towards larger neighbouring pores if local stress prevents their fast annihilation, or are left behind by the moving boundary and can precipitate as intergranular pores.
-0.5
i/
I
-l.ol/
/
//
//
Fig. 3. Normalised curvature eqs. (12) and (16).
References
Y 4Y /
versus dihedral
angle according
J to
4. Conclusions
The proposed model attempts a qualitative explanation of the observed values of the dihedral angles in pores. Of course, the mechanism proposed in the above paragraph might need improvements when compared with more accurate measurements that remain to be
(11 W.D. Kingery and B. Francois, in: Sintering and Related Phenomena, Eds. S.C. Kuczynski, N.A. Hooton and C.F. Gibbon (Gordon and Breach, New York, 1967) p. 471. 121 W.D. Kingery and B. Francois, ibid., p. 499. 131 W.B. Beere, Acta Met. 23 (1975) 131. (41 G.L. Reynolds, W.B. Beere and P.T. Sawbridge, J. Nucl. Mater. 41 (1971) 112. [51 E.N. Hodkin, J. Nucl. Mater. 88 (1980) 7. [61 F. Glodeanu, Doctoral Thesis ( 198 1). S. Nazare, and F. Thiimmler, J. Nucl. [71 P. Nikolopoulos, Mater. 71 (1977) 89.
I.V. Nicolaescu
and F. Glodeanu
Institute for Nuclear Power Reactors, P. 0. Box 78, Pitesti, Romania
Received
9 August
1982; accepted
30 August
1982
253
LETTER TO THE EDITORS PORE
GEOMETRY
AND INTERFACIAL
ENERGY
1. Introduction Pore and grain size and shape are determinant for the development during sintering of ceramic fuel as well as for its irradiation behaviour. The surface energy value ysv is required for any attempt of kinetic calculation and if a more realistic than spherical shape is assumed, the grain boundary energy boa is needed too. From tension equilibrium considerations a simple relation exists between the two energies and the angle + formed at the incidence of grain boudary and the free or pore surface:
k&v
= 2 cos( e/2).
(1)
Kingery and Franqois [l] reported as early as 1965 an important difference between the $I angle values measured on the free surface and those on the internal pores surface, which was attributed to the pressure of the entrapped gas. This explanation was later rejected [2]. For irradiated fuel, fission product contamination was assumed [3], but this does not answer the question of the unirradiated fuel. One also should notice the close values of the dihedral angle between 90 and 100 degrees for pores in such materials as UO,, mixed oxides, carbides both fresh and irradiated while free surface angles cluster roughly around 150 degrees 141. Iron oxide also fits these figures [5]. Our observations
show some dependence of the angle on the number of faces and the sizes of the pore too (see fig. 1). A quantitative description however is difficult to assess due to the uncertainty about the incidence of the fracture surface on the pore. No general explanation has yet been found for this phenomenon.
2. Geometrical model of closed pores In a single crystal isotropic matrix a pore assumes a spherical shape. The pressure exerted by the surface is: $?E,
(2)
r
where r is the radius of the pore. Whether this pressure is compensated or not by the internal gas pressure is immaterial since in the latter case the balance is provided by the matrix stress and we assume the pore contraction is slow enough to allow surface diffusion to maintain equilibrium shape of the pore. In a polycrystalline matrix there will be grain boundaries (GB) intersecting the pore surface. For simplicity we assume a normal incidence of these boundaries. They take over part of the superficial pressure thus modifying the pore shape towards a higher
GB
Fig. 1. Pore in the vicinity of the free surface.
0022-3 115/83/0000-0000/$03.00
0 1983 North-Holland
Fig. 2. Pore surface
geometry.
I. V. Nicolaescu
254
mean curvature
Pore geometry
radius R. Eq. (2) is altered in this case:
2Ysv 2Ysv LYSS
p=R=---
/ F. Glodeanu,
r
4nr2
(3)
’
where r is now an equivalent volumic radius and L is the total length of the surface-boundary intersection of a pore. If eq. (1) is used, eq. (3) becomes: 1 1 _=_R r
L cos( +/2) (4)
’
4or2
We assume the pore surface is shared by n equal-sized grains and focus our attention on a single face and the volume within a sector under it. We consider this sector to be conical rather than pyramidal and an axial crosssection through it is shown in fig. 2 where the left side represents the spherical equivalent pore while at the right the deformed shape is represented. Among the angles defined in fig. 2 an equation exists: rr a ---=__2 2
9 2
P 2’
(5)
which leads to: P siny=-cos
(
2,” 2 2 . 1
(6)
The volume of a sector is:
means:( -) for n < 1 + tan2+/2 and ( +) otherwise. We must admit that in the last term of formula (3) the exact area of the pore should be used rather than the equivalent sphere area in the denominator. But this would complicate to a considerable extent the solution for R/r. Besides, we noticed that the error is negligible for the case of n = 2 where a rigorous solution is easy to compute. For larger n we expect this error to diminish as well as the error of taking circular pore faces instead of polygonal. The relationship between pore curvature and dihedral angle is better expressed by (12) than by the formula proposed in ref. [I].
3. Physical connection of curvature and dihedral angle We mentioned earlier that the matrix is stressed by the pore. This stress is considered isostatic in the immediate vicinity of the surface. Since the grain boundary is formed by a lower atom packing than the matrix, a higher compressive modulus should be expected for it. Since according to (71 the surface energy for UO, is placed around 0.9 J/m2 at elevated temperatures, small pores (R < 10pm) may exert a stress well above 0.1 MN/m’. We assume the thickness b of the grain boundary is increased by Ab which is proportional to the stress p:
The value of L can be computed
as: kss
L=nnRsinz.
11 R
r
(13)
The grain boudary energy may also be assumed proportional to its thickness as a first approximation. This means that the excess energy is:
n-2 n’
and introduced
energ),
p=KAb.
from which results: coscY=2
and interfacial
in eq. (4) gives:
nR.p 4,.2 ‘ln2
’
(14)
= &b/b.
The dihedral unstressed
angle will have a value 9 different
from the
(PO:
+ 2
cos-
We make the notation A = n sin (R/2) cos (e/2) and we notice that by developing (6) and introduction (8) we obtain:
(11) Eq. (10) is solved for R/r:
(12) The maximum value of A is 1 and is reached for: n = 1 + tan2+/2 so that (12) can be used for any combination of n and 9 provided the sign is choosen for a monotonic decrease of curvature for increasing n. This
+ 2
-
cos-
+o= 2b
Ab cos-.+o
-
2
(15)
Using (3) and (13), eq. (15) becomes:
cos$/cos$l=--
~YSV
r
Krb R ’
(16)
Usually b is assumed 5.54 x lo-” m. Then taking for convenience the constant K = 2 X lOI N/m3 we can compute r/R as a function of + and the curves are drawn as descending lines for several equivalent pore radii, in fig. 3. The ascending lines represent the variation of r/R with cp according to eq. (12) for various numbers of faces. Under the above assumption we find dihedral angles varying with the pore size and its number of faces as represented by the intersections in fig. 3.
I. V. Nicolaescu
/ F. Glodeanu, Pore geometry and interfacial
energy
255
done. However we expect the decrease of the angle with increasing curvature will be confirmed. This phenomenon may imply a decrease of the sintering rate of smaller pores compared to the general theory predictions. On the other hand, very small lenticular pores may become unstable on the grain boundary since their dihedral angle tends to zero. Possibly they dissolve in the grain boundary and the vacancies thus generated migrate towards larger neighbouring pores if local stress prevents their fast annihilation, or are left behind by the moving boundary and can precipitate as intergranular pores.
-0.5
i/
I
-l.ol/
/
//
//
Fig. 3. Normalised curvature eqs. (12) and (16).
References
Y 4Y /
versus dihedral
angle according
J to
4. Conclusions
The proposed model attempts a qualitative explanation of the observed values of the dihedral angles in pores. Of course, the mechanism proposed in the above paragraph might need improvements when compared with more accurate measurements that remain to be
(11 W.D. Kingery and B. Francois, in: Sintering and Related Phenomena, Eds. S.C. Kuczynski, N.A. Hooton and C.F. Gibbon (Gordon and Breach, New York, 1967) p. 471. 121 W.D. Kingery and B. Francois, ibid., p. 499. 131 W.B. Beere, Acta Met. 23 (1975) 131. (41 G.L. Reynolds, W.B. Beere and P.T. Sawbridge, J. Nucl. Mater. 41 (1971) 112. [51 E.N. Hodkin, J. Nucl. Mater. 88 (1980) 7. [61 F. Glodeanu, Doctoral Thesis ( 198 1). S. Nazare, and F. Thiimmler, J. Nucl. [71 P. Nikolopoulos, Mater. 71 (1977) 89.
I.V. Nicolaescu
and F. Glodeanu
Institute for Nuclear Power Reactors, P. 0. Box 78, Pitesti, Romania
Received
9 August
1982; accepted
30 August
1982