- Email: [email protected]
Measurement of interfacial energy from extraction replicas of particles on grain boundaries
Metallography
259
Measurement of Interracial Energy from Extraction Replicas of Particles on Grain Boundaries A. G. M A R T I N ~,rD C. M. SELLARS
Department of Metallurgy, University of Sheffield, England
Cumulative frequency distribution curves of apparent dihedral angles expected on extraction replicas are derived from distributions of true dihedral angles by analysis of the projected image shapes of lenses situated on boundaries of random orientation in the solid. The predictions are compared with observations on cementite particles in c~-iron and silica particles in copper. It is shown that mean true dihedral angles can be reliably obtained from the mean apparent dihedral angles measured on extraction replicas.
Die Messung der Grenzfliichenenergie durch Extraktionsabdriicke yon Teilchen auf Korngrenzen Die Summenh~iufigkeitsverteilung der abgebildeten diedrischen Winkel, wie sie bei Extraktionsabdriicken zu erwarten ist, wird aus der wahren diedrischen Winkelverteilung berechnet, indem projizierte Bilder yon linsenf6rmigen Ausscheidungen auf Korngrenzen regelloser Orientierung in Festk6rpern analysiert werden. Die Voraussagen werden mit Beobachtungen an Zementit-Teilchen in ~-Eisen und Kiesels~iure-Teilchen in Kupfer verglichen. Es wird gezeigt, dab die mittleren wahren diedrischen Winkel genau aus den mittleren abgebildeten diedrischen Winkeln in Extraktionsabdriicken bestimmt werden k6nnen.
La mesure de l'dnergie superficielle sur des tirages d' extraction de partlcules prdcipit6es sur les limites de grain Les courbes de distribution des fr6quences cumul6es d'angles di~dres apparents auxquels on s'attend pour les tirages d'extractiort sont d&ermin6es ~ partir des distributions des angles di~dres r6els, et ce ~ l'aide d'une analyse des formes des images projet6es de particules lentiformes qui sont pr6cipit6es sur les limites de grains d'une orientation arbitraire dans le solide. Les pr6visions sont compar6es aux observations faites sur les particules de c6mentite dans le fer c~ et sur des particules de silicum dans le cuivre. Les auteurs montrent que les angles di~dres moyens r~els peuvent ~tre d~termin~s de fa~on certaine h partir des angles di$dres moyens apparents mesur6s sur les tirages d'extraction.
Metallography, 3 (1970) 259-273 Copyright © 1970 by American Elsevier Publishing Company, Inc.
260
A. G. Martin and C. M. Sellars
Introduction
Since the classic work of Smith, z it has been recognized that the structure of polyphase metals annealed at high temperatures approaches an equilibrium morphology determined by the relative interfacial energies. When all the boundaries involved are incoherent, the interfacial energies are expected to be nearly isotropic so that torque terms 2 can be neglected and equilibrium considered simply in terms of a balance of surface tensions. Isolated incoherent particles of a minor phase situated at the grain boundaries of the major phase thus assume a lenticular shape with the plane of the lens in the boundary plane. If the interracial energy (Yi) between the particle and matrix is independent of the misorientation between the particle and matrix lattices, the lens is composed of identical segments of a sphere on each side of the grain boundary. T h e dihedral angle (0) of the lens (Fig. 1) is then given by 2 cos (0/2) = YablY~
(1)
where Ygb is the grain boundary energy. T h e dihedral angle is simply related to the diameter (D) and thickness (t) of the lens as tan (0/2) = 2Dt/(D ~ -- t 2) DIRECTION OF ELECTRON i BEAM I
i
]
"
(2)
GRAIN BOUNDARY PLANE
//
i
i
i / / /!
/
FIG. 1. Geometry of a lenticular grain boundary particle. Section taken through the center of the particle on the plane parallel to the direction of the electron beam (replica norrhal) and perpendicular to the grain boundary plane.
261
Extraction Replicas of Particles on Grain Boundaries
or, equivalently, tan (0/4) = t/D
(3)
These relationships have been used by Ashby and Smith a and Smith and Palmer 4 to determine the interfacial energy of the Cu/SiO 2 interface from observations of internally oxidized copper-silicon alloys by thin foil microscopy. Using this technique enables the true dihedral angle to be measured directly as unsectioned particles are observed and the foil can be tilted to align the boundary plane parallel to the electron beam. The technique is, however, restricted to observations on small particles and, when particles are in the size range 500 to 10,000A, quantitative measurements are more conveniently made from extraction replicas. 5.6 Although in principle such replicas could also be tilted in the microscope to align the diametral section of a particle with the electron beam, by finding the tilt to give the minimum thickness to the image of the lens, in practice this procedure is difficult and inaccurate. This paper shows that reliable measurements of interracial energies can be made by statistical analysis of the image shapes observed from untilted extraction replicas.
Analysis of Image Shapes When a deep (greater than the diameter of the largest grain boundary particles) initial etch is used in the preparation of the specimen for extraction replication, any particles sectioned by initial mechanical polishing should be eliminated. An extraction replica can then be considered as a random plane through the specimen on which all particles that intersect the plane appear in the orientation they had in the solid. It has been shown elsewherO that this condition is obtained experimentally when the axial ratios of particles are relatively close to unity. As grain boundaries are randomly oriented in the solid, the grain boundary planes will intersect the plane of replication at a spectrum of angles from 0 ° to 90 °. The shape of the projected image of the grain boundary particles seen in the electron microscope will therefore depend both on the true dihedral angle (0) and on the angle (X) between the electron beam and the grain boundary plane (Fig. l). The projected diameter is always equal to the true diameter (D), but the projected thickness (1) will be greater than the true thickness (t) when X :?& 0. If l is used instead of t in equation 2, an apparent dihedral angle (0'), which is greater than the true angle, is obtained as tan (0'/2) = 2DI/(D 2 -- l 2)
(4)
From the geometry given in Fig. 1, it is simply shown that l =
D csc (0/2)[1
--
cos (0/2) cos x]
when
X < 0/2
when
X >~0/2
(5) l= Dsin2
A. G. Martin and C. M. Sellars
262
and that the apparent dihedral angle obtained from equation 4 is not the dihedral angle of the projected image (0"), which is given by tan (0"/2) = l sin (O/2)/[D sin (0/2) - - / ]
when
x < 0/2
tan (0"/2) = oo
when
X >~ 0/2
(6)
This angle cannot be calculated from measurements of D and l unless the true dihedral angle is known, and so it is not determinable experimentally. However, equation 6 does show that, as expected, the projected image changes from a lens shape to an ellipse when X ---- 0/2. If D and l are measured for values of X > 0/2 and are substituted into equation 4, the elliptical image is erroneously assumed to be a lens and an apparent dihedral angle < 1 8 0 ° will still be obtained. Substitution from equation 5 into equation 4 gives the relationship between the true dihedral angle (0), the apparent dihedral angle (0'), and the angle X. For convenience in later computation, the relationship has been rearranged to obtain sin X as tan (0/2) sin X -- cos (0'/4~
[ sinsin (0'/2) (0/2)
]
1
when
X < 0/2
when
X ~> 0/2
(7)
sin X = tan (0'/4)
T h e dependence of 0' on X given by this equation is shown in Fig. 2 for true dihedral angles in the range usually observed experimentally. 7 In a specimen with a equiaxed grain structure, the probability (p) that random boundaries of different orientation will intersect the replicated surface is simply p = cos x
(8)
where X is the angle between the normal to the replica and the boundary plane. (Replicas are examined in the microscope with the electron beam parallel to the replica normal.) If particles occupy the same area fraction on all boundaries and particles are measured only if the image of the intersection of the boundary with the replicated surface lies within the image of the particles, u the probability of observing particles on boundaries inclined at different angles is also given by p. T h e probability of measuring particles on boundaries inclined at large angles to the replica normal may be further reduced by imposing a condition that the boundary image must lie within a specified distance from the diametral section of the particle image. If this distance is taken as a constant fraction ( i a ) of the b When X exceeds 0/2, it becomes increasingly possible to extract particles on the boundary for which the image of the boundary lles outside the particles image.
263
Extraction Replicas of Particles on Grain Boundaries 180r 8 DEGREES 170
170
160
150
140
,,=, (~130 L~ CD120
~ z
<[ 110
Q: ~100
~_ 9o 80
70]./
/
/-~
sin X "
20
30
ton
(e'/4)
60 ~
0
J
10
40
"X.
50
60
70
SO
90
DEGREES
FIC. 2. Effect of angle of inclination (X) of the grain boundary to the electron beam on the appar,ent dihedral angle. particle diameter, the probability (P) of measuring particles on boundaries at different angles becomes P=p
when
2a ~ s i n x
P=p-2a/sinx
when
2a < s i n x
(9)
These probability functions are shown in Fig. 3 for values of a = 0.05 or 0.1 in the calculation of P.
264
A. G. Martin and C. M. Sellars
1-0
0.9
0.8
0-7
p 0.5
o o.
P (o = 0.1) 0.4
0-3~-
\P(o
= 0"05)
0-2
0.1
01i 0
10
20
30
40 Z
50
i
i
i
60
70
80
90
DEGREES
FIG. 3. Probability that particles on grain boundaries inclined at different angles to the electron b e a m are measured for dihedral angle determination.
The cumulative frequencies with which particles of different orientation are observed are given by integration of these probabilities as f = sin X
(10)
when all particles with the boundary image within the particle image are measured, and as F--Asinx
when 2 a ) s i n x
F=B(C+lnsinx)
when 2a < s i n x
(11)
when only particles with the boundary image within --LAD of the diametral
Extraction Replicas of Particles on Grain Boundaries
265
section of the particle image are measured. T h e constants A , B, and C d e p e n d on the value of a, and for a ~- 0.05 or 0.1 take the respective values A = 3.028 or 1.916 B = 0.3028 or 0.3832 C = 3.303 or 2.609 C u m u l a t i v e f r e q u e n c y curves of 0' for any true dihedral angle can be c o m p u t e d f r o m equations 7 and 10 or 11. T h e f o r m of c u m u l a t i v e distribution o b t a i n e d is illustrated for 0 = 130 ° by the curves labeled (7 = 0 ° in Fig. 4. It is i m p r o b a b l e that this f o r m of curve w o u l d be o b s e r v e d e x p e r i m e n t a l l y for two reasons. First, the true dihedral angle p r o b a b l y is not a u n i q u e value, but a distribution of 1.0
t
O'=0 O"=5 °o o" =10L cr=1504
08
o= '1 = 130 °
//
>. 0'6
'/ o= ,=,
O'= 5° ___ a.lO =
I//L__
.~ 0'4
////
90
100
110
APPARENT
120
DIHEDRAL
130
ANGLE
140
e'
f
O" • 15 o
150
160
170
110
DEGREES
FIG. 4. Predicted cumulative frequency curves of apparent dihedral angle for a mean true dihedral angle of 130 ° and different standard deviations.
266
.4. G. Martin and C. M. Sellars
values resulting from the dependence of grain boundary energy on misorientation, s Second, the values of O' measured on extraction replicas will be subject to random experimental error. In order to take the first effect into account, it has been assumed that true dihedral angles are normally distributed about a mean value (0) with a standard deviation of a. Cumulative frequency curves of O' have been computed e for a range of values of ~ and a by summing the cumulative 180
170]-
o" = 10 °
16C
150
140
130l-
ONLY PARTICLES WITH BOUNDARY IMAGE WITHIN
PARTICLE~
+ OD OF DIAHETRAL
C)
SECTION OF IMAGE.
120
,,,.J 110 <(
¢:3
W :E
ALL PARTICLES WITH BOUNDARy IMAGE WITHIN PARTICLE IMAGE. 8O
90
100 MEAN
FIG. 5.
110
120
APPARENT
130 DIHEDRAL
140
150
1SO
120
ANGLE 1~' DEGREES
Relationship between mean apparent dihedral angle and mean true dihedral
angle.
e Fortran program with G . E. Time Sharing Computer System.
Extraction Replicas of Particles on Grain Boundaries
267
frequencies obtained for values of 0 taken at 2 ° intervals in the range 0 - - 3a to 0 + 3a. I n the summation the contribution of each interval of 0 was obtained by integration of the normal distribution curve over the appropriate interval. Typical cumulative frequency curves for a = 5 °, 10 °, and 15 ° and 0 ---- 130 ° are shown in Fig. 4. Almost identical curves are obtained if a unique value of 0 equal to 0 is taken and it is assumed that, due to experimental errors in measuring 0', each interval of 2 ° of 0' appears as a normal distribution about the interval mean 0' with a standard deviation equal to that taken previously. T h u s , the computed cumulative frequency curves are appropriate for any combination of normal variation in the true dihedral angle and random error in measurement of 0' values that results in an effective value of a equal to the one chosen. It is apparent from Fig. 4 that the shape of the cumulative frequency distribution depends fairly critically on the standard deviation, but the mean value of 0', marked by vertical lines, hardly changes. T h e general relationships between 0 and 0' for the different conditions of measurement are thus almost independent of a, as shown in Fig. 5, except at high values of 0 when values of a are selected such that 0 q - 3 a is significantly greater than 180 °, leading to truncation of the distribution of 0. T h e values of 0' obtained for selected values of 0 are shown more accurately in Table I. T h e figures given are those for TABLE I RELATIONSHIP BETWEEN MEAN TRUE DIHEDRAL ANGLE AND MEaN APPARENT DIHEDRAL ANGLE
0' (degrees) 0 (degrees)
a = 0.05
a = 0.1
All boundary particles
70 90 110 130 150 170
91.0 4- 0.2 105.2 i 0.2 120.6 4- 0.2 136.8 ::[::0.1 153.9 - - 0.1 (171.6)
96.2 4- 0.2 108.9 4- 0.2 123.1 + 0.2 138.4 ::t::0.1 154.6 - - 0.1 (171.7)
114.6 4- 0.3 123.0 i 0.3 133.2 + 0.2 144.9 4- 0.2 158.1 - - 0.2 (172.5)
cr = 10°, except for 0 = 170 ° when only the value for ~r = 0 ° is given to avoid the problem of truncation. T h e upper limit is that for ~ = 15 ° and the lower limit for o ----0 °, which is identical to the result for a ---- 5 ° within the accuracy reported. T h e analysis thus shows that in most circumstances the mean true dihedral angle of isolated boundary particles can be reliably obtained from the mean apparent dihedral angle measured on extraction replicas, even though the relevant standard deviation is unknown.
268
A. G. Martin and C. M. Sellars
Comparison with Experimental Observations Replicas of Fe3C particles in ~-Fe and of SiOz particles in Cu were prepared as described elsewhere. 5,9 The carbide particles were obtained in a steel of composition 0.15 % C, 0.35% Si, 0.07% Mn, 0.012% A1, 0.012% S, 0.012% P, 0.013 % O, 0.008 % N, by quenching and tempering for 1 hour at 700°C followed by 90% cold work by swaging and rurther tempering for 50 or 75 hours at 700°C to obtain a fully recrystallized ferrite matrix of grain size ,~10/~. The silica particles were obtained by internal oxidation of a Cu-0.25% Si alloy at 1000°C. z° Both materials had nearly spherical particles within the grains and lenticular particles on the grain boundaries. Typical examples of the images obtained from carbide particles on grain boundaries are shown in Fig. 6. The images a, b, and c are all to be expected from symmetrical particles on boundaries inclined at random angles to the replica. Images a and b are on boundaries inclined at X < 0/2, and image c is on a boundary inclined at X > 0/2 and so appears as an ellipse instead of a lens. The image shapes in d and e are observed relatively rarely and could not result by tilting a symmetrical lens. As all particles should attain their equilibrium shapes during the treatment given, these shapes indicate that in all situations the interfacial energy is not entirely independent of misorientation between the particle and matrix lattices (d) and is not necessarily isotropic (e). Such particles have little effect on the measured dihedral angles because of their low frequency of occurrence. Particles were photographed at suitable magnification in an EM6 electron microscope, and measurements of diameter and apparent thickness were obtained on 154 particles of FeaC and 53 particles of SiOz. Only particles with the boundary image within +0.1D of the diametral section of the particle image were measured; that is, particles such as that shown in Fig. 6b were excluded. Apparent dihedral angles were obtained by substituting the measured diameter and apparent thickness into equation 4. The resulting cumulative distributions of 0' are shown in Figs. 7 and 8 together with the curves predicted for a = 0.1 and values of mean true dihedral angle that give a similar mean apparent dihedral angle to that observed experimentally. The values of ~ used for the predicted curves were selected to give equivalent slopes to those of the experimental points over the central part of the curves. The agreement in form of the experimental points and predicted curves is reasonable in both cases, although there is a tendency for fewer observations than predicted to be made at high 0' values. This probably results from the small differences in level between grains produced by etching, which causes the grain boundary images to have apparent width. As the widths are expected to increase as the inclination of the boundary to the surface normal increases, it is probable that more particles on such boundaries are excluded from measurements than predicted.
Extraction Replicas of Particles on Grain Boundaries
269
(a)
(b)
(c)
(a)
(e)
FIG. 6. Typical image shapes of Fe3C particles on grain boundaries observed on extraction replicas of a 0.15 ~o C steel recrystallized and tempered at 700°C. Magnification ~50,000 × .
270
A. G. Martin and C. M. Sellars 1.0 o
o
09
05
0.7
0'6
I.*.
• ~ " 123e° (I° 0 e O'• 132'7°
0.5
0"4
03
0"3
0.1
0
IO0
110
120
1"10
140
150
160
170
190
APPARENT DIHEDRAL ANGLE (~ DEGREES
Fxc. 7. Comparison of experimental points obtained on Fe/FeaC with cumulative distribution curve of apparent dihedral angle predicted to give the observed mean apparent dihedral angle.
The mean experimental values of 0' and the 95 % confidence limits of the means are 132.6 zk 2.0 ° for Fe/F%C and 142.0 ~ 3.2 ° for Cu/SiOz. The mean true dihedral angles derived from Fig. 5 are 122.9 ! 2 .1° and 134.8 :~ 3.3 °, respectively. The slight increase in confidence limit arises because of the uncertainty about the correct value of standard deviation (see Table I). The dihedral angle measured for Fe/FesC is significantly greater than the previously reported 1 value of 115 °. The reason for this discrepancy is not clear. However, in the determination of the interracial energy for Fe/F%C the major uncertainty still remains the value of the grain boundary energy to be used in equation 1. Accurate values of surface energy in body-centered cubic iron have only been reported for the 3-range? 1-13 These show that phosphorus, 11
Extraction Replicas of Particles on Grain Boundaries 1.0
.
.
.
.
271
.
e
'
i
0.9
0.8 o
o
0.7
k0 O W
§
t~
0.5
~
0.4
= 135°
a
0 1
0.3
0.2
EXPERIHENTAL
O* =1~2 "Oe
0'1
o
0 110
120
130
APPARENT
140
DIHEDRAL
1S0
ANGLE
150
e'
170
180
DEGREES
FIc. 8. Comparison of experimental points obtained on Cu/SiO2 with the cumulative distribution curve of apparent dihedral angle predicted to give the observed mean apparent dihedral angle. nitrogen, TM and oxygen a3 are all highly surface-active and significantly reduce the surface energy of pure iron. In the present steel, the oxygen in solution is expected to be reduced to a very low level by the silicon and at least half the nitrogen content is expected to be fixed by the aluminum and traces of vanadium, but the phosphorus is probably all in solution. As phosphorus and nitrogen have similar interfacial activities, 14 their effects have been considered additive--that is, as an effective free nitrogen content of 0.006 to 0.01 wt. %. From the data for surface energy TM and considering that the ratio of grain boundary to surface energy is 0.35,12 the most probable range of grain boundary energies at 1440°C is 700 to 650 ergs/cm z. The temperature dependence of surface energy for 3 - i r o n 0.01 wt.% nitrogen is 1.1 ergs/cm2°K, TM and for pure iron the interfacial
272
A. G. Martin and C. M. Sellars
entropy is probably 14 0.33 erg/cm ~ °K. Considering that the ratio of surface to grain boundary energy is independent of temperature gives a temperature dependence of grain boundary energy for iron--0.01 wt.~/o nitrogen of 0.38 erg/cm ~ °K and, by interpolation, a value of 0.1 erg/cm ~ °K for i r o n - 0.006 wt. % nitrogen. T h e most probable range of grain boundary energy at 700°C is thus 375 to 625 ergs/cm 2. The present dihedral angle measurements lead to a ratio of interracial energies 7gb/Yi = 0.96 4- 0.03, giving the best estimate of Yi = 520 ± 130 ergs/cm ~, with only a very small contribution to the overall uncertainty arising from the dihedral angle measurements. The difference between this result and the value of 700 i 300 ergs/cm ~ obtained by Kramer et al. 15 is of little significance within the combined uncertainties, but is consistent with the somewhat lower content of surface-active elements in their steel. In the case of Cu/SiO~, the ratio of interfacial energies, )'~b/Yi = 0.77 ± 0.05, is obtained from the dihedral angle. T h e grain boundary energy of copper has been particularly well established for the temperature range 800 ° to 1060°C. sas.z7 At 1000°C, Ygb = 550 ± 10 ergs/cm 2 and the derived value of interracial energy is 7i = 715 ± 50ergs/cm 2. In this case the uncertainty in dihedral angle contributes the major part of the overall uncertainty, but the limits are still reasonably narrow. T h e measured interracial energy is significantly less than the value reported from earlier thin foil observations 3 but is in good agreement with the value of 740 ergs/cm ~ obtained in later work. 4
Summary Analysis of the image shapes to be expected from lenticular particles situated on grain boundaries of random orientation shows that reliable values of true dihedral angles can be obtained from measurements of diameter and thickness of particle images on extraction replicas. Experimental measurements of Fe/FesC and Cu/SiO 2 give distributions of apparent dihedral angles in good agreement with the predicted curves. True dihedral angles obtained from these measurements have confidence limits that are sufficiently narrow to contribute a minor part to the overall uncertainty in the derived interracial energy except when the grain boundary energy is known with unusual certainty.
Acknowledgments We are grateful for the use of computing facilities to the Broken Hill Proprietary Co. Ltd., with whom one of us (C.M.S.) spent a year's leave of absence at their Melbourne Research Laboratories; to Dr. E. D. Hondros for helpful discussions about the grain boundary energy in our steel; and to the Science Research Council for provision of financial support for one of us (A.G.M.).
Extraction Replicas of Particles on Grain Boundaries
273
References 1. C. S. Smith, Trans. AIME, 175 (1948) 15. 2. C. Herring, in Physics of Powder Metallurgy (W. E. Kingery, ed.), McGraw-Hill, New York, 1951, p. 143. 3. M. F. Ashby and G. C. Smith, J. Inst. Metals, 91 (1962-63) 182. 4. I. G. Palmer and G. C. Smith, Oxide Dispersion Strengthening, Gordon and Breach, New York, 1968, p. 253. 5. T. Mukherjee, W. E. Stumpf, and C. M. Sellars, J. Mater. Sci., 3 (1968) 127. 6. W. E. Stumpf and C. M. Sellars, Metallography, 1 (1968) 25. 7. C. S. Smith, Met. Rev., 9 (1964) 1. 8. N. A. Gjostein and F. N. Rhines, Acta Met., 7 (1959) 319. 9. T. Mukherjee, W. E. Stumpf, C. M. Sellars, and W. J. McG. Tegart, J. Iron Steel Inst., 207 (1969) 621. 10. S. H. Ghude, Ph.D. Thesis, University of Sheffield, 1968. 11. E. D. Hondros, Proc. Roy. Soc., A286 (1965) 479. 12. E. D. Hondros, Met. Sci. J., 1 (1967) 36. 13. E. D. Hondros, Acta Met., 16 (1968) 1377. 14. E. D. Hondros, Interfaces Conference, Butterworths, Melbourne, 1969, p. 77. 15. J. J. Kramer, G. M. Pound, and R. F. Mehl, Acta Met., 6 (1958) 763. 16. J. C. Fisher and C. G. Dunn, Imperfections in Nearly Perfect Crystals, Wiley, New York, 1952, p. 317. 17. J. E. Hilliard, M. Cohen, and B. L. Averbach, Acta Met., 8 (1951) 251.
Accepted March 11, 1970
259
Measurement of Interracial Energy from Extraction Replicas of Particles on Grain Boundaries A. G. M A R T I N ~,rD C. M. SELLARS
Department of Metallurgy, University of Sheffield, England
Cumulative frequency distribution curves of apparent dihedral angles expected on extraction replicas are derived from distributions of true dihedral angles by analysis of the projected image shapes of lenses situated on boundaries of random orientation in the solid. The predictions are compared with observations on cementite particles in c~-iron and silica particles in copper. It is shown that mean true dihedral angles can be reliably obtained from the mean apparent dihedral angles measured on extraction replicas.
Die Messung der Grenzfliichenenergie durch Extraktionsabdriicke yon Teilchen auf Korngrenzen Die Summenh~iufigkeitsverteilung der abgebildeten diedrischen Winkel, wie sie bei Extraktionsabdriicken zu erwarten ist, wird aus der wahren diedrischen Winkelverteilung berechnet, indem projizierte Bilder yon linsenf6rmigen Ausscheidungen auf Korngrenzen regelloser Orientierung in Festk6rpern analysiert werden. Die Voraussagen werden mit Beobachtungen an Zementit-Teilchen in ~-Eisen und Kiesels~iure-Teilchen in Kupfer verglichen. Es wird gezeigt, dab die mittleren wahren diedrischen Winkel genau aus den mittleren abgebildeten diedrischen Winkeln in Extraktionsabdriicken bestimmt werden k6nnen.
La mesure de l'dnergie superficielle sur des tirages d' extraction de partlcules prdcipit6es sur les limites de grain Les courbes de distribution des fr6quences cumul6es d'angles di~dres apparents auxquels on s'attend pour les tirages d'extractiort sont d&ermin6es ~ partir des distributions des angles di~dres r6els, et ce ~ l'aide d'une analyse des formes des images projet6es de particules lentiformes qui sont pr6cipit6es sur les limites de grains d'une orientation arbitraire dans le solide. Les pr6visions sont compar6es aux observations faites sur les particules de c6mentite dans le fer c~ et sur des particules de silicum dans le cuivre. Les auteurs montrent que les angles di~dres moyens r~els peuvent ~tre d~termin~s de fa~on certaine h partir des angles di$dres moyens apparents mesur6s sur les tirages d'extraction.
Metallography, 3 (1970) 259-273 Copyright © 1970 by American Elsevier Publishing Company, Inc.
260
A. G. Martin and C. M. Sellars
Introduction
Since the classic work of Smith, z it has been recognized that the structure of polyphase metals annealed at high temperatures approaches an equilibrium morphology determined by the relative interfacial energies. When all the boundaries involved are incoherent, the interfacial energies are expected to be nearly isotropic so that torque terms 2 can be neglected and equilibrium considered simply in terms of a balance of surface tensions. Isolated incoherent particles of a minor phase situated at the grain boundaries of the major phase thus assume a lenticular shape with the plane of the lens in the boundary plane. If the interracial energy (Yi) between the particle and matrix is independent of the misorientation between the particle and matrix lattices, the lens is composed of identical segments of a sphere on each side of the grain boundary. T h e dihedral angle (0) of the lens (Fig. 1) is then given by 2 cos (0/2) = YablY~
(1)
where Ygb is the grain boundary energy. T h e dihedral angle is simply related to the diameter (D) and thickness (t) of the lens as tan (0/2) = 2Dt/(D ~ -- t 2) DIRECTION OF ELECTRON i BEAM I
i
]
"
(2)
GRAIN BOUNDARY PLANE
//
i
i
i / / /!
/
FIG. 1. Geometry of a lenticular grain boundary particle. Section taken through the center of the particle on the plane parallel to the direction of the electron beam (replica norrhal) and perpendicular to the grain boundary plane.
261
Extraction Replicas of Particles on Grain Boundaries
or, equivalently, tan (0/4) = t/D
(3)
These relationships have been used by Ashby and Smith a and Smith and Palmer 4 to determine the interfacial energy of the Cu/SiO 2 interface from observations of internally oxidized copper-silicon alloys by thin foil microscopy. Using this technique enables the true dihedral angle to be measured directly as unsectioned particles are observed and the foil can be tilted to align the boundary plane parallel to the electron beam. The technique is, however, restricted to observations on small particles and, when particles are in the size range 500 to 10,000A, quantitative measurements are more conveniently made from extraction replicas. 5.6 Although in principle such replicas could also be tilted in the microscope to align the diametral section of a particle with the electron beam, by finding the tilt to give the minimum thickness to the image of the lens, in practice this procedure is difficult and inaccurate. This paper shows that reliable measurements of interracial energies can be made by statistical analysis of the image shapes observed from untilted extraction replicas.
Analysis of Image Shapes When a deep (greater than the diameter of the largest grain boundary particles) initial etch is used in the preparation of the specimen for extraction replication, any particles sectioned by initial mechanical polishing should be eliminated. An extraction replica can then be considered as a random plane through the specimen on which all particles that intersect the plane appear in the orientation they had in the solid. It has been shown elsewherO that this condition is obtained experimentally when the axial ratios of particles are relatively close to unity. As grain boundaries are randomly oriented in the solid, the grain boundary planes will intersect the plane of replication at a spectrum of angles from 0 ° to 90 °. The shape of the projected image of the grain boundary particles seen in the electron microscope will therefore depend both on the true dihedral angle (0) and on the angle (X) between the electron beam and the grain boundary plane (Fig. l). The projected diameter is always equal to the true diameter (D), but the projected thickness (1) will be greater than the true thickness (t) when X :?& 0. If l is used instead of t in equation 2, an apparent dihedral angle (0'), which is greater than the true angle, is obtained as tan (0'/2) = 2DI/(D 2 -- l 2)
(4)
From the geometry given in Fig. 1, it is simply shown that l =
D csc (0/2)[1
--
cos (0/2) cos x]
when
X < 0/2
when
X >~0/2
(5) l= Dsin2
A. G. Martin and C. M. Sellars
262
and that the apparent dihedral angle obtained from equation 4 is not the dihedral angle of the projected image (0"), which is given by tan (0"/2) = l sin (O/2)/[D sin (0/2) - - / ]
when
x < 0/2
tan (0"/2) = oo
when
X >~ 0/2
(6)
This angle cannot be calculated from measurements of D and l unless the true dihedral angle is known, and so it is not determinable experimentally. However, equation 6 does show that, as expected, the projected image changes from a lens shape to an ellipse when X ---- 0/2. If D and l are measured for values of X > 0/2 and are substituted into equation 4, the elliptical image is erroneously assumed to be a lens and an apparent dihedral angle < 1 8 0 ° will still be obtained. Substitution from equation 5 into equation 4 gives the relationship between the true dihedral angle (0), the apparent dihedral angle (0'), and the angle X. For convenience in later computation, the relationship has been rearranged to obtain sin X as tan (0/2) sin X -- cos (0'/4~
[ sinsin (0'/2) (0/2)
]
1
when
X < 0/2
when
X ~> 0/2
(7)
sin X = tan (0'/4)
T h e dependence of 0' on X given by this equation is shown in Fig. 2 for true dihedral angles in the range usually observed experimentally. 7 In a specimen with a equiaxed grain structure, the probability (p) that random boundaries of different orientation will intersect the replicated surface is simply p = cos x
(8)
where X is the angle between the normal to the replica and the boundary plane. (Replicas are examined in the microscope with the electron beam parallel to the replica normal.) If particles occupy the same area fraction on all boundaries and particles are measured only if the image of the intersection of the boundary with the replicated surface lies within the image of the particles, u the probability of observing particles on boundaries inclined at different angles is also given by p. T h e probability of measuring particles on boundaries inclined at large angles to the replica normal may be further reduced by imposing a condition that the boundary image must lie within a specified distance from the diametral section of the particle image. If this distance is taken as a constant fraction ( i a ) of the b When X exceeds 0/2, it becomes increasingly possible to extract particles on the boundary for which the image of the boundary lles outside the particles image.
263
Extraction Replicas of Particles on Grain Boundaries 180r 8 DEGREES 170
170
160
150
140
,,=, (~130 L~ CD120
~ z
<[ 110
Q: ~100
~_ 9o 80
70]./
/
/-~
sin X "
20
30
ton
(e'/4)
60 ~
0
J
10
40
"X.
50
60
70
SO
90
DEGREES
FIC. 2. Effect of angle of inclination (X) of the grain boundary to the electron beam on the appar,ent dihedral angle. particle diameter, the probability (P) of measuring particles on boundaries at different angles becomes P=p
when
2a ~ s i n x
P=p-2a/sinx
when
2a < s i n x
(9)
These probability functions are shown in Fig. 3 for values of a = 0.05 or 0.1 in the calculation of P.
264
A. G. Martin and C. M. Sellars
1-0
0.9
0.8
0-7
p 0.5
o o.
P (o = 0.1) 0.4
0-3~-
\P(o
= 0"05)
0-2
0.1
01i 0
10
20
30
40 Z
50
i
i
i
60
70
80
90
DEGREES
FIG. 3. Probability that particles on grain boundaries inclined at different angles to the electron b e a m are measured for dihedral angle determination.
The cumulative frequencies with which particles of different orientation are observed are given by integration of these probabilities as f = sin X
(10)
when all particles with the boundary image within the particle image are measured, and as F--Asinx
when 2 a ) s i n x
F=B(C+lnsinx)
when 2a < s i n x
(11)
when only particles with the boundary image within --LAD of the diametral
Extraction Replicas of Particles on Grain Boundaries
265
section of the particle image are measured. T h e constants A , B, and C d e p e n d on the value of a, and for a ~- 0.05 or 0.1 take the respective values A = 3.028 or 1.916 B = 0.3028 or 0.3832 C = 3.303 or 2.609 C u m u l a t i v e f r e q u e n c y curves of 0' for any true dihedral angle can be c o m p u t e d f r o m equations 7 and 10 or 11. T h e f o r m of c u m u l a t i v e distribution o b t a i n e d is illustrated for 0 = 130 ° by the curves labeled (7 = 0 ° in Fig. 4. It is i m p r o b a b l e that this f o r m of curve w o u l d be o b s e r v e d e x p e r i m e n t a l l y for two reasons. First, the true dihedral angle p r o b a b l y is not a u n i q u e value, but a distribution of 1.0
t
O'=0 O"=5 °o o" =10L cr=1504
08
o= '1 = 130 °
//
>. 0'6
'/ o= ,=,
O'= 5° ___ a.lO =
I//L__
.~ 0'4
////
90
100
110
APPARENT
120
DIHEDRAL
130
ANGLE
140
e'
f
O" • 15 o
150
160
170
110
DEGREES
FIG. 4. Predicted cumulative frequency curves of apparent dihedral angle for a mean true dihedral angle of 130 ° and different standard deviations.
266
.4. G. Martin and C. M. Sellars
values resulting from the dependence of grain boundary energy on misorientation, s Second, the values of O' measured on extraction replicas will be subject to random experimental error. In order to take the first effect into account, it has been assumed that true dihedral angles are normally distributed about a mean value (0) with a standard deviation of a. Cumulative frequency curves of O' have been computed e for a range of values of ~ and a by summing the cumulative 180
170]-
o" = 10 °
16C
150
140
130l-
ONLY PARTICLES WITH BOUNDARY IMAGE WITHIN
PARTICLE~
+ OD OF DIAHETRAL
C)
SECTION OF IMAGE.
120
,,,.J 110 <(
¢:3
W :E
ALL PARTICLES WITH BOUNDARy IMAGE WITHIN PARTICLE IMAGE. 8O
90
100 MEAN
FIG. 5.
110
120
APPARENT
130 DIHEDRAL
140
150
1SO
120
ANGLE 1~' DEGREES
Relationship between mean apparent dihedral angle and mean true dihedral
angle.
e Fortran program with G . E. Time Sharing Computer System.
Extraction Replicas of Particles on Grain Boundaries
267
frequencies obtained for values of 0 taken at 2 ° intervals in the range 0 - - 3a to 0 + 3a. I n the summation the contribution of each interval of 0 was obtained by integration of the normal distribution curve over the appropriate interval. Typical cumulative frequency curves for a = 5 °, 10 °, and 15 ° and 0 ---- 130 ° are shown in Fig. 4. Almost identical curves are obtained if a unique value of 0 equal to 0 is taken and it is assumed that, due to experimental errors in measuring 0', each interval of 2 ° of 0' appears as a normal distribution about the interval mean 0' with a standard deviation equal to that taken previously. T h u s , the computed cumulative frequency curves are appropriate for any combination of normal variation in the true dihedral angle and random error in measurement of 0' values that results in an effective value of a equal to the one chosen. It is apparent from Fig. 4 that the shape of the cumulative frequency distribution depends fairly critically on the standard deviation, but the mean value of 0', marked by vertical lines, hardly changes. T h e general relationships between 0 and 0' for the different conditions of measurement are thus almost independent of a, as shown in Fig. 5, except at high values of 0 when values of a are selected such that 0 q - 3 a is significantly greater than 180 °, leading to truncation of the distribution of 0. T h e values of 0' obtained for selected values of 0 are shown more accurately in Table I. T h e figures given are those for TABLE I RELATIONSHIP BETWEEN MEAN TRUE DIHEDRAL ANGLE AND MEaN APPARENT DIHEDRAL ANGLE
0' (degrees) 0 (degrees)
a = 0.05
a = 0.1
All boundary particles
70 90 110 130 150 170
91.0 4- 0.2 105.2 i 0.2 120.6 4- 0.2 136.8 ::[::0.1 153.9 - - 0.1 (171.6)
96.2 4- 0.2 108.9 4- 0.2 123.1 + 0.2 138.4 ::t::0.1 154.6 - - 0.1 (171.7)
114.6 4- 0.3 123.0 i 0.3 133.2 + 0.2 144.9 4- 0.2 158.1 - - 0.2 (172.5)
cr = 10°, except for 0 = 170 ° when only the value for ~r = 0 ° is given to avoid the problem of truncation. T h e upper limit is that for ~ = 15 ° and the lower limit for o ----0 °, which is identical to the result for a ---- 5 ° within the accuracy reported. T h e analysis thus shows that in most circumstances the mean true dihedral angle of isolated boundary particles can be reliably obtained from the mean apparent dihedral angle measured on extraction replicas, even though the relevant standard deviation is unknown.
268
A. G. Martin and C. M. Sellars
Comparison with Experimental Observations Replicas of Fe3C particles in ~-Fe and of SiOz particles in Cu were prepared as described elsewhere. 5,9 The carbide particles were obtained in a steel of composition 0.15 % C, 0.35% Si, 0.07% Mn, 0.012% A1, 0.012% S, 0.012% P, 0.013 % O, 0.008 % N, by quenching and tempering for 1 hour at 700°C followed by 90% cold work by swaging and rurther tempering for 50 or 75 hours at 700°C to obtain a fully recrystallized ferrite matrix of grain size ,~10/~. The silica particles were obtained by internal oxidation of a Cu-0.25% Si alloy at 1000°C. z° Both materials had nearly spherical particles within the grains and lenticular particles on the grain boundaries. Typical examples of the images obtained from carbide particles on grain boundaries are shown in Fig. 6. The images a, b, and c are all to be expected from symmetrical particles on boundaries inclined at random angles to the replica. Images a and b are on boundaries inclined at X < 0/2, and image c is on a boundary inclined at X > 0/2 and so appears as an ellipse instead of a lens. The image shapes in d and e are observed relatively rarely and could not result by tilting a symmetrical lens. As all particles should attain their equilibrium shapes during the treatment given, these shapes indicate that in all situations the interfacial energy is not entirely independent of misorientation between the particle and matrix lattices (d) and is not necessarily isotropic (e). Such particles have little effect on the measured dihedral angles because of their low frequency of occurrence. Particles were photographed at suitable magnification in an EM6 electron microscope, and measurements of diameter and apparent thickness were obtained on 154 particles of FeaC and 53 particles of SiOz. Only particles with the boundary image within +0.1D of the diametral section of the particle image were measured; that is, particles such as that shown in Fig. 6b were excluded. Apparent dihedral angles were obtained by substituting the measured diameter and apparent thickness into equation 4. The resulting cumulative distributions of 0' are shown in Figs. 7 and 8 together with the curves predicted for a = 0.1 and values of mean true dihedral angle that give a similar mean apparent dihedral angle to that observed experimentally. The values of ~ used for the predicted curves were selected to give equivalent slopes to those of the experimental points over the central part of the curves. The agreement in form of the experimental points and predicted curves is reasonable in both cases, although there is a tendency for fewer observations than predicted to be made at high 0' values. This probably results from the small differences in level between grains produced by etching, which causes the grain boundary images to have apparent width. As the widths are expected to increase as the inclination of the boundary to the surface normal increases, it is probable that more particles on such boundaries are excluded from measurements than predicted.
Extraction Replicas of Particles on Grain Boundaries
269
(a)
(b)
(c)
(a)
(e)
FIG. 6. Typical image shapes of Fe3C particles on grain boundaries observed on extraction replicas of a 0.15 ~o C steel recrystallized and tempered at 700°C. Magnification ~50,000 × .
270
A. G. Martin and C. M. Sellars 1.0 o
o
09
05
0.7
0'6
I.*.
• ~ " 123e° (I° 0 e O'• 132'7°
0.5
0"4
03
0"3
0.1
0
IO0
110
120
1"10
140
150
160
170
190
APPARENT DIHEDRAL ANGLE (~ DEGREES
Fxc. 7. Comparison of experimental points obtained on Fe/FeaC with cumulative distribution curve of apparent dihedral angle predicted to give the observed mean apparent dihedral angle.
The mean experimental values of 0' and the 95 % confidence limits of the means are 132.6 zk 2.0 ° for Fe/F%C and 142.0 ~ 3.2 ° for Cu/SiOz. The mean true dihedral angles derived from Fig. 5 are 122.9 ! 2 .1° and 134.8 :~ 3.3 °, respectively. The slight increase in confidence limit arises because of the uncertainty about the correct value of standard deviation (see Table I). The dihedral angle measured for Fe/FesC is significantly greater than the previously reported 1 value of 115 °. The reason for this discrepancy is not clear. However, in the determination of the interracial energy for Fe/F%C the major uncertainty still remains the value of the grain boundary energy to be used in equation 1. Accurate values of surface energy in body-centered cubic iron have only been reported for the 3-range? 1-13 These show that phosphorus, 11
Extraction Replicas of Particles on Grain Boundaries 1.0
.
.
.
.
271
.
e
'
i
0.9
0.8 o
o
0.7
k0 O W
§
t~
0.5
~
0.4
= 135°
a
0 1
0.3
0.2
EXPERIHENTAL
O* =1~2 "Oe
0'1
o
0 110
120
130
APPARENT
140
DIHEDRAL
1S0
ANGLE
150
e'
170
180
DEGREES
FIc. 8. Comparison of experimental points obtained on Cu/SiO2 with the cumulative distribution curve of apparent dihedral angle predicted to give the observed mean apparent dihedral angle. nitrogen, TM and oxygen a3 are all highly surface-active and significantly reduce the surface energy of pure iron. In the present steel, the oxygen in solution is expected to be reduced to a very low level by the silicon and at least half the nitrogen content is expected to be fixed by the aluminum and traces of vanadium, but the phosphorus is probably all in solution. As phosphorus and nitrogen have similar interfacial activities, 14 their effects have been considered additive--that is, as an effective free nitrogen content of 0.006 to 0.01 wt. %. From the data for surface energy TM and considering that the ratio of grain boundary to surface energy is 0.35,12 the most probable range of grain boundary energies at 1440°C is 700 to 650 ergs/cm z. The temperature dependence of surface energy for 3 - i r o n 0.01 wt.% nitrogen is 1.1 ergs/cm2°K, TM and for pure iron the interfacial
272
A. G. Martin and C. M. Sellars
entropy is probably 14 0.33 erg/cm ~ °K. Considering that the ratio of surface to grain boundary energy is independent of temperature gives a temperature dependence of grain boundary energy for iron--0.01 wt.~/o nitrogen of 0.38 erg/cm ~ °K and, by interpolation, a value of 0.1 erg/cm ~ °K for i r o n - 0.006 wt. % nitrogen. T h e most probable range of grain boundary energy at 700°C is thus 375 to 625 ergs/cm 2. The present dihedral angle measurements lead to a ratio of interracial energies 7gb/Yi = 0.96 4- 0.03, giving the best estimate of Yi = 520 ± 130 ergs/cm ~, with only a very small contribution to the overall uncertainty arising from the dihedral angle measurements. The difference between this result and the value of 700 i 300 ergs/cm ~ obtained by Kramer et al. 15 is of little significance within the combined uncertainties, but is consistent with the somewhat lower content of surface-active elements in their steel. In the case of Cu/SiO~, the ratio of interfacial energies, )'~b/Yi = 0.77 ± 0.05, is obtained from the dihedral angle. T h e grain boundary energy of copper has been particularly well established for the temperature range 800 ° to 1060°C. sas.z7 At 1000°C, Ygb = 550 ± 10 ergs/cm 2 and the derived value of interracial energy is 7i = 715 ± 50ergs/cm 2. In this case the uncertainty in dihedral angle contributes the major part of the overall uncertainty, but the limits are still reasonably narrow. T h e measured interracial energy is significantly less than the value reported from earlier thin foil observations 3 but is in good agreement with the value of 740 ergs/cm ~ obtained in later work. 4
Summary Analysis of the image shapes to be expected from lenticular particles situated on grain boundaries of random orientation shows that reliable values of true dihedral angles can be obtained from measurements of diameter and thickness of particle images on extraction replicas. Experimental measurements of Fe/FesC and Cu/SiO 2 give distributions of apparent dihedral angles in good agreement with the predicted curves. True dihedral angles obtained from these measurements have confidence limits that are sufficiently narrow to contribute a minor part to the overall uncertainty in the derived interracial energy except when the grain boundary energy is known with unusual certainty.
Acknowledgments We are grateful for the use of computing facilities to the Broken Hill Proprietary Co. Ltd., with whom one of us (C.M.S.) spent a year's leave of absence at their Melbourne Research Laboratories; to Dr. E. D. Hondros for helpful discussions about the grain boundary energy in our steel; and to the Science Research Council for provision of financial support for one of us (A.G.M.).
Extraction Replicas of Particles on Grain Boundaries
273
References 1. C. S. Smith, Trans. AIME, 175 (1948) 15. 2. C. Herring, in Physics of Powder Metallurgy (W. E. Kingery, ed.), McGraw-Hill, New York, 1951, p. 143. 3. M. F. Ashby and G. C. Smith, J. Inst. Metals, 91 (1962-63) 182. 4. I. G. Palmer and G. C. Smith, Oxide Dispersion Strengthening, Gordon and Breach, New York, 1968, p. 253. 5. T. Mukherjee, W. E. Stumpf, and C. M. Sellars, J. Mater. Sci., 3 (1968) 127. 6. W. E. Stumpf and C. M. Sellars, Metallography, 1 (1968) 25. 7. C. S. Smith, Met. Rev., 9 (1964) 1. 8. N. A. Gjostein and F. N. Rhines, Acta Met., 7 (1959) 319. 9. T. Mukherjee, W. E. Stumpf, C. M. Sellars, and W. J. McG. Tegart, J. Iron Steel Inst., 207 (1969) 621. 10. S. H. Ghude, Ph.D. Thesis, University of Sheffield, 1968. 11. E. D. Hondros, Proc. Roy. Soc., A286 (1965) 479. 12. E. D. Hondros, Met. Sci. J., 1 (1967) 36. 13. E. D. Hondros, Acta Met., 16 (1968) 1377. 14. E. D. Hondros, Interfaces Conference, Butterworths, Melbourne, 1969, p. 77. 15. J. J. Kramer, G. M. Pound, and R. F. Mehl, Acta Met., 6 (1958) 763. 16. J. C. Fisher and C. G. Dunn, Imperfections in Nearly Perfect Crystals, Wiley, New York, 1952, p. 317. 17. J. E. Hilliard, M. Cohen, and B. L. Averbach, Acta Met., 8 (1951) 251.
Accepted March 11, 1970