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The equilibrium morphology of grain boundary phases
METALLOGRAPH Y 6, 457-464 (1973)
457
The Equilibrium Morphology of Grain Boundary Phases*
STEVEN P. TUCKER AND FREDERICK G. HOCHGRAF Department of Mechanical Engineering, Kingsbury Hall, University of New Hampshire, Durham, New Hampshire 03824
The extent of spreading by partially wetting grain boundary phases, i.e., dihedral angles between 0 and 60°, is predicted from the dihedral angle and the volume fraction of the grain boundary phase. A matrix of equiax grains is assumed, modeled as tetrakaidecahedrons, and the grain boundary phase is assumed to have a uniform cross section. The results are compared with published values.
Introduction In a two phase alloy, the minor phase frequently exhibits a preference for the junction between four grains of the major phase. From those points the minor phase may extend outward in the form of prisms along the lines formed by the three grain intersections, and then it may spread across the grain boundaries, thereby covering a fraction of the surface of the grains of the major phase. In many cases the physical properties of metals and ceramics are more closely related to the fraction of boundary area covered by the grain boundary phase than to the relative amount of the phase. At equilibrium this area fraction is believed to be determined by the volume fraction of the second phase and relative values of the surface tensions of the grain boundaries and the interphase interfaces, which define the dihedral angle 20 as shown in Fig. 1. Campbell [1] has written an exact two dimensional solution for the spreading of a boundary phase in a matrix of hexagonal grains. In his extension to three dimensions using a matrix of tetrakaidecahedrons, prisms of uniform cross section are assumed to fill the three grain intersections. The model appears to converge on the exact solution as the volume and area fractions go to zero. T h e present work corrects for overlapping areas and volumes at the intersections between prisms, extending the model's usefulness to higher area and volume fractions. * Sponsored in part by a grant from CURF, University of New Hampshire. © American Elsevier Publishing Company, Inc. 1973 31
458
Steven P. Tucker and Frederick G. Hochgraf
FIG. 1. A cross section is shown of a prism of minor phase in a 120°, three grain intersection. The interfacial tensions ~'zx and yz~ determine the dihedral angle 20. The cross hatched area is one of the six segments making up the prism.
Theory NONWETTING
CASE
For 60 ° < 20 ~< 180 ° the minor phase is distributed as droplets at the four grain intersections. When 20 : 180 °, the droplets are spheres and Campbell's exact solution is F a = 1.515(Fv)~
(1)
where F A is the area fraction and F v is the volume fraction. T h e case for 180 ° < 20 ~< 60 ° is difficult to treat. Smith [2] discusses the difficulty and presents a model for the shape of the minor phase droplet. As 20 becomes smaller, the sphere of minor phase develops four prismatic projections extending outward along the grain edges, oriented, ideally, at 109.5 ° to each other. These prisms appear to be stable to a length which depends upon the amount of second phase present. The angle at the corners of the cross section of the prism is the dihedral angle 29. PARTIAL WETTING
CASE
At 20 = 60 °, the extensions have the cross section of equilateral triangles and are stable at any length. In the case of 0 ° < 20 < 60 °, the minor phase takes on a cusped-triangular prism shape and is continuous in the grain edges. T h e original spheres located at the grain corners have changed, so that their surfaces are concave, becoming fillets between the sharp edges of the prisms where they join at the corners. As 20 becomes less than 60 ° the minor phase begins to spread into the grain boundaries of the major phase. As 2 0 approaches 0 °, and providing that sufficient material is present, the minor phase may spread enough to completely fill all the grain boundaries.
Equilibrium Morphology
459
I n the present model, the dihedral angle is in the range of 0 ° < 20 < 60 °. We assume the minor phase occurs as a continuous network in the three grain intersections of a matrix of tetrakaidecahedrons. All the minor phase is assumed to be in the prisms with none isolated in the faces or trapped within grains of the major phase. T h e system of grains is based upon the tetrakaidecahedron, the T K D , also known as the truncated octahedron. Some justification for the use of the T K D is in order. Smith [3] states that the ideal polyhedron has 13.394 faces, 22.789 corners, and 5.104 edges per face with the edges meeting at the characteristic angle of 109.5 °. T h e tetrakaidecahedron is a space filling regular polyhedron that comes very close to the ideal with 14 faces, 24 corners, an average of 5.143 edges per face and edges meeting at an average angle of 110 °. Although the /3-tetrakaidecahedron is recommended as a model for grain structure [4], its curved edges and faces make mathematical treatment more difficult. Figure 2 shows the T K D with its hexagonal and square faces, all of edge dimension a.
Fro. 2. The grains are modeled as TKD's of edge dimension a. Cross hatched areas are the projections of the prisms of minor phase on the original grain faces. The prisms join with the bevel angle ~ = 30° or 45 °. In this figure, the fraction of grain covered is F~4 = 0.2. T h e cross section of a cusped triangular prism is shown in Fig. 1. T h e parameter h is used as the prime dimension for the minor phase. T h e cross-sectional area of this prism has been given by Campbell [1] as 3h~K/4 where K = ~ / g + 3 cotan ( X ) - - 1--~o
3 X cosec 2 (X),
(1)
and X = 30 ° - O. T h e model is developed as a series of pieces, then the contributions of each piece are summed. F o r calculations of the area fraction, we consider the projection of the prism onto specific faces of the T K D . This projection takes the form of trapezoids h wide by a long at the edges of the faces with the ends beveled at an angle 0c, as shown in Fig. 2.
460
Steven P. Tucker and Frederick G. Hochgraf
For calculations of volume fraction the prism is broken into six prismatic segments, each obtained by rotation and/or reflection of the basic segment. Each segment is associated with a particular face and edge of a given grain as shown in Fig. 1. The volume of this segment is calculated assuming its cross section is uniform, and the ends are beveled at an angle 0c relative to the normal from its axis. T h e angle 0~is 30 ° for the hexagonal faces and 45 ° for the square faces. T h e volume of this segment is Vye ---- (ah2K--h 3 tan (~)M)/8,
(2)
where M = l + 3 v / ' J cotan ( X ) + 2 ~ / 3 cotan 3 ( X ) - - 2 ~ / ] coseca (X). Calculations are made in two regions. T h e first is for h/a < 0.5, before the square face is covered. The second is for 0.5 ~< h/a <. 0.866 after the square face is covered and up to coverage of the hexagonal faces. The area fraction for the first region can be calculated by summing the area from each face, using the appropriate end bevel and dividing by the total surface area of the T K D ; F a = (36h/a--(12q-Sv/~)(h/a)')/(6V~--}-3), h/a < 0.5.
(3)
T h e volume fraction is calculated in the same manner. F v = (9K(h/a)2--(3+2X/3)M(h/a)3)/8~/~, h/a < 0.5.
(4)
Since the model presupposes equal extent of spreading on square and hexagonal faces, the square face is covered first, specifically, at h/a = 0.5. Thereafter the geometry of the minor phase on the square faces of the T K D changes. Figure 3 shows a section of the T K D taken across the plane of a square face, after coverage by the second phase. The new geometry can be described as a set of four seg-
j
0
y~
square face
FIG. 3. Cross section of a square face adjoining hexagonal faces. The~spreading minor phase covers the square faces at h/a ~ 0.5. Thereafter, a truncated pyramid of minor phase lies under the cusps. In this figure, h[a is 0.7 and FA is 0.97.
Equilibrium Morphology
461
ments set upon a truncated pyramid of base a, height L and sides inclined 60 ° with respect to the base. T h e height L can be written as L = hN, where U
(v/g/2 sin (X)) (cos (O)--~/1--(((a/h-2)/~/-J) sin (X)--sin (0))2).
T h e truncated pyramid, together with the cusp shaped pyramids above it are divided into four segments. These four segments can be written in terms of new parameters H and ~, related to h and 0 as H = (~/Ja/2--hN)/~/~ and -- cos -z (~/1--(((a/h--2)/~/~) sin (X)--sin (0)) 2. T o calculate the area fraction of the region where h/a > 0.5, we note that the square faces are covered, and sum them with the covered area on the hexagonal faces and divide by the total surface area of the T K D .
FA
=
(3+24(h/a)--8~/g(h/a)2)/(6~/J+3), 0.5 < h/a < 0.866.
(5)
The volume fraction is calculated by summing the segments from hexagonal faces, the truncated pyramids and the segments from the square faces and dividing by the total volume.
F v = (X/J+6K(h/a)~--Z~/JM(h/a)3+3K(H/a)2--(3M*+8~/J)(H/a)3)/8~/~,
(6)
where M* is expressed in terms of H and $ (0.5 ~< h/a < 0.866).
Discussion The predicted relation between FA, Fv, and 20 is shown in Fig. 4. Equation (1) is plotted for 20 -- 180 °. Equations (3) and (4) are plotted for 0 < h/a < 0.5. Equations (5) and (6) are plotted for 0.5 ~< h/a < 0.866. Following Campbell [1], the region 60 ° < 20 < 180 ° has been interpolated assuming a smooth relation between F A, F v, and 20. The inflection in the curve at F A = 0.861 is due to a strict interpretation of spreading on the faces of the assumed T K D model for equiaxed microstructure. For a given dihedral angle held constant on all faces of the T K D , the square faces are completely covered at h/a = 0.5 or F A = 0.861. Increased volume fraction can only contribute to covering the hexagonal faces, hence an inflection at F A = 0.861. One commonly observes spreading at 20 = 0 which is more extensive than predicted by Fig. 4. This is probably due to an ambiguity in the nature of the statement 20 = 0. Referring to Fig. 1, the case of 20 ~ 0 occurs when 711 > 2712" When 711 = 2712, there is no reduction of total free energy due to a second phase replacing a portion of the grain boundary of the first phase. Spreading will occur until the condition 20 -- 0 is achieved and, in the absence of external effects, will go no further. When YZl > 27zz there is a net reduction of free energy from every element of boundary that is wetted by a second phase. This will tend to drive the second phase until all boundaries have been filled, providing there is sufficient second phase available. Thus, complete spreading
462
Steven P. Tucker and Frederick G. Hochgraf
may or may not occur at 20 : 0. The model is presented only for ~'11 : 2~12" The predicted F A at 20 = 0 should be interpreted as lower bound and similarly F V : 0.25 as an upper bound for completely filled grain boundaries. In Fig. 5, predicted values of F A are compared to observed values. The data in Table 1 were compiled by Campbell [1] from photomicrographs of a variety of annealed alloys [2]. Using the observed values for F z and 20, Fig. 4 was used to predict values of F A. In Fig. 5 these values and Campbell's earlier predicted values are compared to the observed F A. The refinements to the model introduced in the present work appear to have corrected the tendency for the original model to overestimate F A at high fractions. At low fractions the F A predictions converge.
1.0
. __,
__,
__
,
.
.
.
.
FV~0'.5
.
0.8
0.6 't
FA 0,4
~
~-~
~
~
0.05
0,2 0.0I 0.00I
'
i 30
i
t
r
i
i
60
i
i
i
90 DIHEDRAL
ANGLE
J "120
150
180
20
FIG. 4. The covered fraction of grain boundary area FA is plotted as a function of dihedral angle 20 and the volume fraction of the minor phase F v according to the proposed
model. The present model may also be overestimating the extent of spreading. Once equilibrium has been attained, the real shape of the interphase interface would be a surface of minimum energy. If the surface energies are crystallographically isotropic, then the interface would form a so-called minimal surface, subject only to the constraints of grain boundary dimensions, dihedral angle, and minimized surface to volume ratio. The present model has four prismatic bodies meeting with sharp intersections. In adapting it to a minimal surface, the sharp intersections would become smooth fillets. The additional material in the grain corners would be obtained at the expense of material in the boundary prisms, thereby reducing the extent of spreading. It is therefore likely that the present model has overestimated the real extent of spreading.
Equilibrium Morphology
463
1.4
C]
t
0.8
.1.0
x.2
0.8
8
O
c~
C3
u H
o.(,
O
0.4
/ 0.2
0.4
0.6
O[~SERVED COVEK\GE FA
FIG. 5. The fractional grain boundary coverage FA predicted by the model is compared to observed values of Fil. Only particles in grain corners are considered. O predicted values of FA obtained from Fig. 4 using observed values of FV. [] predicted FA values from Campbell's analysis [1].
Summary The model, resulting in Fig. 4, can be used to determine the extent of spreading of a minor phase among equiax grains if the dihedral angle and volume fraction are known. The model is an improvement over an earlier one in that the area fraction predictions fit observed values better, although there may be a tendency to overestimate the extent of spreading. Some of the basic assumptions are as follows. (1) The tetrakaidecahedron is used as the matrix grain in an equiaxed microstructure. This is somewhat artificial because of the complete lack of pentagonal faces which predominate in most microstructures. However, in terms of averages, the T K D represents a reasonable model. (2) The minimum interracial energy condition is represented by using arcs of circles as the cross section of the interphase interface. The intersection of the prisms at grain corners is assumed to have sharp edges; that is, we neglect fillets that would smooth the contour.
464
Steven P. Tucker and Frederick G. Hochgraf TABLE 1 Experimental Measurements of Particles in Grain Corners of Annealed Alloys
Material
Alpha+liquid in leaded brass Cu-30Zn-3Pb Copper+liquid lead Cu-3Pb Beta+liquid in leaded brass Cu-46Zn-3Pb Copper + liquid bismuth Cu-lBi Alpha-beta brass Cu-37Zn Alpha-beta bronze Cu-16Sn Partly melted Cu-15 Ag Iron+liquid Fe-30Cu
Observed FA
Observed FV
0.29
0.032
Liquid vs ~/~ 80°
0.25
0.016
Pb. vs ~[:~ 50-70°
0.17
0.008
Liquid vs 8//3 110°
0.27
0.006
Bi vs Cu/Cu ,~0 ° /3 vs ~/~ 70° /3 vs ~[~ 95° Liquid vs Cu/C ~ 10o Liquid vs y/~, 20°
~0.5
0.09
~0.5
~0.2
,~0.97
~0.1
,-,A).8
~0.2
Observed 20
(3) T h e dihedral angle is independent of crystallographic orientation. This is realistic in most metallic structures. However, in some other materials the surface tensions are not isotropie, thus the prisms may become assymetrical [5]. (4) All the minor phase is assumed to be in the grain edges. I n an equiaxed, well annealed structure the lenticular particles on the grain faces would have been swept into the grain edges or bypassed and entrapped within a grain. Spherical particles trapped inside grains have no effect on the boundary phase, so they may be disregarded. (5) A smooth interpolation is made between 20 = 60 ° and 20 = 180 °. (6) We ignore the issue of m i n i m u m phase thickness, variously reported to be from 5 to 80 atom diameters.
References 1. 2. 3. 4. 5.
J. Campbell, Metallography 4, 269-78 (1971). C. S. Smith, Trans. TMS-AIME 175, 15-51 (1948). C. S. Smith, Sci. Amer. 190, 58-64 (1954). R. E. Williams, Science 161, 276 (1968). W. A. Miller and G. A. Chadwick, Iron Steel Inst. (London) Publ. 110, 49-56 (1967).
Accepted August 16, 1973
457
The Equilibrium Morphology of Grain Boundary Phases*
STEVEN P. TUCKER AND FREDERICK G. HOCHGRAF Department of Mechanical Engineering, Kingsbury Hall, University of New Hampshire, Durham, New Hampshire 03824
The extent of spreading by partially wetting grain boundary phases, i.e., dihedral angles between 0 and 60°, is predicted from the dihedral angle and the volume fraction of the grain boundary phase. A matrix of equiax grains is assumed, modeled as tetrakaidecahedrons, and the grain boundary phase is assumed to have a uniform cross section. The results are compared with published values.
Introduction In a two phase alloy, the minor phase frequently exhibits a preference for the junction between four grains of the major phase. From those points the minor phase may extend outward in the form of prisms along the lines formed by the three grain intersections, and then it may spread across the grain boundaries, thereby covering a fraction of the surface of the grains of the major phase. In many cases the physical properties of metals and ceramics are more closely related to the fraction of boundary area covered by the grain boundary phase than to the relative amount of the phase. At equilibrium this area fraction is believed to be determined by the volume fraction of the second phase and relative values of the surface tensions of the grain boundaries and the interphase interfaces, which define the dihedral angle 20 as shown in Fig. 1. Campbell [1] has written an exact two dimensional solution for the spreading of a boundary phase in a matrix of hexagonal grains. In his extension to three dimensions using a matrix of tetrakaidecahedrons, prisms of uniform cross section are assumed to fill the three grain intersections. The model appears to converge on the exact solution as the volume and area fractions go to zero. T h e present work corrects for overlapping areas and volumes at the intersections between prisms, extending the model's usefulness to higher area and volume fractions. * Sponsored in part by a grant from CURF, University of New Hampshire. © American Elsevier Publishing Company, Inc. 1973 31
458
Steven P. Tucker and Frederick G. Hochgraf
FIG. 1. A cross section is shown of a prism of minor phase in a 120°, three grain intersection. The interfacial tensions ~'zx and yz~ determine the dihedral angle 20. The cross hatched area is one of the six segments making up the prism.
Theory NONWETTING
CASE
For 60 ° < 20 ~< 180 ° the minor phase is distributed as droplets at the four grain intersections. When 20 : 180 °, the droplets are spheres and Campbell's exact solution is F a = 1.515(Fv)~
(1)
where F A is the area fraction and F v is the volume fraction. T h e case for 180 ° < 20 ~< 60 ° is difficult to treat. Smith [2] discusses the difficulty and presents a model for the shape of the minor phase droplet. As 20 becomes smaller, the sphere of minor phase develops four prismatic projections extending outward along the grain edges, oriented, ideally, at 109.5 ° to each other. These prisms appear to be stable to a length which depends upon the amount of second phase present. The angle at the corners of the cross section of the prism is the dihedral angle 29. PARTIAL WETTING
CASE
At 20 = 60 °, the extensions have the cross section of equilateral triangles and are stable at any length. In the case of 0 ° < 20 < 60 °, the minor phase takes on a cusped-triangular prism shape and is continuous in the grain edges. T h e original spheres located at the grain corners have changed, so that their surfaces are concave, becoming fillets between the sharp edges of the prisms where they join at the corners. As 20 becomes less than 60 ° the minor phase begins to spread into the grain boundaries of the major phase. As 2 0 approaches 0 °, and providing that sufficient material is present, the minor phase may spread enough to completely fill all the grain boundaries.
Equilibrium Morphology
459
I n the present model, the dihedral angle is in the range of 0 ° < 20 < 60 °. We assume the minor phase occurs as a continuous network in the three grain intersections of a matrix of tetrakaidecahedrons. All the minor phase is assumed to be in the prisms with none isolated in the faces or trapped within grains of the major phase. T h e system of grains is based upon the tetrakaidecahedron, the T K D , also known as the truncated octahedron. Some justification for the use of the T K D is in order. Smith [3] states that the ideal polyhedron has 13.394 faces, 22.789 corners, and 5.104 edges per face with the edges meeting at the characteristic angle of 109.5 °. T h e tetrakaidecahedron is a space filling regular polyhedron that comes very close to the ideal with 14 faces, 24 corners, an average of 5.143 edges per face and edges meeting at an average angle of 110 °. Although the /3-tetrakaidecahedron is recommended as a model for grain structure [4], its curved edges and faces make mathematical treatment more difficult. Figure 2 shows the T K D with its hexagonal and square faces, all of edge dimension a.
Fro. 2. The grains are modeled as TKD's of edge dimension a. Cross hatched areas are the projections of the prisms of minor phase on the original grain faces. The prisms join with the bevel angle ~ = 30° or 45 °. In this figure, the fraction of grain covered is F~4 = 0.2. T h e cross section of a cusped triangular prism is shown in Fig. 1. T h e parameter h is used as the prime dimension for the minor phase. T h e cross-sectional area of this prism has been given by Campbell [1] as 3h~K/4 where K = ~ / g + 3 cotan ( X ) - - 1--~o
3 X cosec 2 (X),
(1)
and X = 30 ° - O. T h e model is developed as a series of pieces, then the contributions of each piece are summed. F o r calculations of the area fraction, we consider the projection of the prism onto specific faces of the T K D . This projection takes the form of trapezoids h wide by a long at the edges of the faces with the ends beveled at an angle 0c, as shown in Fig. 2.
460
Steven P. Tucker and Frederick G. Hochgraf
For calculations of volume fraction the prism is broken into six prismatic segments, each obtained by rotation and/or reflection of the basic segment. Each segment is associated with a particular face and edge of a given grain as shown in Fig. 1. The volume of this segment is calculated assuming its cross section is uniform, and the ends are beveled at an angle 0c relative to the normal from its axis. T h e angle 0~is 30 ° for the hexagonal faces and 45 ° for the square faces. T h e volume of this segment is Vye ---- (ah2K--h 3 tan (~)M)/8,
(2)
where M = l + 3 v / ' J cotan ( X ) + 2 ~ / 3 cotan 3 ( X ) - - 2 ~ / ] coseca (X). Calculations are made in two regions. T h e first is for h/a < 0.5, before the square face is covered. The second is for 0.5 ~< h/a <. 0.866 after the square face is covered and up to coverage of the hexagonal faces. The area fraction for the first region can be calculated by summing the area from each face, using the appropriate end bevel and dividing by the total surface area of the T K D ; F a = (36h/a--(12q-Sv/~)(h/a)')/(6V~--}-3), h/a < 0.5.
(3)
T h e volume fraction is calculated in the same manner. F v = (9K(h/a)2--(3+2X/3)M(h/a)3)/8~/~, h/a < 0.5.
(4)
Since the model presupposes equal extent of spreading on square and hexagonal faces, the square face is covered first, specifically, at h/a = 0.5. Thereafter the geometry of the minor phase on the square faces of the T K D changes. Figure 3 shows a section of the T K D taken across the plane of a square face, after coverage by the second phase. The new geometry can be described as a set of four seg-
j
0
y~
square face
FIG. 3. Cross section of a square face adjoining hexagonal faces. The~spreading minor phase covers the square faces at h/a ~ 0.5. Thereafter, a truncated pyramid of minor phase lies under the cusps. In this figure, h[a is 0.7 and FA is 0.97.
Equilibrium Morphology
461
ments set upon a truncated pyramid of base a, height L and sides inclined 60 ° with respect to the base. T h e height L can be written as L = hN, where U
(v/g/2 sin (X)) (cos (O)--~/1--(((a/h-2)/~/-J) sin (X)--sin (0))2).
T h e truncated pyramid, together with the cusp shaped pyramids above it are divided into four segments. These four segments can be written in terms of new parameters H and ~, related to h and 0 as H = (~/Ja/2--hN)/~/~ and -- cos -z (~/1--(((a/h--2)/~/~) sin (X)--sin (0)) 2. T o calculate the area fraction of the region where h/a > 0.5, we note that the square faces are covered, and sum them with the covered area on the hexagonal faces and divide by the total surface area of the T K D .
FA
=
(3+24(h/a)--8~/g(h/a)2)/(6~/J+3), 0.5 < h/a < 0.866.
(5)
The volume fraction is calculated by summing the segments from hexagonal faces, the truncated pyramids and the segments from the square faces and dividing by the total volume.
F v = (X/J+6K(h/a)~--Z~/JM(h/a)3+3K(H/a)2--(3M*+8~/J)(H/a)3)/8~/~,
(6)
where M* is expressed in terms of H and $ (0.5 ~< h/a < 0.866).
Discussion The predicted relation between FA, Fv, and 20 is shown in Fig. 4. Equation (1) is plotted for 20 -- 180 °. Equations (3) and (4) are plotted for 0 < h/a < 0.5. Equations (5) and (6) are plotted for 0.5 ~< h/a < 0.866. Following Campbell [1], the region 60 ° < 20 < 180 ° has been interpolated assuming a smooth relation between F A, F v, and 20. The inflection in the curve at F A = 0.861 is due to a strict interpretation of spreading on the faces of the assumed T K D model for equiaxed microstructure. For a given dihedral angle held constant on all faces of the T K D , the square faces are completely covered at h/a = 0.5 or F A = 0.861. Increased volume fraction can only contribute to covering the hexagonal faces, hence an inflection at F A = 0.861. One commonly observes spreading at 20 = 0 which is more extensive than predicted by Fig. 4. This is probably due to an ambiguity in the nature of the statement 20 = 0. Referring to Fig. 1, the case of 20 ~ 0 occurs when 711 > 2712" When 711 = 2712, there is no reduction of total free energy due to a second phase replacing a portion of the grain boundary of the first phase. Spreading will occur until the condition 20 -- 0 is achieved and, in the absence of external effects, will go no further. When YZl > 27zz there is a net reduction of free energy from every element of boundary that is wetted by a second phase. This will tend to drive the second phase until all boundaries have been filled, providing there is sufficient second phase available. Thus, complete spreading
462
Steven P. Tucker and Frederick G. Hochgraf
may or may not occur at 20 : 0. The model is presented only for ~'11 : 2~12" The predicted F A at 20 = 0 should be interpreted as lower bound and similarly F V : 0.25 as an upper bound for completely filled grain boundaries. In Fig. 5, predicted values of F A are compared to observed values. The data in Table 1 were compiled by Campbell [1] from photomicrographs of a variety of annealed alloys [2]. Using the observed values for F z and 20, Fig. 4 was used to predict values of F A. In Fig. 5 these values and Campbell's earlier predicted values are compared to the observed F A. The refinements to the model introduced in the present work appear to have corrected the tendency for the original model to overestimate F A at high fractions. At low fractions the F A predictions converge.
1.0
. __,
__,
__
,
.
.
.
.
FV~0'.5
.
0.8
0.6 't
FA 0,4
~
~-~
~
~
0.05
0,2 0.0I 0.00I
'
i 30
i
t
r
i
i
60
i
i
i
90 DIHEDRAL
ANGLE
J "120
150
180
20
FIG. 4. The covered fraction of grain boundary area FA is plotted as a function of dihedral angle 20 and the volume fraction of the minor phase F v according to the proposed
model. The present model may also be overestimating the extent of spreading. Once equilibrium has been attained, the real shape of the interphase interface would be a surface of minimum energy. If the surface energies are crystallographically isotropic, then the interface would form a so-called minimal surface, subject only to the constraints of grain boundary dimensions, dihedral angle, and minimized surface to volume ratio. The present model has four prismatic bodies meeting with sharp intersections. In adapting it to a minimal surface, the sharp intersections would become smooth fillets. The additional material in the grain corners would be obtained at the expense of material in the boundary prisms, thereby reducing the extent of spreading. It is therefore likely that the present model has overestimated the real extent of spreading.
Equilibrium Morphology
463
1.4
C]
t
0.8
.1.0
x.2
0.8
8
O
c~
C3
u H
o.(,
O
0.4
/ 0.2
0.4
0.6
O[~SERVED COVEK\GE FA
FIG. 5. The fractional grain boundary coverage FA predicted by the model is compared to observed values of Fil. Only particles in grain corners are considered. O predicted values of FA obtained from Fig. 4 using observed values of FV. [] predicted FA values from Campbell's analysis [1].
Summary The model, resulting in Fig. 4, can be used to determine the extent of spreading of a minor phase among equiax grains if the dihedral angle and volume fraction are known. The model is an improvement over an earlier one in that the area fraction predictions fit observed values better, although there may be a tendency to overestimate the extent of spreading. Some of the basic assumptions are as follows. (1) The tetrakaidecahedron is used as the matrix grain in an equiaxed microstructure. This is somewhat artificial because of the complete lack of pentagonal faces which predominate in most microstructures. However, in terms of averages, the T K D represents a reasonable model. (2) The minimum interracial energy condition is represented by using arcs of circles as the cross section of the interphase interface. The intersection of the prisms at grain corners is assumed to have sharp edges; that is, we neglect fillets that would smooth the contour.
464
Steven P. Tucker and Frederick G. Hochgraf TABLE 1 Experimental Measurements of Particles in Grain Corners of Annealed Alloys
Material
Alpha+liquid in leaded brass Cu-30Zn-3Pb Copper+liquid lead Cu-3Pb Beta+liquid in leaded brass Cu-46Zn-3Pb Copper + liquid bismuth Cu-lBi Alpha-beta brass Cu-37Zn Alpha-beta bronze Cu-16Sn Partly melted Cu-15 Ag Iron+liquid Fe-30Cu
Observed FA
Observed FV
0.29
0.032
Liquid vs ~/~ 80°
0.25
0.016
Pb. vs ~[:~ 50-70°
0.17
0.008
Liquid vs 8//3 110°
0.27
0.006
Bi vs Cu/Cu ,~0 ° /3 vs ~/~ 70° /3 vs ~[~ 95° Liquid vs Cu/C ~ 10o Liquid vs y/~, 20°
~0.5
0.09
~0.5
~0.2
,~0.97
~0.1
,-,A).8
~0.2
Observed 20
(3) T h e dihedral angle is independent of crystallographic orientation. This is realistic in most metallic structures. However, in some other materials the surface tensions are not isotropie, thus the prisms may become assymetrical [5]. (4) All the minor phase is assumed to be in the grain edges. I n an equiaxed, well annealed structure the lenticular particles on the grain faces would have been swept into the grain edges or bypassed and entrapped within a grain. Spherical particles trapped inside grains have no effect on the boundary phase, so they may be disregarded. (5) A smooth interpolation is made between 20 = 60 ° and 20 = 180 °. (6) We ignore the issue of m i n i m u m phase thickness, variously reported to be from 5 to 80 atom diameters.
References 1. 2. 3. 4. 5.
J. Campbell, Metallography 4, 269-78 (1971). C. S. Smith, Trans. TMS-AIME 175, 15-51 (1948). C. S. Smith, Sci. Amer. 190, 58-64 (1954). R. E. Williams, Science 161, 276 (1968). W. A. Miller and G. A. Chadwick, Iron Steel Inst. (London) Publ. 110, 49-56 (1967).
Accepted August 16, 1973