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Orientation of coherent interphase boundaries formed by the α to α'H phase transition in belite crystals
CEMENTandCONCRETERESEARCH.Vol.23, pp. 599-602, 1993. Printedin the USA. 0008-8846/93. $6.00+00. Copyright© 1993 PergamonPressLtd.
O R I E N T A T I O N OF C O H E R E N T INTERPHASE BOUNDARIES F O R M E D BY THE ct TO ct' H PHASE TRANSITION IN B E L I T E CRYSTALS
Koichiro Fukuda and lwao Maki Department of Materials Science and Engineering Nagoya Institute of Technology Showa-ku, Nagoya 466, Japan (Communicau~dby H.F.W.Taylor) (ReceivedMay29, 1992)
ABSTRACT The orientation of the Or'H-phase lamellae in the host ct phase in belite (Ca2SiO4 solid solutions) crystals has been derived mathematically on the assumption that the interphase boundaries are completely coherent. The computation is based on the matrix algebra analysis of martensitic transformations. The lamellae intersect (0001)a at 26.82", which is compatible with 27* obtained experimentally.
Introduction The microtextures of belite (Ca2SiO4 solid solutions) crystals have been studied extensively partly because they carry a wealth of information on the processing conditions of portland cement clinker in the kiln. Those textures depend mainly on the temperatures of formation and cooling rates and have been classified into four types: I, Ia, II and III.(1.2) The Type I is characterized by two or more sets of intersecting lamellae under the microscope and has been considered to occur upon transition from tx to ct' n. On subsequent cooling the Ct'H phase is rapidly inverted to the I~ phase passing through the a'Lphase.(3)Fukuda and Maki(4) examined the Type I belite and determined the orientation of the 15-phase lamellae within the host ct phase from combined use of X-ray diffraction and optical microscopy. They found that the lamellae occur in parallel with <210>ct and make an angle of 27" with (O301)a. In the present study we have shown that the orientation of such lamellae can be derived mathematically from the theory of "exact phase boundaries" by Robinson e t al.(5), which assumes complete coherency of interphase boundaries. In this theory the interphase planes remain both undistorted and unrotated after transformation and the orientation of such invariant planes can be computed by the matrix algebra analysis as used in martensitic transformations. (6)
Results On the assumption of complete coherency of the boundaries between the ct and Ct'H phases the present transition, unlike martensitic transformations, requires no additional lattice invariant shear. The description of the transformation therefore includes only two mathematical terms, lattice deformation and rigid body rotation.
599
600
K. Fukuda madI. Maid
Vol. 23, No 3
The lattice deformation accompanying the ct to C~'Htransition can be represented by a matrix, whose elements are derived from the lattice corres~ondence between the two structures and from their cell parameters. An orthohexagonal cell (q3 a = b ) , equivalent to the trigonal cell of the ct phase, has been taken to define an orthonormal basis (o) for the correspondence between the ct and a'H structures. Then, the principal axes of strain will be parallel to the o basis vectors. The principal distortions (110 are thus 1] 1 = a ,~h/a a
along a a axis
112 = b a h / b a
a l o n g b a axis
113 = c a b / c a
alongc a axis.
The matrix representing the lattice deformation in reference to the o basis takes the simple form as follows: 111
0
0
112
0
0
0
113
B =
0)
The lattice parameters used in the present study were those determined at high temperatures for pure Ca2SiO4 (TABLE 1).(7.8) TABLE 1 Cell Parameters for Pure Ca2SiO4 at High Temperatures Phase cz 0.' H
Temperature(°C) 1500 1300
a
b
c
5.526 5.605
9.571 9.543
7.307 6.883
Reference 7 8
The settings of axes differ from those given in the original articles.
From the observation that the lamellae occur in parallel with the ba axis (4), 1.12must be equal to 1 at the transition temperature (b ah=ba). Since 111 and 113 are respectively larger and smaller than 1, these principal distortions meet the general requirement for the occurrence of an undistorted plane(9); that is, the orientation of the plane around the b axis is determined only in terms of the a and c parameters of both phases. Figure 1 illustrates schematically(6) the occurrence of such undistorted planes A'OC' and B'OD' when the initial unit sphere in the ct phase structure is homogeneously distorted into the ellipsoid in the Ct'H phase structure. These planes in the ellipsoid are generated from the planes AOC and BOD in the unit sphere. If a counterclockwise rigid body rotation by an angle of A0 make B'OD' coincide with BOD, the contact plane remains unrotated as well as undistorted. The orientation of these phase boundaries can be defined by the acute angle 01 (between BOD and (001)~) and 02 (between B'OD' and (001)cG) (0 1 > 02>0). The present phase transition transforms vector OB (=p) into OB' ( = p ' ) . This can be expressed as Bp = p'
Vol. 23, No. 3
BELITE, IN'IF_.RPHASEBOUNDAR/F_-S,OR/ENTATION
601
FIG. 1 Schematic illustration of undistorted planes(6).
~'~,111 B'~CD ~
C
'
cos 02
cosO1 where p =
>a
and p ' =
0
0 sin 02
sin01
Expanding the above equation, we obtain TIlCOS0 1 = COS0 2
and
rl3sin0l = sin02.
From these 01 = C O S - I x
and
02 = COS-I(~IX)
where x=[( 1-~132)/(~112-rl32)] 1/2. Substituting the corresponding values for ~11 and ~13 derived from Table 1, we obtain 01=26.82". This is compatible with 27* obtained experimentally.(4) The angle of rigid body rotation is A 0 = 0 1 - 0 2.
The other invariant plane can be formed by a clockwise rotation, by which A'OC' coincides with AOC. This orientation is similarly defined by -01 and -02. That is, the invariant plane intersects (001)~x at :L-01and (001) ah at +02. The ct' H phase structure therefore should be rotated by +A0 to fit perfectly with the parent ct phase structure across the interface. Figure 2 shows the ac-lattice planes of the a and the ct'H phases which are juxtaposed on the invariant plane. Best dimensional fitness can be realized by rotating the Ct'H phase structure by 1.67". The trigonal symmetry of the ct phase produces six symmetrically related sets of the ct' H-phase lamellae in all, which have really been confirmed under the microscope.(4)
602
K.Fuk-udamdI. M~d
Vol.23.No.3
t
.<
phase
0
.... t
J L
I
7
cx~pha~[
qD .t
.§.605A
~
.¢[ W m
---a
L
C FIG. 2 Juxtaposition of the ctand ct' H phases on the invariant plane.
References
1. H. Ins|ey, J. Res. Nat. Bur. Std., 17. 353 (1936). 2. H. Insley, E.P. Flint, E.S. Newman and J.A. Swenson, ibid., 2_L1355 (1938). 3. G.W. Groves, J. Mater. Sci., 18. 1615 (1983). 4. K. Fukuda and I. Maki, Cem. Concr. Res., 19, 913 (1989). 5. P. Robinson, M. Ross, G.L. Nord, Jr., J.R. Smyth and H.W. Jaffe, Am. Mineral., 62, 857 (1977). 6. C. M. Wayman, Introduction to the Crystallography of Martensitic Transformations, The Macmillan Company, New York (1964). 7. M. Regourd, M. Bigar6, J. Forest and A. Guinier, Proc. 5th Int. Symp. Chem. Cement, Tokyo, 44 (1968). 8. G. Yamaguchi, Y. (3no, S. Kawamura and Y. Soda, J. Ceram. Assoc. Jpn., ~ 105 (1963). 9. E.C. Bilby and J.W. Christian, Inst. Met. Monograph No. 18, 121 (1955).
O R I E N T A T I O N OF C O H E R E N T INTERPHASE BOUNDARIES F O R M E D BY THE ct TO ct' H PHASE TRANSITION IN B E L I T E CRYSTALS
Koichiro Fukuda and lwao Maki Department of Materials Science and Engineering Nagoya Institute of Technology Showa-ku, Nagoya 466, Japan (Communicau~dby H.F.W.Taylor) (ReceivedMay29, 1992)
ABSTRACT The orientation of the Or'H-phase lamellae in the host ct phase in belite (Ca2SiO4 solid solutions) crystals has been derived mathematically on the assumption that the interphase boundaries are completely coherent. The computation is based on the matrix algebra analysis of martensitic transformations. The lamellae intersect (0001)a at 26.82", which is compatible with 27* obtained experimentally.
Introduction The microtextures of belite (Ca2SiO4 solid solutions) crystals have been studied extensively partly because they carry a wealth of information on the processing conditions of portland cement clinker in the kiln. Those textures depend mainly on the temperatures of formation and cooling rates and have been classified into four types: I, Ia, II and III.(1.2) The Type I is characterized by two or more sets of intersecting lamellae under the microscope and has been considered to occur upon transition from tx to ct' n. On subsequent cooling the Ct'H phase is rapidly inverted to the I~ phase passing through the a'Lphase.(3)Fukuda and Maki(4) examined the Type I belite and determined the orientation of the 15-phase lamellae within the host ct phase from combined use of X-ray diffraction and optical microscopy. They found that the lamellae occur in parallel with <210>ct and make an angle of 27" with (O301)a. In the present study we have shown that the orientation of such lamellae can be derived mathematically from the theory of "exact phase boundaries" by Robinson e t al.(5), which assumes complete coherency of interphase boundaries. In this theory the interphase planes remain both undistorted and unrotated after transformation and the orientation of such invariant planes can be computed by the matrix algebra analysis as used in martensitic transformations. (6)
Results On the assumption of complete coherency of the boundaries between the ct and Ct'H phases the present transition, unlike martensitic transformations, requires no additional lattice invariant shear. The description of the transformation therefore includes only two mathematical terms, lattice deformation and rigid body rotation.
599
600
K. Fukuda madI. Maid
Vol. 23, No 3
The lattice deformation accompanying the ct to C~'Htransition can be represented by a matrix, whose elements are derived from the lattice corres~ondence between the two structures and from their cell parameters. An orthohexagonal cell (q3 a = b ) , equivalent to the trigonal cell of the ct phase, has been taken to define an orthonormal basis (o) for the correspondence between the ct and a'H structures. Then, the principal axes of strain will be parallel to the o basis vectors. The principal distortions (110 are thus 1] 1 = a ,~h/a a
along a a axis
112 = b a h / b a
a l o n g b a axis
113 = c a b / c a
alongc a axis.
The matrix representing the lattice deformation in reference to the o basis takes the simple form as follows: 111
0
0
112
0
0
0
113
B =
0)
The lattice parameters used in the present study were those determined at high temperatures for pure Ca2SiO4 (TABLE 1).(7.8) TABLE 1 Cell Parameters for Pure Ca2SiO4 at High Temperatures Phase cz 0.' H
Temperature(°C) 1500 1300
a
b
c
5.526 5.605
9.571 9.543
7.307 6.883
Reference 7 8
The settings of axes differ from those given in the original articles.
From the observation that the lamellae occur in parallel with the ba axis (4), 1.12must be equal to 1 at the transition temperature (b ah=ba). Since 111 and 113 are respectively larger and smaller than 1, these principal distortions meet the general requirement for the occurrence of an undistorted plane(9); that is, the orientation of the plane around the b axis is determined only in terms of the a and c parameters of both phases. Figure 1 illustrates schematically(6) the occurrence of such undistorted planes A'OC' and B'OD' when the initial unit sphere in the ct phase structure is homogeneously distorted into the ellipsoid in the Ct'H phase structure. These planes in the ellipsoid are generated from the planes AOC and BOD in the unit sphere. If a counterclockwise rigid body rotation by an angle of A0 make B'OD' coincide with BOD, the contact plane remains unrotated as well as undistorted. The orientation of these phase boundaries can be defined by the acute angle 01 (between BOD and (001)~) and 02 (between B'OD' and (001)cG) (0 1 > 02>0). The present phase transition transforms vector OB (=p) into OB' ( = p ' ) . This can be expressed as Bp = p'
Vol. 23, No. 3
BELITE, IN'IF_.RPHASEBOUNDAR/F_-S,OR/ENTATION
601
FIG. 1 Schematic illustration of undistorted planes(6).
~'~,111 B'~CD ~
C
'
cos 02
cosO1 where p =
>a
and p ' =
0
0 sin 02
sin01
Expanding the above equation, we obtain TIlCOS0 1 = COS0 2
and
rl3sin0l = sin02.
From these 01 = C O S - I x
and
02 = COS-I(~IX)
where x=[( 1-~132)/(~112-rl32)] 1/2. Substituting the corresponding values for ~11 and ~13 derived from Table 1, we obtain 01=26.82". This is compatible with 27* obtained experimentally.(4) The angle of rigid body rotation is A 0 = 0 1 - 0 2.
The other invariant plane can be formed by a clockwise rotation, by which A'OC' coincides with AOC. This orientation is similarly defined by -01 and -02. That is, the invariant plane intersects (001)~x at :L-01and (001) ah at +02. The ct' H phase structure therefore should be rotated by +A0 to fit perfectly with the parent ct phase structure across the interface. Figure 2 shows the ac-lattice planes of the a and the ct'H phases which are juxtaposed on the invariant plane. Best dimensional fitness can be realized by rotating the Ct'H phase structure by 1.67". The trigonal symmetry of the ct phase produces six symmetrically related sets of the ct' H-phase lamellae in all, which have really been confirmed under the microscope.(4)
602
K.Fuk-udamdI. M~d
Vol.23.No.3
t
.<
phase
0
.... t
J L
I
7
cx~pha~[
qD .t
.§.605A
~
.¢[ W m
---a
L
C FIG. 2 Juxtaposition of the ctand ct' H phases on the invariant plane.
References
1. H. Ins|ey, J. Res. Nat. Bur. Std., 17. 353 (1936). 2. H. Insley, E.P. Flint, E.S. Newman and J.A. Swenson, ibid., 2_L1355 (1938). 3. G.W. Groves, J. Mater. Sci., 18. 1615 (1983). 4. K. Fukuda and I. Maki, Cem. Concr. Res., 19, 913 (1989). 5. P. Robinson, M. Ross, G.L. Nord, Jr., J.R. Smyth and H.W. Jaffe, Am. Mineral., 62, 857 (1977). 6. C. M. Wayman, Introduction to the Crystallography of Martensitic Transformations, The Macmillan Company, New York (1964). 7. M. Regourd, M. Bigar6, J. Forest and A. Guinier, Proc. 5th Int. Symp. Chem. Cement, Tokyo, 44 (1968). 8. G. Yamaguchi, Y. (3no, S. Kawamura and Y. Soda, J. Ceram. Assoc. Jpn., ~ 105 (1963). 9. E.C. Bilby and J.W. Christian, Inst. Met. Monograph No. 18, 121 (1955).