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Chapter 76 Elasticity, Dislocations and their Motion in Quasicrystals
CHAPTER 76
Elasticity, Dislocations and their Motion in Quasicrystals K. EDAGAWA Institute of Industrial Science, The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8505, Japan and
S. TAKEUCHI Department of Materials Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan
© 2007 Elsevier B.V. All rights reserved
Dislocations in Solids Edited by F. R. N. Nabarro and J. P. Hirth
Contents 1. Introduction 367 2. Elastic properties of quasicrystals 368 2.1. Phonon and phason degrees of freedom 368 2.2. Generalized elastic theory 376 2.3. Phonon elasticity 381 2.4. Phason elasticity 383 2.5. Phonon-phason coupling 387 3. Dislocations in quasicrystals 390 3.1. Perfect dislocations in quasicrystalline lattice 390 3.2. Displacement field and self free energy of dislocations 393 4. Dislocation motion in quasicrystals 398 4.1. Dislocation mechanism of deformation 398 4.2. Dislocation glide process 399 4.3. Dislocation climb process 405 4.4. Plastic homology 407 Appendix A: Irreducible strain components 411 Appendix B: Derivation of eqs (47) and (48) by the generalized Eshelby's method References 413
412
1. Introduction Quasicrystals have a peculiar type of ordered structure characterized by crystallographically disallowed rotational symmetry and by quasiperiodic translational order [ 1-3]. Since the discovery of the quasicrystal by Shechtman et al. in 1984 [4], physical properties of quasicrystals have attracted much attention by solid-state physicists and materials scientists. The mechanical property is one of them. Due to its peculiar structural order, mechanical properties, including elastic and plastic properties, of quasicrystals exhibit characteristic features different from those of crystalline matter. Originating in the quasiperiodic translational order, quasicrystals have a special type of elastic degrees of freedom, termed as phason degrees of freedom [3,5]. Quasicrystals are accompanied by the phason elastic field in addition to the phonon (conventional) elastic field. The generalized elasticity of quasicrystals is described in terms of the two types of elastic fields [6-8]. Within linear elasticity, the elastic free energy of quasicrystals consists of three types of quadratic terms: phonon-phonon, phason-phason and phonon-phason coupling terms. Correspondingly, elasticity of quasicrystals comprises the three parts: pure phonon elasticity, pure phason elasticity and phonon-phason coupling. In relation to the generalized elasticity of quasicrystals, the concept of dislocations in quasicrystals was theoretically established [5,6,9] in an early stage of the research on quasicrystals. Experimentally, since the first demonstration by Shibuya et al. [10] in 1990 of the plastic deformation of an A1-Cu-Ru icosahedral quasicrystal at high temperatures, it has been shown that quasicrystals, both icosahedral and decagonal quasicrystals, are generally deformable at high temperatures above 0.8Tin (Tm: melting temperature), as reviewed in [11-13]. Wollgarten et al. [14,15] have shown by transmission electron microscopy (TEM) that high temperature deformation of icosahedral A1-Pd-Mn is brought by a dislocation process. Until recent years, it has been generally believed that the plasticity of quasicrystals is carried by a glide process of dislocations, and various models of the deformation mechanism have been proposed based on the dislocation glide [16-20]. However, recently, Caillard et al. [21-23] have shown by TEM that the high temperature deformation of icosahedral A1-Pd-Mn is brought not by a glide process but by a pure climb process. In this article, we review the present understandings of characteristic features of elasticity, dislocations and their motion in quasicrystals. This article is organized as follows. In Section 2, we review elastic properties of quasicrystals. Here, we show in Section 2.1 that quasicrystals have the phason elastic degrees of freedom in addition to the phonon (conventional) elastic degrees of freedom. In Section 2.2, we review the generalized elastic theory constructed by incorporating the two types of elastic degrees of freedom. Here, explicit formulae are presented for the two most important classes of quasicrystals, that is, icosahedral and decagonal quasicrystals. Sections 2.3, 2.4 and 2.5 are devoted to the review of experimental and theoretical works related, respectively, to the three parts of elasticity of quasicrystals, that is, pure phonon elasticity, pure phason elasticity and phonon-phason
K. Edagawa and S. Takeuchi
368
Ch. 76
coupling. In Section 3, we present characteristic features of dislocations in quasicrystals. In Section 3.1, we show the definition of a perfect dislocation in quasicrystals and review which types of dislocations have been experimentally observed. In Section 3.2, we demonstrate a calculation of displacement field and self free energy of a dislocation. Using a general formula of the self free energy, relative stability among dislocations with various Burgers vectors is discussed. Section 4 is devoted to dislocation motion. After reviewing briefly the experimental results about dislocation mechanism of deformation of quasicrystals in Section 4.1, we discuss detailed mechanisms of dislocation glide and dislocation climb in Section 4.2 and 4.3, respectively. Finally, in Section 4.4, we show a homologous nature in the temperature dependence of the yield stress for icosahedral quasicrystals and discuss it in terms of microscopic deformation mechanism common to icosahedral quasicrystals.
2. E l a s t i c p r o p e r t i e s o f q u a s i c r y s t a l s
2.1. Phonon and phason degrees of freedom In general, the diffraction intensity function I (q) of solid is given by I (q) _----IS(q)l2,
s(q) - f
p ( r ) e x p ( - 2 : r i q - r)dr.
(1)
Here, p (r) is the atomic-density function in real space. The function I (q) observed experimentally for the quasicrystal has the following characteristics [1-3]: (1) It consists of 6-functions. (2) The number of basis vectors necessary for indexing the positions of the 6-functions exceeds the number of dimensions. (3) It shows a rotational symmetry forbidden in conventional crystallography. Conversely, we define the quasicrystal as the material with a p(r) which gives a diffraction intensity function I (q) satisfying these conditions. The condition (1) implies that this material has a kind of long-range translational order. The conditions (2) and (3) indicate that the order is not periodic. The translational order defined by the conditions (1) and (2) is called quasiperiodicity. According to the definition, every d-dimensional quasiperiodic atomic-density function p (r) can be expressed as p(r) -- ~ / g m l miEI
.....
mN exp[27riGm1 .....mN " r],
(2)
§2.1
Elasticity, dislocations and their motion in quasicrystals
w h e r e tom1 . . . . .
mN
369
is the Fourier component associated with the reciprocal vector N
G m l ..... mN --- Y ~ m n q n .
n--1
Here, {qn } (n = l, 2 . . . . . N) (N > d) are the reciprocal basis vectors. Now, let us define an N-dimensional function ph (x 1. . . . . xu) as
mNeXPI2 iZmnxn ] N
. . . . .
.....
miEl
(3)
n=l
This function is periodic in every eqs (2) and (3), we find
Xn
p ( r ) = p h ( x l . . . . . XN),
(n -- 1, 2 . . . . . N) with the period of unity. Comparing
(4)
where
xn=qn'r
(n=l,2
. . . . . N).
(5)
This indicates that the d-dimensional quasiperiodic function p(r) can be described as a d-dimensional section of an N-dimensional periodic function. As an example, Fig. l(a) presents a typical one-dimensional quasiperiodic structure known as a Fibonacci lattice, which is described as a one-dimensional section of a twodimensional periodic function (d = 1 and N --2). Here, Ell and E± denote the physical space and the complementary space perpendicular to it, respectively. In this case, the two-dimensional periodic structure consists of a periodic arrangement of a line segment extending in the direction of E±. The line segment is called atomic surface. A point sequence is obtained on the Ell section, comprising an arrangement of two spacings L and S. Here, the slope of Ell with respect to the two-dimensional lattice is an irrational number r (the golden mean: (1 + V/5)/2). In this case, the resultant sequence of the two spacings becomes equivalent to the Fibonacci sequence and therefore it is called a Fibonacci lattice. The irrational slope indicates lack of periodicity in the arrangement of L and S. In the description of the Fibonacci lattice in Fig. 1(a), let us consider a translation of the two-dimensional periodic structure by a vector U with respect to the origin of the physical space Ell (Fig. 1(b)). The two structures on Ell before and after the displacement U shown in Figs l(a) and (b) can be overlapped out to arbitrarily large finite distances by a finite translation in Ell. Two structures satisfying this condition are said to belong to the same local isomorphism class (LI class) [1-3]. Thus, the displacement U represents the degrees of freedom of generating a series of structures belonging to the same LI class. Obviously from the definition of the LI class, a series of structures in the same LI class
K. Edagawa and S. Takeuchi
370
E±
(a)
(b)
T
(
±
Ch. 76
El
-
T
-
T
v
(d)
Fig. 1. A Fibonacci lattice (a), and a structure resulting from a displacement of U (b), that of u (c) and that of w (d).
are geometrically indistinguishable on any finite scale and thus they are also physically indistinguishable: they give the same diffraction intensity function I (q) and have the same energy. The vector U can be decomposed into u in Ell and w in E±: U = u + w.
(6)
Here, u represents the degrees of freedom of d-dimensional translation in physical space, which conventional crystals also possess, and w represents (N - d) degrees of freedom characteristic of quasiperiodic system. As shown in Figs l(c) and (d), while u results in translation of Fibonacci lattice in Ell, w generates a rearrangement of L and S. The two kinds of degrees of freedom are called phonon and phason degrees of freedom, and u and w phonon and phason displacements, respectively [3,5]. When these displacements vary spatially, the gradients of them yield distortion or strain. More specifically, while the gradient of u, i.e.
Vu(r)-
Oui Orj
gives conventional distortion (or strain), that of w, i.e. Ot~ i
Vw(r)- Orj
§2.1
Elasticity, dislocations and their motion in quasicrystals
371
E.t.
(a)
~-EII
(b)
~
v
-
~
(c) J
J
I !
Fig. 2. A Fibonacci lattice (a), and its phonon-strained (b) and phason-strained (c) structures.
yields distortion (strain) called phason distortion (strain) .1 Here, u and w are the functions of only r c Ell and ~-/ denotes a spatial derivative in Ell Figs 2(b) and (c) illustrate, respectively, a phonon-strained and a phason-strained structure of the Fibonacci lattice in Fig. 2(a). In Fig. 2(b), uniform phonon strain is introduced by a compression deformation of the two-dimensional structure. On the other hand, uniform phason strain is introduced by a shear deformation of the two-dimensional structure in Fig. 2(c). It is noteworthy that the phonon and phason stain fields should have quite different dynamical properties. As described above, phason displacement results in a local rearrangement of points (atoms) such as LS ~-+ SL, which is called phason flip. Examples of phason l In general, the gradient of displacement vector is called 'distortion.' In the one-dimensional case, the distorOw also represent strains in the same form. In the case of two-dimensional and three-dimensional tions ~Ou and -gT Oui quasicrystals, the phonon strain should be defined as a symmetrical form 1 ( ~ + ~Ouj ) to remove the component .
/
Owi as the distortion (see Section 2.2). of rigid rotation while the phason strain should have the same form ~ ~,,j
K. Edagawa and S. Takeuchi
372
Ch. 76 A
Fig. 3. Examples of phason flip in a two-dimensional Penrose lattice.
flip in a two-dimensional Penrose lattice, which is known as a typical two-dimensional decagonal quasicrystal, are shown in Fig. 3. In this case, a lattice point in the hexagon makes a transition between two (meta-) stable positions. Introduction or relaxation of phason strain requires a combination of phason flips. Generally, the phason flip is a thermally activated process and thus relaxation of phason strain whose elementary process is phason flip must proceed relatively slowly like atomic diffusion in solids [3,5,8]. This is in sharp contrast to conventional phonon strain, which can be relaxed instantaneously via displacive phonon modes. Experimentally, thermally-induced phason flips have been investigated by neutron scattering [24-27], by Moessbauer spectroscopy [28], by NMR [29, 30] and specific heat measurements [31,32]. By in-situ high-temperature high-resolution transmission electron microscopy, direct observations of thermally-induced phason flips [33-35] and those of the phason-strain relaxation process [36] have been performed. The activation enthalpy of a phason flip or collective phason flips has been estimated to be ~1 eV in an A1-Pd-Mn icosahedral quasicrystal by radiotracer diffusion experiments [37]. In general, to decompose properly the N total degrees of freedom into d phonon and (N - d) phason degrees of freedom, or equivalently, to embed properly a given ddimensional quasiperiodic structure into an N-dimensional hypercrystal, the information on the point group symmetry of the system is needed. Below, the method of embedding is briefly reviewed for the two most important classes of quasicrystals: icosahedral and decagonal quasicrystals [3,38-41 ]. To describe the structure of a three-dimensional icosahedral quasicrystal, we use a sixdimensional hypercubic lattice spanned by di (i -- 1. . . . . 6): di ~- M i j e j ,
r M
a ,22~r , + 2 ' ~- 1 -
1 0 0
1 -1 0 r r
0 0 r 1 -1
1 1 -r 0 0
-r r 0 1 1
0 0 1 -r r
1
0
-r
-r
0
-1
'
(7)
§2.1
Elasticity, dislocations and their motion in quasicrystals
373
where r is the golden mean ( = (1 + ~/-5)/2) and a is the lattice constant, ei (i - 1 . . . . . 6) are the orthonormal unit vectors of the six-dimensional space. The icosahedral point group Y is generated by a fivefold rotation C5 and a threefold rotation C3. The lattice spanned by d i (i = 1 . . . . . 6) is invariant under these operations. The action of these operations o n d i (i = 1 . . . . . 6) is given as
F(C5)--
F(C3)--
1 0
0 0
0 1
0 0
0 0
0 0
0 0 0 0
0 0 0 1
1 0 0 0
0 0 0 0
0 1 0 0
0 0 1 0
0 1
0 0
0 o
0 0
0 0
0 0
0
0
o
-~
0 1
0 0
'
0 0
1 0
0
1
0
o
o
o
-1 0
0 0
0 0
•
(8)
The character table for the icosahedral group Y is presented in Table 1. Here, the irreducible representations of the group Y are a one-dimensional F l, two distinct three-dimensional 1F2 and F 3, a four-dimensional F 4, and a five-dimensional F 5. The six-dimensional representation in eq. (8) is reducible and can be decomposed into the sum of the irreducible representations as F = F 2 + F 3. The subspace spanned by el, e2 and e3, and the subspace spanned by e4, e5 and e6 are the eigenspaces of the representations [,2 and F 3, and correspond to the physical space Ell and its complementary space E_k, respectively. Shown in Fig. 4(a) are the projections dl I (i - 1 . . . . . 6) and d/± (i - 1 . . . . . 6) of the basis vectors di (i = 1 . . . . . 6) onto Ell and E_k, respectively. Because the sixdimensional hypercubic lattice is invariant under the inversion Ci, it has the maximum icosahedral point group symmetry Yh -- Y x Ci. By placing atomic surfaces spreading in E± (in this case they are three-dimensional objects) at appropriate positions in the six-dimensional lattice, a three-dimensional icosahedral quasicrystalline structure with the symmetry Y or Yh is obtained as a section on Ell. Here, to preserve the symmetry, each of the atomic surfaces must satisfy the site symmetry of its position in the corresponding symmetry group. Similarly to the case illustrated in Fig. 1, a six-dimensional Table 1 Character table for the icosahedral group Y Y
E
12C 5
12C 2
20C3
15C2
r 1
1
1
1
1
r 2
3
r
-r -]
0
-1 -1
F3
3
-r -j
r
0
r4
4
-1
-1
1
F5
5
0
0
-1
1
0 1
374
K. Edagawa and S. Takeuchi
E,,
Ch. 76
E,
e3
e6 ,3
,)' ',
l--e~
\
"/
Is \
'
,1
e2
e;
Z es
V
(a)
E, ,I
E, I
3 /
5 ~e~ 3
""'"'
1 ~e3
e5
0 0
4
"
,
(b) (. i = l .... 6) of the Fig. 4. (a) The projections d iII ( i = l . ... 6) andd/L . . basis. vectorsdi ( i = l ... 6) in eq. (7) for icosahedral quasicrystals onto Ell and EL, respectively; (b) the projections d iII ( i = l , ... , 5) andd{ (i = 1..... 5) of the basis vectors di (i = 1..... 5) in eq. (9) for decagonal quasicrystals onto El, E~I and EL, respectively. Here, di (i = 1..... 4) have no components of E~I while d 5 has the component of E~I only.
displacement U can be defined, which can be decomposed into u ~ Ell and w e E ± , representing the three-dimensional phonon and the three-dimensional phason displacements. It is noteworthy that the hypercubic lattice is not a unique choice for embedding icosahedral quasicrystalline structure. In fact, we can use any lattice spanned by the basis vectors of the form d I - d l . l + cd/x (c" an arbitrary scale factor)instead of the hypercubic lattice spanned by di - d l I -+-d/x in embedding any given structure. In other words, we can determine arbitrarily the length scale in E±. This fact should always be born in mind in discussing the magnitudes of the physical-property parameters related to the length scale in E x such as phason displacement, phason strain, phason elastic constants, etc. (see Sections 2.4, 3.1 and 3.2). To describe a three-dimensional decagonal quasicrystal, we use a five-dimensional lattice spanned by di (i = 1 . . . . . 5): di - M i j e j ,
§2.1
Elasticity, dislocations and their motion in quasicrystals
a ~(Cl-
1)
a ~Sl
~(C2-
1)
a ~$2
~
a (C2 -M
a
m
a a
a
a 1)
~(C4-
1)
0-
a (C4 -- 1)
a
~$4
0
a
a ~s4
a ~(c3-
0
0
0
a ~$2
375
1)
(9)
a ~s3
0
0
c-
where a and c are the lattice constants, and ci = cos(2rri/5) and Si = sin(2rri/5), ei (i = 1 . . . . . 5) are the orthonormal unit vectors of the five-dimensional space. The character table for the decagonal group Cloy is shown in Table 2. The generators of this group are a tenfold rotation C10 and a mirror av. The lattice spanned by di (i -- 1 . . . . . 5) is invariant under these operations. The action of these operations on di (i = 1 . . . . . 5) is given by
F(C10)--
0 0 0 -1 0
1 1 1 1 0
-1 0 0 0 0
0 -1 0 0 0
0 0 0 0 1
,
F(av)--
0 0 0 1 0
0 0 1 0 0
0 1 0 0 0
1 0 0 0 0
0 0 0 0 1
.
(10)
This five-dimensional representation is reducible: F -- F 5 + F 7 -t- 1-'1 . The subspace Ell1 spanned by el and e2, the subspace E± spanned by e3 and e4, and the subspace E~ along e5 are the eigenspaces of the representations F 5, 1-`7 and F l, respectively. Eli is parallel to the quasiperiodic plane and E~ is along the tenfold periodic directions in the physical space Ell -- E] + E~. In Fig. 4(b), the projections dl I (i - 1 . . . . . 5) and d/± (i - 1 . . . . . 5) of the basis vectors di (i = 1 . . . . . 5) onto each space are illustrated. Because the lattice spanned by d i (i - - 1 . . . . . 5) in eq. (9) is invariant under the inversion Ci, it has the maximum decagonal point group symmetry Dloh = Cloy × Ci. There are seven point groups in the decagonal system: C 1 0 , C5h, ClOh, Cloy, D 1 0 , D5h and DlOh. By placing atomic surfaces spreading in E± at appropriate positions in the five-dimensional lattice, a threeTable 2 Character table for the decagonal group ClOy C10v
E
2C10
2C5
2C~0
2C2
C2
F1
1
1
1
1
1
1
F2
1
1
1
1
1
1
5av
5a d
1 -1
1 -1
F3
1
-1
1
-1
1
-1
F4
1
-1
1
-1
1
-1
F5
2
r
r - 1
1- r
-r
-2
0
0
F6 F7
2 2
r -- 1 1- r
--r -r
--r r
r- 1 r - 1
2 -2
0 0
0 0
F8
2
-r
r-
2
0
0
1
r-
1
-r
1 -1
-1 1
K. Edagawaand S. Takeuchi
376
Ch. 76
dimensional decagonal quasicrystalline structure with any of those decagonal point group symmetries can be constructed. Here, to preserve the symmetry, each atomic surface must satisfy the site symmetry of its position in the given point group. A five-dimensional displacement U can be defined, which can be decomposed into u 6 Eli and w 6 E_L, representing the three-dimensional phonon and the two-dimensional phason displacements. As in the case of icosahedral quasicrystal, we can use any five-dimensional lattice spanned by the basis vectors of the form d I - dl.I + cd~ (c" an arbitrary scale factor)in embedding any given structure.
2.2. Generalized elastic theory As in the preceding section, in quasicrystals there exist two types of elastic degrees of freedom, that is, phonon and phason degrees of freedom. In view of this fact, a generalized elastic theory can be formulated for the quasicrystals [6-9,42-45]. In this section, we review the theory and present the explicit formulae for the icosahedral and decagonal quasicrystals. The spatial variation of u, i.e. Vu(r) - Oui yields phonon (conventional) distortion while that of w, i.e. Vw(r) - Owi yields phason distortion. Here, u and w are the functions of only r 6 Eli and ~-~Tdenotes a spatial derivative in Eli. The phonon strain should be defined as a symmetrical form
l ( Oui Ouj ) Uij~-~ ~rj +-~ri to remove the component of the rigid rotation which does not change the elastic free energy. On the other hand, the phason strain should be defined as w i j z Owi Orj . The elastic free energy density f is given as a function of the phonon strain U ij and the phason strain Wij. The function f can be expanded into the Taylor series in the vicinity o f uij -- 0 and Wij = O. In the regime with ]Uijl, IllOij[ ~ 1, we would omit the third and higher order terms, leaving only the quadratic terms (linear elasticity):
f =fu-u+fw-w+fu-w 1 ~_u 1 w-w U--//3 - -~Cij~l uijukl + -~Cij~l tOij tOkl -[- Cijkl uij tOkl,
(11)
where fu-u, fw-w and fu-w are a pure phonon, pure phason and phonon-phason coupling U--U W--W U--W terms, respectively, and Cij~l, Cijkl and Cijkt are the corresponding second-order elastic constant tensors. The explicit form of f depends on the symmetry of the system and we derive those for icosahedral [6,9,42-45] and decagonal [46,47] quasicrystals below. For the icosahedral quasicrystal with the point group Y, the six components uij transform under (F 2 × I-'2)sym. - - F 1 ~ F 5, corresponding to dilatation and shear, respectively (see Table 1). The nine components wij transform under F 2 × F 3 - F 4 + F 5. Now let Ul and u5 be the irreducible components of the phonon strain for 1-'1 and F 5, respectively,
§2.2
Elasticity, dislocations and their motion in quasicrystals
377
and W4 and w5 be those of the phason strain for [,4 and F 5, respectively. These irreducible strain components are given explicitly in Appendix A. Then, we find that there exist five quadratic combinations of strains that transform as scalars, i.e. under F 1" Ul • Ul, u5 .us, w4 • W4, W5 • W5 and u5 • ws. This fact indicates that the elastic free energy density f can be rewritten as
f-f,-u+fw-w+fu-w, 1 1 fu-u - -~klUl "Ul -Jr- ~k2u5 - u s , 1 fw-w
1
fu-w-ksus'w5.
-- ~k3w4 • w4 + ~k4w5 • w5,
(12)
Here, ki (i = 1 . . . . . 5) are scalar constants representing the elastic constants. In many works done so far on the elasticity of icosahedral quasicrystals, a different set of elastic constants have been used: )~ and # (Lame constants) for pure phonon elasticity; K1 and K2 for pure phason elasticity; and Ks for phonon-phason coupling [48-57]. They are related to ki (i -- 1 . . . . . 5) as [44,58] 1 )v -- x-(kl -- k2), 3 1
1 /z -- 7k2, it
K1 - 7(4k3 -+- 5k4),
1
K2 -- 7(k3 - k4), 3
K3 --
1
--~ks. ,/6
(13)
Inserting eq. (13) into eq. (12) leads to
f =f~-u+fw-w+fu-~, 1 f~_. - -~x.~ + l z t t i j u i j ,
1
1
I
4
f~-~ - ~K,~u~u + ~tc2. ~
- ~u~u
+ [(~12 + ~-'~2,)2
+ cyclic permutations] }, fu-w
-- K3{ [tt)ll (Ul,-+- r - l u 2 2
-
ru33)-+- 2 u 2 3 ( r -1
+ cyclic permutations }.
~23
-
~32)]
(14)
It should be noted that the equations presented in [34,35] can be obtained from eq. (14) by the substitution of tt)ij by tt3ji and the coordinate transformation (x -+ y, y --+ - x , z --+ z) [44,58]. Comparing eq. (14) with eq. (11), we finally obtain the elastic constant tensors, as fo1U--b/ lows. The pure phonon elastic constant tensor C i j k l is identical to that for conventional isotropic solids:
c,";7; -xaua~, + u,(~s,~j, + ~,,ajk),
(is)
K. Edagawa and S. Takeuchi
378
Ch. 76 tO--tO
w h e r e (~ij is the K r o n e c k e r delta. T h e p u r e p h a s o n elastic c o n s t a n t t e n s o r Cijkl
a n d the
p h o n o n - p h a s o n c o u p l i n g t e n s o r Cij-£] v are s h o w n in Tables 3 a n d 4, r e s p e c t i v e l y , u s i n g a 9 x 9 r e p r e s e n t a t i o n . B e c a u s e b o t h uij a n d toij are c e n t r o s y m m e t r i c a l , i.e. i n v a r i a n t u n d e r the a c t i o n o f the i n v e r s i o n o p e r a t i o n , elastic p r o p e r t i e s s h o u l d p o s s e s s an intrinsic c e n t r o s y m m e t r y . T h i s fact i n d i c a t e s that all the p o i n t g r o u p s b e l o n g i n g to the s a m e L a u e class h a v e e x a c t l y the s a m e elastic p r o p e r t i e s . T h u s , the a b o v e results d e r i v e d for the g r o u p Y also a p p l y to the g r o u p Yh -- Y x Ci.
Table 3 1/3-//3 for icosahedral quasicrystals with the point The phason elastic constant tensor C ijkl group Y or Yh. A - D are defined as" a = K 1 - ~__Z2,B = K 1 + K2(r - 1), C = K1 -t- K2( 2 - r), D = K 2 cWmtO
kl-- 11
22
33
23
31
12
32
13
21
ij=ll
A
D
D
0
0
0
0
0
0
22
D
A
D
0
0
0
0
0
0
33
D
D
A
0
0
0
0
0
0
23
0
0
0
B
0
0
D
0
0
31
0
0
0
0
B
0
0
D
0
12
0
0
0
0
0
B
0
0
D
32
0
0
0
D
0
0
C
0
0
13
0
0
0
0
D
0
0
C
0
21
0
0
0
0
0
D
0
0
C
i.]kl
Table 4 - W for icosahedral quasicrystals The phonon-phason coupling constant tensor c Uijkl with the point group Y or Yh. A - C are defined as: A = K3, B = - r K3, C -- r - 1 K3
cUBW
kl-- 11
22
33
23
31
12
32
13
21
ij - - 11
A
B
C
0
0
0
0
0
0
22
C
A
B
0
0
0
0
0
0
33
B
C
A
0
0
0
0
0
0
23
0
0
0
C
0
0
B
0
0
31
0
0
0
0
C
0
0
B
0
12
0
0
0
0
0
C
0
0
B
32
0
0
0
C
0
0
B
0
0
13
0
0
0
0
C
0
0
B
0
21
0
0
0
0
0
C
0
0
B
ijkl
§2.2
379
Elasticity, d i s l o c a t i o n s a n d t h e i r m o t i o n in q u a s i c r y s t a l s
For three-dimensional decagonal quasicrystals, U ij is a 3 x 3 symmetrical tensor and Cloy, the six components of uij and wij transform under {(F 5 + F 1) x (F 5 + F1)}sym.- 2F 1 + F 5 + 1-'6 and (F 5 + F 1) x I-'7 = 1 - ' 6 + 1-'7 -Jr- 1-'8, respectively (see Table 2). Let ul and u'1 be the two irreducible components of the phonon strain for F 1, and u5 and u6 be those for F 5 and F 6, respectively. Let w6, w7 and w8 be those of the phason strain for F 6, 1-'7 and F 8, respectively. These irreducible strain components are given explicitly in Appendix A. Then, we find the following quadratic combinations of strains that transform as scalars, constituting the elastic free energy density f : tOij is a 2 x 3 tensor, respectively. For the point group
f =fu-u+f~-w+fu-w, 1
fu-, - -~klUl
,
1
,
I
1
1
"Ul + k2Ul "u 1 -t- ~k3u 1 "u 1 "t'- ~k4u5 "u5 -+- ~k5u6 • u6,
1
1
1
fw-w
-
~ k 6 w 6 • w6 -Jr- ~ k 7 w 7 • w7 -Jr- ~ksw8 • ws,
fu-w
--
k9u6 • w6.
(16)
Here, ki (i -- 1 . . . . . 9) are scalar constants representing the elastic constants. Eq. (16) can be written explicitly as:
f =f~-u+fw-w+f~-w, S -u -
1
(.21 + u 2)+
+
nt- 2C44(U23 -Jr- U71 ) J r - ( C l l -
1 f w - w -- -~ KI
+
+
1
2
Cl2)U 212,
(~71 -[- tU22 -I- 1/)72-it- 1/)21)
1 qt- K 2 ( t O l l 1/322- tO21 tVl2)-Jr- ~ K 3 ( ~ 2 3
nt- t023),
f . - w - K 4 { ( U l l - u 2 2 ) ( W l l qt- t022) --{- 2u12(to21 - w 1 2 ) } ,
(17)
where a different set of elastic constants are used, which are: Cll
=
kl + k5,
K l=k6+kg,
C12 = kl - k5, c13 = k2, c33 = k3, C44 --
K2=k6-kg,
k4
4'
(18)
K3=k7, K4=k9.
Comparing eq. (17) with eq. (11), we obtain the three elastic constant tensors that are presented in Tables 5-7. We note that the form of the phonon elastic constant tensor is identical to that for the hexagonal crystals. The point groups Dlo, D5h and Dloh belong to the same Laue class as that of C10v and therefore the above results also apply to all of them. In the decagonal system, there is another Laue class to which the point groups C10, C5h and Cloh belong. Similar analysis reveals that Cij-£~ and C/W.~Wfor the latter Laue class are J
K. Edagawa and S. Takeuchi
380
Ch. 76
Table 5 The phonon elastic constant tensor c U . ~ for decagonal quasicrystals J
with the point group C10, C5h, ClOh, D10, D5h, ClOy or DIOh. There are five independent elastic constants: Cll , c12 , c13 , c33 and Cll --c22 c44. c66 is given by c66 = 2
Ci,lklu
k l = 11
22
33
23
31
12
ij = 11
Cll
c12
c13
0
0
0
22
c12
Cll
c13
0
0
0
33
c13
c13
c33
0
0
0
23
0
0
0
c44
0
0
31
0
0
0
0
c44
0
12
0
0
0
0
0
c66
Table 6 .-,w-w f o r decagonal quasicrystals The phason elastic constant tensor Cijkl with the point group C10, C5h, ClOh, D10, D5h, ClOy or DIOh
CW-W ijkl
kl = 11
ij = 11
K1
K2
0
0
0
0
22
K2
K1
0
0
0
0
23
0
0
K3
0
0
0
12
0
0
0
K1
0
13
0
0
0
0
K3
0
21
0
0
0
-K 2
0
K1
22
23
12
13
21
K2
Table 7 - 1/) for decagonal The phonon-phason coupling constant tensor c Uijkl quasicrystals with the point group D10, D5h, Cloy, or DIOh
Ci~k]v
kl = 11
22
23
12
13
21
ij = 11
K4
K4
0
0
0
0
-K 4
0
0
0
0
22
K4
33
0
0
0
0
0
0
23
0
0
0
0
0
0
31
0
0
0
0
0
0
12
0
0
0
-K 4
0
K4
Cijkl
U--tO
identical to those for the former and t h a t for the latter is slightly different; there are two independent elastic constants K4 and K5 in this tensor, as shown in Table 8.
Elasticity, dislocations and their motion in quasicrystals
§2.3
381
Table 8 The phonon-phason coupling constant tensor cU-tO ijkl for decagonal quasicrystals with the point group C10, C5h or CIO h
cU--tU ijkl
kl -- 11
22
23
12
13
ij = 11
K4
K4
0
K5
0
-K4
0
-K 5
0
K5
22
K5
33
0
0
0
0
0
0
23
0
0
0
0
0
0
31
0
0
0
0
0
0
12
K5
K5
0
K4
0
-
We define the phason O'ij __ m 0__~f OUij.
K4
21
stress
K4
Of in addition to the conventional phonon stress crijw --_ Owi----]'
From eq. (11) these stresses can be written as
lg U-- U It--tO Crij - - C i j k l Ukl -~- C i j k l tOkl,
/13
U--to
O'ij - - C i j k l
tO--//)
Ukl @ C i j k l
tOkl,
(19)
which represents a generalized Hooke's law. Introducing a phason body-force density fi w as well as the conventional phonon body-force density fi u, we obtain a generalized form of the elastic equations of balance: 0 Orj O'iju @ f iu - - O,
0
~ F j O'ijw -Jr- f tw _
0"
(20)
Inserting eq. (19) into (20) leads to u-u 02Uk nt_ C u - w 02tOk + f u -0, Cijkl Orj Orl ijkl Orj Orl 0 2 tt) k u-w 02Uk w-w ~ -F f.w _ O, Cijkl Orj Orl -+- Cijkl Orj Orl
(21)
where we take into consideration the relation C ijkl u-u auk; 02Uk " Orj - - C i uj k- lu OrjOrl
2.3. Phonon elasticity As shown in eq. (11), the elastic free energy density f of quasicrystals has the three quadratic terms, which represent a pure phonon elasticity, a pure phason elasticity and phonon-phason coupling, respectively. As described in Section 2.1, relaxation of phason strain proceeds diffusively with thermally activated phason flips [3,5,8], indicating that d (t) --0. Even at high at low temperatures the phason mode should be frozen, i.e. -d-[Wij
382
Ch. 76
K. Edagawa and S. Takeuchi
Table 9 Phonon elastic constants of various icosahedral quasicrystals: )~ and/z are Lame constants; G =/z, B = 3~.+2/x 3 and v = 20.+lz) are shear modulus, bulk modulus and Poisson's ratio, respectively.The unit of)~,/z and B is GPa Alloy system
)~
#(G)
A1-Li-Cu A1-Li-Cu A1-Cu-Fe A1-Cu-Fe-Ru A1-Pd-Mn A1-Pd-Mn Ti-Zr-Ni Cd-Yb Zn-Mg-Y
30 30.4 59.1 48.4 74.9 74.2 85.5 35.28 33.0
35 40.9 68.1 57.9 72.4 70.4 38.3 25.28 46.5
B 53 57.7 104 87.0 123 121 111 52.13 64.0
v
Ref.
0.23 0.213 0.232 0.228 0.254 0.256 0.345 0.2913 0.208
[60] [62] [65] [65] [65] [66] [67] [64] [68]
temperatures, it can be regarded as being effectively frozen under applied stress oscillating with a much shorter period than the relaxation time of phason strain. If additionally the residual phason strain is zero, i.e. Wij(t) = 0 - - const., the elasticity of quasicrystals consists entirely of the phonon elasticity. As shown in eq. (15), icosahedral quasicrystals should behave as an isotropic elastic body. 2 Actually, isotropic elasticity has been observed experimentally for icosahedral A1-Li-Cu [60-62], A1-Pd-Mn [63] and Cd-Yb [64]. Using various ultrasonic methods, the elastic constants have been measured for icosahedral quasicrystals of A1-Li-Cu [60,62], A1-Cu-Fe [65], A1-Cu-Fe-Ru [65], A1-Pd-Mn [65,66], Ti-Zr-Ni [67], Cd-Yb [64] and Mg-Zn-Y [68]. The evaluated elastic constants at room temperature are summarized in Table 9, where G - # B -- 3~.+2# and v x , 3 2(z+~z) are shear modulus, bulk modulus and Poisson's ratio, respectively. First, we note that the bulk moduli differ considerably for different alloy systems. We find that they correlate well with the melting temperature, which is generally observed for conventional crystals. With the exception of Ti-Zr-Ni, the main feature characteristic of the quasicrystals is that the Poisson's ratio is relatively small, or equivalently, that the ratio G - 3(1-2v) 2(1+v) is relatively large. For example, the Poisson's ratios of the elemental metals of A1, Cu and Pd are about 0.35, 0.37 and 0.39, respectively, at room temperature. Except for Ti-Zr-Ni, the Poisson's ratios of the quasicrystals in Table 9 are much smaller than those values and close to those of typical covalent crystals such as Si (0.22) and Ge (0.21). Tanaka et al. [65] have pointed out that the small v or equivalently the large G / B ratio is indicative of the existence of directional atomic bonds. The bulk modulus B is a measure of the resistance against the deformation in which atomic bond lengths change with keeping bond angles unchanged. In contrast, the shear modulus G is a measure of the resistance against the deformation in which bond angles change without changing bond lengths. Therefore, the G / B ratio should become large if the atomic bonds are directional, as in covalent crystals. In relation to this fact, covalent nature of the atomic bonds has been revealed experimentally in some quasicrystals and their crystal approximants [69-71]. 2In the third-order elastic constants described as a sixth-rank tensor, anisotropy is theoretically expected and a sign of anisotropy has actually been observed in the study of nonlinear propagation of acoustic waves in an A1-Pd-Mn icosahedral quasicrystal [59].
Elasticity, dislocations and their motion in quasicrystals
§2.4
383
Table 10 Phonon elastic constants of an A1-Ni-Co decagonal quasicrystal [72]: Cll, c33, c44, c12 and c13 are the five independent elastic constants of this system; B and G are the bulk and shear moduli calculated as Voigt averages of the moduli cij ; v is the Poisson's ratio calculated from the B and G values. The unit is GPa, except for v Alloy system
Cl 1
c33
c44
c12
c13
B
G
v
A1-Ni-Co
234.33
232.22
70.19
57.41
66.63
120.25
79.78
0.228
As shown in Table 5, there are five independent phonon elastic constants c11, c33, C44, Cl2 and Cl3 in the decagonal quasicrystals and the form of the elastic constant tensor is identical to that of the hexagonal crystal. Isotropic elasticity is expected in the basal plane, which has been confirmed experimentally for an A1-Ni-Co decagonal quasicrystal by Chernikov et al. [72]. The same group has reported the values of a complete set of elastic moduli, which are presented in Table 10. Here, a striking feature to be noted is that the deviation from complete elastic isotropy is rather small. The compressional anisotropy and shear anisotropy, measured by the ratios C 3 3 / C l l and c 4 4 / C 6 6 = 2 c 4 4 / ( C l l -- c12), respectively, are 0.99 and 0.79, indicating a high degree of elastic isotropy. Inelastic neutron scattering measurements have been done for a decagonal A1-Ni-Co [73] and considerably isotropic phonon dispersion relations have been obtained, which is consistent with the result of the elastic constant measurement. So far, many structural models have been proposed for decagonal quasicrystals, as reviewed by Ranganathan et al. [74]. Most of them have a layered structure, for which one would expect anisotropic elastic properties. Actually, phonon dispersion relations have been calculated for one of such a layered structure model, exhibiting quite large anisotropy [75]. Dmitrienko [76] has suggested the possibility of both icosahedral and decagonal quasicrystals having common local atomic ordering with icosahedral symmetry. The highly isotropic elastic properties may be attributable to such an isotropic local atomic arrangement. In Table 10, the bulk modulus B and shear modulus G are calculated as Voigt averages of the moduli cij, and the Poisson's ratio v is obtained from the B and G values. As in icosahedral quasicrystal, Poisson's ratio is small and the ratio G/B is large, indicating the existence of directional atomic bonds also in decagonal quasicrystals.
2.4. Phason elasticity Phason elasticity is closely related to the problem of what is the physical origin of quasicrystalline structural order and has attracted much attention since the discovery of the quasicrystal. Despite years of study, there remain two plausible competing models that explain the realization of quasicrystalline long-range order. One is a perfectly ordered model [1-3,77]. In this model, the existence of local matching rules that force the formation of the quasicrystalline order is assumed. When atomic interactions are in accordance with the local matching rules, the quasicrystal is formed as an energetically favorable structure. Introduction of phason strain induces local atomic rearrangements and thus brings matching rule violations, which increase the elastic free energy by increasing the energetic part of the free energy. The other model is a random tiling [48], in which the quasicrystal is assumed
384
K. Edagawa and S. Takeuchi
Ch. 76
to be stabilized by a configurational entropy originating in many nearly degenerate ways of packing structural units. Such a configurational entropy decreases with increasing average phason strain and thus the phason strain contributes to the elastic free energy f through its entropic part. Theoretically, quasicrystals in both models show g-functional diffraction peaks and projection of the structure along a symmetry axis in both states preserves perfect quasicrystalline order unless the projection thickness is too small [48,78]. These facts make it difficult to distinguish between the two states by a simple diffraction experiment or by high-resolution transmission electron microscopy. In early studies [48,79,80], it was pointed out that in the perfectly ordered model the phasonic elastic free energy fw-w should show a linear dependence on the phason strain, that is, fw-w c~ IVw(r)l, instead of a quadratic form given in eq. (11). This arises from the conjecture that the number of the matching rule violation is in proportion to IVw(r) I in the vicinity of IVw(r) l = 0 and that each such defect independently costs a constant energy. In this case, the radius of the curvature of the energy surface is arbitrarily small at IVw(r) I = 0 and therefore introduction of phason strain must be strongly suppressed especially at low temperatures. Such a state is called a state of locked phason or simply a locked state. In contrast, the elastic free energy in random tiling quasicrystals is shown to have a standard quadratic form [48,79,81]. The state with such a quadratic form of elastic free energy is called a state of unlocked phason or an unlocked state. Monte Carlo simulations using various filing models have shown that a locked state is indeed realized as a ground state and that the locked state undergoes a transition to an unlocked state at some temperature. In general, such a transition occurs at 0 K in two-dimensional systems and at some finite temperature in three-dimensional systems [79-81 ]. Due to all these studies, it may be widely believed that the form of the phason strain dependence of elastic free energy can distinguish between the energetically stabilized perfectly ordered model and the entropically stabilized random tiling model, and that a quadratic form experimentally evidenced by phasonic diffuse scattering (described below) is in favor of the latter model. However, based on the calculations using model quasicrystals with interatomic interactions, Koschella et al. [82] and Trebin et al. [83] have recently claimed that even energetically stabilized quasicrystals should have a quadratic form of elastic free energy, indicating that the phasonic diffuse scattering does not immediately justify the random tiling model. The quadratic form of phasonic elastic free energy generates phason fluctuations. Such phason fluctuations should induce diffuse scattering around Bragg peaks in diffraction spectra just as phonon fluctuations generate conventional thermal diffuse scattering [48, 84]. Phasonic diffuse scattering has indeed been observed experimentally for quasicrystals of various alloy systems, which evidences the quadratic form of phasonic elastic free energy. The shape and the intensity of the diffuse scattering changes depending on the shape of the basin of the energy surface, which is characterized by the elastic constants. Therefore, in principle, by examining the phasonic diffuse scattering one can evaluate the phason elastic constants. The evaluation of the phason elastic constants by diffuse scattering measurements has been reported for icosahedral A1-Pd-Mn [51-55], Zn-Mg-Sc [57] and A1-Cu-Fe [56]. In [54] and [57], absolute-scale measurements of the diffuse scattering intensity have been reported for the A1-Pd-Mn and Zn-Mg-Sc systems, respectively. From the results of the absolute-scale measurements, the phasonic elastic constants K1 and K2 have been eval-
§2.4
Elasticity, dislocations and their motion in quasicrystals
385
Table 11 Phason elastic constants of various icosahedral quasicrystals evaluated by diffuse scattering measurements of X-ray, neutron and electron Alloy system
Source
A1-Pd-Mn A1-Pd-Mn A1-Pd-Mn A1-Pd-Mn A1-Pd-Mn Zn-Mg-Sc A1-Cu-Fe
X-ray neutron neutron neutron electron X-ray X-ray
Meas. temp.
kB Tt
kB T t
K!
K2
(atom- 1)
K1
(atom- 1)
K2
(MPa)
(MPa)
R.T. R.T. 1043 K R.T. R.T. R.T. R.T.
0.06 0.1 0.13 0.5 -
-0.031 -0.052 -0.052 . . . . 0.075 -
43 72 125 . . 300 -
-22 -37 -50 . . -45 -
K1
K2
Ref.
-0.52 -0.52 -0.4 0.5 0.5 -0.15 0.5
[54] [54] [54] [51 ] [73] [57] [56]
uated on the assumption that the phonon-phason coupling K3 is negligible (see eq. (14), and Tables 3, 4). On the other hand, in [51,55,56], only the shapes of the diffuse scatterings have been analyzed and the ratios Kz/K1 have been evaluated on the assumption of K3 = 0. The evaluated values of the elastic constants are summarized in Table 11. In general, the absolute-scale measurements first give the elastic constants in the form of kB/(iTt , where k8 denotes the Boltzmann constant and T' the temperature at which the phason fluctuation equilibrates. To evaluate K1 and K2, we assume T' -- 773 K [54] for the room temperature measurements and T' = 1043 K for the measurements at 1043 K in Table 11. We notice that the phason elastic constants in Table 11 is by about three orders of magnitude smaller than the phonon elastic constants in Table 9. It should be noted, however, that the magnitudes of the phason elastic constants change depending on the arbitrarily chosen scale factor for E±, as pointed out in Section 2.1 and that the ratio K±/KII (K± = K1, K2; KII = )v, #) by itself has no physical meaning. For the icosahedral quasicrystalline state to be thermodynamically stable, the elastic free energy density f in eq. (14) should be positive definite, imposing on the elastic constants the following condition [49]:
5
K 1 - - I - ~ K 2 > 0,
/z
(4) K1-~
K2
> 3 K 2.
(22)
This indicates that K1 should always be positive. Dividing the inequalities by K1 leads to K2
3
K2
K1
5'
K1
~.
3
9K 2
4
4#K1
(23)
In addition to this thermodynamic stability, the system should satisfy another stability condition, which is of the hydrodynamic stability [49,85]. This condition is for the relaxation rate of phason fluctuation to be finite, which requires [49] K2
3
3K2
K2
a
18 K 2
K1
4
()~ + 2#)K1 '
K1
b
b#K1
386
K. Edagawa and S. Takeuchi
Ch. 76
(a -- 27 - 9x/5 ,~ 6.9, b - 15~/5 - 27 ~ 6.5), K2 K1
3 > --
4
K2 +
3(~. + 2/z)K1
•
(24)
It should be noted that the hydrodynamic stability condition in eq. (24) is less restrictive than the thermodynamic one in eq. (23). If we assume K3 --0, eqs (23) and (24) reduce to 3 5
K2
<
3 < K1 4
(25)
and 3 4
K2
<
3 < K1 4'
(26)
respectively. As in Table 11, the A1-Pd-Mn system has been studied most intensively. For this system, the following facts should be noted: (1) K1 and K2 are positive and negative, respectively, (2) the values ~IKll and ~IK2I are smaller at room temperature (or at T z ~ 773 K) than those at T ( ~ T ~) = 1043 K, (3) the decrease in IK~I with the temperature change from T z -- 1043 to 773 K is more IKzl significant than the decrease in kBY" (4) as a result, the ratio K 2 / K 1 also changes slightly; it changes from - 0 . 4 to - 0 . 5 .
First, the signs and the values of K1 and K2 in (1) and (4) are consistent with the stability conditions. Generally speaking, Igll
Ig21
-1
is expected when the phasonic part of the elastic free energy f w - w consists entirely of the energetic contribution. On the other hand, if it consists entirely of the entropic contribution, as in the case of an ideal random tiling, IKll
IK21
(v,)o
kB T ~' kB T ~
is expected. The fact (2) implies IKll
IK21
kB T ~' kB T ~
(T')
(z, > o),
which is closer to the latter. The facts (3) and (4) indicate that the system is close to the instability points K 2 / K 1 = - 0 . 6 (thermodynamic instability) in eq. (25) and K 2 / K 1 =
§2.5
Elasticity, dislocations and their motion in quasicrystals
387
- 0 . 7 5 (hydrodynamic instability) in eq. (26) and that it approaches to them with decreasing temperature. Ishii [85] and Widom [49] have shown that the hydrodynamic instability at K2/K1 = - 0 . 7 5 leads to the symmetry breaking from Yh to D3d. In relation to this result, the formation of a modulated icosahedral phase with the modulation along a threefold direction has been reported in an A1-Pd-Mn alloy with the composition close to that of the icosahedral quasicrystal [86]. For Zn-Mg-Sc, K1 is considerably larger than that for the A1-Pd-Mn system while K2 is roughly the same, resulting in K2/K1 = - 0 . 1 5 that is far from instability points. For A1-Cu-Fe [56], a positive ratio of K2/K1 = 0.5 has been reported, which is close to another instability point K2/K1 = 0.75 that leads to the symmetry breaking to Dsd [49,56,85]. Apart from the alloy systems listed in Table 11, for A1-Pd-Re [87] qualitatively different shapes of diffuse scatterings from those for A1-Pd-Mn have been reported, although the K2/K1 ratio has not been evaluated quantitatively. The intensity and the shape of diffuse scatterings for Cd-Yb [87] have been shown to be similar to those for A1-Pd-Mn. For a A1-Ni-Co decagonal quasicrystal, anisotropic diffuse scattering has been observed in synchrotron X-ray diffraction measurements [88]. It has been shown that the observed diffuse scattering can be attributed partly to the phasonic origin, although no quantitative evaluation of K2/K1 have been reported.
2.5. Phonon-phason coupling In contrast to pure phonon and pure phason elasticities described in the preceding sections, the number of studies so far done on the phonon-phason coupling is limited and we still have little knowledge about it. Henley [48] has discussed the microscopic origin of the phonon-phason coupling by using a toy atomic model of a quasicrystal, in which the atoms interact with a given set of pair potentials. In general, not all neighbor pairs can have optimal distances simultaneously; this results in frustration: some linkages are under compression and others under tension. The introduction of phason strain causes a statistical imbalance between the frequencies of the appearance of various symmetry-related linkages under compression or tension, resulting in the introduction of a global phonon strain. Such a mechanism implies that the phonon-phason coupling strength can be evaluated by measuring the magnitude of the phonon strain spontaneously introduced in crystal approximants to quasicrystals. Fig. 5(a) illustrates schematically an elastic free energy density f of a quasicrystal as a function of phonon strain u and phason strain w, in the vicinity of u = w = 0 (see eq. (11)). Here, the principal axes of the quadratic surface of f (u, w) do not coincide with the u and w axes, indicating a non-zero phonon-phason coupling. In the quasicrystalline state characterized by w = 0, the minimum of f lies at u -- 0, as shown in the curve 1 in Fig. 2(b). In general, each crystal approximant structure is characterized by a nonzero uniform phason strain with a fixed value w0. Now, let us consider an atomistic model quasicrystal with a certain set of atomic interactions. We can construct any crystal approximant structure by introducing the uniform phason strain w0 characterizing the crystal approximant. Then, the dependence of the elastic energy of the crystal approximant structure on the phonon strain u should be given as
K. Edagawa and S. Takeuchi
388
Ch. 76
(b) 2
f
1
u
Fig. 5. A schematic illustration of an elastic free energy density f of a quasicrystal as a function of phonon strain u and phason strain w, in the vicinity of u = w = 0 (a). The cuts along the line w = 0 and w -- w0 in (a) are drawn as the curves 1 and 2 in (b), respectively. The curve 3 in (b) represents the elastic free energy of the approximant phase, which is assumed to be identical to the curve 2 except for the constant term.
f (u, w0) at T = 0, as shown in the curve 2 in Fig. 5(b). Therefore, if we relax the structure while keeping the global phason strain of w = w0, a spontaneous deformation u0 should take place, where u - u0 gives the minimum of f (u, w0). From the quantity u0, we can evaluate the phonon-phason coupling constant because u0 is proportional to the coupling constant. By use of such a method, the coupling constants in a model decagonal quasicrystal and in model icosahedral quasicrystals have been evaluated by Koschella et al. [82], and Zhu and Henley [50], respectively. Koschella et al. [82] have used a two-dimensional binary tiling model with decagonal symmetry, where the atoms interact with pair potentials of the Lennard-Jones type. The coupling constant evaluated is about 3% of the shear modulus. Zhu and Henley [50] have calculated the phonon-phason coupling constants for realistic models of icosahedral A1-Mn and A1-Cu-Li, which represent the two major classes of quasicrystals. The results show that (1) the sign of the coupling constant K3 in eq. (14) is positive for A1-Mn and negative for A1-Cu-Li and (2) IK3] is 0.003-0.02# for A1-Mn and 0.02-0.1# for A1-Cu-Li, where/z is the shear modulus. Here, it would be interesting to examine whether the magnitude of K3 is large enough to effect the elastic instability of the quasicrystal discussed in the preceding section. Comparing eqs (23) and (24) with eqs (25) and (26), we notice that
()~ + 2#)K1 ' #K1
<<1
§2.5
Elasticity, dislocations and their motion in quasicrystals
389
is the condition for K3 to be neglected in the discussion of the elastic instability. The followings are crude estimations: for A1-Mn, we assume [K3] -- 0.01# and adopt K1 = 70 MPa and # = 70 GPa that are for A1-Pd-Mn in Tables 9 and 11, respectively, leading to K2 ~ 0.1" for A1-Cu-Li, we use IK31 -- 0.05# and adopt K1 -- 300 MPa and # -- 40 GPa
#K1
that are for Zn-Mg-Sc and A1-Li-Cu, respectively, giving ~ ~ 0.3. These values suggest that K3 values are not small enough to be completely neglected. We may be allowed to apply the above method to real crystal approximants. Here, it should be noted that we have to make the following assumption to apply the method to the real systems. When we evaluate experimentally the phonon strain introduced in a given crystal approximant phase, the phase should be in the stable state. This indicates that the elastic free energy f of this state must be lower than that of the corresponding quasicrystalline state characterized by u = w = 0. In fact, the phase transition from the quasicrystalline state to the crystal approximant state is generally induced by the decrease in the phason elastic constants, i.e. phason softening and also by the effect of higher-order terms in the expansion of f [49,85]. This fact indicates that the functional form of f in the vicinity of u -- w = 0 should be modified considerably on the transition. Nevertheless, we may be allowed to assume that such a change in the functional form of f occurs in its w-dependence only, considering the nature of the quasicrystal-to-approximant phase transition. On this assumption, the u-dependence of the elastic free energy g(u) of the approximant phase should be identical to the u-dependence of the elastic free energy f of the quasicrystalline phase at w = w0, i.e. f (u, w0) except for the constant term, as shown in the curve 3 in Fig. 5(b). We find in eq. (12) that the phonon and phason strain components u5 and w5 couple in icosahedral quasicrystals. This indicates that (1) only the crystal approximants with w5 ~ 0 induces phonon strain, and that (2) the induced phonon strain has only the u5 component, i.e. pure shear strain. The fact (1) poses a rather severe limitation in choosing the crystal approximant for the experiment. Most of the crystal approximants so far found are of the cubic system, which have only the w4 component, i.e. w5 -- 0. There have been only two non-cubic approximants known so far, which have good structural quality suitable for the present experiments: one is an orthorhombic approximant in the Mg-Ga-A1-Zn system [89] and the other is a rhombohedral one in the A1-Cu-Fe system [90-92]. Recently, Edagawa [93] has measured the phonon strain induced in the crystal approximant of the 3_2_2 orthorhombic type in the Mg-Ga-A1-Zn system and estimated the phonon-phason coupling constant K3, as follows. In the composition Mg39.sGa16.4A14.1Zn40.0, a metastable icosahedral phase produced by melt-spinning underwent a transition to the stable 3_2_2 orthorhombic approximant phase on heating. By X-ray diffraction measurements, the quasilattice parameter aq of the icosahedral phase and the lattice parameters a, b and c of the orthorhombic approximant have been evaluated. Without induced phonon 2_ 2 orthorhombic approximant should be strain, the lattice constants of the 3- T 1
a0--2r4aq(l+r 2) 2 and bo-co-ao/v.
K. Edagawa and S. Takeuchi
390
Ch. 76
Comparing (a, b, c) with (a0, b0, co), the induced phonon strain components, Ull = (a etc., were evaluated. Finally, the coupling constant K3 has been estimated to be ~ - 0 . 0 3 # . Using/z = 46 GPa for a Zn-Mg-Y in Table 9, we obtain K3 = - 1 . 4 GPa. In a following paper [94], it has been pointed out that an unnecessary constraint was imposed in the analysis of the data in [93]. The correction by removing this constraint has been shown to give a slightly larger values of K3 = (-0.04 ± 0.01)# and K3 -- - 1 . 8 4- 0.4 GPa. The icosahedral Mg-Ga-A1-Zn is of the Frank-Kasper type and should be compared with the results of the calculation for A1-Cu-Li by Zhu and Henley [50]. The negative sign of K3 for the Mg-Ga-A1-Zn system is in agreement with the calculation. The IK3] value is also in good agreement with the calculation, although the calculation has yielded a considerably large range of values. Recently, Edagawa and So [94] have performed the measurements of the phonon strain induced in a rhombohedral approximant in the A1-Cu-Fe system [90-92]. They have reported a small positive K3 value of about 0.004# for this system.
ao)/ao,
3. Dislocations in quasicrystals 3.1. Perfect dislocations in quasicrystalline lattice
Because of lack of periodicity, we cannot introduce a perfect dislocation into a quasicrystal simply by applying the conventional definition of crystalline dislocation. However, a perfect dislocation can also be defined in quasicrystal by extending the definition of crystalline dislocation [5,6,9]; a perfect dislocation in quasicrystal is a line defect in the physical space Ell, satisfying
fc dU- B ~ LR,
(27)
for any loop C surrounding the dislocation line. Here, U denotes a high-dimensional displacement vector (see Fig. 1), B a high-dimensional Burgers vector and LR a set of highdimensional lattice translational vectors. Using eq. (6), eq. (27) can be decomposed into the two equations:
cdU-
fc
bll ,
dw-b±,
(28)
where bll and b± are the Ell and E± components of B, i.e. B = bll + b±.
(29)
Eq. (28) indicates that in the presence of dislocations the displacement fields u(r) and w(r) are multi-valued. The dislocations defined here generally have the following properties:
§3.1
Elasticity, dislocations and their motion in quasicrystals o o
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Fig. 6. A perfect dislocation introduced in a two-dimensional Penrose lattice. The Burgers vector bll indicated by an arrow is [0110], where the indices are referred to the basis vectors di (i = 1 . . . . . 4) in eq. (9) (see Fig. 4(b)).
(1) It is not accompanied by any planar fault because B is a lattice translational vector in the high-dimensional space. (2) The magnitude of distortion or strain is in inverse proportion to the distance from the dislocation center. These properties are common to conventional crystalline dislocations. However, the perfect dislocation in quasicrystal is different from crystalline perfect dislocation in that the former is inevitably accompanied by the phason strain because every high-dimensional lattice vector necessarily has a phason component. Fig. 6 presents an example of a perfect dislocation in a quasicrystal. Here, a perfect dislocation with the Burgers vector [0110] is introduced in a two-dimensional Penrose lattice that is known as a typical example of a two-dimensional decagonal quasicrystal. Here, the indices of the Burgers vector are referred to the basis vectors di (i = 1 . . . . . 4) in eq. (9) (Fig. 4(b)). In this case, the dislocation is a point defect because of the twodimensional system. Here, as a simple example, the displacement vector is given by
(0) ~
B,
(0)
bll,
U(r,O)or equivalently,
u(r, 0 ) -
~
(30)
392
w(r, 0 ) - -
(0) ~
K. Edagawa and S. Takeuchi
b±,
Ch. 76
(31)
where r is the distance from the dislocation core at the center and 0 is the polar angle measured about the core. If you observe carefully the figure at grazing angle, you will find distortions and jogs in lattice planes in the vicinity of the dislocation core. Here, the distortions of lattice planes correspond to the phonon distortion or strain. On the other hand, the jogs in the lattice planes correspond to the phason one. In real quasicrystals, the displacement fields around dislocation should be calculated on the basis of the generalized elastic theory described in Section 2.2 and must be more complex than the form of eq. (31). The displacement fields around real dislocations will be discussed in more detail in the next subsection. Since dislocations in quasicrystals are accompanied by a strain field as well as those in conventional crystals, they can be observed as a diffraction contrast image by transmission electron microscopy. For dislocations in conventional crystals, the direction of the Burgers vector b can be determined by use of the invisibility condition g. b = 0 [95], where g is a reciprocal vector used for imaging. For dislocations in quasicrystals, this condition is generalized to [96,97] G . B =gll "bll + g ± "b± = 0 ,
(32)
where G = gll + g±.
(33)
Here, G is a high-dimensional reciprocal vector, and gll and g± are its Ell and E± components, respectively. There are two cases in which this invisibility condition is satisfied. One is the case gll "bll : g± "b± = O,
(34)
which is called 'strong' invisibility condition. Because g± • b± = 0 is automatically satisfied when gll" bll = 0 is satisfied, the strong invisibility condition is written just as gJl" bll = 0. This condition is the same as the invisibility condition for conventional crystals. The other case is gll" bll -- - g ±
"b± ~ O.
(35)
This condition is called 'weak' invisibility condition, which is characteristic of dislocations in quasicrystals. By use of the convergent-beam electron diffraction (CBED) technique, not only the direction but also the magnitude of the Burgers vector can be determined [98]. In the defocus CBED pattern, zeroth-order Laue zone line splits when it crosses a dislocation line, where the number n of splitting is given by [99] n = G . B = gll "bll -+- g± "b±.
(36)
§3.2
Elasticity, dislocations and their motion in quasicrystals
393
Using such methods, the Burgers vector determination has been attempted for icosahedral [99-102] and decagonal [103-108] quasicrystals. Rosenfelt et al. [102] have analyzed in detail dislocations in deformed and undeformed samples of icosahedral A1-Pd-Mn. It has been found that about 90% of the dislocations in both deformed and undeformed samples are the perfect dislocations with Burgers vector pointing along twofold direction. The rest are partial dislocations with Burgers vector pointing along fivefold, threefold and pseudotwofold directions. Four types of twofold Burgers vectors have been observed: (001111), (001221), (002332) and (003553), where the indices are referred to the basis vectors di in eq. (7) with a - 6 . 4 5 A (see Fig. 4(a)). 3 For the four Burgers vectors, the ratios Ib±l/Ibll I are r 3, r 5, r 7 and r 9, respectively. Of the four, (001221) is most frequently observed; about 65% of the twofold Burgers vectors observed are (001221). The major slip planes are shown to be fivefold and threefold planes. For decagonal quasicrystals, the dislocations with the Burgers vector parallel to the periodic direction [103-106], in the quasiperiodic plane [106-108] and with both periodic and quasiperiodic components [106] have been reported. The Burgers vectors of the second type so far reported are (10000), (01000), and (01110), whose Ib±l/lbll I ratios are r, r -1 and r -3, respectively. Here, the indices are refereed to the basis vectors in eq. (9) with a - 3.39 A. All of them are perfect dislocations with twofold Burgers vectors.
3.2. Displacement field and self free energy of dislocations The phonon and phason displacement fields u(r) and w(r) around a dislocation in quasicrystals should satisfy the definition of the dislocation given by eq. (28) and elastic equations of balance by eq. (21). In the absence of body forces, eq. (21) reduces to
02Uk u-w 02113k u-u -Jr-Cijkl = O, Cijkl Orj Orl Orj Orl
u-w 02Uk w-w 02~k =0. Cijkl -Jr- Cijkl Orj Orl Orj Orl
(37)
In principle, the displacement fields u(r) and w(r) around a dislocation in quasicrystals can be fully determined by eqs (28) and (37). The calculations of the fields around a dislocation in pentagonal [109], decagonal [110-113], octagonal [111], dodecagonal [111, 114] and icosahedral [58,115] quasicrystals have been reported. In the calculations, various mathematical methods have been applied, which are summarized in the review by Hu et al. [116]. Because eqs (28) and (37) have the forms of a simple generalization of the corresponding equations for conventional crystalline dislocations, the methods used for crystalline dislocations can be generalized easily for the application to the quasicrystalline dislocations. In the following, we briefly review the Eshelby's method [117] generalized by Ding et al. [113]. By use of the generalized Eshelby's method, we demonstrate a calculation of the displacement fields around a dislocation in an icosahedral quasicrystal and deduce self free energy of the dislocation. 3In this case, the lattice spanned by di in eq. (7) with a = 6.45 A has a F-type superlattice order; the structure 6 nidi with )--~6 has the translational symmetry only for the vectors R = }--~4=1 i=1 ni - - e v e n . We confirm that }-~6 1n i = even is satisfied for all the four Burgers vectors and therefore they are perfect dislocations.
394
Ch. 76
K. Edagawa and S. Takeuchi
We consider a straight dislocation parallel to the x3 axis. We define
ot/~ 2
(38)
a °~(p) -- Bill 4- (Bl~2 4- B2~I)p 4- B22 p , where
B jr -- ~,ot(~k~Cijk? 4- Sk(~_3)Cijk~ °) 4- Si(ot_3)(~k~Cijkl w 4- ~k(~_3)CijklW).
(39)
Here, oe and/3 take 1, 2 . . . . or 6, and i, j, k and 1 take 1, 2 or 3. We solve the equation det(aC~ (p)) - 0 .
(40)
This is a twelfth-order equation for p and we obtain twelve roots pl, p2 . . . . and p12. These roots are necessarily complex and consist of six conjugate pairs. We pick up p] . . . . and p6, one from each pair of complex conjugates. Solve the equations
a °~ (Pn) A/~ (n)
= 0
(41)
to obtain A/~ (n) for each of the six roots of mine the constants D (n) satisfying
Pn. Using
A~ (n) (n -- 1, 2 . . . . . 6), we deter-
(42)
and
Re
(B21~ + B22
(n) D (n)
n=l
]
(43)
- 0
where bll = [bl, b2, b3] and b± = [b4, b5, b6] are the Burgers vectors. Then, we finally obtain the displacement fields
U(Xl, x2) -
I U1 (Xl, x2) ] U2(xl, x2)
and
W(Xl, X2) --
U3 (x l , x2)
U5(xl, x2) U6(xl, x2)
with
U~(xl, x2) - Re
~4-27ri A~(n)D(n) n=l
ln(xl +
pnx2) 1 ,
(44)
Elasticity, dislocations and their motion in quasicrystals
§3.2
395
where the sign + is taken to be the same as the sign of the imaginary part of Pn. Following the Foreman's method [ 118], the self free energy F per unit length of the dislocation has the form
Kb2 I n ( R ) F=
4rr
(45)
ro
where
Kb 2 - b~ Im
-4--(B2~+ B2~2 Pn)A~(n)D(n)
(46)
.
n--1
Here, K is a factor called energy factor, and r0 and R are inner and outer cut-off radii, respectively. As described before, the dislocations with the Burgers vector along the twofold direction are by far the most frequently observed in icosahedral quasicrystals. As explained in more detail later, such a Burgers vector can be written as bll -- [0, 0, bll] and b± = [0, 0, b±], where the indices are referred to el, e2 and e3 for bll, and e4, e5 and e6 for b±, respectively (see Fig. 4(a)). As an example, we consider a simple case of a dislocation with such a Burgers vector and with the line direction along the x3 axis, i.e. the e3 direction. If we assume that the phonon-phason coupling is negligible, i.e. K3 --0, we can deduce easily the displacement fields u(r) and w(r) and the self free energy F. The results are
U(Xl, X2) - -
0 bll t a n - 1 ~-y
W ( X l , X2) - X2
M
'
Xl
0 b±
~-y t a n -
1
X2
'
(47)
--
Xl
and =
4rr
In
~0 '
(48)
where B -- K1 + K2(r - ½) and C - K1 + K2(~- - r) (see Table 3). The derivation of eqs (47) and (48) is given in Appendix B. Because K3 - - 0 is assumed, u(r) is identical to the field around a screw dislocation in conventional isotropic elastic body, which has the same form as in eq. (31). On the other hand, w(r) deviates from the form in eq. (31). Assuming 1
4zr as often assumed for conventional crystalline dislocations, we obtain from eq. (48): F ~/zb~ + ~-B~b 2.
(49)
396
K. Edagawa and S. Takeuchi
Ch. 76
More generally, the self free energy F of a dislocation in icosahedral quasicrystals can be written as F -- a#[bll
12+ bY1 ]b±[ 2,
(50)
where a and b are positive constants of the order unity, which change depending on the Burgers vector direction and the dislocation line direction. Here, we note that K] is always positive and that [Kzl/K] and )v/# are of the order unity. Now, let us discuss about relative stability among dislocations with various Burgers vectors, based on eq. (50). Here, we restrict our discussion to the dislocations with twofold Burgers vectors. In general, the Burgers vectors with bll parallel to a twofold direction e3 in Fig. 4(a) can be written as R1 - [0, 0, 0, 1, i, 0],
R2 - [ 0 , 0, 1, 0, 0, i],
B - - m R 1 + nR2 - [O, O, n, m, rh, ~].
(51)
Here, the indices are referred to the basis vectors di in eq. (7). We note that all these B belong to the translational symmetry vectors for the F-type superlattice order (see footnote 3 in Section 3.1). The same Burgers vectors can be rewritten as
R] - [0, 0, p, 0, 0 , - z p ] ,
R2 - [0, 0, rp, 0, 0, p]
B - mR1 + nR2 -- [0, 0, p ( m + nr), 0, 0, p(n - mr)],
(
2a ) P-
r~/6~-2r
' (52)
where the indices are referred to the basis vectors ei in Fig. 4(a). This shows that for those B, the b± component is always parallel to the e6 direction in E±. In other words, all those B lie on the two-dimensional subspace spanned by e3 and e6 in the six-dimensional space. More specifically, eq. (52) indicates that the Burgers vectors B form a square lattice in the e3-e6 plane, as shown in Fig. 7. Here, the square lattice is tilted with respect to the e3-e6 coordinates by 0 - tan -] (r) and the projections of B onto Ell and E± give the bll and b± components, respectively. Now, we consider the problem as to which B gives the minimum self free energy F in eq. (50). For example, the lattice point P with (m, n) - (1, 3) can never be the minimum F point because there are other points, e.g., the point Q with (m, n) - (1, 2), whose ]bll ] and ]b±] are simultaneously smaller than those for the point E In contrast, the point Q can be the minimum F point because there are no other points whose ]bll ] and ]b±] are simultaneously smaller than those for the point Q. The lattice points satisfying this condition are those with ( m , n ) - . . . , (3, 2,), (2, 1), (1, 1), (1, 0), (0, 1), (1, 1), (1, 2), (2, 3) . . . . . which are indicated by open circle in Fig. 7, and their inversion counterparts indicated by solid circle. Here, the open circle points can be expressed as B~ - (Fk, Fk+]) (k --0, 1, 2 . . . . ) and B~ -- (F-k, F - k - l ) (k -- - 1 , - 2 . . . . ) . . . . where Fn(n ~> 0) is the Fibonacci numbers defined by F n + 2 - Fn+] + Fn with F 0 - 0 and F] - 1. The solid circle points can simply be expressed as { - B ~ l k - 0, 4-1, 4-2...}. Combining the two groups, we obtain {-+-B~Ik -- 0, +1, + 2 . . . } .
§3.2
397
Elasticity, dislocations and their motion in quasicrystals
X6,~
o
o
o
r
X3 II
o
1
•
c
Fig. 7. The two-dimensional plane spanned by e3 and e6 in Fig. 4(a). The Burgers vectors of the type eq. (51) form a square lattice on this plane. The open and solid circles represent the subsets {+B~lk = 0, + l , 4-2...} and {-B~ ]k = 0, 4-1, 4-2... }, respectively (see text).
The Burgers vectors {B~ [k -- 0, 4-1, 4-2... } satisfy Ibl~kI -
Ibl~olr ~,
Ib~_~ I- b~_olr -k- Ib,~olT-<~+I>,
(53)
where b*Ilk and b*±k are the Ell and E± components of B k, * respectively. From eq. (53), we obtain the following relations:
bl~kl" Ibm_,I - bl]0 I" Ibm_01-
const.
(54)
and Ib*l k [ Ib,Ilkl
=r
-2k- 1
(55)
As shown above, the minimum F point necessarily lies in the subset {4-B~Jk0, 4-1, 4-2...} (open and solid circles in Fig. 7) of all the Burgers vectors B defined by eq. (51) or (52). Then, which point in the subset {-+-B~ Ik - 0, +1, + 2 . . . } is the minimum F point? Using the relation of eq. (54), we find that the self free energy F in eq. (50) becomes the minimum at
±kl Ilk]
(
a# bK1
(56)
K. Edagawaand S. Takeuchi
398
Ch. 76
For the A1-Pd-Mn system, we find # ~ 70 GPa and K1 ~ 70 MPa in Tables 9 and 11, respectively. Thus, we obtain #/K1 ~ 1000. Because of a/b ~ 1, eq. (56) gives
Ib*l ±~ Ib*Ilkl
~ 30.
As described in Section 3.2, the twofold Burgers vectors (001111), (001221), (002332) and (003553) have been reported in the A1-Pd-Mn system. All of them belong to the subset { + B ~ l k - 0, +l,-+-2...}; they correspond to B* 2, B*_3, B* 4 and B*_5, respectively. The most frequently observed Burgers vector is B*_3. The ratios Ib±l/lbll I for the four Burgers vectors are z 3 -- 4.2, r 5 -- 11, Z" 7 - - 29 and Z" 9 - - 76, respectively, agreeing roughly with Ib~_kI the above estimation ~ ~ 30 for the minimum F Ibllkl For decagonal quasicrystals, twofold Burgers vectors (10000), (01000), and (01110) have been reported, whose Ib±l/lbll I ratios are r - 1.6, r -1 - 0 . 6 2 and r -3 - 0 . 2 3 , respectively. All of them belong to the subset that can be defined for the decagonal system similarly to the subset {4-B~lk- 0, 4-1, 4-2...} for the icosahedral system. In contrast to the case of icosahedral quasicrystals, no measurements of the phason elastic constants have been reported for decagonal quasicrystals. The observed Ib±l/lbll I ratios for decagonal quasicrystals may allow us to make a crude estimation of the magnitude of the phason elastic constants" K± ~ ~.~ # ~ 3# ~ 240 GPa.
4. D i s l o c a t i o n m o t i o n in q u a s i c r y s t a l s 4.1. Dislocation mechanism of deformation
In general, quasicrystals are brittle at room temperature and can only be plastically deformed at high temperatures above about 0.8Tm (Tm: melting temperature). In the high temperature range, systematic deformation experiments and transmission electron microscopic (TEM) observations of the deformation microstructures have so far been performed for quasicrystals of various alloy systems, as reviewed in [11-13]. In 1993 [14], Wollgarten et al. have revealed by transmission electron microscopy (TEM) that by plastic deformation the dislocation density increases to the order of 108/cm 2 from 106/cm 2 in the as-grown state, suggesting a dislocation mechanism for plastic deformation. In icosahedral A1-Cu-Fe, a twin-like structure and planar faults were observed after plastic deformation [119-121], but later high density dislocations and stacking faults were observed also in this alloy [ 122] as in icosahedral A1-Pd-Mn. The most direct evidence for the dislocation mechanism of plastic deformation has been obtained by an in-situ stretching experiment on a high temperature tensile stage in an electron microscope [15,122]. Under an applied stress, straight dislocations have been observed to move steadily and continuously, just like the dislocation glide process observed in-situ by transmission electron microscopy for bcc metals at low temperatures and for semiconducting crystals at high temperatures. Until recent years, it has been generally believed that the plasticity of quasicrystals is carried by a
§4.2
Elasticity, dislocations and their motion in quasicrystals
399
glide process of dislocations, and various models of the deformation mechanism have been proposed based on the dislocation glide [16-20]. However, recently, Caillard et al. [2123] have shown by TEM that high temperature deformation of icosahedral A1-Pd-Mn is brought not by a glide process but by a pure climb process. On the other hand, at low temperatures Mompiou et al. [23], Caillard et al. [123] and Texier et al. [124,125] have performed deformation of icosahedral A1-Pd-Mn phases under a high pressure and investigated the deformation microstructures by TEM observation. Mompiou et al. [23] and Caillard et al. [ 123] have shown that for single-crystals of icosahedral A1-Pd-Mn deformation at 573 K is exclusively due to dislocation climb. On the other hand, Texier et al. [124,125] have conducted deformation experiments of polycrystalline A1-Pd-Mn icosahedral quasicrystals in the temperature range between room temperature and 573 K. They have shown the following facts: (1) at room temperature the deformation is mostly brought by dislocation glide but dislocation climb events are also occasionally observed and (2) the frequency of the climb events increases as the deformation temperature is raised. Saito et al. [ 126] have recently performed deformation experiments of MgZn-Y icosahedral quasicrystals at low temperatures under a hydrostatic confining pressure superimposed to an applied uniaxial stress. They have found an irregular behavior in the temperature dependence of the yield stress. The origin of the irregular temperature dependence has been discussed in terms of possible transition between glide process at low temperatures and climb process at high temperatures. In the following two subsections, we discuss theoretically the two important dislocation processes in quasicrystals, i.e. glide and climb processes.
4.2. Dislocation glide process For a phason strain field to be produced or relaxed, phason jumps must occur via short range atomic diffusion, which takes place only as a thermally activated process, as mentioned in Section 2.1. Thus, in a low temperature range below half the melting point where the atomic diffusion is practically prohibited, the perfect dislocation glide is impossible and only the partial dislocation with the Burgers vector of bll can glide, leaving behind the stacking fault (also called the phason fault) with the fault vector of b±. Let the energy of the phason fault be F, the dragging stress necessary to move the dislocation is ra = F/bll,
(57)
where r represents the resolved shear stress acting on the glide plane in the direction of bll, and bll is the strength of bll. In contrast, at high enough temperature at which atomic mobility by diffusion is faster than the dislocation velocity, perfect dislocations can migrate accompanying both phonon and phason strains. Fig. 8 shows the structural change that should take place when a perfect dislocation moves in the two-dimensional Penrose lattice. Here, the patterns indicated by broken lines show the phason flips (see Fig. 3) that should take place when the perfect dislocation is translated to the left by the distance represented by an arrow. In actual quasicrystals, the quasicrystalline lattice is decorated by atomic clusters; depending on the
400
K. Edagawa and S. Takeuchi
Ch. 76
Fig. 8. Structural changes taking place when a perfect dislocation moves in a two-dimensional Penrose lattice; the patterns indicated by broken lines show the phason flips (see Fig. 3) that should take place when the perfect dislocation is translated to the left by the distance represented by an arrow.
way of the decoration, different types of quasicrystals are produced such as the FrankKasper type, the Mackay-icosahedron type and the Cd6Yb type for icosahedral quasicrystals. Fig. 9(a) shows an example of a realistic quasicrystalline structure containing a perfect dislocation: the atomic structure is a model decagonal quasicrystal (Burkov model [127]) viewed from the ten-fold axis [128]. The perfect lattice is composed of decagonal columnar-clusters with glue atoms among them. In contrast to the case of undecorated Penrose lattice in Fig. 8, we can see clearly a phason strain field by a distribution of imperfect cluster columns in Fig. 9(a). Fig. 9(b) shows the atomic structure for the dislocation position at the left end in the figure. By comparing Figs 9(a) and (b) we see a change of the cluster pattern as a result of the change of the phason strain field. As is evident from Figs 8 and 9, two types of the structural change should take place as a perfect dislocation translates on the glide plane; one is the re-tiling structural change outside the glide plane and the other is the intra-tile structural change. The energy release due to the latter structural change must be much higher than that due to the former type of the structural change, because without the latter relaxation severe nearest neighbor violation occurs, while without the former relaxation no nearest neighbor violation occurs and the energy increase is only due to a long range interaction. Therefore, in an intermediate temperature region where the short range diffusion can take place with the dislocation translation, a partial relaxation of the phason strain should occur only near the glide plane by healing intra-tile structural violation. Fig. 10 shows schematically the three cases of the structure change as a result of dislocation glide from the left end to the center: (a) low temperature case, (b) intermediate temperature case and (c) high temperature case.
Elasticity, dislocations and their motion in quasicrystals
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Fig. 9. An example of a realistic quasicrystalline structure containing a perfect dislocation: the atomic structure is a model decagonal quasicrystal (Burkov model [127]) viewed from the ten-fold axis [128]. The dislocation position is at the center in (a) and at the left end in (b).
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=b±
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(b)
(o)
Fig. 10. Three cases of the structure of a dislocation gliding from left to right in a quasicrystal: low temperature case (a), intermediate temperature case (b) and high temperature case (c).
In addition to the glide resistance due to the phason defect production, dislocations in quasicrystals should be accompanied by a Peierls potential barrier as the dislocations in crystals. The Peierls potential is the self-energy change with the change in dislocation position due to the discreteness of the lattice; the Peierls potential in crystals is periodic but that in quasicrystals quasiperiodic. For crystals, it is established theoretically [ 129,130] and experimentally [131,132] that the simpler the crystal structure having a larger d/b ratio (d: the glide plane spacing, b: the strength of the Burgers vector), the higher is the Peierls potential. It is naturally considered that the Peierls potential is high for dislocations in quasicrystals due to the complex atomic structures.
K. Edagawa and S. Takeuchi
402 '
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Unless the phason relaxation occurs, the change of the potential energy with the translation of a dislocation comprises the phason fault energy and the Peierls potential which is superimposed on the former. Examples of the potential energy profile calculated for two different perfect dislocations in a realistic model decagonal quasicrystal [133] are presented in Fig. 11. It is seen that the Peierls potential is quite sensitive to the type of the dislocation as for crystal dislocations. For type (a) dislocation in the figure, the calculated stress to move the dislocation at 0 K is as high as 0.159G (G: the shear modulus) which consists of the phason production stress component of 0.048G and the Peierls stress component of 0.111 G, the latter being twice as large as the former. The so-called F-surfaces (the energy surface of the plane fault created by a rigid displacement parallel to the glide plane of the upper half crystal (quasicrystal) with respect to the lower half) for the model quasiperiodic lattice in the two Burgers vector directions in Fig. 11 are shown in Fig. 12. It is to be noted that in multicomponent quasicrystals the F-surface energy consists not
Elasticity, dislocations and their motion in quasicrystals
§4.2
(a)
403
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Fig. 13. (a): Kink-pairformation process; (b): potential profile for a kink-pair formationprocess under no applied stress (dashed curve) and under a stress higher than F/bll (solid curve).
only of the phonon and phason strains but also of chemical disordering energy, as shown in the simulation of a model decagonal quasicrystal [134]. As seen in Fig. 12, due to the quasiperiodicity of the lattice, the F-surface also oscillates quasiperiodically. Essentially the same results have been obtained for other models [135,136]. Such an oscillation of the F-surface means that if a group of dislocations glide together on the same plane, the necessary phason dragging stress which is given by rd = F / ~ b i becomes negligibly small as the number of dislocation increases [133,137]. Thus, the Peierls potential plays a dominant role in the dislocation glide rather than the phason dragging stress. At high temperatures, we would expect thermally activated dislocation glide. Because dislocations in quasicrystals are subject to high Peierls potential, they should tend to lie along a Peierls valley. This has been verified experimentally by in-situ electron microscopy observation of dislocation motion, where dislocations are observed to move keeping a straight form oriented in a particular direction [15,122]. The thermally activated glide process for such a dislocation is the kink pair formation followed by motion of kinks along the Peierls valley, as illustrated in Fig. 13(a) [138]. For the thermally activated kink-pair formation to occur a phason fault relaxation should take place to some extent; otherwise the enthalpy of the final state (kink-pair state) cannot be lower than the enthalpy of the initial state (straight form lying in a Peierls valley) under a reasonably low stress that does not lead to fracture. As mentioned previously, two kinds of phason fault should be produced with dislocation translation, the re-tiling fault and the intra-tile fault. The re-tiling fault has
K. Edagawa and S. Takeuchi
404
Ch. 76
much lower energy than the intra-tile fault and furthermore the relaxation of the re-tiling requires cooperative jumps of a number of atoms leading to a much longer relaxation time than the relaxation of the intra-tile fault. For these reasons, it seems reasonable to assume that in the thermally activated kink-pair formation process and the kink migration process, the relaxation only of the intra-tile fault occurs. There are two regimes for the kink-pair formation process in crystals, one is the smooth kink regime and the other is the abrupt kink regime [139]. In the case of relatively low Peierls potential such as that in bcc metals or NaC1 crystals, the kink shape is smooth, and in the case of high Peierls potential such as that in covalent crystals of Si, the kink shape is abrupt. Due to the lattice periodicity along the Peierls valley, there exists a periodic potential also for the kink translation, which is called the Peierls potential of the second kind. For the smooth kink, the Peierls potential of the second kind is generally negligible and the kink energy can well be represented by the line tension approximation of the bowing-out dislocation. The dislocation mobility is determined by the kink-pair formation enthalpy, which has been given by Celli et al. [140], and Dorn and Rajnak [141]. For the abrupt kink, the Peierls potential of the second kind is also involved in the kink-pair formation enthalpy and also in the kink migration process. The kink-pair formation in the abrupt kink case has been treated in terms of the kink diffusion theory by Hirth and Lothe [142], which has been applied to the interpretation of the dislocation mobility in semiconducting crystals. As mentioned above, the thermally activated dislocation migration should be accompanied by the intra-tile phason relaxation in the close vicinity of the glide phane, which requires thermally activated short range atomic jumps. Thus, even if we neglect the Peierls potential of the second kind, the kink-pair formation of a dislocation in quasicrystals has to be treated by a kink diffusion theory in a similar manner to the kink-pair formation in the abrupt kink regime. The overall potential profile for the kink-pair formation process under zero applied stress is depicted by a dashed line in Fig. 13(b). As the kink separation 1 of the kink-pair increases, the self-energy of the kink-pair increases towards twice the single kink energy EK. Constantly increasing component with a slope F is due to the unrelaxed phason strain outside the glide plane. The potential hills superimposed aperiodically on the curve are due to the activation barrier for the atomic jumps in the intra-tile relaxation process. In this potential profile, the thermally activated kink-pair formation can take place only under a stress higher than rd = F/bll. The solid curve in Fig. 13(b) shows the enthalpy profile for the kink-pair formation under a stress higher than rd. The rate of the kink-pair formation under a stress r* -- ra - rd (r*: the effective stress, ra: the applied stress) is determined by the rate of the kink diffusion surmounting the potential barrier H~p towards larger l value. This rate can be formulated in perfect analogy to the kink-pair formation of the abrupt kink regime in crystal dislocations with the substitution of the kink migration enthalpy by the activation enthalpy of the atomic jump and the period of the Peierls potential of the second kind by the separation of the potential hills H' in Fig. 13(b). The frequency of the kink-par formation per unit length of a dislocation at low stress is given by T*blld
[
V~:p - 2-£-~8T v~exp -
kBT
(58)
Elasticity, dislocations and their motion in quasicrystals
§4.3
405
where vk is the kink vibration frequency. The kink velocity is written as
vk --
C*blldd'2 (H') kBT exp - k ' ~
'
(59)
where d' is the average distance between the phason jumps. For an infinite length of the dislocation, the equilibrium density of kinks is determined by the balance between the rate of the kink-pair formation and the rate of annihilation of positive and negative kinks, and the equilibrium kink separation is given by
[/~ -- x/2d' exp 2k8 T
"
(60)
Let the dislocation segment length lying in a Peierls valley be L. Depending on the relative magnitude of L and lk, there are two cases for the dislocation mobility; one is the kinkcollision case and the other is the kink-collisionless case. For a typical value of H~*p = 3 eV and d' = 0.5 nm, [ at 1000 K is calculated to be 3 cm, which is much larger than the dislocation segment length L, leading to the kink-collisionless case. Therefore, the dislocation glide velocity Vg is controlled only by the rate of kink pair formation on the segment length L, which is followed by the kink motion over the distance L, and is written as
r*bll d2____~L
(H~*p(r*) + H') ksT "
Vg= 2k~T vkexp -
(61)
4.3. Dislocation climb process
For a dislocation confined in a deep Peierls valley, the climb process should be the jog-pair formation followed by the jog motion, in a similar manner to the kink-pair formation and the kink motion, as illustrated in Fig. 14(a) [138,143]. Unlike the kink-pair formation by glide, the climb motion occurs by vacancy absorption or emission at jogs, i.e., accompanying atomic diffusion to or from the jog sites. As a result, the intra-tile relaxation is expected to occur simultaneously with the dislocation climb. Due to unrelaxed phason defects outside close vicinity of the climbing plane, there should be a constant increase of the phason strain energy with the jog motion, as in the case of the kink motion. Thus, the potential energy profile under no applied stress for the jog-pair formation is like that depicted by a dashed line in Fig. 14(b) and that under an applied stress T > F/bjj is shown by a solid line in the figure. In the case of the kink-pair formation process, the kink velocity is determined by the short range atomic diffusion to relax the intra-tile phason relaxation, whereas in the case of the jog-pair formation process, the jog velocity is controlled by the rate of vacancy absorption or emission at the jog, i.e. the self diffusion. Taking account of the fact that the formation energy at the dislocation core and the migration energy along the dislocation
406
Ch. 76
K. Edagawa and S. Takeuchi
(a) ,--
'....
1
--q
zSH
(b) 2~ f
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p
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Fig. 14. (a): Jog-pair formation process; (b): potential profile for a jog-pair under no applied stress (dashed curve) and under a stress higher then I'/bll (solid curve).
core of a vacancy are considerably lower than those in the lattice, the jog velocity under an effective resolved stress a* for the climb is written as [ 142]
vj =
4rcDsa*blla
exp
(AHs)
,
(62)
where a is the atomic distance along the dislocation line, Ds is the self-diffusion coefficient, A Hs is the difference between the activation enthalpy of the self-diffusion in the bulk and that along the dislocation core and ~ is the mean life length of a core vacancy along the dislocation line given by
~,=~/2aexp
(A.s) 2kBT
"
(63)
As in the case of the kink-pair formation, there are two cases for the dislocation velocity expression, the jog-collision case and the jog-collisionless case, depending on the relative magnitude of the equilibrium jog spacing [j - a exp(Ej/kB T) and the dislocation segment length L lying in a Peierls valley. Since E j is of the order of an eV, fj > L is generally satisfied. Thus, as in the case of the kink-pair formation, the jog-collisionless case is real-
Elasticity, dislocations and their motion in quasicrystals
§4.4
407
ized. For L < [j, the dislocation climb velocity Vc is determined by the rate of the jog-pair formation on the segment length L and is given by
4zcLDsvaCr*
(
Hj*p(Cr*)- AHs/2) Vc -- aS--~B-~-~nU/~l) exp -kBT '
(64)
Hjp is the activation enthalpy of the jog-pair formation. Writing Ds -- aZvD e x p ( - H s / k8 T) (Hs: the activation enthalpy of the self-diffusion; vo: the Debye frequency) and approximating Va = a 3, eq. (64) is rewritten as where
1) a
is the atomic volume,
4rcLa3cr * (Hj*p(~*) + (Hs - AHs/2) ) gc = kgT1--n(~-f/~ll) vDexp -kgT "
(65)
As described in Section 4. l, detailed TEM observations have revealed that in icosahedral A1-Pd-Mn climb process is the dominant deformation process over the glide process at high temperatures. This indicates Vc > Vg. Below, we make a crude estimation about relative magnitude of the velocities, which indeed justifies the fact Vc > Vg. From eqs (61) and (65), the ratio of Vg/Vc is given by
For a typical case of T = 1000 K, Hs = 2 eV and AHs -- Hs/2, the pre-exponential factor of eq. (66) is of the order of unity. H t may be comparable to the migration enthalpy of vacancy which is about Hs/2, and hence the second term of the exponent is small. Thus, the ratio Vg/Vc is determined essentially by the relative magnitude of H~p and Hj*p or the relative magnitude of the kink energy E~ and the jog energy Ej. Let r p / G --/3, the kink energy is given by Ek = 0.5~/-fiGdb 2 [139]. E j -- GbZh/{4rc(1 - v)} ~ O.1GbZh [142]. Assuming fl = 10-1 .,, 10 -2, Ek Ej
d = (0.5 ^-. 1.5)=.
h
(67)
Since the kink height d (period of the Peierls potential) is generally larger than the jog height h (atomic spacing), Ek can be larger than E j , and hence Vc can be larger than Vg.
4.4. Plastic homology Although the climb controlled deformation has been confirmed only in A1-Pd-Mn icosahedral quasicrystals, we assume that the high temperature plasticity of any icosahedral quasicrystals is governed by a common dislocation climb process. Then, we expect that some homologous relation should hold for icosahedral quasicrystals, as already established in bcc metals [144] and tetrahedrally coordinated crystals [145].
408
Ch. 76
K. Edagawa and S. Takeuchi
900
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1200
TEMPERATURE (K) (a) Fig. 15. (a)" Temperature dependence of the upper yield stress for icosahedral quasicrystals of A1-Pd-Mn [16, 146], A1-Cu-Fe [147], Cd-Yb [148], Mg-Zn-Y [149] and A1-Li-Cu [150]. (b): Normalized upper yield stress vs. normalized temperature replotted from (a).
Fig. 15(a) shows the temperature dependence of the upper yield stress for various icosahedral quasicrystals [16,146-150]. The temperature range of the plastic deformation differs largely among the icosahedral quasicrystals. In Fig. 15(b), we replotted the data to the normalized upper yield stress vs. the normalized temperature relations [ 143], where the upper yield stress is normalized with respect to Young's modulus E and the temperature with respect to the material parameters of Young's modulus times cube of the average atomic diameter fi divided by the Boltzmann constant, i.e. E~3/ks. As seen in Fig. 15(b), a homologous relation holds approximately.
Elasticity, dislocations and their motion in quasicrystals
§4.4
409
B
A1-Cu-Fe A1-Pd-Mn 4Mg-Zn-Y
Cd-Yb
3-
2-
A1-Li-Cu
1-
0
0
I
I
1
2
I
I
3 4 T/(Efi3/kB) ( 10 -3) (b)
-
•
I
5
Fig. 15. (Continued).
For
AHs -- Hs/2, the climb velocity is written
Vc -
-k-~8T4rCLa3°*VD I H* + (3/4)H~I ~-n~;~i) exp Jp kgT
as (68)
Assuming Hs ~ 3 eV, a - b l l - 0.3 nm, L - 10 ~am and VD -- 1014 s -1, the preexponential factor at T ~ 1000 K of eq. (68) is estimated to be 7 x 107 ms -1 at a low stress of o-* = 10 MPa ( ~ 1 0 - 4 E ) . Assuming the mobile dislocation density at the upper yield point is of the order of 109 cm -2 [151], the dislocation velocity at the yielding stage is 3 x 10 -8 ms-1 for the usual strain rate of 10 -4 s -1 . At the yielding stage at high temperature and low stress ( ~ 1 0 - 4 E ) , the value of exponent of eq. (68) is estimated to be 35. The activation enthalpy of dislocation motion has been evaluated experimentally from the temperature dependence of the yield stress and the stress dependence of the activation
K. Edagawa and S. Takeuchi
410
Ch. 76
0
w
50
Mo 0 J
40
d=7.5 ~x
>
30 Ta Oo Cr 0
Pt 201--
Feo oPd O
O
v
Nb
I0 Opb
0 o'
0
I
I
I
I
I
I
I
1
2
3
4
5
6
7
Hd (eV) Fig. 16. The relation between E a 3 and the activation energy of self-diffusion E d for cubic metals.
volume obtained by the stress relaxation experiments. The obtained values of the exponent are largely scattered: 40 [152], 60 and 43 [146], 45 [153], 65 [154], 20 [155] and 35 [148]. However, taking account of a variety of uncertainty in the experimental evaluation due to effects of work softening, recovery during the relaxation test and temperature dependent dragging stress etc., the above estimated value of 35 does not seem to be inconsistent with the experimental results. From the homologous plot in Fig. 15(b), the deformation temperature at low stress satisfies T ~ 4.2 x 10 -3 E~ 3/ k~. This indicates
Hjp + (3/4)Hs 4.2 x 10 -3 E~ 3
= 35.
(69)
Elasticity, dislocations and their motion in quasicrystals
411
H~p ~ 2Ej (Ej: the jog energy) at low stress,
Since
3
2Ej + -~H~ ~ 0.15Eft 3
(70)
The energy of a jog with a height h is written as [142]
Gb2h Eb2h Ej = 4rr(1 - v) = 87r(1 - v2)"
(71)
Approximating b ,~ a and h ~ ~, and inserting v ~ 0.25 [62,64,65,68], one obtains
Ej -- 0.04Eft 3.
(72)
In Fig. 16, we plot Ea 3 values against Hs for 16 cubic metals, showing a correlation Ea3/Hs ~ 7.5 except two transition metals V and Nb. Since the same vacancy mechanism has been suggested for quasicrystals as for usual metals by diffusion experiments for specific atomic species in quasicrystals [156,157], we assume here that the same correlation holds also for icosahedral quasicrystals, i.e. Eg~3/Hs = 7.5. Then,
3H s -- 0.10Eh 3
(73)
m
4
From eqs (72) and (73), 3
2Ej + -~Hs -- 0.08Eft 3 + 0.10Eft 3 -- 0.18E~ 3
(74)
Taking various simplified assumptions made in deriving eq. (74), the agreement between the experimental relation of eq. (70) and the estimated relation of eq. (74) seems reasonable. The above results indicate that about a half of the activation enthalpy of the dislocation climb is due to the kink-pair energy and the other half to the activation enthalpy of self-diffusion.
Appendix A:
Irreducible strain components
The irreducible strain components in the icosahedral system are given below [42,58].
+
1
1 (/7-1
1 u5 =
+
u 11 -- r u 2 2 -3r- u33) ~/-2u 12 V/2u23 x//2u 31
m
/
412
Ch. 76
K. Edagawa and S. Takeuchi
~/~(1011 -- 10/222) _ w4
1 ~
T-11021 -~- "61012 r_11032 _+_"61023
,
w5--
1
~/~(.c1021 _ .c_11012) ~/r2( "61032 _ r-11023)
~
.
(A.1)
The irreducible strain components in the decagonal system are given below [47].
Ul = Ull nt- U22, w6--
u 1-u33,
I I/)l 1 nt- 10221 1021 - 10/212
u5--
iu31] u23
W7--[1013 1'w23
,
w8--
u6--
lUllu22] 2u12
,
I t/)l 1 - 1022 ] W21 -Jr-1012
(A.2)
Appendix B: Derivation of eqs (47) and (48) by the generalized Eshelby's method U -- U
llJ --
By use of Cijkl in eq. (15) and Cijkl
aC~ (p) =
tO
in Table 3, we obtain
(1. 4- 2#) 4- #p2 (1. 4- #)p
(1. 4- #)p # 4- (I. 4- 2#)p 2
0 0
0 0
0 0
0 0
0
0
0
0
# 4- #p2
0
0
A 4- Bp 2
0 0
2Dp 0
0 2Dp C 4- Ap 2 0
0 0 0 B 4-Cp 2
0 0
0 0
"
(B.1) The determinant of a c~t~(p) reduces to the product of four subdeterminants: ]a °e'8(p)]- D1. D2. D 3 - D 4 , Ol --
()~ + 2#) + #p2 0~ + ~)p kt + ()~ + 2 # ) p 2 (~ +U)P
03 =
A + Bp 2 2Dp
D2 = # + / z p 2,
2Dp C + Ap 2
(B.2)
D4 = B + Cp 2.
Of them, D2 and D4 are relevant to the Burgers vectors bll = [0, 0, bll] and b± = [0, 0, b±], respectively. The roots of D2- D4 - - 0 are
p 1 - i,
p2 -
,
(B.3)
Elasticity, dislocations and their motion in quasicrystals
413
and p7 - P~ and p8 -- P~. For Pl and p2, eq. (41) gives, respectively, 0 0 1
A(1)-
0 and
A (2) --
o
(B.4)
i "
0
Eqs (42) and (43) become, respectively, Re[D(1)] -- bll,
Re[D(2)] -- b±
(B.5)
and Re[#iD(1)] - 0 ,
Re[~/BCiD(2)]--0.
(B.6)
From eqs (B.5) and (B.7), we obtain simply D(1) - bll,
D(2) -- b±.
(B.7)
Inserting eqs (B.3), (B.4) and (B.7) into eq. (44) and into eq. (46) lead to eq. (47) and eq. (48), respectively.
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K. Edagawa and S. Takeuchi
Z. Zhang, Y.E Yan, H. Zhang and R. Wang, Mater. Sci. Forum 150/151 (1994) 335. R De and RA. Pelcovits, Phys. Rev. B35 (1987) 8609. D. Ding, R. Wang, W. Yang and C. Hu, J. Phys.: Condens. Matter 7 (1995) 5423. Y. Qin, R. Wang, D. Ding and J. Lei, J. Phys.: Condens. Matter 9 (1997) 859. X.E Li, X.Y. Duan, T.Y. Fan and Y.E Sun, J. Phys. Condens. Matter 11 (1999) 703. D.H. Ding, Y.L. Qin, R. Wang, C.Z. Hu and W.G. Yang, Acta Phys. Sin., Overseas Ed. 4 (1995) 816. W. Yang, J. Lei, D. Ding, R. Wang and C. Hu, Phys. Lett. A200 (1995) 177. W. Yang, M. Feuerbacher, N. Tamura, D. Ding, R. Wang and K. Urban, Phil. Mag. A77 (1998) 1481. C. Hu, R. Wang and D. Ding, Rep. Prog. Phys. 63 (2000) 1. J.D. Eshelby, W.T. Read and W. Shockley, Acta Metall. 1 (1953) 251. A.J.E. Foreman, Acta Metall. 3 (1955) 322. J.E. Shield and M.J. Kramer, Phil. Mag. Lett. 69 (1994) 115. J.E. Shield and M.J. Kramer, J Mater. Res. 12 (1997) 300. U. Koester, X.L. Ma, J. Greiser and H. Liebertz, in: Proceedings of the 6th International Conference on Quasicrystals, eds S. Takeuchi and T. Fujiwara (World Scientific, Singapore, 1998) p. 505. U. Messerschmidt, B. Geyer, M. Bartsch, M. Feuerbacher and K. Urban, in: Proceedings of the 6th International Conference on Quasicrystals, eds S. Takeuchi and T. Fujiwara (World Scientific, Singapore, 1998) p. 509. D. Caillard, E Mompiou, L. Bresson, D. Gratias, Scripta Mater. 49 (2003) 11. M. Texier, A. Proult, J. Bonneville, J. Rabier, N. Baluc and R Cordier, Philos. Mag. Lett 82 (2002) 659. M. Texier, A. Proult, J. Bonneville, J. Rabier, N. Baluc and R Cordier, Scripta Mater. 49 (2003) 47. T. Saito, K. Miyaki, Y. Kamimura, K. Edagawa and S. Takeuchi, Mater. Trans. 46 (2005) 369. S.E. Burkov, Phys. Rev. Lett. 67 (1991) 614. S. Takeuchi, K. Shinoda and K. Edagawa, Philos. Mag. A 79 (1999) 317. R.E. Peierls, Proc. Phys. Soc. 52 (1940) 23. F.R.N. Nabarro, Proc. Phys. Soc. 59 (1947) 256. T. Suzuki and S. Takeuchi, Rev. Phys. Appl. 23 (1988) 405. J.N. Wang, Mater. Sci. Eng. A 206 (1996) 259. R. Tamura, S. Takeuchi and K. Edagawa, in: Quasicrystals - Preparation, Properties and Applications (MRS Symp. Proc., Vol. 643), eds E. Belin-Ferfe, RA. Thiel, A.-R Tsai and K. Urban (Mater. Res. Soc., Warrendale, 2001) K6.4. S. Takeuchi, K. Shinoda and K. Edagawa, Philos. Mag. A 79 (1999) 317. R. Mikulla, R Gumbsch and H.-R. Trebin, Philos. Mag. Lett. 78 (1998) 369. D. Schaaf, J. Roth, H.-R. Trebin and R. Mikulla, Philos. Mag. A 80 (2000) 1657. R. Tamura, S. Takeuchi and K. Edagawa, Mater. Sci. Eng. A 309-310 (2001) 552. S. Takeuchi, Mater. Sci. and Eng. A 400-401 (2005) 306. T. Suzuki, S. Takeuchi and H. Yoshinaga, Dislocation Dynamics and Plasticity (Springer, Berlin, 1991). V. Celli, M. Kabler, T. Ninomiya and R. Thomson, Phys. Rev. 131 (1963) 58. J.E. Dorn and S. Rajnak, Trans. Metall. Soc. AIME 230 (1965) 1052. J.R Hirth and J. Lothe, Theory of Dislocations (Wiley-Interscience, New York, 1982). S. Takeuchi, Philos. Mag. 86 (2006) 1007. T. Suzuki, Y. Kamimura and H.O.K. Kirchner, Philos. Mag. A 79 (1999) 1629. H.O.K. Kirchner and T. Suzuki, Acta Mater. 46 (1998) 305. D. Brunner, D. Plachke and H.D. Carstanjen, Mater. Sci. Eng. A 234-236 (1997) 310. E. Giacometti, N. Baluc, J. Bonneville and J.-E Jeanneret, in: Proceedings of the 6th International Conference on Quasicrystals, eds S. Takeuchi and T. Fujiwara (World Scientific, Singapore, 1998) p. 525. Y. Imai, T. Shibuya, R. Tamura and S. Takeuchi, J. Non-Cryst. Solids 334-335 (2004) 444. S. Takeuchi, K. Shinoda, Y. Yoshida and K. Kakegawa, in: Proceedings of the 6th International Conference on Quasicrystals, eds S. Takeuchi and T. Fujiwara (World Scientific, Singapore, 1998) p. 541. E Semadeni, N. Baluc and J. Bonneville, in: Proceedings of the 6th International Conference on Quasicrystals, eds S. Takeuchi and T. Fujiwara (World Scientific, Singapore, 1998) p. 513. R Schall, M. Feuerbacher, M. Bartsch, U. Messerschmidt and K. Urban, Mater. Sci. and Eng. A 294-296 (2000) 765.
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Elasticity, Dislocations and their Motion in Quasicrystals K. EDAGAWA Institute of Industrial Science, The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8505, Japan and
S. TAKEUCHI Department of Materials Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan
© 2007 Elsevier B.V. All rights reserved
Dislocations in Solids Edited by F. R. N. Nabarro and J. P. Hirth
Contents 1. Introduction 367 2. Elastic properties of quasicrystals 368 2.1. Phonon and phason degrees of freedom 368 2.2. Generalized elastic theory 376 2.3. Phonon elasticity 381 2.4. Phason elasticity 383 2.5. Phonon-phason coupling 387 3. Dislocations in quasicrystals 390 3.1. Perfect dislocations in quasicrystalline lattice 390 3.2. Displacement field and self free energy of dislocations 393 4. Dislocation motion in quasicrystals 398 4.1. Dislocation mechanism of deformation 398 4.2. Dislocation glide process 399 4.3. Dislocation climb process 405 4.4. Plastic homology 407 Appendix A: Irreducible strain components 411 Appendix B: Derivation of eqs (47) and (48) by the generalized Eshelby's method References 413
412
1. Introduction Quasicrystals have a peculiar type of ordered structure characterized by crystallographically disallowed rotational symmetry and by quasiperiodic translational order [ 1-3]. Since the discovery of the quasicrystal by Shechtman et al. in 1984 [4], physical properties of quasicrystals have attracted much attention by solid-state physicists and materials scientists. The mechanical property is one of them. Due to its peculiar structural order, mechanical properties, including elastic and plastic properties, of quasicrystals exhibit characteristic features different from those of crystalline matter. Originating in the quasiperiodic translational order, quasicrystals have a special type of elastic degrees of freedom, termed as phason degrees of freedom [3,5]. Quasicrystals are accompanied by the phason elastic field in addition to the phonon (conventional) elastic field. The generalized elasticity of quasicrystals is described in terms of the two types of elastic fields [6-8]. Within linear elasticity, the elastic free energy of quasicrystals consists of three types of quadratic terms: phonon-phonon, phason-phason and phonon-phason coupling terms. Correspondingly, elasticity of quasicrystals comprises the three parts: pure phonon elasticity, pure phason elasticity and phonon-phason coupling. In relation to the generalized elasticity of quasicrystals, the concept of dislocations in quasicrystals was theoretically established [5,6,9] in an early stage of the research on quasicrystals. Experimentally, since the first demonstration by Shibuya et al. [10] in 1990 of the plastic deformation of an A1-Cu-Ru icosahedral quasicrystal at high temperatures, it has been shown that quasicrystals, both icosahedral and decagonal quasicrystals, are generally deformable at high temperatures above 0.8Tin (Tm: melting temperature), as reviewed in [11-13]. Wollgarten et al. [14,15] have shown by transmission electron microscopy (TEM) that high temperature deformation of icosahedral A1-Pd-Mn is brought by a dislocation process. Until recent years, it has been generally believed that the plasticity of quasicrystals is carried by a glide process of dislocations, and various models of the deformation mechanism have been proposed based on the dislocation glide [16-20]. However, recently, Caillard et al. [21-23] have shown by TEM that the high temperature deformation of icosahedral A1-Pd-Mn is brought not by a glide process but by a pure climb process. In this article, we review the present understandings of characteristic features of elasticity, dislocations and their motion in quasicrystals. This article is organized as follows. In Section 2, we review elastic properties of quasicrystals. Here, we show in Section 2.1 that quasicrystals have the phason elastic degrees of freedom in addition to the phonon (conventional) elastic degrees of freedom. In Section 2.2, we review the generalized elastic theory constructed by incorporating the two types of elastic degrees of freedom. Here, explicit formulae are presented for the two most important classes of quasicrystals, that is, icosahedral and decagonal quasicrystals. Sections 2.3, 2.4 and 2.5 are devoted to the review of experimental and theoretical works related, respectively, to the three parts of elasticity of quasicrystals, that is, pure phonon elasticity, pure phason elasticity and phonon-phason
K. Edagawa and S. Takeuchi
368
Ch. 76
coupling. In Section 3, we present characteristic features of dislocations in quasicrystals. In Section 3.1, we show the definition of a perfect dislocation in quasicrystals and review which types of dislocations have been experimentally observed. In Section 3.2, we demonstrate a calculation of displacement field and self free energy of a dislocation. Using a general formula of the self free energy, relative stability among dislocations with various Burgers vectors is discussed. Section 4 is devoted to dislocation motion. After reviewing briefly the experimental results about dislocation mechanism of deformation of quasicrystals in Section 4.1, we discuss detailed mechanisms of dislocation glide and dislocation climb in Section 4.2 and 4.3, respectively. Finally, in Section 4.4, we show a homologous nature in the temperature dependence of the yield stress for icosahedral quasicrystals and discuss it in terms of microscopic deformation mechanism common to icosahedral quasicrystals.
2. E l a s t i c p r o p e r t i e s o f q u a s i c r y s t a l s
2.1. Phonon and phason degrees of freedom In general, the diffraction intensity function I (q) of solid is given by I (q) _----IS(q)l2,
s(q) - f
p ( r ) e x p ( - 2 : r i q - r)dr.
(1)
Here, p (r) is the atomic-density function in real space. The function I (q) observed experimentally for the quasicrystal has the following characteristics [1-3]: (1) It consists of 6-functions. (2) The number of basis vectors necessary for indexing the positions of the 6-functions exceeds the number of dimensions. (3) It shows a rotational symmetry forbidden in conventional crystallography. Conversely, we define the quasicrystal as the material with a p(r) which gives a diffraction intensity function I (q) satisfying these conditions. The condition (1) implies that this material has a kind of long-range translational order. The conditions (2) and (3) indicate that the order is not periodic. The translational order defined by the conditions (1) and (2) is called quasiperiodicity. According to the definition, every d-dimensional quasiperiodic atomic-density function p (r) can be expressed as p(r) -- ~ / g m l miEI
.....
mN exp[27riGm1 .....mN " r],
(2)
§2.1
Elasticity, dislocations and their motion in quasicrystals
w h e r e tom1 . . . . .
mN
369
is the Fourier component associated with the reciprocal vector N
G m l ..... mN --- Y ~ m n q n .
n--1
Here, {qn } (n = l, 2 . . . . . N) (N > d) are the reciprocal basis vectors. Now, let us define an N-dimensional function ph (x 1. . . . . xu) as
mNeXPI2 iZmnxn ] N
. . . . .
.....
miEl
(3)
n=l
This function is periodic in every eqs (2) and (3), we find
Xn
p ( r ) = p h ( x l . . . . . XN),
(n -- 1, 2 . . . . . N) with the period of unity. Comparing
(4)
where
xn=qn'r
(n=l,2
. . . . . N).
(5)
This indicates that the d-dimensional quasiperiodic function p(r) can be described as a d-dimensional section of an N-dimensional periodic function. As an example, Fig. l(a) presents a typical one-dimensional quasiperiodic structure known as a Fibonacci lattice, which is described as a one-dimensional section of a twodimensional periodic function (d = 1 and N --2). Here, Ell and E± denote the physical space and the complementary space perpendicular to it, respectively. In this case, the two-dimensional periodic structure consists of a periodic arrangement of a line segment extending in the direction of E±. The line segment is called atomic surface. A point sequence is obtained on the Ell section, comprising an arrangement of two spacings L and S. Here, the slope of Ell with respect to the two-dimensional lattice is an irrational number r (the golden mean: (1 + V/5)/2). In this case, the resultant sequence of the two spacings becomes equivalent to the Fibonacci sequence and therefore it is called a Fibonacci lattice. The irrational slope indicates lack of periodicity in the arrangement of L and S. In the description of the Fibonacci lattice in Fig. 1(a), let us consider a translation of the two-dimensional periodic structure by a vector U with respect to the origin of the physical space Ell (Fig. 1(b)). The two structures on Ell before and after the displacement U shown in Figs l(a) and (b) can be overlapped out to arbitrarily large finite distances by a finite translation in Ell. Two structures satisfying this condition are said to belong to the same local isomorphism class (LI class) [1-3]. Thus, the displacement U represents the degrees of freedom of generating a series of structures belonging to the same LI class. Obviously from the definition of the LI class, a series of structures in the same LI class
K. Edagawa and S. Takeuchi
370
E±
(a)
(b)
T
(
±
Ch. 76
El
-
T
-
T
v
(d)
Fig. 1. A Fibonacci lattice (a), and a structure resulting from a displacement of U (b), that of u (c) and that of w (d).
are geometrically indistinguishable on any finite scale and thus they are also physically indistinguishable: they give the same diffraction intensity function I (q) and have the same energy. The vector U can be decomposed into u in Ell and w in E±: U = u + w.
(6)
Here, u represents the degrees of freedom of d-dimensional translation in physical space, which conventional crystals also possess, and w represents (N - d) degrees of freedom characteristic of quasiperiodic system. As shown in Figs l(c) and (d), while u results in translation of Fibonacci lattice in Ell, w generates a rearrangement of L and S. The two kinds of degrees of freedom are called phonon and phason degrees of freedom, and u and w phonon and phason displacements, respectively [3,5]. When these displacements vary spatially, the gradients of them yield distortion or strain. More specifically, while the gradient of u, i.e.
Vu(r)-
Oui Orj
gives conventional distortion (or strain), that of w, i.e. Ot~ i
Vw(r)- Orj
§2.1
Elasticity, dislocations and their motion in quasicrystals
371
E.t.
(a)
~-EII
(b)
~
v
-
~
(c) J
J
I !
Fig. 2. A Fibonacci lattice (a), and its phonon-strained (b) and phason-strained (c) structures.
yields distortion (strain) called phason distortion (strain) .1 Here, u and w are the functions of only r c Ell and ~-/ denotes a spatial derivative in Ell Figs 2(b) and (c) illustrate, respectively, a phonon-strained and a phason-strained structure of the Fibonacci lattice in Fig. 2(a). In Fig. 2(b), uniform phonon strain is introduced by a compression deformation of the two-dimensional structure. On the other hand, uniform phason strain is introduced by a shear deformation of the two-dimensional structure in Fig. 2(c). It is noteworthy that the phonon and phason stain fields should have quite different dynamical properties. As described above, phason displacement results in a local rearrangement of points (atoms) such as LS ~-+ SL, which is called phason flip. Examples of phason l In general, the gradient of displacement vector is called 'distortion.' In the one-dimensional case, the distorOw also represent strains in the same form. In the case of two-dimensional and three-dimensional tions ~Ou and -gT Oui quasicrystals, the phonon strain should be defined as a symmetrical form 1 ( ~ + ~Ouj ) to remove the component .
/
Owi as the distortion (see Section 2.2). of rigid rotation while the phason strain should have the same form ~ ~,,j
K. Edagawa and S. Takeuchi
372
Ch. 76 A
Fig. 3. Examples of phason flip in a two-dimensional Penrose lattice.
flip in a two-dimensional Penrose lattice, which is known as a typical two-dimensional decagonal quasicrystal, are shown in Fig. 3. In this case, a lattice point in the hexagon makes a transition between two (meta-) stable positions. Introduction or relaxation of phason strain requires a combination of phason flips. Generally, the phason flip is a thermally activated process and thus relaxation of phason strain whose elementary process is phason flip must proceed relatively slowly like atomic diffusion in solids [3,5,8]. This is in sharp contrast to conventional phonon strain, which can be relaxed instantaneously via displacive phonon modes. Experimentally, thermally-induced phason flips have been investigated by neutron scattering [24-27], by Moessbauer spectroscopy [28], by NMR [29, 30] and specific heat measurements [31,32]. By in-situ high-temperature high-resolution transmission electron microscopy, direct observations of thermally-induced phason flips [33-35] and those of the phason-strain relaxation process [36] have been performed. The activation enthalpy of a phason flip or collective phason flips has been estimated to be ~1 eV in an A1-Pd-Mn icosahedral quasicrystal by radiotracer diffusion experiments [37]. In general, to decompose properly the N total degrees of freedom into d phonon and (N - d) phason degrees of freedom, or equivalently, to embed properly a given ddimensional quasiperiodic structure into an N-dimensional hypercrystal, the information on the point group symmetry of the system is needed. Below, the method of embedding is briefly reviewed for the two most important classes of quasicrystals: icosahedral and decagonal quasicrystals [3,38-41 ]. To describe the structure of a three-dimensional icosahedral quasicrystal, we use a sixdimensional hypercubic lattice spanned by di (i -- 1. . . . . 6): di ~- M i j e j ,
r M
a ,22~r , + 2 ' ~- 1 -
1 0 0
1 -1 0 r r
0 0 r 1 -1
1 1 -r 0 0
-r r 0 1 1
0 0 1 -r r
1
0
-r
-r
0
-1
'
(7)
§2.1
Elasticity, dislocations and their motion in quasicrystals
373
where r is the golden mean ( = (1 + ~/-5)/2) and a is the lattice constant, ei (i - 1 . . . . . 6) are the orthonormal unit vectors of the six-dimensional space. The icosahedral point group Y is generated by a fivefold rotation C5 and a threefold rotation C3. The lattice spanned by d i (i = 1 . . . . . 6) is invariant under these operations. The action of these operations o n d i (i = 1 . . . . . 6) is given as
F(C5)--
F(C3)--
1 0
0 0
0 1
0 0
0 0
0 0
0 0 0 0
0 0 0 1
1 0 0 0
0 0 0 0
0 1 0 0
0 0 1 0
0 1
0 0
0 o
0 0
0 0
0 0
0
0
o
-~
0 1
0 0
'
0 0
1 0
0
1
0
o
o
o
-1 0
0 0
0 0
•
(8)
The character table for the icosahedral group Y is presented in Table 1. Here, the irreducible representations of the group Y are a one-dimensional F l, two distinct three-dimensional 1F2 and F 3, a four-dimensional F 4, and a five-dimensional F 5. The six-dimensional representation in eq. (8) is reducible and can be decomposed into the sum of the irreducible representations as F = F 2 + F 3. The subspace spanned by el, e2 and e3, and the subspace spanned by e4, e5 and e6 are the eigenspaces of the representations [,2 and F 3, and correspond to the physical space Ell and its complementary space E_k, respectively. Shown in Fig. 4(a) are the projections dl I (i - 1 . . . . . 6) and d/± (i - 1 . . . . . 6) of the basis vectors di (i = 1 . . . . . 6) onto Ell and E_k, respectively. Because the sixdimensional hypercubic lattice is invariant under the inversion Ci, it has the maximum icosahedral point group symmetry Yh -- Y x Ci. By placing atomic surfaces spreading in E± (in this case they are three-dimensional objects) at appropriate positions in the six-dimensional lattice, a three-dimensional icosahedral quasicrystalline structure with the symmetry Y or Yh is obtained as a section on Ell. Here, to preserve the symmetry, each of the atomic surfaces must satisfy the site symmetry of its position in the corresponding symmetry group. Similarly to the case illustrated in Fig. 1, a six-dimensional Table 1 Character table for the icosahedral group Y Y
E
12C 5
12C 2
20C3
15C2
r 1
1
1
1
1
r 2
3
r
-r -]
0
-1 -1
F3
3
-r -j
r
0
r4
4
-1
-1
1
F5
5
0
0
-1
1
0 1
374
K. Edagawa and S. Takeuchi
E,,
Ch. 76
E,
e3
e6 ,3
,)' ',
l--e~
\
"/
Is \
'
,1
e2
e;
Z es
V
(a)
E, ,I
E, I
3 /
5 ~e~ 3
""'"'
1 ~e3
e5
0 0
4
"
,
(b) (. i = l .... 6) of the Fig. 4. (a) The projections d iII ( i = l . ... 6) andd/L . . basis. vectorsdi ( i = l ... 6) in eq. (7) for icosahedral quasicrystals onto Ell and EL, respectively; (b) the projections d iII ( i = l , ... , 5) andd{ (i = 1..... 5) of the basis vectors di (i = 1..... 5) in eq. (9) for decagonal quasicrystals onto El, E~I and EL, respectively. Here, di (i = 1..... 4) have no components of E~I while d 5 has the component of E~I only.
displacement U can be defined, which can be decomposed into u ~ Ell and w e E ± , representing the three-dimensional phonon and the three-dimensional phason displacements. It is noteworthy that the hypercubic lattice is not a unique choice for embedding icosahedral quasicrystalline structure. In fact, we can use any lattice spanned by the basis vectors of the form d I - d l . l + cd/x (c" an arbitrary scale factor)instead of the hypercubic lattice spanned by di - d l I -+-d/x in embedding any given structure. In other words, we can determine arbitrarily the length scale in E±. This fact should always be born in mind in discussing the magnitudes of the physical-property parameters related to the length scale in E x such as phason displacement, phason strain, phason elastic constants, etc. (see Sections 2.4, 3.1 and 3.2). To describe a three-dimensional decagonal quasicrystal, we use a five-dimensional lattice spanned by di (i = 1 . . . . . 5): di - M i j e j ,
§2.1
Elasticity, dislocations and their motion in quasicrystals
a ~(Cl-
1)
a ~Sl
~(C2-
1)
a ~$2
~
a (C2 -M
a
m
a a
a
a 1)
~(C4-
1)
0-
a (C4 -- 1)
a
~$4
0
a
a ~s4
a ~(c3-
0
0
0
a ~$2
375
1)
(9)
a ~s3
0
0
c-
where a and c are the lattice constants, and ci = cos(2rri/5) and Si = sin(2rri/5), ei (i = 1 . . . . . 5) are the orthonormal unit vectors of the five-dimensional space. The character table for the decagonal group Cloy is shown in Table 2. The generators of this group are a tenfold rotation C10 and a mirror av. The lattice spanned by di (i -- 1 . . . . . 5) is invariant under these operations. The action of these operations on di (i = 1 . . . . . 5) is given by
F(C10)--
0 0 0 -1 0
1 1 1 1 0
-1 0 0 0 0
0 -1 0 0 0
0 0 0 0 1
,
F(av)--
0 0 0 1 0
0 0 1 0 0
0 1 0 0 0
1 0 0 0 0
0 0 0 0 1
.
(10)
This five-dimensional representation is reducible: F -- F 5 + F 7 -t- 1-'1 . The subspace Ell1 spanned by el and e2, the subspace E± spanned by e3 and e4, and the subspace E~ along e5 are the eigenspaces of the representations F 5, 1-`7 and F l, respectively. Eli is parallel to the quasiperiodic plane and E~ is along the tenfold periodic directions in the physical space Ell -- E] + E~. In Fig. 4(b), the projections dl I (i - 1 . . . . . 5) and d/± (i - 1 . . . . . 5) of the basis vectors di (i = 1 . . . . . 5) onto each space are illustrated. Because the lattice spanned by d i (i - - 1 . . . . . 5) in eq. (9) is invariant under the inversion Ci, it has the maximum decagonal point group symmetry Dloh = Cloy × Ci. There are seven point groups in the decagonal system: C 1 0 , C5h, ClOh, Cloy, D 1 0 , D5h and DlOh. By placing atomic surfaces spreading in E± at appropriate positions in the five-dimensional lattice, a threeTable 2 Character table for the decagonal group ClOy C10v
E
2C10
2C5
2C~0
2C2
C2
F1
1
1
1
1
1
1
F2
1
1
1
1
1
1
5av
5a d
1 -1
1 -1
F3
1
-1
1
-1
1
-1
F4
1
-1
1
-1
1
-1
F5
2
r
r - 1
1- r
-r
-2
0
0
F6 F7
2 2
r -- 1 1- r
--r -r
--r r
r- 1 r - 1
2 -2
0 0
0 0
F8
2
-r
r-
2
0
0
1
r-
1
-r
1 -1
-1 1
K. Edagawaand S. Takeuchi
376
Ch. 76
dimensional decagonal quasicrystalline structure with any of those decagonal point group symmetries can be constructed. Here, to preserve the symmetry, each atomic surface must satisfy the site symmetry of its position in the given point group. A five-dimensional displacement U can be defined, which can be decomposed into u 6 Eli and w 6 E_L, representing the three-dimensional phonon and the two-dimensional phason displacements. As in the case of icosahedral quasicrystal, we can use any five-dimensional lattice spanned by the basis vectors of the form d I - dl.I + cd~ (c" an arbitrary scale factor)in embedding any given structure.
2.2. Generalized elastic theory As in the preceding section, in quasicrystals there exist two types of elastic degrees of freedom, that is, phonon and phason degrees of freedom. In view of this fact, a generalized elastic theory can be formulated for the quasicrystals [6-9,42-45]. In this section, we review the theory and present the explicit formulae for the icosahedral and decagonal quasicrystals. The spatial variation of u, i.e. Vu(r) - Oui yields phonon (conventional) distortion while that of w, i.e. Vw(r) - Owi yields phason distortion. Here, u and w are the functions of only r 6 Eli and ~-~Tdenotes a spatial derivative in Eli. The phonon strain should be defined as a symmetrical form
l ( Oui Ouj ) Uij~-~ ~rj +-~ri to remove the component of the rigid rotation which does not change the elastic free energy. On the other hand, the phason strain should be defined as w i j z Owi Orj . The elastic free energy density f is given as a function of the phonon strain U ij and the phason strain Wij. The function f can be expanded into the Taylor series in the vicinity o f uij -- 0 and Wij = O. In the regime with ]Uijl, IllOij[ ~ 1, we would omit the third and higher order terms, leaving only the quadratic terms (linear elasticity):
f =fu-u+fw-w+fu-w 1 ~_u 1 w-w U--//3 - -~Cij~l uijukl + -~Cij~l tOij tOkl -[- Cijkl uij tOkl,
(11)
where fu-u, fw-w and fu-w are a pure phonon, pure phason and phonon-phason coupling U--U W--W U--W terms, respectively, and Cij~l, Cijkl and Cijkt are the corresponding second-order elastic constant tensors. The explicit form of f depends on the symmetry of the system and we derive those for icosahedral [6,9,42-45] and decagonal [46,47] quasicrystals below. For the icosahedral quasicrystal with the point group Y, the six components uij transform under (F 2 × I-'2)sym. - - F 1 ~ F 5, corresponding to dilatation and shear, respectively (see Table 1). The nine components wij transform under F 2 × F 3 - F 4 + F 5. Now let Ul and u5 be the irreducible components of the phonon strain for 1-'1 and F 5, respectively,
§2.2
Elasticity, dislocations and their motion in quasicrystals
377
and W4 and w5 be those of the phason strain for [,4 and F 5, respectively. These irreducible strain components are given explicitly in Appendix A. Then, we find that there exist five quadratic combinations of strains that transform as scalars, i.e. under F 1" Ul • Ul, u5 .us, w4 • W4, W5 • W5 and u5 • ws. This fact indicates that the elastic free energy density f can be rewritten as
f-f,-u+fw-w+fu-w, 1 1 fu-u - -~klUl "Ul -Jr- ~k2u5 - u s , 1 fw-w
1
fu-w-ksus'w5.
-- ~k3w4 • w4 + ~k4w5 • w5,
(12)
Here, ki (i = 1 . . . . . 5) are scalar constants representing the elastic constants. In many works done so far on the elasticity of icosahedral quasicrystals, a different set of elastic constants have been used: )~ and # (Lame constants) for pure phonon elasticity; K1 and K2 for pure phason elasticity; and Ks for phonon-phason coupling [48-57]. They are related to ki (i -- 1 . . . . . 5) as [44,58] 1 )v -- x-(kl -- k2), 3 1
1 /z -- 7k2, it
K1 - 7(4k3 -+- 5k4),
1
K2 -- 7(k3 - k4), 3
K3 --
1
--~ks. ,/6
(13)
Inserting eq. (13) into eq. (12) leads to
f =f~-u+fw-w+fu-~, 1 f~_. - -~x.~ + l z t t i j u i j ,
1
1
I
4
f~-~ - ~K,~u~u + ~tc2. ~
- ~u~u
+ [(~12 + ~-'~2,)2
+ cyclic permutations] }, fu-w
-- K3{ [tt)ll (Ul,-+- r - l u 2 2
-
ru33)-+- 2 u 2 3 ( r -1
+ cyclic permutations }.
~23
-
~32)]
(14)
It should be noted that the equations presented in [34,35] can be obtained from eq. (14) by the substitution of tt)ij by tt3ji and the coordinate transformation (x -+ y, y --+ - x , z --+ z) [44,58]. Comparing eq. (14) with eq. (11), we finally obtain the elastic constant tensors, as fo1U--b/ lows. The pure phonon elastic constant tensor C i j k l is identical to that for conventional isotropic solids:
c,";7; -xaua~, + u,(~s,~j, + ~,,ajk),
(is)
K. Edagawa and S. Takeuchi
378
Ch. 76 tO--tO
w h e r e (~ij is the K r o n e c k e r delta. T h e p u r e p h a s o n elastic c o n s t a n t t e n s o r Cijkl
a n d the
p h o n o n - p h a s o n c o u p l i n g t e n s o r Cij-£] v are s h o w n in Tables 3 a n d 4, r e s p e c t i v e l y , u s i n g a 9 x 9 r e p r e s e n t a t i o n . B e c a u s e b o t h uij a n d toij are c e n t r o s y m m e t r i c a l , i.e. i n v a r i a n t u n d e r the a c t i o n o f the i n v e r s i o n o p e r a t i o n , elastic p r o p e r t i e s s h o u l d p o s s e s s an intrinsic c e n t r o s y m m e t r y . T h i s fact i n d i c a t e s that all the p o i n t g r o u p s b e l o n g i n g to the s a m e L a u e class h a v e e x a c t l y the s a m e elastic p r o p e r t i e s . T h u s , the a b o v e results d e r i v e d for the g r o u p Y also a p p l y to the g r o u p Yh -- Y x Ci.
Table 3 1/3-//3 for icosahedral quasicrystals with the point The phason elastic constant tensor C ijkl group Y or Yh. A - D are defined as" a = K 1 - ~__Z2,B = K 1 + K2(r - 1), C = K1 -t- K2( 2 - r), D = K 2 cWmtO
kl-- 11
22
33
23
31
12
32
13
21
ij=ll
A
D
D
0
0
0
0
0
0
22
D
A
D
0
0
0
0
0
0
33
D
D
A
0
0
0
0
0
0
23
0
0
0
B
0
0
D
0
0
31
0
0
0
0
B
0
0
D
0
12
0
0
0
0
0
B
0
0
D
32
0
0
0
D
0
0
C
0
0
13
0
0
0
0
D
0
0
C
0
21
0
0
0
0
0
D
0
0
C
i.]kl
Table 4 - W for icosahedral quasicrystals The phonon-phason coupling constant tensor c Uijkl with the point group Y or Yh. A - C are defined as: A = K3, B = - r K3, C -- r - 1 K3
cUBW
kl-- 11
22
33
23
31
12
32
13
21
ij - - 11
A
B
C
0
0
0
0
0
0
22
C
A
B
0
0
0
0
0
0
33
B
C
A
0
0
0
0
0
0
23
0
0
0
C
0
0
B
0
0
31
0
0
0
0
C
0
0
B
0
12
0
0
0
0
0
C
0
0
B
32
0
0
0
C
0
0
B
0
0
13
0
0
0
0
C
0
0
B
0
21
0
0
0
0
0
C
0
0
B
ijkl
§2.2
379
Elasticity, d i s l o c a t i o n s a n d t h e i r m o t i o n in q u a s i c r y s t a l s
For three-dimensional decagonal quasicrystals, U ij is a 3 x 3 symmetrical tensor and Cloy, the six components of uij and wij transform under {(F 5 + F 1) x (F 5 + F1)}sym.- 2F 1 + F 5 + 1-'6 and (F 5 + F 1) x I-'7 = 1 - ' 6 + 1-'7 -Jr- 1-'8, respectively (see Table 2). Let ul and u'1 be the two irreducible components of the phonon strain for F 1, and u5 and u6 be those for F 5 and F 6, respectively. Let w6, w7 and w8 be those of the phason strain for F 6, 1-'7 and F 8, respectively. These irreducible strain components are given explicitly in Appendix A. Then, we find the following quadratic combinations of strains that transform as scalars, constituting the elastic free energy density f : tOij is a 2 x 3 tensor, respectively. For the point group
f =fu-u+f~-w+fu-w, 1
fu-, - -~klUl
,
1
,
I
1
1
"Ul + k2Ul "u 1 -t- ~k3u 1 "u 1 "t'- ~k4u5 "u5 -+- ~k5u6 • u6,
1
1
1
fw-w
-
~ k 6 w 6 • w6 -Jr- ~ k 7 w 7 • w7 -Jr- ~ksw8 • ws,
fu-w
--
k9u6 • w6.
(16)
Here, ki (i -- 1 . . . . . 9) are scalar constants representing the elastic constants. Eq. (16) can be written explicitly as:
f =f~-u+fw-w+f~-w, S -u -
1
(.21 + u 2)+
+
nt- 2C44(U23 -Jr- U71 ) J r - ( C l l -
1 f w - w -- -~ KI
+
+
1
2
Cl2)U 212,
(~71 -[- tU22 -I- 1/)72-it- 1/)21)
1 qt- K 2 ( t O l l 1/322- tO21 tVl2)-Jr- ~ K 3 ( ~ 2 3
nt- t023),
f . - w - K 4 { ( U l l - u 2 2 ) ( W l l qt- t022) --{- 2u12(to21 - w 1 2 ) } ,
(17)
where a different set of elastic constants are used, which are: Cll
=
kl + k5,
K l=k6+kg,
C12 = kl - k5, c13 = k2, c33 = k3, C44 --
K2=k6-kg,
k4
4'
(18)
K3=k7, K4=k9.
Comparing eq. (17) with eq. (11), we obtain the three elastic constant tensors that are presented in Tables 5-7. We note that the form of the phonon elastic constant tensor is identical to that for the hexagonal crystals. The point groups Dlo, D5h and Dloh belong to the same Laue class as that of C10v and therefore the above results also apply to all of them. In the decagonal system, there is another Laue class to which the point groups C10, C5h and Cloh belong. Similar analysis reveals that Cij-£~ and C/W.~Wfor the latter Laue class are J
K. Edagawa and S. Takeuchi
380
Ch. 76
Table 5 The phonon elastic constant tensor c U . ~ for decagonal quasicrystals J
with the point group C10, C5h, ClOh, D10, D5h, ClOy or DIOh. There are five independent elastic constants: Cll , c12 , c13 , c33 and Cll --c22 c44. c66 is given by c66 = 2
Ci,lklu
k l = 11
22
33
23
31
12
ij = 11
Cll
c12
c13
0
0
0
22
c12
Cll
c13
0
0
0
33
c13
c13
c33
0
0
0
23
0
0
0
c44
0
0
31
0
0
0
0
c44
0
12
0
0
0
0
0
c66
Table 6 .-,w-w f o r decagonal quasicrystals The phason elastic constant tensor Cijkl with the point group C10, C5h, ClOh, D10, D5h, ClOy or DIOh
CW-W ijkl
kl = 11
ij = 11
K1
K2
0
0
0
0
22
K2
K1
0
0
0
0
23
0
0
K3
0
0
0
12
0
0
0
K1
0
13
0
0
0
0
K3
0
21
0
0
0
-K 2
0
K1
22
23
12
13
21
K2
Table 7 - 1/) for decagonal The phonon-phason coupling constant tensor c Uijkl quasicrystals with the point group D10, D5h, Cloy, or DIOh
Ci~k]v
kl = 11
22
23
12
13
21
ij = 11
K4
K4
0
0
0
0
-K 4
0
0
0
0
22
K4
33
0
0
0
0
0
0
23
0
0
0
0
0
0
31
0
0
0
0
0
0
12
0
0
0
-K 4
0
K4
Cijkl
U--tO
identical to those for the former and t h a t for the latter is slightly different; there are two independent elastic constants K4 and K5 in this tensor, as shown in Table 8.
Elasticity, dislocations and their motion in quasicrystals
§2.3
381
Table 8 The phonon-phason coupling constant tensor cU-tO ijkl for decagonal quasicrystals with the point group C10, C5h or CIO h
cU--tU ijkl
kl -- 11
22
23
12
13
ij = 11
K4
K4
0
K5
0
-K4
0
-K 5
0
K5
22
K5
33
0
0
0
0
0
0
23
0
0
0
0
0
0
31
0
0
0
0
0
0
12
K5
K5
0
K4
0
-
We define the phason O'ij __ m 0__~f OUij.
K4
21
stress
K4
Of in addition to the conventional phonon stress crijw --_ Owi----]'
From eq. (11) these stresses can be written as
lg U-- U It--tO Crij - - C i j k l Ukl -~- C i j k l tOkl,
/13
U--to
O'ij - - C i j k l
tO--//)
Ukl @ C i j k l
tOkl,
(19)
which represents a generalized Hooke's law. Introducing a phason body-force density fi w as well as the conventional phonon body-force density fi u, we obtain a generalized form of the elastic equations of balance: 0 Orj O'iju @ f iu - - O,
0
~ F j O'ijw -Jr- f tw _
0"
(20)
Inserting eq. (19) into (20) leads to u-u 02Uk nt_ C u - w 02tOk + f u -0, Cijkl Orj Orl ijkl Orj Orl 0 2 tt) k u-w 02Uk w-w ~ -F f.w _ O, Cijkl Orj Orl -+- Cijkl Orj Orl
(21)
where we take into consideration the relation C ijkl u-u auk; 02Uk " Orj - - C i uj k- lu OrjOrl
2.3. Phonon elasticity As shown in eq. (11), the elastic free energy density f of quasicrystals has the three quadratic terms, which represent a pure phonon elasticity, a pure phason elasticity and phonon-phason coupling, respectively. As described in Section 2.1, relaxation of phason strain proceeds diffusively with thermally activated phason flips [3,5,8], indicating that d (t) --0. Even at high at low temperatures the phason mode should be frozen, i.e. -d-[Wij
382
Ch. 76
K. Edagawa and S. Takeuchi
Table 9 Phonon elastic constants of various icosahedral quasicrystals: )~ and/z are Lame constants; G =/z, B = 3~.+2/x 3 and v = 20.+lz) are shear modulus, bulk modulus and Poisson's ratio, respectively.The unit of)~,/z and B is GPa Alloy system
)~
#(G)
A1-Li-Cu A1-Li-Cu A1-Cu-Fe A1-Cu-Fe-Ru A1-Pd-Mn A1-Pd-Mn Ti-Zr-Ni Cd-Yb Zn-Mg-Y
30 30.4 59.1 48.4 74.9 74.2 85.5 35.28 33.0
35 40.9 68.1 57.9 72.4 70.4 38.3 25.28 46.5
B 53 57.7 104 87.0 123 121 111 52.13 64.0
v
Ref.
0.23 0.213 0.232 0.228 0.254 0.256 0.345 0.2913 0.208
[60] [62] [65] [65] [65] [66] [67] [64] [68]
temperatures, it can be regarded as being effectively frozen under applied stress oscillating with a much shorter period than the relaxation time of phason strain. If additionally the residual phason strain is zero, i.e. Wij(t) = 0 - - const., the elasticity of quasicrystals consists entirely of the phonon elasticity. As shown in eq. (15), icosahedral quasicrystals should behave as an isotropic elastic body. 2 Actually, isotropic elasticity has been observed experimentally for icosahedral A1-Li-Cu [60-62], A1-Pd-Mn [63] and Cd-Yb [64]. Using various ultrasonic methods, the elastic constants have been measured for icosahedral quasicrystals of A1-Li-Cu [60,62], A1-Cu-Fe [65], A1-Cu-Fe-Ru [65], A1-Pd-Mn [65,66], Ti-Zr-Ni [67], Cd-Yb [64] and Mg-Zn-Y [68]. The evaluated elastic constants at room temperature are summarized in Table 9, where G - # B -- 3~.+2# and v x , 3 2(z+~z) are shear modulus, bulk modulus and Poisson's ratio, respectively. First, we note that the bulk moduli differ considerably for different alloy systems. We find that they correlate well with the melting temperature, which is generally observed for conventional crystals. With the exception of Ti-Zr-Ni, the main feature characteristic of the quasicrystals is that the Poisson's ratio is relatively small, or equivalently, that the ratio G - 3(1-2v) 2(1+v) is relatively large. For example, the Poisson's ratios of the elemental metals of A1, Cu and Pd are about 0.35, 0.37 and 0.39, respectively, at room temperature. Except for Ti-Zr-Ni, the Poisson's ratios of the quasicrystals in Table 9 are much smaller than those values and close to those of typical covalent crystals such as Si (0.22) and Ge (0.21). Tanaka et al. [65] have pointed out that the small v or equivalently the large G / B ratio is indicative of the existence of directional atomic bonds. The bulk modulus B is a measure of the resistance against the deformation in which atomic bond lengths change with keeping bond angles unchanged. In contrast, the shear modulus G is a measure of the resistance against the deformation in which bond angles change without changing bond lengths. Therefore, the G / B ratio should become large if the atomic bonds are directional, as in covalent crystals. In relation to this fact, covalent nature of the atomic bonds has been revealed experimentally in some quasicrystals and their crystal approximants [69-71]. 2In the third-order elastic constants described as a sixth-rank tensor, anisotropy is theoretically expected and a sign of anisotropy has actually been observed in the study of nonlinear propagation of acoustic waves in an A1-Pd-Mn icosahedral quasicrystal [59].
Elasticity, dislocations and their motion in quasicrystals
§2.4
383
Table 10 Phonon elastic constants of an A1-Ni-Co decagonal quasicrystal [72]: Cll, c33, c44, c12 and c13 are the five independent elastic constants of this system; B and G are the bulk and shear moduli calculated as Voigt averages of the moduli cij ; v is the Poisson's ratio calculated from the B and G values. The unit is GPa, except for v Alloy system
Cl 1
c33
c44
c12
c13
B
G
v
A1-Ni-Co
234.33
232.22
70.19
57.41
66.63
120.25
79.78
0.228
As shown in Table 5, there are five independent phonon elastic constants c11, c33, C44, Cl2 and Cl3 in the decagonal quasicrystals and the form of the elastic constant tensor is identical to that of the hexagonal crystal. Isotropic elasticity is expected in the basal plane, which has been confirmed experimentally for an A1-Ni-Co decagonal quasicrystal by Chernikov et al. [72]. The same group has reported the values of a complete set of elastic moduli, which are presented in Table 10. Here, a striking feature to be noted is that the deviation from complete elastic isotropy is rather small. The compressional anisotropy and shear anisotropy, measured by the ratios C 3 3 / C l l and c 4 4 / C 6 6 = 2 c 4 4 / ( C l l -- c12), respectively, are 0.99 and 0.79, indicating a high degree of elastic isotropy. Inelastic neutron scattering measurements have been done for a decagonal A1-Ni-Co [73] and considerably isotropic phonon dispersion relations have been obtained, which is consistent with the result of the elastic constant measurement. So far, many structural models have been proposed for decagonal quasicrystals, as reviewed by Ranganathan et al. [74]. Most of them have a layered structure, for which one would expect anisotropic elastic properties. Actually, phonon dispersion relations have been calculated for one of such a layered structure model, exhibiting quite large anisotropy [75]. Dmitrienko [76] has suggested the possibility of both icosahedral and decagonal quasicrystals having common local atomic ordering with icosahedral symmetry. The highly isotropic elastic properties may be attributable to such an isotropic local atomic arrangement. In Table 10, the bulk modulus B and shear modulus G are calculated as Voigt averages of the moduli cij, and the Poisson's ratio v is obtained from the B and G values. As in icosahedral quasicrystal, Poisson's ratio is small and the ratio G/B is large, indicating the existence of directional atomic bonds also in decagonal quasicrystals.
2.4. Phason elasticity Phason elasticity is closely related to the problem of what is the physical origin of quasicrystalline structural order and has attracted much attention since the discovery of the quasicrystal. Despite years of study, there remain two plausible competing models that explain the realization of quasicrystalline long-range order. One is a perfectly ordered model [1-3,77]. In this model, the existence of local matching rules that force the formation of the quasicrystalline order is assumed. When atomic interactions are in accordance with the local matching rules, the quasicrystal is formed as an energetically favorable structure. Introduction of phason strain induces local atomic rearrangements and thus brings matching rule violations, which increase the elastic free energy by increasing the energetic part of the free energy. The other model is a random tiling [48], in which the quasicrystal is assumed
384
K. Edagawa and S. Takeuchi
Ch. 76
to be stabilized by a configurational entropy originating in many nearly degenerate ways of packing structural units. Such a configurational entropy decreases with increasing average phason strain and thus the phason strain contributes to the elastic free energy f through its entropic part. Theoretically, quasicrystals in both models show g-functional diffraction peaks and projection of the structure along a symmetry axis in both states preserves perfect quasicrystalline order unless the projection thickness is too small [48,78]. These facts make it difficult to distinguish between the two states by a simple diffraction experiment or by high-resolution transmission electron microscopy. In early studies [48,79,80], it was pointed out that in the perfectly ordered model the phasonic elastic free energy fw-w should show a linear dependence on the phason strain, that is, fw-w c~ IVw(r)l, instead of a quadratic form given in eq. (11). This arises from the conjecture that the number of the matching rule violation is in proportion to IVw(r) I in the vicinity of IVw(r) l = 0 and that each such defect independently costs a constant energy. In this case, the radius of the curvature of the energy surface is arbitrarily small at IVw(r) I = 0 and therefore introduction of phason strain must be strongly suppressed especially at low temperatures. Such a state is called a state of locked phason or simply a locked state. In contrast, the elastic free energy in random tiling quasicrystals is shown to have a standard quadratic form [48,79,81]. The state with such a quadratic form of elastic free energy is called a state of unlocked phason or an unlocked state. Monte Carlo simulations using various filing models have shown that a locked state is indeed realized as a ground state and that the locked state undergoes a transition to an unlocked state at some temperature. In general, such a transition occurs at 0 K in two-dimensional systems and at some finite temperature in three-dimensional systems [79-81 ]. Due to all these studies, it may be widely believed that the form of the phason strain dependence of elastic free energy can distinguish between the energetically stabilized perfectly ordered model and the entropically stabilized random tiling model, and that a quadratic form experimentally evidenced by phasonic diffuse scattering (described below) is in favor of the latter model. However, based on the calculations using model quasicrystals with interatomic interactions, Koschella et al. [82] and Trebin et al. [83] have recently claimed that even energetically stabilized quasicrystals should have a quadratic form of elastic free energy, indicating that the phasonic diffuse scattering does not immediately justify the random tiling model. The quadratic form of phasonic elastic free energy generates phason fluctuations. Such phason fluctuations should induce diffuse scattering around Bragg peaks in diffraction spectra just as phonon fluctuations generate conventional thermal diffuse scattering [48, 84]. Phasonic diffuse scattering has indeed been observed experimentally for quasicrystals of various alloy systems, which evidences the quadratic form of phasonic elastic free energy. The shape and the intensity of the diffuse scattering changes depending on the shape of the basin of the energy surface, which is characterized by the elastic constants. Therefore, in principle, by examining the phasonic diffuse scattering one can evaluate the phason elastic constants. The evaluation of the phason elastic constants by diffuse scattering measurements has been reported for icosahedral A1-Pd-Mn [51-55], Zn-Mg-Sc [57] and A1-Cu-Fe [56]. In [54] and [57], absolute-scale measurements of the diffuse scattering intensity have been reported for the A1-Pd-Mn and Zn-Mg-Sc systems, respectively. From the results of the absolute-scale measurements, the phasonic elastic constants K1 and K2 have been eval-
§2.4
Elasticity, dislocations and their motion in quasicrystals
385
Table 11 Phason elastic constants of various icosahedral quasicrystals evaluated by diffuse scattering measurements of X-ray, neutron and electron Alloy system
Source
A1-Pd-Mn A1-Pd-Mn A1-Pd-Mn A1-Pd-Mn A1-Pd-Mn Zn-Mg-Sc A1-Cu-Fe
X-ray neutron neutron neutron electron X-ray X-ray
Meas. temp.
kB Tt
kB T t
K!
K2
(atom- 1)
K1
(atom- 1)
K2
(MPa)
(MPa)
R.T. R.T. 1043 K R.T. R.T. R.T. R.T.
0.06 0.1 0.13 0.5 -
-0.031 -0.052 -0.052 . . . . 0.075 -
43 72 125 . . 300 -
-22 -37 -50 . . -45 -
K1
K2
Ref.
-0.52 -0.52 -0.4 0.5 0.5 -0.15 0.5
[54] [54] [54] [51 ] [73] [57] [56]
uated on the assumption that the phonon-phason coupling K3 is negligible (see eq. (14), and Tables 3, 4). On the other hand, in [51,55,56], only the shapes of the diffuse scatterings have been analyzed and the ratios Kz/K1 have been evaluated on the assumption of K3 = 0. The evaluated values of the elastic constants are summarized in Table 11. In general, the absolute-scale measurements first give the elastic constants in the form of kB/(iTt , where k8 denotes the Boltzmann constant and T' the temperature at which the phason fluctuation equilibrates. To evaluate K1 and K2, we assume T' -- 773 K [54] for the room temperature measurements and T' = 1043 K for the measurements at 1043 K in Table 11. We notice that the phason elastic constants in Table 11 is by about three orders of magnitude smaller than the phonon elastic constants in Table 9. It should be noted, however, that the magnitudes of the phason elastic constants change depending on the arbitrarily chosen scale factor for E±, as pointed out in Section 2.1 and that the ratio K±/KII (K± = K1, K2; KII = )v, #) by itself has no physical meaning. For the icosahedral quasicrystalline state to be thermodynamically stable, the elastic free energy density f in eq. (14) should be positive definite, imposing on the elastic constants the following condition [49]:
5
K 1 - - I - ~ K 2 > 0,
/z
(4) K1-~
K2
> 3 K 2.
(22)
This indicates that K1 should always be positive. Dividing the inequalities by K1 leads to K2
3
K2
K1
5'
K1
~.
3
9K 2
4
4#K1
(23)
In addition to this thermodynamic stability, the system should satisfy another stability condition, which is of the hydrodynamic stability [49,85]. This condition is for the relaxation rate of phason fluctuation to be finite, which requires [49] K2
3
3K2
K2
a
18 K 2
K1
4
()~ + 2#)K1 '
K1
b
b#K1
386
K. Edagawa and S. Takeuchi
Ch. 76
(a -- 27 - 9x/5 ,~ 6.9, b - 15~/5 - 27 ~ 6.5), K2 K1
3 > --
4
K2 +
3(~. + 2/z)K1
•
(24)
It should be noted that the hydrodynamic stability condition in eq. (24) is less restrictive than the thermodynamic one in eq. (23). If we assume K3 --0, eqs (23) and (24) reduce to 3 5
K2
<
3 < K1 4
(25)
and 3 4
K2
<
3 < K1 4'
(26)
respectively. As in Table 11, the A1-Pd-Mn system has been studied most intensively. For this system, the following facts should be noted: (1) K1 and K2 are positive and negative, respectively, (2) the values ~IKll and ~IK2I are smaller at room temperature (or at T z ~ 773 K) than those at T ( ~ T ~) = 1043 K, (3) the decrease in IK~I with the temperature change from T z -- 1043 to 773 K is more IKzl significant than the decrease in kBY" (4) as a result, the ratio K 2 / K 1 also changes slightly; it changes from - 0 . 4 to - 0 . 5 .
First, the signs and the values of K1 and K2 in (1) and (4) are consistent with the stability conditions. Generally speaking, Igll
Ig21
-1
is expected when the phasonic part of the elastic free energy f w - w consists entirely of the energetic contribution. On the other hand, if it consists entirely of the entropic contribution, as in the case of an ideal random tiling, IKll
IK21
(v,)o
kB T ~' kB T ~
is expected. The fact (2) implies IKll
IK21
kB T ~' kB T ~
(T')
(z, > o),
which is closer to the latter. The facts (3) and (4) indicate that the system is close to the instability points K 2 / K 1 = - 0 . 6 (thermodynamic instability) in eq. (25) and K 2 / K 1 =
§2.5
Elasticity, dislocations and their motion in quasicrystals
387
- 0 . 7 5 (hydrodynamic instability) in eq. (26) and that it approaches to them with decreasing temperature. Ishii [85] and Widom [49] have shown that the hydrodynamic instability at K2/K1 = - 0 . 7 5 leads to the symmetry breaking from Yh to D3d. In relation to this result, the formation of a modulated icosahedral phase with the modulation along a threefold direction has been reported in an A1-Pd-Mn alloy with the composition close to that of the icosahedral quasicrystal [86]. For Zn-Mg-Sc, K1 is considerably larger than that for the A1-Pd-Mn system while K2 is roughly the same, resulting in K2/K1 = - 0 . 1 5 that is far from instability points. For A1-Cu-Fe [56], a positive ratio of K2/K1 = 0.5 has been reported, which is close to another instability point K2/K1 = 0.75 that leads to the symmetry breaking to Dsd [49,56,85]. Apart from the alloy systems listed in Table 11, for A1-Pd-Re [87] qualitatively different shapes of diffuse scatterings from those for A1-Pd-Mn have been reported, although the K2/K1 ratio has not been evaluated quantitatively. The intensity and the shape of diffuse scatterings for Cd-Yb [87] have been shown to be similar to those for A1-Pd-Mn. For a A1-Ni-Co decagonal quasicrystal, anisotropic diffuse scattering has been observed in synchrotron X-ray diffraction measurements [88]. It has been shown that the observed diffuse scattering can be attributed partly to the phasonic origin, although no quantitative evaluation of K2/K1 have been reported.
2.5. Phonon-phason coupling In contrast to pure phonon and pure phason elasticities described in the preceding sections, the number of studies so far done on the phonon-phason coupling is limited and we still have little knowledge about it. Henley [48] has discussed the microscopic origin of the phonon-phason coupling by using a toy atomic model of a quasicrystal, in which the atoms interact with a given set of pair potentials. In general, not all neighbor pairs can have optimal distances simultaneously; this results in frustration: some linkages are under compression and others under tension. The introduction of phason strain causes a statistical imbalance between the frequencies of the appearance of various symmetry-related linkages under compression or tension, resulting in the introduction of a global phonon strain. Such a mechanism implies that the phonon-phason coupling strength can be evaluated by measuring the magnitude of the phonon strain spontaneously introduced in crystal approximants to quasicrystals. Fig. 5(a) illustrates schematically an elastic free energy density f of a quasicrystal as a function of phonon strain u and phason strain w, in the vicinity of u = w = 0 (see eq. (11)). Here, the principal axes of the quadratic surface of f (u, w) do not coincide with the u and w axes, indicating a non-zero phonon-phason coupling. In the quasicrystalline state characterized by w = 0, the minimum of f lies at u -- 0, as shown in the curve 1 in Fig. 2(b). In general, each crystal approximant structure is characterized by a nonzero uniform phason strain with a fixed value w0. Now, let us consider an atomistic model quasicrystal with a certain set of atomic interactions. We can construct any crystal approximant structure by introducing the uniform phason strain w0 characterizing the crystal approximant. Then, the dependence of the elastic energy of the crystal approximant structure on the phonon strain u should be given as
K. Edagawa and S. Takeuchi
388
Ch. 76
(b) 2
f
1
u
Fig. 5. A schematic illustration of an elastic free energy density f of a quasicrystal as a function of phonon strain u and phason strain w, in the vicinity of u = w = 0 (a). The cuts along the line w = 0 and w -- w0 in (a) are drawn as the curves 1 and 2 in (b), respectively. The curve 3 in (b) represents the elastic free energy of the approximant phase, which is assumed to be identical to the curve 2 except for the constant term.
f (u, w0) at T = 0, as shown in the curve 2 in Fig. 5(b). Therefore, if we relax the structure while keeping the global phason strain of w = w0, a spontaneous deformation u0 should take place, where u - u0 gives the minimum of f (u, w0). From the quantity u0, we can evaluate the phonon-phason coupling constant because u0 is proportional to the coupling constant. By use of such a method, the coupling constants in a model decagonal quasicrystal and in model icosahedral quasicrystals have been evaluated by Koschella et al. [82], and Zhu and Henley [50], respectively. Koschella et al. [82] have used a two-dimensional binary tiling model with decagonal symmetry, where the atoms interact with pair potentials of the Lennard-Jones type. The coupling constant evaluated is about 3% of the shear modulus. Zhu and Henley [50] have calculated the phonon-phason coupling constants for realistic models of icosahedral A1-Mn and A1-Cu-Li, which represent the two major classes of quasicrystals. The results show that (1) the sign of the coupling constant K3 in eq. (14) is positive for A1-Mn and negative for A1-Cu-Li and (2) IK3] is 0.003-0.02# for A1-Mn and 0.02-0.1# for A1-Cu-Li, where/z is the shear modulus. Here, it would be interesting to examine whether the magnitude of K3 is large enough to effect the elastic instability of the quasicrystal discussed in the preceding section. Comparing eqs (23) and (24) with eqs (25) and (26), we notice that
()~ + 2#)K1 ' #K1
<<1
§2.5
Elasticity, dislocations and their motion in quasicrystals
389
is the condition for K3 to be neglected in the discussion of the elastic instability. The followings are crude estimations: for A1-Mn, we assume [K3] -- 0.01# and adopt K1 = 70 MPa and # = 70 GPa that are for A1-Pd-Mn in Tables 9 and 11, respectively, leading to K2 ~ 0.1" for A1-Cu-Li, we use IK31 -- 0.05# and adopt K1 -- 300 MPa and # -- 40 GPa
#K1
that are for Zn-Mg-Sc and A1-Li-Cu, respectively, giving ~ ~ 0.3. These values suggest that K3 values are not small enough to be completely neglected. We may be allowed to apply the above method to real crystal approximants. Here, it should be noted that we have to make the following assumption to apply the method to the real systems. When we evaluate experimentally the phonon strain introduced in a given crystal approximant phase, the phase should be in the stable state. This indicates that the elastic free energy f of this state must be lower than that of the corresponding quasicrystalline state characterized by u = w = 0. In fact, the phase transition from the quasicrystalline state to the crystal approximant state is generally induced by the decrease in the phason elastic constants, i.e. phason softening and also by the effect of higher-order terms in the expansion of f [49,85]. This fact indicates that the functional form of f in the vicinity of u -- w = 0 should be modified considerably on the transition. Nevertheless, we may be allowed to assume that such a change in the functional form of f occurs in its w-dependence only, considering the nature of the quasicrystal-to-approximant phase transition. On this assumption, the u-dependence of the elastic free energy g(u) of the approximant phase should be identical to the u-dependence of the elastic free energy f of the quasicrystalline phase at w = w0, i.e. f (u, w0) except for the constant term, as shown in the curve 3 in Fig. 5(b). We find in eq. (12) that the phonon and phason strain components u5 and w5 couple in icosahedral quasicrystals. This indicates that (1) only the crystal approximants with w5 ~ 0 induces phonon strain, and that (2) the induced phonon strain has only the u5 component, i.e. pure shear strain. The fact (1) poses a rather severe limitation in choosing the crystal approximant for the experiment. Most of the crystal approximants so far found are of the cubic system, which have only the w4 component, i.e. w5 -- 0. There have been only two non-cubic approximants known so far, which have good structural quality suitable for the present experiments: one is an orthorhombic approximant in the Mg-Ga-A1-Zn system [89] and the other is a rhombohedral one in the A1-Cu-Fe system [90-92]. Recently, Edagawa [93] has measured the phonon strain induced in the crystal approximant of the 3_2_2 orthorhombic type in the Mg-Ga-A1-Zn system and estimated the phonon-phason coupling constant K3, as follows. In the composition Mg39.sGa16.4A14.1Zn40.0, a metastable icosahedral phase produced by melt-spinning underwent a transition to the stable 3_2_2 orthorhombic approximant phase on heating. By X-ray diffraction measurements, the quasilattice parameter aq of the icosahedral phase and the lattice parameters a, b and c of the orthorhombic approximant have been evaluated. Without induced phonon 2_ 2 orthorhombic approximant should be strain, the lattice constants of the 3- T 1
a0--2r4aq(l+r 2) 2 and bo-co-ao/v.
K. Edagawa and S. Takeuchi
390
Ch. 76
Comparing (a, b, c) with (a0, b0, co), the induced phonon strain components, Ull = (a etc., were evaluated. Finally, the coupling constant K3 has been estimated to be ~ - 0 . 0 3 # . Using/z = 46 GPa for a Zn-Mg-Y in Table 9, we obtain K3 = - 1 . 4 GPa. In a following paper [94], it has been pointed out that an unnecessary constraint was imposed in the analysis of the data in [93]. The correction by removing this constraint has been shown to give a slightly larger values of K3 = (-0.04 ± 0.01)# and K3 -- - 1 . 8 4- 0.4 GPa. The icosahedral Mg-Ga-A1-Zn is of the Frank-Kasper type and should be compared with the results of the calculation for A1-Cu-Li by Zhu and Henley [50]. The negative sign of K3 for the Mg-Ga-A1-Zn system is in agreement with the calculation. The IK3] value is also in good agreement with the calculation, although the calculation has yielded a considerably large range of values. Recently, Edagawa and So [94] have performed the measurements of the phonon strain induced in a rhombohedral approximant in the A1-Cu-Fe system [90-92]. They have reported a small positive K3 value of about 0.004# for this system.
ao)/ao,
3. Dislocations in quasicrystals 3.1. Perfect dislocations in quasicrystalline lattice
Because of lack of periodicity, we cannot introduce a perfect dislocation into a quasicrystal simply by applying the conventional definition of crystalline dislocation. However, a perfect dislocation can also be defined in quasicrystal by extending the definition of crystalline dislocation [5,6,9]; a perfect dislocation in quasicrystal is a line defect in the physical space Ell, satisfying
fc dU- B ~ LR,
(27)
for any loop C surrounding the dislocation line. Here, U denotes a high-dimensional displacement vector (see Fig. 1), B a high-dimensional Burgers vector and LR a set of highdimensional lattice translational vectors. Using eq. (6), eq. (27) can be decomposed into the two equations:
cdU-
fc
bll ,
dw-b±,
(28)
where bll and b± are the Ell and E± components of B, i.e. B = bll + b±.
(29)
Eq. (28) indicates that in the presence of dislocations the displacement fields u(r) and w(r) are multi-valued. The dislocations defined here generally have the following properties:
§3.1
Elasticity, dislocations and their motion in quasicrystals o o
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Fig. 6. A perfect dislocation introduced in a two-dimensional Penrose lattice. The Burgers vector bll indicated by an arrow is [0110], where the indices are referred to the basis vectors di (i = 1 . . . . . 4) in eq. (9) (see Fig. 4(b)).
(1) It is not accompanied by any planar fault because B is a lattice translational vector in the high-dimensional space. (2) The magnitude of distortion or strain is in inverse proportion to the distance from the dislocation center. These properties are common to conventional crystalline dislocations. However, the perfect dislocation in quasicrystal is different from crystalline perfect dislocation in that the former is inevitably accompanied by the phason strain because every high-dimensional lattice vector necessarily has a phason component. Fig. 6 presents an example of a perfect dislocation in a quasicrystal. Here, a perfect dislocation with the Burgers vector [0110] is introduced in a two-dimensional Penrose lattice that is known as a typical example of a two-dimensional decagonal quasicrystal. Here, the indices of the Burgers vector are referred to the basis vectors di (i = 1 . . . . . 4) in eq. (9) (Fig. 4(b)). In this case, the dislocation is a point defect because of the twodimensional system. Here, as a simple example, the displacement vector is given by
(0) ~
B,
(0)
bll,
U(r,O)or equivalently,
u(r, 0 ) -
~
(30)
392
w(r, 0 ) - -
(0) ~
K. Edagawa and S. Takeuchi
b±,
Ch. 76
(31)
where r is the distance from the dislocation core at the center and 0 is the polar angle measured about the core. If you observe carefully the figure at grazing angle, you will find distortions and jogs in lattice planes in the vicinity of the dislocation core. Here, the distortions of lattice planes correspond to the phonon distortion or strain. On the other hand, the jogs in the lattice planes correspond to the phason one. In real quasicrystals, the displacement fields around dislocation should be calculated on the basis of the generalized elastic theory described in Section 2.2 and must be more complex than the form of eq. (31). The displacement fields around real dislocations will be discussed in more detail in the next subsection. Since dislocations in quasicrystals are accompanied by a strain field as well as those in conventional crystals, they can be observed as a diffraction contrast image by transmission electron microscopy. For dislocations in conventional crystals, the direction of the Burgers vector b can be determined by use of the invisibility condition g. b = 0 [95], where g is a reciprocal vector used for imaging. For dislocations in quasicrystals, this condition is generalized to [96,97] G . B =gll "bll + g ± "b± = 0 ,
(32)
where G = gll + g±.
(33)
Here, G is a high-dimensional reciprocal vector, and gll and g± are its Ell and E± components, respectively. There are two cases in which this invisibility condition is satisfied. One is the case gll "bll : g± "b± = O,
(34)
which is called 'strong' invisibility condition. Because g± • b± = 0 is automatically satisfied when gll" bll = 0 is satisfied, the strong invisibility condition is written just as gJl" bll = 0. This condition is the same as the invisibility condition for conventional crystals. The other case is gll" bll -- - g ±
"b± ~ O.
(35)
This condition is called 'weak' invisibility condition, which is characteristic of dislocations in quasicrystals. By use of the convergent-beam electron diffraction (CBED) technique, not only the direction but also the magnitude of the Burgers vector can be determined [98]. In the defocus CBED pattern, zeroth-order Laue zone line splits when it crosses a dislocation line, where the number n of splitting is given by [99] n = G . B = gll "bll -+- g± "b±.
(36)
§3.2
Elasticity, dislocations and their motion in quasicrystals
393
Using such methods, the Burgers vector determination has been attempted for icosahedral [99-102] and decagonal [103-108] quasicrystals. Rosenfelt et al. [102] have analyzed in detail dislocations in deformed and undeformed samples of icosahedral A1-Pd-Mn. It has been found that about 90% of the dislocations in both deformed and undeformed samples are the perfect dislocations with Burgers vector pointing along twofold direction. The rest are partial dislocations with Burgers vector pointing along fivefold, threefold and pseudotwofold directions. Four types of twofold Burgers vectors have been observed: (001111), (001221), (002332) and (003553), where the indices are referred to the basis vectors di in eq. (7) with a - 6 . 4 5 A (see Fig. 4(a)). 3 For the four Burgers vectors, the ratios Ib±l/Ibll I are r 3, r 5, r 7 and r 9, respectively. Of the four, (001221) is most frequently observed; about 65% of the twofold Burgers vectors observed are (001221). The major slip planes are shown to be fivefold and threefold planes. For decagonal quasicrystals, the dislocations with the Burgers vector parallel to the periodic direction [103-106], in the quasiperiodic plane [106-108] and with both periodic and quasiperiodic components [106] have been reported. The Burgers vectors of the second type so far reported are (10000), (01000), and (01110), whose Ib±l/lbll I ratios are r, r -1 and r -3, respectively. Here, the indices are refereed to the basis vectors in eq. (9) with a - 3.39 A. All of them are perfect dislocations with twofold Burgers vectors.
3.2. Displacement field and self free energy of dislocations The phonon and phason displacement fields u(r) and w(r) around a dislocation in quasicrystals should satisfy the definition of the dislocation given by eq. (28) and elastic equations of balance by eq. (21). In the absence of body forces, eq. (21) reduces to
02Uk u-w 02113k u-u -Jr-Cijkl = O, Cijkl Orj Orl Orj Orl
u-w 02Uk w-w 02~k =0. Cijkl -Jr- Cijkl Orj Orl Orj Orl
(37)
In principle, the displacement fields u(r) and w(r) around a dislocation in quasicrystals can be fully determined by eqs (28) and (37). The calculations of the fields around a dislocation in pentagonal [109], decagonal [110-113], octagonal [111], dodecagonal [111, 114] and icosahedral [58,115] quasicrystals have been reported. In the calculations, various mathematical methods have been applied, which are summarized in the review by Hu et al. [116]. Because eqs (28) and (37) have the forms of a simple generalization of the corresponding equations for conventional crystalline dislocations, the methods used for crystalline dislocations can be generalized easily for the application to the quasicrystalline dislocations. In the following, we briefly review the Eshelby's method [117] generalized by Ding et al. [113]. By use of the generalized Eshelby's method, we demonstrate a calculation of the displacement fields around a dislocation in an icosahedral quasicrystal and deduce self free energy of the dislocation. 3In this case, the lattice spanned by di in eq. (7) with a = 6.45 A has a F-type superlattice order; the structure 6 nidi with )--~6 has the translational symmetry only for the vectors R = }--~4=1 i=1 ni - - e v e n . We confirm that }-~6 1n i = even is satisfied for all the four Burgers vectors and therefore they are perfect dislocations.
394
Ch. 76
K. Edagawa and S. Takeuchi
We consider a straight dislocation parallel to the x3 axis. We define
ot/~ 2
(38)
a °~(p) -- Bill 4- (Bl~2 4- B2~I)p 4- B22 p , where
B jr -- ~,ot(~k~Cijk? 4- Sk(~_3)Cijk~ °) 4- Si(ot_3)(~k~Cijkl w 4- ~k(~_3)CijklW).
(39)
Here, oe and/3 take 1, 2 . . . . or 6, and i, j, k and 1 take 1, 2 or 3. We solve the equation det(aC~ (p)) - 0 .
(40)
This is a twelfth-order equation for p and we obtain twelve roots pl, p2 . . . . and p12. These roots are necessarily complex and consist of six conjugate pairs. We pick up p] . . . . and p6, one from each pair of complex conjugates. Solve the equations
a °~ (Pn) A/~ (n)
= 0
(41)
to obtain A/~ (n) for each of the six roots of mine the constants D (n) satisfying
Pn. Using
A~ (n) (n -- 1, 2 . . . . . 6), we deter-
(42)
and
Re
(B21~ + B22
(n) D (n)
n=l
]
(43)
- 0
where bll = [bl, b2, b3] and b± = [b4, b5, b6] are the Burgers vectors. Then, we finally obtain the displacement fields
U(Xl, x2) -
I U1 (Xl, x2) ] U2(xl, x2)
and
W(Xl, X2) --
U3 (x l , x2)
U5(xl, x2) U6(xl, x2)
with
U~(xl, x2) - Re
~4-27ri A~(n)D(n) n=l
ln(xl +
pnx2) 1 ,
(44)
Elasticity, dislocations and their motion in quasicrystals
§3.2
395
where the sign + is taken to be the same as the sign of the imaginary part of Pn. Following the Foreman's method [ 118], the self free energy F per unit length of the dislocation has the form
Kb2 I n ( R ) F=
4rr
(45)
ro
where
Kb 2 - b~ Im
-4--(B2~+ B2~2 Pn)A~(n)D(n)
(46)
.
n--1
Here, K is a factor called energy factor, and r0 and R are inner and outer cut-off radii, respectively. As described before, the dislocations with the Burgers vector along the twofold direction are by far the most frequently observed in icosahedral quasicrystals. As explained in more detail later, such a Burgers vector can be written as bll -- [0, 0, bll] and b± = [0, 0, b±], where the indices are referred to el, e2 and e3 for bll, and e4, e5 and e6 for b±, respectively (see Fig. 4(a)). As an example, we consider a simple case of a dislocation with such a Burgers vector and with the line direction along the x3 axis, i.e. the e3 direction. If we assume that the phonon-phason coupling is negligible, i.e. K3 --0, we can deduce easily the displacement fields u(r) and w(r) and the self free energy F. The results are
U(Xl, X2) - -
0 bll t a n - 1 ~-y
W ( X l , X2) - X2
M
'
Xl
0 b±
~-y t a n -
1
X2
'
(47)
--
Xl
and =
4rr
In
~0 '
(48)
where B -- K1 + K2(r - ½) and C - K1 + K2(~- - r) (see Table 3). The derivation of eqs (47) and (48) is given in Appendix B. Because K3 - - 0 is assumed, u(r) is identical to the field around a screw dislocation in conventional isotropic elastic body, which has the same form as in eq. (31). On the other hand, w(r) deviates from the form in eq. (31). Assuming 1
4zr as often assumed for conventional crystalline dislocations, we obtain from eq. (48): F ~/zb~ + ~-B~b 2.
(49)
396
K. Edagawa and S. Takeuchi
Ch. 76
More generally, the self free energy F of a dislocation in icosahedral quasicrystals can be written as F -- a#[bll
12+ bY1 ]b±[ 2,
(50)
where a and b are positive constants of the order unity, which change depending on the Burgers vector direction and the dislocation line direction. Here, we note that K] is always positive and that [Kzl/K] and )v/# are of the order unity. Now, let us discuss about relative stability among dislocations with various Burgers vectors, based on eq. (50). Here, we restrict our discussion to the dislocations with twofold Burgers vectors. In general, the Burgers vectors with bll parallel to a twofold direction e3 in Fig. 4(a) can be written as R1 - [0, 0, 0, 1, i, 0],
R2 - [ 0 , 0, 1, 0, 0, i],
B - - m R 1 + nR2 - [O, O, n, m, rh, ~].
(51)
Here, the indices are referred to the basis vectors di in eq. (7). We note that all these B belong to the translational symmetry vectors for the F-type superlattice order (see footnote 3 in Section 3.1). The same Burgers vectors can be rewritten as
R] - [0, 0, p, 0, 0 , - z p ] ,
R2 - [0, 0, rp, 0, 0, p]
B - mR1 + nR2 -- [0, 0, p ( m + nr), 0, 0, p(n - mr)],
(
2a ) P-
r~/6~-2r
' (52)
where the indices are referred to the basis vectors ei in Fig. 4(a). This shows that for those B, the b± component is always parallel to the e6 direction in E±. In other words, all those B lie on the two-dimensional subspace spanned by e3 and e6 in the six-dimensional space. More specifically, eq. (52) indicates that the Burgers vectors B form a square lattice in the e3-e6 plane, as shown in Fig. 7. Here, the square lattice is tilted with respect to the e3-e6 coordinates by 0 - tan -] (r) and the projections of B onto Ell and E± give the bll and b± components, respectively. Now, we consider the problem as to which B gives the minimum self free energy F in eq. (50). For example, the lattice point P with (m, n) - (1, 3) can never be the minimum F point because there are other points, e.g., the point Q with (m, n) - (1, 2), whose ]bll ] and ]b±] are simultaneously smaller than those for the point E In contrast, the point Q can be the minimum F point because there are no other points whose ]bll ] and ]b±] are simultaneously smaller than those for the point Q. The lattice points satisfying this condition are those with ( m , n ) - . . . , (3, 2,), (2, 1), (1, 1), (1, 0), (0, 1), (1, 1), (1, 2), (2, 3) . . . . . which are indicated by open circle in Fig. 7, and their inversion counterparts indicated by solid circle. Here, the open circle points can be expressed as B~ - (Fk, Fk+]) (k --0, 1, 2 . . . . ) and B~ -- (F-k, F - k - l ) (k -- - 1 , - 2 . . . . ) . . . . where Fn(n ~> 0) is the Fibonacci numbers defined by F n + 2 - Fn+] + Fn with F 0 - 0 and F] - 1. The solid circle points can simply be expressed as { - B ~ l k - 0, 4-1, 4-2...}. Combining the two groups, we obtain {-+-B~Ik -- 0, +1, + 2 . . . } .
§3.2
397
Elasticity, dislocations and their motion in quasicrystals
X6,~
o
o
o
r
X3 II
o
1
•
c
Fig. 7. The two-dimensional plane spanned by e3 and e6 in Fig. 4(a). The Burgers vectors of the type eq. (51) form a square lattice on this plane. The open and solid circles represent the subsets {+B~lk = 0, + l , 4-2...} and {-B~ ]k = 0, 4-1, 4-2... }, respectively (see text).
The Burgers vectors {B~ [k -- 0, 4-1, 4-2... } satisfy Ibl~kI -
Ibl~olr ~,
Ib~_~ I- b~_olr -k- Ib,~olT-<~+I>,
(53)
where b*Ilk and b*±k are the Ell and E± components of B k, * respectively. From eq. (53), we obtain the following relations:
bl~kl" Ibm_,I - bl]0 I" Ibm_01-
const.
(54)
and Ib*l k [ Ib,Ilkl
=r
-2k- 1
(55)
As shown above, the minimum F point necessarily lies in the subset {4-B~Jk0, 4-1, 4-2...} (open and solid circles in Fig. 7) of all the Burgers vectors B defined by eq. (51) or (52). Then, which point in the subset {-+-B~ Ik - 0, +1, + 2 . . . } is the minimum F point? Using the relation of eq. (54), we find that the self free energy F in eq. (50) becomes the minimum at
±kl Ilk]
(
a# bK1
(56)
K. Edagawaand S. Takeuchi
398
Ch. 76
For the A1-Pd-Mn system, we find # ~ 70 GPa and K1 ~ 70 MPa in Tables 9 and 11, respectively. Thus, we obtain #/K1 ~ 1000. Because of a/b ~ 1, eq. (56) gives
Ib*l ±~ Ib*Ilkl
~ 30.
As described in Section 3.2, the twofold Burgers vectors (001111), (001221), (002332) and (003553) have been reported in the A1-Pd-Mn system. All of them belong to the subset { + B ~ l k - 0, +l,-+-2...}; they correspond to B* 2, B*_3, B* 4 and B*_5, respectively. The most frequently observed Burgers vector is B*_3. The ratios Ib±l/lbll I for the four Burgers vectors are z 3 -- 4.2, r 5 -- 11, Z" 7 - - 29 and Z" 9 - - 76, respectively, agreeing roughly with Ib~_kI the above estimation ~ ~ 30 for the minimum F Ibllkl For decagonal quasicrystals, twofold Burgers vectors (10000), (01000), and (01110) have been reported, whose Ib±l/lbll I ratios are r - 1.6, r -1 - 0 . 6 2 and r -3 - 0 . 2 3 , respectively. All of them belong to the subset that can be defined for the decagonal system similarly to the subset {4-B~lk- 0, 4-1, 4-2...} for the icosahedral system. In contrast to the case of icosahedral quasicrystals, no measurements of the phason elastic constants have been reported for decagonal quasicrystals. The observed Ib±l/lbll I ratios for decagonal quasicrystals may allow us to make a crude estimation of the magnitude of the phason elastic constants" K± ~ ~.~ # ~ 3# ~ 240 GPa.
4. D i s l o c a t i o n m o t i o n in q u a s i c r y s t a l s 4.1. Dislocation mechanism of deformation
In general, quasicrystals are brittle at room temperature and can only be plastically deformed at high temperatures above about 0.8Tm (Tm: melting temperature). In the high temperature range, systematic deformation experiments and transmission electron microscopic (TEM) observations of the deformation microstructures have so far been performed for quasicrystals of various alloy systems, as reviewed in [11-13]. In 1993 [14], Wollgarten et al. have revealed by transmission electron microscopy (TEM) that by plastic deformation the dislocation density increases to the order of 108/cm 2 from 106/cm 2 in the as-grown state, suggesting a dislocation mechanism for plastic deformation. In icosahedral A1-Cu-Fe, a twin-like structure and planar faults were observed after plastic deformation [119-121], but later high density dislocations and stacking faults were observed also in this alloy [ 122] as in icosahedral A1-Pd-Mn. The most direct evidence for the dislocation mechanism of plastic deformation has been obtained by an in-situ stretching experiment on a high temperature tensile stage in an electron microscope [15,122]. Under an applied stress, straight dislocations have been observed to move steadily and continuously, just like the dislocation glide process observed in-situ by transmission electron microscopy for bcc metals at low temperatures and for semiconducting crystals at high temperatures. Until recent years, it has been generally believed that the plasticity of quasicrystals is carried by a
§4.2
Elasticity, dislocations and their motion in quasicrystals
399
glide process of dislocations, and various models of the deformation mechanism have been proposed based on the dislocation glide [16-20]. However, recently, Caillard et al. [2123] have shown by TEM that high temperature deformation of icosahedral A1-Pd-Mn is brought not by a glide process but by a pure climb process. On the other hand, at low temperatures Mompiou et al. [23], Caillard et al. [123] and Texier et al. [124,125] have performed deformation of icosahedral A1-Pd-Mn phases under a high pressure and investigated the deformation microstructures by TEM observation. Mompiou et al. [23] and Caillard et al. [ 123] have shown that for single-crystals of icosahedral A1-Pd-Mn deformation at 573 K is exclusively due to dislocation climb. On the other hand, Texier et al. [124,125] have conducted deformation experiments of polycrystalline A1-Pd-Mn icosahedral quasicrystals in the temperature range between room temperature and 573 K. They have shown the following facts: (1) at room temperature the deformation is mostly brought by dislocation glide but dislocation climb events are also occasionally observed and (2) the frequency of the climb events increases as the deformation temperature is raised. Saito et al. [ 126] have recently performed deformation experiments of MgZn-Y icosahedral quasicrystals at low temperatures under a hydrostatic confining pressure superimposed to an applied uniaxial stress. They have found an irregular behavior in the temperature dependence of the yield stress. The origin of the irregular temperature dependence has been discussed in terms of possible transition between glide process at low temperatures and climb process at high temperatures. In the following two subsections, we discuss theoretically the two important dislocation processes in quasicrystals, i.e. glide and climb processes.
4.2. Dislocation glide process For a phason strain field to be produced or relaxed, phason jumps must occur via short range atomic diffusion, which takes place only as a thermally activated process, as mentioned in Section 2.1. Thus, in a low temperature range below half the melting point where the atomic diffusion is practically prohibited, the perfect dislocation glide is impossible and only the partial dislocation with the Burgers vector of bll can glide, leaving behind the stacking fault (also called the phason fault) with the fault vector of b±. Let the energy of the phason fault be F, the dragging stress necessary to move the dislocation is ra = F/bll,
(57)
where r represents the resolved shear stress acting on the glide plane in the direction of bll, and bll is the strength of bll. In contrast, at high enough temperature at which atomic mobility by diffusion is faster than the dislocation velocity, perfect dislocations can migrate accompanying both phonon and phason strains. Fig. 8 shows the structural change that should take place when a perfect dislocation moves in the two-dimensional Penrose lattice. Here, the patterns indicated by broken lines show the phason flips (see Fig. 3) that should take place when the perfect dislocation is translated to the left by the distance represented by an arrow. In actual quasicrystals, the quasicrystalline lattice is decorated by atomic clusters; depending on the
400
K. Edagawa and S. Takeuchi
Ch. 76
Fig. 8. Structural changes taking place when a perfect dislocation moves in a two-dimensional Penrose lattice; the patterns indicated by broken lines show the phason flips (see Fig. 3) that should take place when the perfect dislocation is translated to the left by the distance represented by an arrow.
way of the decoration, different types of quasicrystals are produced such as the FrankKasper type, the Mackay-icosahedron type and the Cd6Yb type for icosahedral quasicrystals. Fig. 9(a) shows an example of a realistic quasicrystalline structure containing a perfect dislocation: the atomic structure is a model decagonal quasicrystal (Burkov model [127]) viewed from the ten-fold axis [128]. The perfect lattice is composed of decagonal columnar-clusters with glue atoms among them. In contrast to the case of undecorated Penrose lattice in Fig. 8, we can see clearly a phason strain field by a distribution of imperfect cluster columns in Fig. 9(a). Fig. 9(b) shows the atomic structure for the dislocation position at the left end in the figure. By comparing Figs 9(a) and (b) we see a change of the cluster pattern as a result of the change of the phason strain field. As is evident from Figs 8 and 9, two types of the structural change should take place as a perfect dislocation translates on the glide plane; one is the re-tiling structural change outside the glide plane and the other is the intra-tile structural change. The energy release due to the latter structural change must be much higher than that due to the former type of the structural change, because without the latter relaxation severe nearest neighbor violation occurs, while without the former relaxation no nearest neighbor violation occurs and the energy increase is only due to a long range interaction. Therefore, in an intermediate temperature region where the short range diffusion can take place with the dislocation translation, a partial relaxation of the phason strain should occur only near the glide plane by healing intra-tile structural violation. Fig. 10 shows schematically the three cases of the structure change as a result of dislocation glide from the left end to the center: (a) low temperature case, (b) intermediate temperature case and (c) high temperature case.
Elasticity, dislocations and their motion in quasicrystals
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Fig. 9. An example of a realistic quasicrystalline structure containing a perfect dislocation: the atomic structure is a model decagonal quasicrystal (Burkov model [127]) viewed from the ten-fold axis [128]. The dislocation position is at the center in (a) and at the left end in (b).
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=b±
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(b)
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Fig. 10. Three cases of the structure of a dislocation gliding from left to right in a quasicrystal: low temperature case (a), intermediate temperature case (b) and high temperature case (c).
In addition to the glide resistance due to the phason defect production, dislocations in quasicrystals should be accompanied by a Peierls potential barrier as the dislocations in crystals. The Peierls potential is the self-energy change with the change in dislocation position due to the discreteness of the lattice; the Peierls potential in crystals is periodic but that in quasicrystals quasiperiodic. For crystals, it is established theoretically [ 129,130] and experimentally [131,132] that the simpler the crystal structure having a larger d/b ratio (d: the glide plane spacing, b: the strength of the Burgers vector), the higher is the Peierls potential. It is naturally considered that the Peierls potential is high for dislocations in quasicrystals due to the complex atomic structures.
K. Edagawa and S. Takeuchi
402 '
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Ch. 76
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Position of the first dislocation ( A ) Fig. 11. Energy variation with the translation of a straight dislocation in two different orientations in a realistic model quasicrystal [133]. Peierls potential is dependent on the dislocation orientation. '
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Unless the phason relaxation occurs, the change of the potential energy with the translation of a dislocation comprises the phason fault energy and the Peierls potential which is superimposed on the former. Examples of the potential energy profile calculated for two different perfect dislocations in a realistic model decagonal quasicrystal [133] are presented in Fig. 11. It is seen that the Peierls potential is quite sensitive to the type of the dislocation as for crystal dislocations. For type (a) dislocation in the figure, the calculated stress to move the dislocation at 0 K is as high as 0.159G (G: the shear modulus) which consists of the phason production stress component of 0.048G and the Peierls stress component of 0.111 G, the latter being twice as large as the former. The so-called F-surfaces (the energy surface of the plane fault created by a rigid displacement parallel to the glide plane of the upper half crystal (quasicrystal) with respect to the lower half) for the model quasiperiodic lattice in the two Burgers vector directions in Fig. 11 are shown in Fig. 12. It is to be noted that in multicomponent quasicrystals the F-surface energy consists not
Elasticity, dislocations and their motion in quasicrystals
§4.2
(a)
403
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only of the phonon and phason strains but also of chemical disordering energy, as shown in the simulation of a model decagonal quasicrystal [134]. As seen in Fig. 12, due to the quasiperiodicity of the lattice, the F-surface also oscillates quasiperiodically. Essentially the same results have been obtained for other models [135,136]. Such an oscillation of the F-surface means that if a group of dislocations glide together on the same plane, the necessary phason dragging stress which is given by rd = F / ~ b i becomes negligibly small as the number of dislocation increases [133,137]. Thus, the Peierls potential plays a dominant role in the dislocation glide rather than the phason dragging stress. At high temperatures, we would expect thermally activated dislocation glide. Because dislocations in quasicrystals are subject to high Peierls potential, they should tend to lie along a Peierls valley. This has been verified experimentally by in-situ electron microscopy observation of dislocation motion, where dislocations are observed to move keeping a straight form oriented in a particular direction [15,122]. The thermally activated glide process for such a dislocation is the kink pair formation followed by motion of kinks along the Peierls valley, as illustrated in Fig. 13(a) [138]. For the thermally activated kink-pair formation to occur a phason fault relaxation should take place to some extent; otherwise the enthalpy of the final state (kink-pair state) cannot be lower than the enthalpy of the initial state (straight form lying in a Peierls valley) under a reasonably low stress that does not lead to fracture. As mentioned previously, two kinds of phason fault should be produced with dislocation translation, the re-tiling fault and the intra-tile fault. The re-tiling fault has
K. Edagawa and S. Takeuchi
404
Ch. 76
much lower energy than the intra-tile fault and furthermore the relaxation of the re-tiling requires cooperative jumps of a number of atoms leading to a much longer relaxation time than the relaxation of the intra-tile fault. For these reasons, it seems reasonable to assume that in the thermally activated kink-pair formation process and the kink migration process, the relaxation only of the intra-tile fault occurs. There are two regimes for the kink-pair formation process in crystals, one is the smooth kink regime and the other is the abrupt kink regime [139]. In the case of relatively low Peierls potential such as that in bcc metals or NaC1 crystals, the kink shape is smooth, and in the case of high Peierls potential such as that in covalent crystals of Si, the kink shape is abrupt. Due to the lattice periodicity along the Peierls valley, there exists a periodic potential also for the kink translation, which is called the Peierls potential of the second kind. For the smooth kink, the Peierls potential of the second kind is generally negligible and the kink energy can well be represented by the line tension approximation of the bowing-out dislocation. The dislocation mobility is determined by the kink-pair formation enthalpy, which has been given by Celli et al. [140], and Dorn and Rajnak [141]. For the abrupt kink, the Peierls potential of the second kind is also involved in the kink-pair formation enthalpy and also in the kink migration process. The kink-pair formation in the abrupt kink case has been treated in terms of the kink diffusion theory by Hirth and Lothe [142], which has been applied to the interpretation of the dislocation mobility in semiconducting crystals. As mentioned above, the thermally activated dislocation migration should be accompanied by the intra-tile phason relaxation in the close vicinity of the glide phane, which requires thermally activated short range atomic jumps. Thus, even if we neglect the Peierls potential of the second kind, the kink-pair formation of a dislocation in quasicrystals has to be treated by a kink diffusion theory in a similar manner to the kink-pair formation in the abrupt kink regime. The overall potential profile for the kink-pair formation process under zero applied stress is depicted by a dashed line in Fig. 13(b). As the kink separation 1 of the kink-pair increases, the self-energy of the kink-pair increases towards twice the single kink energy EK. Constantly increasing component with a slope F is due to the unrelaxed phason strain outside the glide plane. The potential hills superimposed aperiodically on the curve are due to the activation barrier for the atomic jumps in the intra-tile relaxation process. In this potential profile, the thermally activated kink-pair formation can take place only under a stress higher than rd = F/bll. The solid curve in Fig. 13(b) shows the enthalpy profile for the kink-pair formation under a stress higher than rd. The rate of the kink-pair formation under a stress r* -- ra - rd (r*: the effective stress, ra: the applied stress) is determined by the rate of the kink diffusion surmounting the potential barrier H~p towards larger l value. This rate can be formulated in perfect analogy to the kink-pair formation of the abrupt kink regime in crystal dislocations with the substitution of the kink migration enthalpy by the activation enthalpy of the atomic jump and the period of the Peierls potential of the second kind by the separation of the potential hills H' in Fig. 13(b). The frequency of the kink-par formation per unit length of a dislocation at low stress is given by T*blld
[
V~:p - 2-£-~8T v~exp -
kBT
(58)
Elasticity, dislocations and their motion in quasicrystals
§4.3
405
where vk is the kink vibration frequency. The kink velocity is written as
vk --
C*blldd'2 (H') kBT exp - k ' ~
'
(59)
where d' is the average distance between the phason jumps. For an infinite length of the dislocation, the equilibrium density of kinks is determined by the balance between the rate of the kink-pair formation and the rate of annihilation of positive and negative kinks, and the equilibrium kink separation is given by
[/~ -- x/2d' exp 2k8 T
"
(60)
Let the dislocation segment length lying in a Peierls valley be L. Depending on the relative magnitude of L and lk, there are two cases for the dislocation mobility; one is the kinkcollision case and the other is the kink-collisionless case. For a typical value of H~*p = 3 eV and d' = 0.5 nm, [ at 1000 K is calculated to be 3 cm, which is much larger than the dislocation segment length L, leading to the kink-collisionless case. Therefore, the dislocation glide velocity Vg is controlled only by the rate of kink pair formation on the segment length L, which is followed by the kink motion over the distance L, and is written as
r*bll d2____~L
(H~*p(r*) + H') ksT "
Vg= 2k~T vkexp -
(61)
4.3. Dislocation climb process
For a dislocation confined in a deep Peierls valley, the climb process should be the jog-pair formation followed by the jog motion, in a similar manner to the kink-pair formation and the kink motion, as illustrated in Fig. 14(a) [138,143]. Unlike the kink-pair formation by glide, the climb motion occurs by vacancy absorption or emission at jogs, i.e., accompanying atomic diffusion to or from the jog sites. As a result, the intra-tile relaxation is expected to occur simultaneously with the dislocation climb. Due to unrelaxed phason defects outside close vicinity of the climbing plane, there should be a constant increase of the phason strain energy with the jog motion, as in the case of the kink motion. Thus, the potential energy profile under no applied stress for the jog-pair formation is like that depicted by a dashed line in Fig. 14(b) and that under an applied stress T > F/bjj is shown by a solid line in the figure. In the case of the kink-pair formation process, the kink velocity is determined by the short range atomic diffusion to relax the intra-tile phason relaxation, whereas in the case of the jog-pair formation process, the jog velocity is controlled by the rate of vacancy absorption or emission at the jog, i.e. the self diffusion. Taking account of the fact that the formation energy at the dislocation core and the migration energy along the dislocation
406
Ch. 76
K. Edagawa and S. Takeuchi
(a) ,--
'....
1
--q
zSH
(b) 2~ f
J
p
_r -'-'------..__.~-cy*b II+F
Fig. 14. (a): Jog-pair formation process; (b): potential profile for a jog-pair under no applied stress (dashed curve) and under a stress higher then I'/bll (solid curve).
core of a vacancy are considerably lower than those in the lattice, the jog velocity under an effective resolved stress a* for the climb is written as [ 142]
vj =
4rcDsa*blla
exp
(AHs)
,
(62)
where a is the atomic distance along the dislocation line, Ds is the self-diffusion coefficient, A Hs is the difference between the activation enthalpy of the self-diffusion in the bulk and that along the dislocation core and ~ is the mean life length of a core vacancy along the dislocation line given by
~,=~/2aexp
(A.s) 2kBT
"
(63)
As in the case of the kink-pair formation, there are two cases for the dislocation velocity expression, the jog-collision case and the jog-collisionless case, depending on the relative magnitude of the equilibrium jog spacing [j - a exp(Ej/kB T) and the dislocation segment length L lying in a Peierls valley. Since E j is of the order of an eV, fj > L is generally satisfied. Thus, as in the case of the kink-pair formation, the jog-collisionless case is real-
Elasticity, dislocations and their motion in quasicrystals
§4.4
407
ized. For L < [j, the dislocation climb velocity Vc is determined by the rate of the jog-pair formation on the segment length L and is given by
4zcLDsvaCr*
(
Hj*p(Cr*)- AHs/2) Vc -- aS--~B-~-~nU/~l) exp -kBT '
(64)
Hjp is the activation enthalpy of the jog-pair formation. Writing Ds -- aZvD e x p ( - H s / k8 T) (Hs: the activation enthalpy of the self-diffusion; vo: the Debye frequency) and approximating Va = a 3, eq. (64) is rewritten as where
1) a
is the atomic volume,
4rcLa3cr * (Hj*p(~*) + (Hs - AHs/2) ) gc = kgT1--n(~-f/~ll) vDexp -kgT "
(65)
As described in Section 4. l, detailed TEM observations have revealed that in icosahedral A1-Pd-Mn climb process is the dominant deformation process over the glide process at high temperatures. This indicates Vc > Vg. Below, we make a crude estimation about relative magnitude of the velocities, which indeed justifies the fact Vc > Vg. From eqs (61) and (65), the ratio of Vg/Vc is given by
For a typical case of T = 1000 K, Hs = 2 eV and AHs -- Hs/2, the pre-exponential factor of eq. (66) is of the order of unity. H t may be comparable to the migration enthalpy of vacancy which is about Hs/2, and hence the second term of the exponent is small. Thus, the ratio Vg/Vc is determined essentially by the relative magnitude of H~p and Hj*p or the relative magnitude of the kink energy E~ and the jog energy Ej. Let r p / G --/3, the kink energy is given by Ek = 0.5~/-fiGdb 2 [139]. E j -- GbZh/{4rc(1 - v)} ~ O.1GbZh [142]. Assuming fl = 10-1 .,, 10 -2, Ek Ej
d = (0.5 ^-. 1.5)=.
h
(67)
Since the kink height d (period of the Peierls potential) is generally larger than the jog height h (atomic spacing), Ek can be larger than E j , and hence Vc can be larger than Vg.
4.4. Plastic homology Although the climb controlled deformation has been confirmed only in A1-Pd-Mn icosahedral quasicrystals, we assume that the high temperature plasticity of any icosahedral quasicrystals is governed by a common dislocation climb process. Then, we expect that some homologous relation should hold for icosahedral quasicrystals, as already established in bcc metals [144] and tetrahedrally coordinated crystals [145].
408
Ch. 76
K. Edagawa and S. Takeuchi
900
u24Fe12.5
800
A1705PdziMn8 5
700 t
~'~ 600
A170.4Pdzo.8Mn8.8
I21 C
r~ 500 Mg36Zns6Y8
400
Cd85Yb15
300
200
|
- A16oLi3oCUlo
100
.... I . . . . . .
400
I
500
........
I
600
.................
I
700
~ i
800
.
i
............
900
I
1000
.... I . . . . . . .
1100
I
1200
TEMPERATURE (K) (a) Fig. 15. (a)" Temperature dependence of the upper yield stress for icosahedral quasicrystals of A1-Pd-Mn [16, 146], A1-Cu-Fe [147], Cd-Yb [148], Mg-Zn-Y [149] and A1-Li-Cu [150]. (b): Normalized upper yield stress vs. normalized temperature replotted from (a).
Fig. 15(a) shows the temperature dependence of the upper yield stress for various icosahedral quasicrystals [16,146-150]. The temperature range of the plastic deformation differs largely among the icosahedral quasicrystals. In Fig. 15(b), we replotted the data to the normalized upper yield stress vs. the normalized temperature relations [ 143], where the upper yield stress is normalized with respect to Young's modulus E and the temperature with respect to the material parameters of Young's modulus times cube of the average atomic diameter fi divided by the Boltzmann constant, i.e. E~3/ks. As seen in Fig. 15(b), a homologous relation holds approximately.
Elasticity, dislocations and their motion in quasicrystals
§4.4
409
B
A1-Cu-Fe A1-Pd-Mn 4Mg-Zn-Y
Cd-Yb
3-
2-
A1-Li-Cu
1-
0
0
I
I
1
2
I
I
3 4 T/(Efi3/kB) ( 10 -3) (b)
-
•
I
5
Fig. 15. (Continued).
For
AHs -- Hs/2, the climb velocity is written
Vc -
-k-~8T4rCLa3°*VD I H* + (3/4)H~I ~-n~;~i) exp Jp kgT
as (68)
Assuming Hs ~ 3 eV, a - b l l - 0.3 nm, L - 10 ~am and VD -- 1014 s -1, the preexponential factor at T ~ 1000 K of eq. (68) is estimated to be 7 x 107 ms -1 at a low stress of o-* = 10 MPa ( ~ 1 0 - 4 E ) . Assuming the mobile dislocation density at the upper yield point is of the order of 109 cm -2 [151], the dislocation velocity at the yielding stage is 3 x 10 -8 ms-1 for the usual strain rate of 10 -4 s -1 . At the yielding stage at high temperature and low stress ( ~ 1 0 - 4 E ) , the value of exponent of eq. (68) is estimated to be 35. The activation enthalpy of dislocation motion has been evaluated experimentally from the temperature dependence of the yield stress and the stress dependence of the activation
K. Edagawa and S. Takeuchi
410
Ch. 76
0
w
50
Mo 0 J
40
d=7.5 ~x
>
30 Ta Oo Cr 0
Pt 201--
Feo oPd O
O
v
Nb
I0 Opb
0 o'
0
I
I
I
I
I
I
I
1
2
3
4
5
6
7
Hd (eV) Fig. 16. The relation between E a 3 and the activation energy of self-diffusion E d for cubic metals.
volume obtained by the stress relaxation experiments. The obtained values of the exponent are largely scattered: 40 [152], 60 and 43 [146], 45 [153], 65 [154], 20 [155] and 35 [148]. However, taking account of a variety of uncertainty in the experimental evaluation due to effects of work softening, recovery during the relaxation test and temperature dependent dragging stress etc., the above estimated value of 35 does not seem to be inconsistent with the experimental results. From the homologous plot in Fig. 15(b), the deformation temperature at low stress satisfies T ~ 4.2 x 10 -3 E~ 3/ k~. This indicates
Hjp + (3/4)Hs 4.2 x 10 -3 E~ 3
= 35.
(69)
Elasticity, dislocations and their motion in quasicrystals
411
H~p ~ 2Ej (Ej: the jog energy) at low stress,
Since
3
2Ej + -~H~ ~ 0.15Eft 3
(70)
The energy of a jog with a height h is written as [142]
Gb2h Eb2h Ej = 4rr(1 - v) = 87r(1 - v2)"
(71)
Approximating b ,~ a and h ~ ~, and inserting v ~ 0.25 [62,64,65,68], one obtains
Ej -- 0.04Eft 3.
(72)
In Fig. 16, we plot Ea 3 values against Hs for 16 cubic metals, showing a correlation Ea3/Hs ~ 7.5 except two transition metals V and Nb. Since the same vacancy mechanism has been suggested for quasicrystals as for usual metals by diffusion experiments for specific atomic species in quasicrystals [156,157], we assume here that the same correlation holds also for icosahedral quasicrystals, i.e. Eg~3/Hs = 7.5. Then,
3H s -- 0.10Eh 3
(73)
m
4
From eqs (72) and (73), 3
2Ej + -~Hs -- 0.08Eft 3 + 0.10Eft 3 -- 0.18E~ 3
(74)
Taking various simplified assumptions made in deriving eq. (74), the agreement between the experimental relation of eq. (70) and the estimated relation of eq. (74) seems reasonable. The above results indicate that about a half of the activation enthalpy of the dislocation climb is due to the kink-pair energy and the other half to the activation enthalpy of self-diffusion.
Appendix A:
Irreducible strain components
The irreducible strain components in the icosahedral system are given below [42,58].
+
1
1 (/7-1
1 u5 =
+
u 11 -- r u 2 2 -3r- u33) ~/-2u 12 V/2u23 x//2u 31
m
/
412
Ch. 76
K. Edagawa and S. Takeuchi
~/~(1011 -- 10/222) _ w4
1 ~
T-11021 -~- "61012 r_11032 _+_"61023
,
w5--
1
~/~(.c1021 _ .c_11012) ~/r2( "61032 _ r-11023)
~
.
(A.1)
The irreducible strain components in the decagonal system are given below [47].
Ul = Ull nt- U22, w6--
u 1-u33,
I I/)l 1 nt- 10221 1021 - 10/212
u5--
iu31] u23
W7--[1013 1'w23
,
w8--
u6--
lUllu22] 2u12
,
I t/)l 1 - 1022 ] W21 -Jr-1012
(A.2)
Appendix B: Derivation of eqs (47) and (48) by the generalized Eshelby's method U -- U
llJ --
By use of Cijkl in eq. (15) and Cijkl
aC~ (p) =
tO
in Table 3, we obtain
(1. 4- 2#) 4- #p2 (1. 4- #)p
(1. 4- #)p # 4- (I. 4- 2#)p 2
0 0
0 0
0 0
0 0
0
0
0
0
# 4- #p2
0
0
A 4- Bp 2
0 0
2Dp 0
0 2Dp C 4- Ap 2 0
0 0 0 B 4-Cp 2
0 0
0 0
"
(B.1) The determinant of a c~t~(p) reduces to the product of four subdeterminants: ]a °e'8(p)]- D1. D2. D 3 - D 4 , Ol --
()~ + 2#) + #p2 0~ + ~)p kt + ()~ + 2 # ) p 2 (~ +U)P
03 =
A + Bp 2 2Dp
D2 = # + / z p 2,
2Dp C + Ap 2
(B.2)
D4 = B + Cp 2.
Of them, D2 and D4 are relevant to the Burgers vectors bll = [0, 0, bll] and b± = [0, 0, b±], respectively. The roots of D2- D4 - - 0 are
p 1 - i,
p2 -
,
(B.3)
Elasticity, dislocations and their motion in quasicrystals
413
and p7 - P~ and p8 -- P~. For Pl and p2, eq. (41) gives, respectively, 0 0 1
A(1)-
0 and
A (2) --
o
(B.4)
i "
0
Eqs (42) and (43) become, respectively, Re[D(1)] -- bll,
Re[D(2)] -- b±
(B.5)
and Re[#iD(1)] - 0 ,
Re[~/BCiD(2)]--0.
(B.6)
From eqs (B.5) and (B.7), we obtain simply D(1) - bll,
D(2) -- b±.
(B.7)
Inserting eqs (B.3), (B.4) and (B.7) into eq. (44) and into eq. (46) lead to eq. (47) and eq. (48), respectively.
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[66] [67] [68] [69]
[70] [71] [72] [73] [741 [751 [761
[77] [78] [79]
[80] [81] [82]
[83] [841
[85] [861 [87] [881 [89] [90]
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