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Shapes of quasicrystals
Prog. Crystal Growth and Charact. Vol. 34, pp. 237-249, 1997 © 1997 Publishedby Elsevier Science Ltd Printed in Great Britain. All rights reserved 0960-8974/97 $32.00
Pergamon
PII: S0960-8974(97100018-1
SHAPES OF QUASlCRYSTALS K. C h a t t o p a d h y a y , N. Ravishankar and R. G o s w a m i Centre for Advanced Study, Departmentof Metallurgy, Indian Instituteof Science, BangaJore560 012, India
ABSTRACT In tile present article, we review tile present status of our understanding of the shapes of quasicrystals. Although theoretical efforts exist to determine the equilibrium shape of quasicrystals, very little experimental information exists. We report some preliminary results of our work ill this direction. On the other hand, beautiful growth shapes of quasicrystals are reported in the literature. However, only recently theoretical attention is paid to the problem of atomistic mechanism of growth. Ill spite of linfited studies, comparison of theory and experiments reveal a difference in growth behaviour between crystal and quasicrystal.
KEYWORDS Equilibrium Crystal Shape, Roughening Transition, Quasicrystals
INTRODUCTION We shall, in this article primarily deal with the morphology and growth behaviour of quasicrystals. Since tile point group symmetry of a crystal dictates the morphology of a crystal, the noncrystallographic point group symmetry of qnasi,:rystals (Shechtman et al., 1984) promises distinctive morphological features. On the other hand, qnasicrystals can be treated just like any other phase in a l)hase diagram. Thus all known phase transformations, in principle, can be expected with actual occurrenceguided by free energy considerations. Therefore, quasicrystals can form as a single phase as well as a primary phase in multiphase mixtures. It can exist as eutectic mixtures and also be a product of solid state transformations.
MORPHOLOGY OF QUASICRYSTALS (EXPERIMENTAL STATUS) Several investigations in the early stages were carried out to characterise tile nlorphology of quasicrystals. Most of the initial work was carried out for AI-Mn and AI-Mn-Si qua.sicrystals. Although one often obtains a dendritic microstr,wtm'e, an early work by Chattopadhyay el, al. (1985) indicated a well-facetted morphology for the quasicrystals. The basic shape of the small icosahedral quasicrystals in AI- Mn alloys was investigated bv several investigators by Transmission Electron Microscopy (Nishitani et al., 1986, Thangaraj et al., 1987). One of the initial difficulties was the problem of identification among the possil>le different shapes consistent with the icosahedral point group symmetry. It was found that the shapes are indistinguishable when projected along higher symmetry axes like five-fold and three-fi)hl axes. tlowev~'r, as shown in Fig. l due to Thangaraj et al. (1987), the projection along the two-fold axes enable dislinction between different shapes and establish the pentagonal dodecahedron as the shape for the AI:Mn type of quasicrystal. The dis~'ow-ry of a sta237
238
K. Chattopadhyay et al.
ICOSAHEDRON
DODECA HEDRON
TRIACONTAHEI]RON
@
L..,'\ ]
L,...I _ ~
• 6)4°t I
\~,V"-") ,,i ,s'~/
Ia
~'.iI/ 6..~.4e / ~,L41
Fig.l. Shapes of an icosahedron, dode,'ahe(h'on and triacontahedron under different projections al general view, h) aloug 5-fold axis, c) along 3-fi~hl axis. (I) along 2-fi)hl axis. (Thangaraj. 1987)
Shapes of Quasicrystals
239
ble quasicrystal with large pentagonal dodecahedral solidification morphology followed (Ohashi and Spaepen, 1987). In contrast, AI-Cu-Li quasicrystals were found to have rhombic triaeontahedral morphology. (Dubost et al., 1986). With the recent discovery of equilibrium icosahedral quasicrystal in AL-TM ternary system of Al-Cu-Fe and AI-Pd-Mn (Tsai et al., 1987, Tsai et al., 1990), large crystals having well-defined shapes of pentagonal dodecahedron and icosidodecahedr(2n can be routinely observed. Recently an equilibrium quasicrystal in Mg-based alloys (Mg-Zn-RE, RE-rare earth element) has been discovered. These also give the peutagonal dodecahedral morphology (Tsai et al., 1994). The decagona] quasicrysta] in A1-Cu-Co also exhibits well defined decagonal prism morphology. (He et al., 1990). There also exists report of a pencil shape of a one dimensional quasicrystal (Tsai et al., 1992). Thus tbe morphology of a quasicrystal often reflects the inherent non-crystallographic symmetry. For any detailed discussion of the crystal morphology, two situations need to be distinguisbed. The equilibrium shape of a crystal is determined by a surface energy principle and can be different from the growth morphology which depends on the kinetics of the growth and processing conditions. We now examine the morphology of quasicrystals under these two conditions.
EQUILIBRIUM SHAPE OF A CRYSTAL Before we consider the problem of equilibrium shape of a quasicrystal, a brief introduction to the subject with reference to the crystalline case is helpful. Surface energy is defined as the work done to create unit area of a surface. In case of fluids, this is isotropic since it does not depend on direction. On the other hand, this is anisotropic for the case of crystals. This is because the number of bonds cut per unit area depends on the orientation of the surface. This can be understood by considering a simple two-dimensional square lattice (Fig. 2). We wish to calculate the number of 1)onds cut per unit length as a function of orientation. Along any arbitrary direction which passes through the origin and another lattice point (x,y), the total number of bonds cut is given by x + y. Thus, the number of bonds cut per unit length would be given by N = (x + y) / ~ y 2 ) t
.....
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240
K. Chattopadhyay et al.
thermodynamics. This work established tlle existence of the relation between surface energy and the equilibrium crystal shape (ECS). The statement of the problem of equilibrium shape is as follows. Given a particular volume of material, what is the shape it umst adopt in order that the total surface energy is a oonstant. This condition can he expressed as r a~ dA, = rain, V = constant. Wulff (1901) proposed a solution to this p,'oblem. Wulff's theorem states that if planes are drawn perpendicular to the radius vectors where they cut the gamma-plot, then the ironerenvelopeof these planes is geometrically similar to the equilibrium shape. Many important and interesting features of the nature of the gamma-plot and the ECS were pointed out by Herring (1951). It was shown that when only nearest-neighbour interactions were considered and relaxation effects were neglected, then the gmmna-plot consisted of segmeuts of spheres which when extended passed through the origin. Such a plot led to cusps appearing along some special (low-index) orientations. The corresponding equilibrium shape would consist of flat facets of finite extent for these orientations. As pointed out by Herring, to ensure that the surface of the equilibrimn shape has a flat portion with art orientation normal to a given vector OX of the plot (Fig. 3), l) it, is necessary and sufficient that the plot has a pointed cusp at X, and 2) it should be possible to draw a sphere between O and X which lies entirely within the gamma-plot. The regions outside the Wulff envelope do not contribute to the equilibrium shape and remain passive. For crystals, there exist sharp cusps in the high-symmetry directions at low temperatures. Thus, the E(~S reflect the symmetry of the crystal. Ideally at 0 K, the normal crystals should be 1)ounded only by flat surfaces. Thus any orientation which is not a part of the equilibrimn shape will undergo a reconstuction to a hill-and-valley structure. This behaviour is marginal if we were to consider first nearest-neighbour interactions alone because the energy of any arbitrary orientation is identical to the em,rgy of the hill-and- valley structure which comprises it. In the abow• analysis, the tenll)erature has been assumed to be absolute zero to avoid the effects of thermal disordering and entropy effects. The structure of surfaces can be described in terms of the TLK model (Gruber and Mullins, l!l(i7). In lhis model, surfaces close to a low-index plane can l)e described as containing terraces of the low-index orientation separated by steps whose density depends on the extent of deviation from the low-index plane. For more complex 1)lanes, kinks have to introduced in the steps to obtain the overall orientation of the surface. The energy of such a surface can I)e deeoml)osed in to the sum of the terra,'e energy per unit area and step energy per unit length. fi(O) --et cos(O) + (es/a)(sin(O))
,
~
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) / r (h)
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Fig.3. Wulff constructiou to obtain tile crystal shape r(h) from tile polar plot of surface energy f~(m). (Herring. 1951)
Shapes of Quasicrystals EFFECT
OF
TEMPERATURE
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24"
EQUILIBRIUM
SHAPE
The effect of temperature is to increase tile disorder in the surface. The main effects are as follows, For higher-index planes, the main effect is that the density of kinks on the steps increases until all long range order is destroyed. Figure 4 shows a typical stepped structure at some high temperature. The step free energy is given as a combination of the usual energy and entropy terms f~ = e, - TS~
a ~z.__._._.: ::..::: ::___Z_.___. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Fig.4. Excitations oil the steps leading to tbrmation of kinks on steps and islands on tile terraces. As the density of kinks increases, the entropy contribution increases and at a critical temperature called the roughening transition temperature, f~ goes to zero. Figure 5 shows the typical variation of the step free energy as a function of temperature. Each hkl face has its characterstic roughening temperature with the close-packed planes having the highest roughening temperature. Macroscopically, this marks the transition fi'om a facetted ])lane to a smoothly curved one. For low-index planes, the scenario is much more complicated. The surface disorders by formation of vacancies and adatoms on the surface which leads to the formation of islands on the surface. The statistics of kinks on the steps is simple because the [)roblem is one-dimensional. On the other hand, the problem of surface disordering is complicated since it is a co-operative phenomenon analogous to ferromagnetism. The exact sohttion to the two-dimensional problem has been given by Onsager (1944) in context of the two-dimensional lsing model for ferromagnetism. This is just a two level problem while the actual surface atoms are not confined to two levels alone. Burton at al. (1951) have extended this to a Bethe type of approximation to investigate the effect of the higher levels.
fs(T)
T 1
. temperature Fig.5. Variation in step free energy as a function of temperature. The step free energy vanishes at the roughening temperature Tn. (Wortis, 1988)
242
K. Chattopadhyay et al.
For a crystalline lattice, the roughening transition temperature in general scales with lattice parameter. Thus a crystal with larger lattice parameter shows more facetting tendency. The sharp edges and corners of tile crystal at 0 K start rounding as the temperature is increased. Thus the crystal shape consists of sharp facets separated by rounded edges and corners at higher temperatures. Andreev (1982) pointed out that the Wulff construction is just the two- dimensional Legendre transformation from the free energy to the crystal shape. This means that the crystal shape can be treated as a free energy. In that case the transition from facetted to rough can be treated as a two-dimensional phase transition. There are two types of thermal ewllution that have been observed in crystals. The first type of thermal evolution which has been designated as Type A evolution is as follows : At T = 0, the crystal is polyhedral and as the temperature is increased, rounded regions appear between the facets at arbitrarily low temperatures. There is no slope discontinuity at the point where the rounded region starts. The proportion of the rounded regions increase on increasing the temperature aud each fiat facet disappears at its characteristic roughening temperature unless bulk melting intervenes this transition. This type of evolution has been observed in the case of hcp 4He in contact with superfluid helium below 1.4 K. as well a.s in tile case of metals crystals like Pb and In. This temperature evolution can be convenieutly represented in a phase diagram. Such a diagram has been shown in Fig.6 . Since the crystal has tbur fold symmetry, it is sufficient to consider only angles between 0 and 90. At T = 0, the crystal has the (100) and (010) faces separated by a sharp edge. As the temperature is raised, there appears a rough region in the phase diagram (corresponding to the rounded region in the shape). The angular range of this rough region increases until the temperature TR where the (100) type face vanishes.
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- phase ~u
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Fig.6. Phase Diagram for Type A thermal evolution of the crystal shape. The diagram is tile locus of the crystal edges in the equitorial plane only. (Wortis, 1988) Tile free, energy expression near the critical point of a second-order transition exhibits power-law behaviour (Wortis, 1988). y = x" , where n is typically a non-integer. Since it has been shown that tile crystal shape can be treated to be a free, energy, we can expect that tile shape of the crystal near a smooth edge would folh)w a power-law behaviour (for Type A thermal
Shapes of Quasicrystals
243
evolution). There are good experimental evidences to believe that this behaviour corresponds to the Pokrovsky-Talapov x3/2 law. y = A (x- xc)" , where xc is the critical edge and n, an universal exponent bas a value of 1.5 . Tile other type of thermal evolution (designated Type B) is tile first-order transformation analogue which has a slope discontinuity. This has been experimentally observed in Au. The Type B evolution can change to Type A evolution at higher temperatures. Tile Type A thermal evolution implies that the T = 0 Wulff plot is degenerate in the sense tbat a whole bunch of normals pass through the same corner. This is because of the assumption of nearest-neighbour interactions alone. At infinitesimally small temperatures, thermal fluctuations break the degeneracy and thus gives rise to Type A evolution. Oil adding higher neighbour interactions, a) additional faces may be added to the equilibrium shape and the transition may still be Type A or b) tile transition may follow type B behaviour depending on tile whether the higher neighbour interactions are attractive or repulsive. For the case of crystals, new faces cannot be stabilised by second-nearest neighbour repulsive interactions. Tile equilibrium shape obtained by pure attractive interactions contains the minimum possible facets among all the equilibrium shapes possible.
EQUILIBRIUM SHAPE OF QUASICRYSTALS THEORETICAL D E V E L O P M E N T Starting with Kepler, polyhedral shal)es are always associated with crystalline structures having periodic arraugement of atoms. Thus, it was not clear whether quasicrystals can have facets and definite polyhedral shapes. He et al. (1987) adol)ted a bond oriented quasiglass model to determine whether facets are possible in such a case using only attractive potential. The results clearly indicate a strong tendency for facetting. The possible set of shapes computed by these investigators for different bond strengths is shown ill Fig.7. One interesting conclusion of this work is the exclusion of pentagonal dodecalmdron as a possible ES. l!sing Penrose tiling. Garg and Levine (1987) obtained a facetted morphology at 0 K for quasicrystals. In a further development, Ingersent and Steinhardt (1989) made a detailed study of the shapes of (luasicrystals by considering the combination of both attractive and repulsive interactions. They established that the equilibrium shapes of quasicrystals had some important differences from the crystal case. The main difference is that the shape obtained using purely attractive interactions contains more than the nfinima] number of facets. The shape with minimum number of facets consisite,~t with the symmetry considered is obtained by including higher neighbour repulsive interactions. This may ]lave a bearing on the deternfination of the atomic structure of the quasicystals. This work also established firmly that the pentagonal dodecahedron can appear as an equilibrium shape when a more realistic situatiou of both attractive and repulsive potentials are taken into account. Tytfical shapes obtained by these investigators are shown in Fig.8. Tile first realistic calculation of quasicrystalline phase is due to Lei and Henley (1991) who used a cluster model. These clusters have icosahedra[ synnnetry, similar to that observed in AI-Mn-Si and Al-(',u-Fe and arranged in a cubic :~/2 crystalline approxilnant structure with 32 clusters per cell The results of the ES calculation suggests that the planes normal to five-fold and two-fold axes have the lowest surface energies and shouhl dominate the shape. However, there exists I)resence of small planes normal to three-fold direction. The phenomenon of roughening of the corners and edges at high temperatures has to be considered. This aspect is more ameuable to experiments. There have been attempts to theoretically study the roughening transition. Tile argument that roughening scales with lattice parameter implies that true roughening temperature of a quasicrysta] will be infinity. However, it has been shown that a pseudo-roughening transition does indeed exist for a two-dimensional Penrose lattice although the magnitude of the exponent was found to be much smaller. (Garg and Levine (1987), Lipowski and Henley (1987)). Lipowski and [lenl¢~y also studied this aspect for the icosahedral as well as
244
K. Chattopadhyay et al.
the decagonal quasicrystal. They concluded that a roughening transition exists for the decagonal quasicrystal along planes normal to the ten-fold direction whereas in the case of the icosahedral quasicrystal, the five-fold facets remain flat at all temperatures (Table 1). They also found that the rounding exponent for a Penros~lattice type model of the icosahedral quasicrystal was lower than 3/2 (Table 2). Table 1. A smnmary of the expected nature of tile roughening transition temperatures of the surfaces in crystals and quasicrystals Nature of the Roughening Transition Crystal
Finite scales with the lattice parameter
Decagonal 10-fold plane
Finite
Rectangular facets
Infinite
(Quasiperiodic planes) Icosahedral
Infinite
Table 2. The theoretically predicted rounding exponent A (z(x) = x 'x) for the decagonal and icosahedral quasicrystal Quasicrystal Decagonal Facet
A <
3/2
2.2 (random tile model) Rectangular Facet lcosahedral
<
3/2
EXPERIMENTAL STUDY
Tile experimental study of equilibrium shapes is beset with problems of equilibrating large size crystals/qua.sicrystals. There may I)e a narrow window of realistic temperature for carrying out equilibration heat treatment abow~ which it melts aud below which it takes impractically long for equilibriation (Wortis, 1988). There art~ problems associated with temperature control as well as problems of adsorption. Another w%, of studying these shapes is by nucleation of these phases in a liquid. It is well known that during the initial stages of nucleation, interracial energy effects predonfinate and hence the critical nucleus that forms is identical to the equilibrium shape since this shape has tile minimmn surface energy per m6t volume. This, however may not be valid at later stages when substantial growth has taken place. Nucleation can be promoted by undercooling the melt sufficiently. Rapid solidification of dilute AI-Mn alloys provide substantial undercooling promoting the nucleation of the quasicrystal in the melt. Later the Al nucleates and grows very fast trapping the small quasicrystallites. This provides an excellent opportunity to study the equilibrium shape and the roughening behaviour of the quasicrystals. It is to be noted that the shape reflects the shal)e al the trapping temperature. A log-log plot of the boundary co-ordinates is used to analyse the critical exponent in case of the shal)es that are observed experimentally. Rottman et al. have discussed about the region from which the actual data is to be taken to avoid effects due to atomic scale rounding or higher-order terms due to scaling corrections. Fig.9 shows the actual crystal profile in comparison to the ideal profile and the window region flom which data can be used for determining the index.
Shapes of Quasicrystals
245
Fig.10 shows the results for such an analysis from all A1-5% Mn alloy. The slope in this case is found to be close to 3. Ananlysis of results from various samples indicate that the slope in this case generally is between 2 and 3.
G R O W T H M O R P H O L O G Y OF QUASICRYSTALS At a first glance, the quasiperiodicity does not seem to yield any dramatic difference in the growth behaviour of quasicrystals in comparison to crystals. It acts like any other crystalline phase in a phase diagram. One, therefore, observes primary, eutectic as well as peritectic growth in quasierystals. However, the quasiperiodicity may influence the atomic growth mechanism, For example, the faceted growth of a crystal takes place below tile roughening transition temperature. As mentioned earlier, the latter scal~ with lattice parameter. Since the quasicrystal can be considered as a periodic crystal with infinite periodicity, the roughening transition temperature should be infinity. Practically this means that quasicrystals are expected to have faeeted growth. The expectation is borne out by a large number of reported observations. All tile slowly grown equilibrium quasicrystals show a pronounced facets in the growth form (Duhost et al., 1986; Tsai et al., 1987, 1990, 1994). Because of the faceted growth, the growth form reflects the 1)oint group sylnmetry of the quasicrystal and one often observes shapes like pentagonal dodecahedron. [n such a situation, the shape also reflects the slowest growing planes which has highest atomic density. Abundance of pentagonal facets in as grown quassicrystal indicate that these planes have highest density and therefore, slowest growth rate. The dendritic growth of quasicrystals is common in rapidly solidified alloys. The early reports on AI-Mn quasicrystals often reported well-rounded dendrites. It was pointed out that even if this quasierystal can grow in a faceted manner in twin-rolled samples which yield a lower cooling rate in comparison to melt spinning (Chattopadhyay et al., 1985). It is, therefore, clear that the non-faceted growth of quasicrystal is due to dynamic roughening at high undercooling of the melt. All faceted crystals can undergo a dynamic roughening below a critical undercooling of the melt. This leads to a continuous growth and yields a rounded growth form. The reasoning is equally valid for quasicrystals having a quasiperiodic potential. The first theoretical attempt to analyse the atolnistic growth behaviour of quasicrystal in comparison to crystals is due to Toner (1990). lie analysed the stepped growth of faceted quasicrystalline interface. It was found that the qua,sicrystals grow 1)y nucleating a step height (h~) which is related to tile chemical potential difference across the interface (A u) h~ ~ (A u) -'/3 Thus as the chemical potential decreases, tile height of the step needed to grow a quasicrysta] increases. On the other hand, the step height for growth in a crystal is related to the lattice parameter and is independent of the difference in chemical potential. Further for crystals, the critical potential barrier for nucleation of steps vanishes at the roughening transition temperature while for quasicrystals, it always remanis finite. One of the direct consequences of this analysis is the nature of the growth step in a quasicrystal. It is predicted to l)e much larger than the crystalline case. Thus, the step growth of quasicrystalline interface can easily he observed by normal hot-stage transmission electron microscot)y without resorting to high-resolution microscopy. The growth of decagonal quasicrystals in a icosahedra] quasicrystalline matrix in the solid state is ideal for testing the above predictions. This is because of the fact that the decagonal phase has one periodic direction which should behave like a n(Jrmal crystal. The other directions will experience only the qua,siperiodic potentials and will grow 1)y the movements of large steps. Schaefer and Bendersky (1986) have earlier shown an epitaxial relatiolL l)etween icosahedral and decagonal crystal with tenfold symmetry plane or the decagonal plane (the periodic plane) always nucleating on the five-fold plane of the icosahedral phase. High-resolution imaging of these interfaces reveals step heights of the order of the spacing of the periodic planes (Thangaraj et al., 1987). On the other hand, in-situ electron microscol)ic observation of the icosahedral to decagona] quasierystalline transformation in
246
K. Chattopadhyay et al. center of crysta I
Ideal crystal profile
'" \ Z i i
on0,,0c.,
x
x
"crillcoI reqion'L----~,
i"-eck:je \
..,
Actual c r y s t a l p r o f i l e
I/R rounding-
noncritical
~
1
~
Fig.9. a) A log-log plot of ideal crystal shat)~, coor(linates to determine tile critical eXl)[ment for the transition, b) shows tile window from which data can be used for a,alysis.
3.1( --
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Shapes of Quasicn~stals
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the quasiperiodic directions by Kim et al., (1990) reveals the growth to occur by the movement of large steps. A schematic diagram from their work is shown in Fig.11. Thus, there exixts differences in the nature of the growth of quasicrystals and crystals. More experimentM studies in this direction is clearly necessary.
Ledge GrowtN Dir.
Facet __,..5rov~ th dir. 2f A
Fig.11. A schelnatic plot showing the stepped structure ill a quasicrystal. (Kim et al., 1990) CONCLUSIONS
Although there exists extensive literature on the structure and properties of quasicrystals, the study of the morplmlogical evolution of quasicrystMs is limited. Since morphology will depend on the nature of the qnasiperiodic potential, tile symmetry and tile atom attachment kinetics, such a study will yiels rich information on the nature of the quasicrystals. It is hoped that more such studies will be undertaken in the near future. ACKNOWLEDGMENTS
It is a plea.sure to acknowledge tile discussions with Professor S. t{anganathan regarding the problems of shat)e and symmetry. REFERENCES
Andreev A.F., (1982), Faceting Phase Transitions of (',rystals. Sov. Phys. JETP., 5:1, 1063. Burton, W.K.. N. ('abrera and F.(~. Frank, (1951). The growth of crystals and the equilibrium strncture of their surfaces., Phil. Trans. A, 243,299-358 (:hattopadbyay, K., S. Ranganathan, (I.N. Subhanna and N. Thangaraj (1985). Electron Microscopy of quasicrystals ill rapidly solidified AI-Mn alloys. Scripta Metall., 19, 767 I)ubost B...I.M. Lang, M. Tanaka, P. Sainfort, and M. Audier (1986). Large Al-(;u-Li single qna,~icrystal with triac,ontahedral solidification lnorphology. Nature (London)., 324, 48 (larg A and D. Levine, (1987), Faceting and Roughening in Quasicrystals, Phys. Rev. Lett., 59, 1683 (Iruber E.E. and W.W. Mullins. (1967). On the Theory of Anisotropic (',rystalline Surface Tension., J. Phys. (',hem. Solids., 28, 875-887. He L.X., Y.K. Lu, X.M. Mang and K.H. Kuo., (1990), Philos. Mag. Lett., 61, 15. Herring (~., (1951). Some theorems outhe free energies of crystal surfaces. Phys. Rev., 82, 83
Shapes o! Quasicrystals
249
Ho T., .I.A..Iaszezak, Y.H. Li and W.F. Saam (1989), Faceting in bond-oriented glass and quasicrystals, Phys. Rev. Lett., 59, 1116. lngersent K. aud P.J. Steinhardt (1989), Equilibrium Faceting Shapes for Quasicrystals, Phys. Rev. B., 39, 980 Kim D.H., K. Chattopadhyay and B. Cantor, (1990), An In-situ Electron Microscopic Investigation of tile Icosahedral to Decagonal Quasicrystalline Transformation, Phil. Mag. A., 62, 157 Lei T. and C.L. Henley, (199l), Equilibrium Faceting Shape of Quasicrystals at Low Temperatures : Cluster Model, Phil. Mag. B, 63, 677 Lipowski R. and C.L. Henley. (1987). Equilihrium Shapes for Ideal and Random Quasierystals, Phys. Rev. Lett., 59, 2394 Nishitani, S.R., H. Kawara, K.F. Kohayashi and P.H. Shingu (1986). J. Cryst. Growth., 76, 209 Ohashi W. and F. Spaepen., (1987), Nature, 330, 555 Onsager, L., (1944), Phys. Rev., 65, 117-149 Schae.fer R.J., and L. A. Bendersky, (1986), Replacement of Icosahedral AI-Mn by Decagonal Phase, Scripta Metall., 20, 745 Shechtman. D., I. Blech. D. (h'atias and .].W. (~ahn (1984). Metallic phase with Long-Ranged Orientationa] Order and No Translational Symmetry., Phys. Rev. Lett., 53, 1951 Thangaraj, N., K. (~hattopadhyay, E.S.R. (;opal and S. Ranganathan (1987). On tim Morphology of lcosahedral qua~sicrystals in AI-Mn alloys. Key Engineering Materials., 13-15, 245 Thangaraj, N.. G.N. Suhbanlla, S. Ranganathan and K. Chattopadhyay, (1987), Electron Microscopy and Diffraction of Icosahedral and Decagonal Quasicrystals in AI-Mn Alloys, J. Micros., 146. 7 Toner .]., (1990). Growth and Kinetic Roughening of Quasicrystals and Other Incommensurate Systems, Phys. Rev. Lett., 930. Proc. Adriatico Res. Conf., Eds. Jaric M.V., and S. Lundquist, World Scientific, Singal)ore, 250. Tsai A.P., A. lnoue and T. Masumoto (1987). Preparation of a new AI-Cu-Fe qua,si(:rysta] with large grain sizes by rai)id solidification..I. Mater. Sci Lett., 6, 1403 Tsai A.P., A. Inoue, Y. Yokoyama and T. Masumoto (1990). New icosahedral alloys with superlattice order in the A]-Pd-Mn system prepared by Rapid Solidification. Phil. Mag. Lett., 61, 9
Tsai A.P., A. Sato, A. Yamamota, A. Inoue and T. Masumoto (1992). Stable one-dimensional Quasicrystal in a Al-(:u-Fe-Mn System..]pn..I. Appl. Phys., 31,970 Tsai A.P.. A. Nikura, A. Inoue, T. Masumoto, Y. Nishida, K. Tsuda and M. Tanaka (1994). Highly Ordered Structure of Icosahedral Quasicrystals in Zn-Mg-RE (RE - rare earth metals) Systenls. Phil. Mag. Left., 70, 169. Wortis M., (1988), Equilibrium (Irystal Shapes and Interracial Phase Transitions, in (:henfistry and Physics of Solid Surfaces, ~, Ed. R. Vanselow, Springer Verlag, Berlin Wulff G., (1901). Zur frage (let' geschwindigkeit des wachsthums und der aufloesung der krystallflaechen. Z. Kristallogr., 34,449
Pergamon
PII: S0960-8974(97100018-1
SHAPES OF QUASlCRYSTALS K. C h a t t o p a d h y a y , N. Ravishankar and R. G o s w a m i Centre for Advanced Study, Departmentof Metallurgy, Indian Instituteof Science, BangaJore560 012, India
ABSTRACT In tile present article, we review tile present status of our understanding of the shapes of quasicrystals. Although theoretical efforts exist to determine the equilibrium shape of quasicrystals, very little experimental information exists. We report some preliminary results of our work ill this direction. On the other hand, beautiful growth shapes of quasicrystals are reported in the literature. However, only recently theoretical attention is paid to the problem of atomistic mechanism of growth. Ill spite of linfited studies, comparison of theory and experiments reveal a difference in growth behaviour between crystal and quasicrystal.
KEYWORDS Equilibrium Crystal Shape, Roughening Transition, Quasicrystals
INTRODUCTION We shall, in this article primarily deal with the morphology and growth behaviour of quasicrystals. Since tile point group symmetry of a crystal dictates the morphology of a crystal, the noncrystallographic point group symmetry of qnasi,:rystals (Shechtman et al., 1984) promises distinctive morphological features. On the other hand, qnasicrystals can be treated just like any other phase in a l)hase diagram. Thus all known phase transformations, in principle, can be expected with actual occurrenceguided by free energy considerations. Therefore, quasicrystals can form as a single phase as well as a primary phase in multiphase mixtures. It can exist as eutectic mixtures and also be a product of solid state transformations.
MORPHOLOGY OF QUASICRYSTALS (EXPERIMENTAL STATUS) Several investigations in the early stages were carried out to characterise tile nlorphology of quasicrystals. Most of the initial work was carried out for AI-Mn and AI-Mn-Si qua.sicrystals. Although one often obtains a dendritic microstr,wtm'e, an early work by Chattopadhyay el, al. (1985) indicated a well-facetted morphology for the quasicrystals. The basic shape of the small icosahedral quasicrystals in AI- Mn alloys was investigated bv several investigators by Transmission Electron Microscopy (Nishitani et al., 1986, Thangaraj et al., 1987). One of the initial difficulties was the problem of identification among the possil>le different shapes consistent with the icosahedral point group symmetry. It was found that the shapes are indistinguishable when projected along higher symmetry axes like five-fold and three-fi)hl axes. tlowev~'r, as shown in Fig. l due to Thangaraj et al. (1987), the projection along the two-fold axes enable dislinction between different shapes and establish the pentagonal dodecahedron as the shape for the AI:Mn type of quasicrystal. The dis~'ow-ry of a sta237
238
K. Chattopadhyay et al.
ICOSAHEDRON
DODECA HEDRON
TRIACONTAHEI]RON
@
L..,'\ ]
L,...I _ ~
• 6)4°t I
\~,V"-") ,,i ,s'~/
Ia
~'.iI/ 6..~.4e / ~,L41
Fig.l. Shapes of an icosahedron, dode,'ahe(h'on and triacontahedron under different projections al general view, h) aloug 5-fold axis, c) along 3-fi~hl axis. (I) along 2-fi)hl axis. (Thangaraj. 1987)
Shapes of Quasicrystals
239
ble quasicrystal with large pentagonal dodecahedral solidification morphology followed (Ohashi and Spaepen, 1987). In contrast, AI-Cu-Li quasicrystals were found to have rhombic triaeontahedral morphology. (Dubost et al., 1986). With the recent discovery of equilibrium icosahedral quasicrystal in AL-TM ternary system of Al-Cu-Fe and AI-Pd-Mn (Tsai et al., 1987, Tsai et al., 1990), large crystals having well-defined shapes of pentagonal dodecahedron and icosidodecahedr(2n can be routinely observed. Recently an equilibrium quasicrystal in Mg-based alloys (Mg-Zn-RE, RE-rare earth element) has been discovered. These also give the peutagonal dodecahedral morphology (Tsai et al., 1994). The decagona] quasicrysta] in A1-Cu-Co also exhibits well defined decagonal prism morphology. (He et al., 1990). There also exists report of a pencil shape of a one dimensional quasicrystal (Tsai et al., 1992). Thus tbe morphology of a quasicrystal often reflects the inherent non-crystallographic symmetry. For any detailed discussion of the crystal morphology, two situations need to be distinguisbed. The equilibrium shape of a crystal is determined by a surface energy principle and can be different from the growth morphology which depends on the kinetics of the growth and processing conditions. We now examine the morphology of quasicrystals under these two conditions.
EQUILIBRIUM SHAPE OF A CRYSTAL Before we consider the problem of equilibrium shape of a quasicrystal, a brief introduction to the subject with reference to the crystalline case is helpful. Surface energy is defined as the work done to create unit area of a surface. In case of fluids, this is isotropic since it does not depend on direction. On the other hand, this is anisotropic for the case of crystals. This is because the number of bonds cut per unit area depends on the orientation of the surface. This can be understood by considering a simple two-dimensional square lattice (Fig. 2). We wish to calculate the number of 1)onds cut per unit length as a function of orientation. Along any arbitrary direction which passes through the origin and another lattice point (x,y), the total number of bonds cut is given by x + y. Thus, the number of bonds cut per unit length would be given by N = (x + y) / ~ y 2 ) t
.....
.'
. . . . o. . .
t
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I I
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Fig.2. Square lattice to calculate the density of bonds cut as a function of orientation. Thus the surface energy is given by E = (cos0 + sin0)¢ where ¢ is the energy per bond. It is to be noted that only first nearest neighbour interactions have been considered in the present case. A polar plot of the surface energy as a function of the orientation is known as tbe gamma-plot. The gamnla-plot for the case considered above is shown in Fig.3 . The problem of determining the equilibrium shape of a crystal from the ganmla-plot was first resolved in a predictive manner by Wulff (1901). This was further developed by Herring using
240
K. Chattopadhyay et al.
thermodynamics. This work established tlle existence of the relation between surface energy and the equilibrium crystal shape (ECS). The statement of the problem of equilibrium shape is as follows. Given a particular volume of material, what is the shape it umst adopt in order that the total surface energy is a oonstant. This condition can he expressed as r a~ dA, = rain, V = constant. Wulff (1901) proposed a solution to this p,'oblem. Wulff's theorem states that if planes are drawn perpendicular to the radius vectors where they cut the gamma-plot, then the ironerenvelopeof these planes is geometrically similar to the equilibrium shape. Many important and interesting features of the nature of the gamma-plot and the ECS were pointed out by Herring (1951). It was shown that when only nearest-neighbour interactions were considered and relaxation effects were neglected, then the gmmna-plot consisted of segmeuts of spheres which when extended passed through the origin. Such a plot led to cusps appearing along some special (low-index) orientations. The corresponding equilibrium shape would consist of flat facets of finite extent for these orientations. As pointed out by Herring, to ensure that the surface of the equilibrimn shape has a flat portion with art orientation normal to a given vector OX of the plot (Fig. 3), l) it, is necessary and sufficient that the plot has a pointed cusp at X, and 2) it should be possible to draw a sphere between O and X which lies entirely within the gamma-plot. The regions outside the Wulff envelope do not contribute to the equilibrium shape and remain passive. For crystals, there exist sharp cusps in the high-symmetry directions at low temperatures. Thus, the E(~S reflect the symmetry of the crystal. Ideally at 0 K, the normal crystals should be 1)ounded only by flat surfaces. Thus any orientation which is not a part of the equilibrimn shape will undergo a reconstuction to a hill-and-valley structure. This behaviour is marginal if we were to consider first nearest-neighbour interactions alone because the energy of any arbitrary orientation is identical to the em,rgy of the hill-and- valley structure which comprises it. In the abow• analysis, the tenll)erature has been assumed to be absolute zero to avoid the effects of thermal disordering and entropy effects. The structure of surfaces can be described in terms of the TLK model (Gruber and Mullins, l!l(i7). In lhis model, surfaces close to a low-index plane can l)e described as containing terraces of the low-index orientation separated by steps whose density depends on the extent of deviation from the low-index plane. For more complex 1)lanes, kinks have to introduced in the steps to obtain the overall orientation of the surface. The energy of such a surface can I)e deeoml)osed in to the sum of the terra,'e energy per unit area and step energy per unit length. fi(O) --et cos(O) + (es/a)(sin(O))
,
~
(fi(m) "wuzf~ Plot"
) / r (h)
"crystal
Fig.3. Wulff constructiou to obtain tile crystal shape r(h) from tile polar plot of surface energy f~(m). (Herring. 1951)
Shapes of Quasicrystals EFFECT
OF
TEMPERATURE
ON
THE
24"
EQUILIBRIUM
SHAPE
The effect of temperature is to increase tile disorder in the surface. The main effects are as follows, For higher-index planes, the main effect is that the density of kinks on the steps increases until all long range order is destroyed. Figure 4 shows a typical stepped structure at some high temperature. The step free energy is given as a combination of the usual energy and entropy terms f~ = e, - TS~
a ~z.__._._.: ::..::: ::___Z_.___. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Fig.4. Excitations oil the steps leading to tbrmation of kinks on steps and islands on tile terraces. As the density of kinks increases, the entropy contribution increases and at a critical temperature called the roughening transition temperature, f~ goes to zero. Figure 5 shows the typical variation of the step free energy as a function of temperature. Each hkl face has its characterstic roughening temperature with the close-packed planes having the highest roughening temperature. Macroscopically, this marks the transition fi'om a facetted ])lane to a smoothly curved one. For low-index planes, the scenario is much more complicated. The surface disorders by formation of vacancies and adatoms on the surface which leads to the formation of islands on the surface. The statistics of kinks on the steps is simple because the [)roblem is one-dimensional. On the other hand, the problem of surface disordering is complicated since it is a co-operative phenomenon analogous to ferromagnetism. The exact sohttion to the two-dimensional problem has been given by Onsager (1944) in context of the two-dimensional lsing model for ferromagnetism. This is just a two level problem while the actual surface atoms are not confined to two levels alone. Burton at al. (1951) have extended this to a Bethe type of approximation to investigate the effect of the higher levels.
fs(T)
T 1
. temperature Fig.5. Variation in step free energy as a function of temperature. The step free energy vanishes at the roughening temperature Tn. (Wortis, 1988)
242
K. Chattopadhyay et al.
For a crystalline lattice, the roughening transition temperature in general scales with lattice parameter. Thus a crystal with larger lattice parameter shows more facetting tendency. The sharp edges and corners of tile crystal at 0 K start rounding as the temperature is increased. Thus the crystal shape consists of sharp facets separated by rounded edges and corners at higher temperatures. Andreev (1982) pointed out that the Wulff construction is just the two- dimensional Legendre transformation from the free energy to the crystal shape. This means that the crystal shape can be treated as a free energy. In that case the transition from facetted to rough can be treated as a two-dimensional phase transition. There are two types of thermal ewllution that have been observed in crystals. The first type of thermal evolution which has been designated as Type A evolution is as follows : At T = 0, the crystal is polyhedral and as the temperature is increased, rounded regions appear between the facets at arbitrarily low temperatures. There is no slope discontinuity at the point where the rounded region starts. The proportion of the rounded regions increase on increasing the temperature aud each fiat facet disappears at its characteristic roughening temperature unless bulk melting intervenes this transition. This type of evolution has been observed in the case of hcp 4He in contact with superfluid helium below 1.4 K. as well a.s in tile case of metals crystals like Pb and In. This temperature evolution can be convenieutly represented in a phase diagram. Such a diagram has been shown in Fig.6 . Since the crystal has tbur fold symmetry, it is sufficient to consider only angles between 0 and 90. At T = 0, the crystal has the (100) and (010) faces separated by a sharp edge. As the temperature is raised, there appears a rough region in the phase diagram (corresponding to the rounded region in the shape). The angular range of this rough region increases until the temperature TR where the (100) type face vanishes.
T-O
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~
/
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Fig.6. Phase Diagram for Type A thermal evolution of the crystal shape. The diagram is tile locus of the crystal edges in the equitorial plane only. (Wortis, 1988) Tile free, energy expression near the critical point of a second-order transition exhibits power-law behaviour (Wortis, 1988). y = x" , where n is typically a non-integer. Since it has been shown that tile crystal shape can be treated to be a free, energy, we can expect that tile shape of the crystal near a smooth edge would folh)w a power-law behaviour (for Type A thermal
Shapes of Quasicrystals
243
evolution). There are good experimental evidences to believe that this behaviour corresponds to the Pokrovsky-Talapov x3/2 law. y = A (x- xc)" , where xc is the critical edge and n, an universal exponent bas a value of 1.5 . Tile other type of thermal evolution (designated Type B) is tile first-order transformation analogue which has a slope discontinuity. This has been experimentally observed in Au. The Type B evolution can change to Type A evolution at higher temperatures. Tile Type A thermal evolution implies that the T = 0 Wulff plot is degenerate in the sense tbat a whole bunch of normals pass through the same corner. This is because of the assumption of nearest-neighbour interactions alone. At infinitesimally small temperatures, thermal fluctuations break the degeneracy and thus gives rise to Type A evolution. Oil adding higher neighbour interactions, a) additional faces may be added to the equilibrium shape and the transition may still be Type A or b) tile transition may follow type B behaviour depending on tile whether the higher neighbour interactions are attractive or repulsive. For the case of crystals, new faces cannot be stabilised by second-nearest neighbour repulsive interactions. Tile equilibrium shape obtained by pure attractive interactions contains the minimum possible facets among all the equilibrium shapes possible.
EQUILIBRIUM SHAPE OF QUASICRYSTALS THEORETICAL D E V E L O P M E N T Starting with Kepler, polyhedral shal)es are always associated with crystalline structures having periodic arraugement of atoms. Thus, it was not clear whether quasicrystals can have facets and definite polyhedral shapes. He et al. (1987) adol)ted a bond oriented quasiglass model to determine whether facets are possible in such a case using only attractive potential. The results clearly indicate a strong tendency for facetting. The possible set of shapes computed by these investigators for different bond strengths is shown ill Fig.7. One interesting conclusion of this work is the exclusion of pentagonal dodecalmdron as a possible ES. l!sing Penrose tiling. Garg and Levine (1987) obtained a facetted morphology at 0 K for quasicrystals. In a further development, Ingersent and Steinhardt (1989) made a detailed study of the shapes of (luasicrystals by considering the combination of both attractive and repulsive interactions. They established that the equilibrium shapes of quasicrystals had some important differences from the crystal case. The main difference is that the shape obtained using purely attractive interactions contains more than the nfinima] number of facets. The shape with minimum number of facets consisite,~t with the symmetry considered is obtained by including higher neighbour repulsive interactions. This may ]lave a bearing on the deternfination of the atomic structure of the quasicystals. This work also established firmly that the pentagonal dodecahedron can appear as an equilibrium shape when a more realistic situatiou of both attractive and repulsive potentials are taken into account. Tytfical shapes obtained by these investigators are shown in Fig.8. Tile first realistic calculation of quasicrystalline phase is due to Lei and Henley (1991) who used a cluster model. These clusters have icosahedra[ synnnetry, similar to that observed in AI-Mn-Si and Al-(',u-Fe and arranged in a cubic :~/2 crystalline approxilnant structure with 32 clusters per cell The results of the ES calculation suggests that the planes normal to five-fold and two-fold axes have the lowest surface energies and shouhl dominate the shape. However, there exists I)resence of small planes normal to three-fold direction. The phenomenon of roughening of the corners and edges at high temperatures has to be considered. This aspect is more ameuable to experiments. There have been attempts to theoretically study the roughening transition. Tile argument that roughening scales with lattice parameter implies that true roughening temperature of a quasicrysta] will be infinity. However, it has been shown that a pseudo-roughening transition does indeed exist for a two-dimensional Penrose lattice although the magnitude of the exponent was found to be much smaller. (Garg and Levine (1987), Lipowski and Henley (1987)). Lipowski and [lenl¢~y also studied this aspect for the icosahedral as well as
244
K. Chattopadhyay et al.
the decagonal quasicrystal. They concluded that a roughening transition exists for the decagonal quasicrystal along planes normal to the ten-fold direction whereas in the case of the icosahedral quasicrystal, the five-fold facets remain flat at all temperatures (Table 1). They also found that the rounding exponent for a Penros~lattice type model of the icosahedral quasicrystal was lower than 3/2 (Table 2). Table 1. A smnmary of the expected nature of tile roughening transition temperatures of the surfaces in crystals and quasicrystals Nature of the Roughening Transition Crystal
Finite scales with the lattice parameter
Decagonal 10-fold plane
Finite
Rectangular facets
Infinite
(Quasiperiodic planes) Icosahedral
Infinite
Table 2. The theoretically predicted rounding exponent A (z(x) = x 'x) for the decagonal and icosahedral quasicrystal Quasicrystal Decagonal Facet
A <
3/2
2.2 (random tile model) Rectangular Facet lcosahedral
<
3/2
EXPERIMENTAL STUDY
Tile experimental study of equilibrium shapes is beset with problems of equilibrating large size crystals/qua.sicrystals. There may I)e a narrow window of realistic temperature for carrying out equilibration heat treatment abow~ which it melts aud below which it takes impractically long for equilibriation (Wortis, 1988). There art~ problems associated with temperature control as well as problems of adsorption. Another w%, of studying these shapes is by nucleation of these phases in a liquid. It is well known that during the initial stages of nucleation, interracial energy effects predonfinate and hence the critical nucleus that forms is identical to the equilibrium shape since this shape has tile minimmn surface energy per m6t volume. This, however may not be valid at later stages when substantial growth has taken place. Nucleation can be promoted by undercooling the melt sufficiently. Rapid solidification of dilute AI-Mn alloys provide substantial undercooling promoting the nucleation of the quasicrystal in the melt. Later the Al nucleates and grows very fast trapping the small quasicrystallites. This provides an excellent opportunity to study the equilibrium shape and the roughening behaviour of the quasicrystals. It is to be noted that the shape reflects the shal)e al the trapping temperature. A log-log plot of the boundary co-ordinates is used to analyse the critical exponent in case of the shal)es that are observed experimentally. Rottman et al. have discussed about the region from which the actual data is to be taken to avoid effects due to atomic scale rounding or higher-order terms due to scaling corrections. Fig.9 shows the actual crystal profile in comparison to the ideal profile and the window region flom which data can be used for determining the index.
Shapes of Quasicrystals
245
Fig.10 shows the results for such an analysis from all A1-5% Mn alloy. The slope in this case is found to be close to 3. Ananlysis of results from various samples indicate that the slope in this case generally is between 2 and 3.
G R O W T H M O R P H O L O G Y OF QUASICRYSTALS At a first glance, the quasiperiodicity does not seem to yield any dramatic difference in the growth behaviour of quasicrystals in comparison to crystals. It acts like any other crystalline phase in a phase diagram. One, therefore, observes primary, eutectic as well as peritectic growth in quasierystals. However, the quasiperiodicity may influence the atomic growth mechanism, For example, the faceted growth of a crystal takes place below tile roughening transition temperature. As mentioned earlier, the latter scal~ with lattice parameter. Since the quasicrystal can be considered as a periodic crystal with infinite periodicity, the roughening transition temperature should be infinity. Practically this means that quasicrystals are expected to have faeeted growth. The expectation is borne out by a large number of reported observations. All tile slowly grown equilibrium quasicrystals show a pronounced facets in the growth form (Duhost et al., 1986; Tsai et al., 1987, 1990, 1994). Because of the faceted growth, the growth form reflects the 1)oint group sylnmetry of the quasicrystal and one often observes shapes like pentagonal dodecahedron. [n such a situation, the shape also reflects the slowest growing planes which has highest atomic density. Abundance of pentagonal facets in as grown quassicrystal indicate that these planes have highest density and therefore, slowest growth rate. The dendritic growth of quasicrystals is common in rapidly solidified alloys. The early reports on AI-Mn quasicrystals often reported well-rounded dendrites. It was pointed out that even if this quasierystal can grow in a faceted manner in twin-rolled samples which yield a lower cooling rate in comparison to melt spinning (Chattopadhyay et al., 1985). It is, therefore, clear that the non-faceted growth of quasicrystal is due to dynamic roughening at high undercooling of the melt. All faceted crystals can undergo a dynamic roughening below a critical undercooling of the melt. This leads to a continuous growth and yields a rounded growth form. The reasoning is equally valid for quasicrystals having a quasiperiodic potential. The first theoretical attempt to analyse the atolnistic growth behaviour of quasicrystal in comparison to crystals is due to Toner (1990). lie analysed the stepped growth of faceted quasicrystalline interface. It was found that the qua,sicrystals grow 1)y nucleating a step height (h~) which is related to tile chemical potential difference across the interface (A u) h~ ~ (A u) -'/3 Thus as the chemical potential decreases, tile height of the step needed to grow a quasicrysta] increases. On the other hand, the step height for growth in a crystal is related to the lattice parameter and is independent of the difference in chemical potential. Further for crystals, the critical potential barrier for nucleation of steps vanishes at the roughening transition temperature while for quasicrystals, it always remanis finite. One of the direct consequences of this analysis is the nature of the growth step in a quasicrystal. It is predicted to l)e much larger than the crystalline case. Thus, the step growth of quasicrystalline interface can easily he observed by normal hot-stage transmission electron microscot)y without resorting to high-resolution microscopy. The growth of decagonal quasicrystals in a icosahedra] quasicrystalline matrix in the solid state is ideal for testing the above predictions. This is because of the fact that the decagonal phase has one periodic direction which should behave like a n(Jrmal crystal. The other directions will experience only the qua,siperiodic potentials and will grow 1)y the movements of large steps. Schaefer and Bendersky (1986) have earlier shown an epitaxial relatiolL l)etween icosahedral and decagonal crystal with tenfold symmetry plane or the decagonal plane (the periodic plane) always nucleating on the five-fold plane of the icosahedral phase. High-resolution imaging of these interfaces reveals step heights of the order of the spacing of the periodic planes (Thangaraj et al., 1987). On the other hand, in-situ electron microscol)ic observation of the icosahedral to decagona] quasierystalline transformation in
246
K. Chattopadhyay et al. center of crysta I
Ideal crystal profile
'" \ Z i i
on0,,0c.,
x
x
"crillcoI reqion'L----~,
i"-eck:je \
..,
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~
1
~
Fig.9. a) A log-log plot of ideal crystal shat)~, coor(linates to determine tile critical eXl)[ment for the transition, b) shows tile window from which data can be used for a,alysis.
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O
/
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3.30 3.40 In (X) Fig.10. Experimental plot fl'om all AI-Mn alloy which shows the critical index to be about 3.0.
Shapes of Quasicn~stals
@
247
u~
8 E
¢d
,.c
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~-
,
,LT.."=
248
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the quasiperiodic directions by Kim et al., (1990) reveals the growth to occur by the movement of large steps. A schematic diagram from their work is shown in Fig.11. Thus, there exixts differences in the nature of the growth of quasicrystals and crystals. More experimentM studies in this direction is clearly necessary.
Ledge GrowtN Dir.
Facet __,..5rov~ th dir. 2f A
Fig.11. A schelnatic plot showing the stepped structure ill a quasicrystal. (Kim et al., 1990) CONCLUSIONS
Although there exists extensive literature on the structure and properties of quasicrystals, the study of the morplmlogical evolution of quasicrystMs is limited. Since morphology will depend on the nature of the qnasiperiodic potential, tile symmetry and tile atom attachment kinetics, such a study will yiels rich information on the nature of the quasicrystals. It is hoped that more such studies will be undertaken in the near future. ACKNOWLEDGMENTS
It is a plea.sure to acknowledge tile discussions with Professor S. t{anganathan regarding the problems of shat)e and symmetry. REFERENCES
Andreev A.F., (1982), Faceting Phase Transitions of (',rystals. Sov. Phys. JETP., 5:1, 1063. Burton, W.K.. N. ('abrera and F.(~. Frank, (1951). The growth of crystals and the equilibrium strncture of their surfaces., Phil. Trans. A, 243,299-358 (:hattopadbyay, K., S. Ranganathan, (I.N. Subhanna and N. Thangaraj (1985). Electron Microscopy of quasicrystals ill rapidly solidified AI-Mn alloys. Scripta Metall., 19, 767 I)ubost B...I.M. Lang, M. Tanaka, P. Sainfort, and M. Audier (1986). Large Al-(;u-Li single qna,~icrystal with triac,ontahedral solidification lnorphology. Nature (London)., 324, 48 (larg A and D. Levine, (1987), Faceting and Roughening in Quasicrystals, Phys. Rev. Lett., 59, 1683 (Iruber E.E. and W.W. Mullins. (1967). On the Theory of Anisotropic (',rystalline Surface Tension., J. Phys. (',hem. Solids., 28, 875-887. He L.X., Y.K. Lu, X.M. Mang and K.H. Kuo., (1990), Philos. Mag. Lett., 61, 15. Herring (~., (1951). Some theorems outhe free energies of crystal surfaces. Phys. Rev., 82, 83
Shapes o! Quasicrystals
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