Amplitude expansion for the Grinfeld instability due to uniaxial stress at a solid surface

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Amplitude expansion for the Grinfeld instability due to uniaxial stress at a solid surface

P. Nozières

To cite this version:

P. Nozières. Amplitude expansion for the Grinfeld instability due to uniaxial stress at a solid surface.

Journal de Physique I, EDP Sciences, 1993, 3 (3), pp.681-686. �10.1051/jp1:1993108�. �jpa-00246748�

J. Pltys. I France 3

(1993)

681-686 _MARCH 1993, PAGE 681

Classification Physics Abstracts

68.45

Short Communication

Amplitude expansion for the Grinfeld instability due to uniaxial stress at

_a

solid surface

P. Nozilres

Institut

Laue-Langevin,

B-P. 156, 38042 Grenoble Cedex 9, France

(Received

8 December1992, accepted 5

January1993)

Abstract The physical origin of the Grinfeld melting instability at the surface ofa uniaxially strained solid is briefly discussed. The calculation of the energy is pushed to the fourth order in interface displacement h. It is thus shown that the instability is usually subcritical

(first order).

Moreover, harmonic generation leads to grooves on the back side

(that

of the

solid).

As shown

by

Grinfeld in 1986

[ii,

the surface of _a solid

subject

to _a

non-hydrostatic

stress may

develop

instabilities: _a

melting-freezing

_wave

develops

in such _{a way as} to reduce the elastic _energy. Note that the

instability implies

_a transfer of matter, either from _a

liquid phase

or

through

surface diffusion. The interface

displacement

is due _to

growth,

not to elastic strain.

The

physical origin

of that

instability

is

easily

understood [2, 3]. ^Consider a flat interface

z = 0 with _a

liquid

at pressure pL. A uniaxial _stress is

applied

to the

solid,

such that

?zz = -PL, ?z~ = 0, _{a~~ = -pL} + ao

(I)

The "effective chemical

potential"

that controls solid

melting

is

fs

_«zz

~s = ~~~

Ps

where

fs

is the free _{energy per} unit volume _{and ps} the

specific

_mass. For _a

hydrostatic

situation

(ao

"

0),

the

phase equilibrium corresponds

to the nominal _pressure

p[.

^When we stress the

solid,

_we increase its energy, and thus _we make it less favourable _as

compared

to the

liquid.

The

equilibrium liquid

_pressure thus increases

by 6pL

the interface _goes down [4]. ^The

explicit

calculation is

straightforward.

Let aij, uij be the _stress and strain tensors in the solid. In _an infinitesimal transformation _~s varies

by

6~~ ₌

j,»6uu if

_6U«

^ll~

-

II

^~"~~

^ll~

(2)

is

easily integrated. Assuming

that u~y =

0,

we find

6ps

₌

~~ (~ ^al

+

~~ (3)

Ps Ps

where E and _{a are}

Young's

modulus and Poisson's coefficient. At

equilibrium

spa

₌

6pL

₌ ^~~~

(4)

PL

from which _we infer

6pL

^{and the}

hydrostatic

surface

displacement.

Assume _now that the interface is

displaced by

_an amount

h(z),

_as shown in

figure

I. The uniaxial stress _ao is reduced in the

bumps:

the

equilibrium 6pL

is smaller than the actual

one

the solid _grows further.

Conversely~

_ao increases in the

troughs: 6pL

^is too small in order to maintain

equilibrium

the solid melts. Hence the Grinfeld

instability,

countered

by gravity

at

long wavelength

and

by capillarity

at short

wavelength.

ds

dF~

i

I

~

x

Fig. 1. Geometry of the Grinfeld instabiity.

In order to describe the bifurcation it is

simpler

to calculate

directly

the free

enthalpy

of the distorted state,

G[h(x)].

The latter contains trivial

gravity

and

capillarity

terms, which _are both

positive (I.e. stabilizing)

Go

₌

f

_dz

I( ^{(ps pL) gh2}

⁺

~7h'2 (7h'~ ⁽⁵⁾

the surface _energy involves

@).

_{The stress}

_gives

_{rise to}

a

negative

elastic

contribution, leading

to

instability(~).

The

origin

of this contribution is _once

again

clear. If the

growth

occurred at constant strain, the cost of energy would be _zero

(both phases

_are in

equilibrium

to start

with).

But we

thereby

^break mechanical

equilibrium

at the surface: _a force

dF~

= aodz

must be

applied

in order to compensate the bulk stress. If _we do not

apply

that

force,

the solid relaxes

elastically, thereby lowering

its

enthalpy by

_an amount

(I

These three contributions _are additive. It is easily shown that _cross terms _are

negligible

_as long

as each contribution is small compared to internal energies _+~ E _per unit volume

(e.g.,

the change in density due to

capillary

_pressure is _very small !).

N°3 AMPLITUDE EXPANSION FOR GRINFELD INSTABILITY 683

tie,

₌

f _aodx

uz +

f

_dr

_a;ju;j

=

-( f

_dr

_a;juu

(61 where d, fi, are the

changes

due to relaxation.

We limit ourselves to the

simplest

_case of _a semi-infinite solid medium. The calculation to order h2 for _a

plane

_wave distortion _was carried _out in

(2).

The net

enthalpy

cost

is(2)

AG

=

(ps

+

pL)

_g 2^~~

~~~ «(k

₊

k2j

(7)

The bifurcation threshold _{occurs at} finite momentum

l~

^~~

(Ps PL) j

^~~~

7

(8)

?#

_"

~~

[(Ps

PL)g7)~~~

The obvious

question

is the nature of this bifurcation: is it first _or second order? For that

purpose, we must

push

the calculation of AG to order h~ _{this is the}

purpose of the present

note.

As usual in similar

problems,

the calculation

only

makes _sense if _we allow for harmonic

generation

due to non-linearities. We thus write the

displacement

_as

h(z)

_{= a cos} kz ₊

fl

_cos 2kz _{+ 7 cog} 3kz Due to translational

invariance,

AG will contain terms

a~, a~, fl~, li~..

,

a~fl, a~7, all?

Thus,

harmonics

automatically

_appear, with

amplitudes fl

+~

a~,

₇

+~

a~ _{etc. The}

leading

_non-

linearity

contains

a direct part a~ and _an induced part ^due to the second harmonic

coupling a~fl.

What _we need is _a detailed calculation of each term in

AG(a, fl).

We then minimize with

respect to

fl

in order to obtain

AG(a).

1)

Calculation _{of the} elastic _energy: We _use the standard

representation

of stress in terms of

an

Airy

function

x(z, z).

The extra stress due to relaxation is

~"

0~'

_~~~

0~'

_~~~

OZ~Z' ~~~

_~

For _a

given displacement h(z),

_we choose _x such that the

boundary

conditions _are

satisfied,

?nn " ~PLI ?nt " 0

where _n and _t denote normal and tangent to the local interface.

Up

to order

h~,

these

boundary

conditions read

lazz

^{+ h'2 jw~~}

^fizz]

^{2wm hi +,o h,2 = o}

₍₉₎

azz + hi

jazz azz]

2azz h'2 +,o h' 11 ⁺

h'2)

₌ o

(~)

For _an absorbed epitaxial layer with small thickness

d(kd

_{< 1),} the role of gravity is played by

van der Waals interactions.

The Airy function contains both harmonics I and 2

x = cos kz e~~

[al

_z + ao) + cos2kx e~~~

[biz

_{+ bo)}

(10)

Matching

the _cos kx and _cos 2kz terms in the two

boundary

conditions

(9) provide

four

equations

that fix the four constants in terms of _a and

fl.

The calculation is

heavy

_as

(9)

must be written at

height

h and _{not at}

heigth

0. But it is

straightforward, yielding

° ^{)~ ~"~}

~

i _~~

al = -ao« + _ao

(k~«~

+

(k ^Pi ii)

bo =

ao~, bi

_{= ao}

~)~ lj

It remains _to be checked that the harmonic ₀

(constant term)

in

(9)

is satisfied: it is indeed

identically, ensuring

that the interface

displacement

does not induce uniform _stress

(which

would penetrate

throughout

the

sample).

We thus have

h(z)

and aij

(r).

It remains to calculate the free

enthalpy (6).

We _can

actually

calculate the two terms

separately~

and check that the

identify (6)

is

obeyed (it

follows from the

linearity

of Hooke's

law):

_we

thereby

have _a check of _our

algebra.

The first term of

AGej

is

AG[)~

_{= -ao}

f

_dz

_{h'u~ (12)}

The second term is

AG[)~

=

~@ f

_dr

(2&)~

+

(l a)

_{[&(~ +}

_d)~] 2a&~~

&zz This _can be transformed into

~~S~

"

~@ /

~~

l(~ ')i~Xi~

_{+ 2lX11}

_{Xix Xizl}

Integrating

the last _term

by

_{part, we}

finally

obtain

AG[)~

=

~

)~ / _dr[Ax]~ l' f _x[

dz,

dsi (13)

The steps ^{of the} calculation _{are now} obvious

(I)

From &ij ^{infer the strain}

flu

₌

jj' _ldu

a d«

Sol (I

_{= xi}

z)

(it)

Then calculate the

displacement

_vi at interface _z

= h.

We thus obtain

AG[)~

₌

-2AG[)~,

_as it should.

Altogether

The

quadratic

terms were known

(their

coefficient _are

+~

k).

The

quartic

terms _{are new.}

N°3 AMPLITUDE EXPANSION FOR GRINFELD INSTABILITY 685

2)

Minimization of the total

enerjy:

With the _same

notation,

the

gravity

and

capillary energies

are

Ggr

=

~~~

~~~~

_~

la~

+

fl~l

~2 3 ~~~~

~~~~~

_~

4

~~

~

~~~ 6~~"~~

We _must minimize the total AG.

(I)

At the

bifurcation,

k

=

kc,

_{ao = ac,} ^{AG becomes}

AG =

~~~ ~~~_~

fl~

+

~~

k~a~

₊

ka~flj

4 16

The minimum is achieved for

fl

= -2ka2

~~ (Ps

pL)

_g 43

~ ~

(16)

~

jk

a

We conclude that the bifurcation is first order

(subcritical),

_a somewhat

unexpected

result.

Note the

importance

of second harmonic

hybridization:

without it the bifurcation would be

supercritical.

We also note that b is

negative:

the interface thus flattens _on the

liquid

side and

develops

_{grooves on} the solid side~ _as shown in

figure

2.

, ,

' '

'

x

,

Fig. 2. A negative b develops _grooves in the _rear.

(it)

At

large

stress aoj

gravity

is

negligible.

^All the modes between km and

2km

_are

unstable,

with

~~'

'~ ~~

~~~~

(the instability

is strongest at k

=

km).

^If we fix

k,

_we find that minimization

corresponds

to

lka~ ^~

²_k2^{k kmkm}

⁽¹⁸⁾

2km 2 4

3

km 3

k$

~~ _~

4

~~ ~

k ~ ~ _~

2 k 16 k

(k km)

(fl diverges

at k ₌ km because the second harmonic is

"soft").

The

quartic

term is

negative

if k _< 1.87

km, becoming positive

_very ^close to

2km.

3)

Evolution at

large amplitude:

the

expansion

in _powers of _o is of _no avail. One

point, however,

should be stressed: _as it

displaces~

^{the interface} cannot overcome the level _z

= 0 it had in the absence of

strain,

_ao

= 0. If the interface is

locally flat,

the distance

hi

is such that

11

,2)

(ps pL) ghi

₌

al ₍₁₉₎

2E

where _al is the residual uniaxial stress on the upper ^terrace. hi

always implies

_a

lowering

as

compared

to _{z =} 0.

Qualitatively,

it _seems

likely

that rather flat parts ^should

develop

near z =

0, separated by

_{narrow grooves}

(whose

role is to release the _excess

a~~).

A

really

reliable

description

^will

necessarily

^{be numerical. Note that}

experiments

_seem to confirm such

a behaviour

(5).

Acknowledgements.

The author is indebted to S. Balibar for _many fruitful discussions

(especially

for the

suggestion

that the interface cannot exceed its _zero stress

position).

A related calculation _was carried out

by

J. Grilhd [6], ^{who did not include the effect of harmonics: discussion with him} was very

helpful.

References

[ii

GRINFELD M.Ya, Dokl. Akad. Nauk SSSR 283

(1985)

1139.

[Sov. Phys. Dokl.

31(1956)

831].

[2] ^NOzIERES P., Lectures _at Collbge de France

(1988),

unpublished notes.

Lectures at the Beg Rohu Summer School

(1989),

in "Solids far flom equilibrium" C. Godrkche Ed.

(Cambridge

Univ.

Press,1991).

[3] ^SROLOVITz D.J., Acta Metall. 37

(1989)

621. An extension to finite thickness layers is given in SPENCER B-J-, VOORHEES P-W-, DAVIS S-H-, Phys. Rev. Lent. 67

(1991)

3696.

[4j ^BALIBAR S., EDWARDS D-O-, SAAM W-F-, J. Low Temp. Pllys. 82

(1991)

l19.

[5] ^{See for instance TORII} R-H., BALIBAR S., to be published in J. Low Temp. PA _ys., and references therein.

[6] GRILHE J., to be published in Acta Metall.

J. Phys. I France 3

(1993)

687-692 MARCH 1993, PAGE 687

Classification Physics Abstracts

61.50E 61.55H

Short Communication

Icosahedral quasicrystals: tilings of icosahedral clusters

Marek Mihalkovit and Peter Mrall~o

Institute of Physics, Slovak Academy of Sciences, CS-842 28 Bratislava, Czecho-Slovakia

(Received

13 October 1992, accepted in final form 10 December

1992)

Abstract Tiling of four canonical cells, proposed by Henley

[1],is

a

promising

alternative to the Penrose-tiling and six-dimensional approaches to the structure of icosahedral quasicrystals.

In this _{paper we} briefly summarize the firts results of the structural modeling in the I-Almnsi and I-AlcuLi systems. ^The understanding of the structure of the stable quasicrystals, like I-

AlCuFe and I-AlPdmn, is of _course of greatest interest; _a possible canonical-cell approach is outlined.

1 Introduction,

The

advantage (and disadvantage)

of the canonical cell concept

is,

that it does not

anticipate

any

simple description

in

higher (six)

dimensional _{space, nor} within the 3D Penrose

tiling scheme,

^but is derived from the

physically grounded requirements

_on the cluster-based icosa- hedral network. It _seems that the two criteria

(I)

^maximal

density

of the network and

(it)

the two allowed kinds of icosalledral

linkages

b and _{c =}

bvi12,

which _are inferred from the familiar

structures of

Frank-Kasper phases (a-Almnsi, (AlZn)49Mg~~)

_are

enough

to force _{a reason-}

ably simple

solution:

tiling

of four canonical cells

(TCC). Although

the constructive

proof

of the

quasiperiodicity

of such

tiling,

as well _as the

proof,

that TCC is the densest bc-network

(where

b and _{c are} the two icosahedral

linkages)

still does not

exits,

tranfer matrix

technique

generates all

examples

of the TOG'S _up to

5/3

_[3] and the Monte Carlo maximization of the

density

at small

background phason

strain [4]

produce random,

but

perfect (without defects)

TCC

approximants

_up to

13/8

_case

(2440-2472

vertices _per cubic

cell).

2.

Tiling

of canonical cells.

There _are four kinds of canonical cells denoted A

(a

'bcc

tetralledron'),

B

(half

of _a

trigonal

antiprism),

C

(triangular pyramid)

and D

(trigonal prism).

In _an

arbitrary

TCC structure,

the B and C cells _are

always paired

and heretofore

they

will be treated _{as a} virtual BC cell.

An

important point

is that the cells and bonds

(b

^and

c)

_can be decorated

by

the Penrose rhombohedra without gaps and

overlaps (the drawings

of cells and Penrose rhombohedra dec-

orating

them _can be found in Ref.

[I])

and this decoration defines _a set of

packing

rules for the cells. The nodes of the TCC become 12-fold vertices of the inscribed rhombohedral

tiling,

but these _are not identified with the _set of12-fold vertices in the

perfect

Penrose

tiling.

^{With the}

gain

of10il in the

density

of the TCC

compared

to the 3DPT based 12-fold

packing

_[2], the

simplicity

of the

higher-dimensional description

is lost.

An

unique

property ^{of the TCC is}

adjustability

of the nodes

(and c-bonds) density,

which is controlled

by

the parameter

(,

related to _a ratio of A and D cells. At _zero

background phason strain,

50il of the volume is

occupied by

^the BC

cells,

but it is

conjectured

[4], ^that D-cells volume fraction is

adjustable

from 8A to 15.8il

(A

cells fill the rest of the

volume).

Considering

the known structures of

approximant phases,

one can choose to maximize either A

or D cell type ^decoration at

(close to)

_zero

background phason

strain. Besides

this,

TCC is _an ideal candidate for the

description

of

approximant

structures: from all

possible

rhombohedral

packings,

it selects

only

few

physically

_most relevant

examples

with maximal number of12-fold

vertices without short b-fold

linkages

between them.

Thus, given

the decoration of canonical cells and

assuming

that

chemical/topological properties (atomic

local environments, chemical

composition)

of the decorated

A,

^BC and D cells _{are never}

exactly identical,

the relation between

quasiperiodic

and

periodically

locked icosahedral cluster

packings

_can be studied in terms of the relative

density

of the cells.

From the

modeling point

of

view,

the canonical cells _are viewed _{as an} interstitials in the

packing

of

(interpenetrating)

icosahedral

clusters, decorating

TCC vertices. The rhombohedral decoration of the TCC

helps

_one to _see how the atoms are shared among

neighboring

clusters and _to fill the voids

by

_an additional atoms in _a

rigid

_manner.

The

following

brief reports on the structural models and their agreement ^with

X-ray /neutron

diffraction _are based _on the

816

^TCC

approximant

with 592 nodes per cubic cell and Pa3 _space symmetry [4, 5].

Decorating

it

by

the icosahedral

clusters,

one

straightforwardly

obtains 97- 98it of the atomic

positions

in the structure. When

completed by a'glue'

atom, the models

contain

43908,

51752 and 40356 atoms per cubic cell for the _cases

I-Almnsi,

I-AlcuLi and

I-Alcufe, respectively.

3. Model of I-Almnsi.

There _{are many} indications that the structure of I-Almnsi h

intimately

related to the o-Almnsi Frank

Kasper phase [6-9],

^{which is}

nothing

but the bcc

packing

of

(Mackay)

icosahedral

clusters,

or a pure

periodic packing

of A-cells with ico clusters centered at the

tiling

nodes. To define the

quasicrystal (and arbitrary approximant crystal

based _on

TCC)

model _{we now}

need, only

to

specify

the decoration of the

remaining

types of

cells,

BC and D.

This _can be done

quite easily, utilizing

the refinement of a-Almnsi structure [10] ^and a

combination of

large

126-atom three-shell and 54-atom double-shell icosahedral clusters that

occupy

respectively

^{'even'and 'odd'TCC} nodes

[iii.

The 3-shell clusters _are

composed

of

a small

Mackay

icosahedron

(MI) (12 atoms),

_an icosidodecahedron

(30 atoms),

_a

large

MI

(12 atoms),

_a triacontahedron truncated

along

3-fold _axes

(60 atoms)

and _a r-inflated

large

MI

(12 atoms),

where _r

=

Ii

₊

v$)) /2

is the

golden

_mean. The details

concerning

atomic decoration will be

published elsewhere,

but it _can be

shown,

^{that in} such model 97il of the atomic

positions

_are

directly

determined

by

the a-Almnsi structure, as

compared

with about 70it in the 6D models

[12,

13]. ^Due to

(weak)

even-odd

ordering,

inherited from

a-phase,

the

N°3 ICOSAHEDRAL QUASICRYSTALS 689

quasicrystal

has fcc Bravais lattice in 6D. The calculated

density

_agree with the

experimental

value

(see

Tab.

I)

^{and the}

reliability

factors _are 9it for

X-ray

and lsit _{for neutron} diffraction

(experimental

data _were taken from Ref.

[14]).

Table I.

Density

in the canonical-cells based models.

Note that for the first two systems, where there exists _a cubic 'I

Ii' approximant Frank-Kasper phme

^with lattice parameter

equal

to b

(the

_pure

packing

of

A-cells,

12 A-cells _per unit

cube,

A-cells in the model have the _same

density. Experimental

densities _are taken from references

[24],

[18] ^and [25]

respectively.

^'Glue'are called the atoms, not

belonging

to the icosahedral

clusters, defining

_a model.

ALLOY _{PA pBc pD}

calc.j ^(exp.)

^{GLUE ~BOND}

tat/b~l tat/b~l tat/b~l g/cm3 g/cm3j

li~l iii

138.0 136.0 130.7 3.63 3.63 3.3 12.66

A157CuiiL132

160.0 161.0 162.7 2.44 2.47 2.2 13.90

A163Cu25Fe12

126.0 125.2 122.7 4.44 4Al 3.6 12.30

4. Model of I-AlcuLi.

The

I-phase belongs

to

I-AlmgZn family

of

quasicrystals.

Like in previous _case, its local structure is

obviously

related to the structure of cubic

Frank-Kasper phase (R-AlcuLi), though

the 5-shell icosahedral clusters [15] are of different type so called Samson

complexes capped by

rhombic triacontahedra. A structural refinement [16] ^leads to _a

following description: (I)

a small MI

(12 atoms); (2)

_a

pentagonal

dodecahedron

(20 atoms); (3)

_a double-size MI

(12 atoms); (4)

_a 'soccer ball'

(60 atoms) (5)

a triacontahedron

(32 atoms).

^Shells

(2)+(3)

form

the _so called small

triacontahedron,

r-times smaller than the _one in 5-th shell.

Utilizing

the network of canonical cells and

positioning

these

large

clusters _seen in the cubic

R-phase

on the TCC vertices leads

again

to _a model that agrees well with the

experimental density

and

composition (see

Tab.

I).

^{Glue decoration is}

easily

obtained

by

_an

algorithm, decorating

rhombohedral vertices,

edges

and

body-diagonal points

of

prolate

rhombohedra

[iii, decorating

BC and D cells.

The model

predicts

_a

slight

decrease in the Li atomic concentration in the

quasicrystal:

R-phase,

which is _{a pure}

packing

of

A-cells~

contains 32.Sit of

Li,

and in the present ^{model BC} and D cells

yield

31.I and 29.Sit

respectively (on

_average 31.6il

Li).

Another

advantage

is~

that with the icosahedral-cluster

description,

there is _a well defined

correspondence

between the classes of atomic sites of the

R-phase

and the

quasicrystal

that allows _us to

optimize Al/Cu ordering~ minimizing

the R-factor of the

calculatedlexperimental single-crystal

diffraction data [18]. ^The

resulting reliability

factors _are 6.lit for

X-rays (56 independent reflections)

and 6.6il

for neutron

(40 reflections)

diffraction. The details and

comparison

with other structural

models will be

published

elsewhere

[19].

In both I-Almnsi and I-AlcuLi _structures

a very small fraction of

glue

atoms can be _con- sidered _{as a} direct

proof

that the

large

icosahedral clusters,

experimentally

found in related Frank

Kasper phases~

can be

arranged nonperiodically nearly

_as

efficiently

as

periodically.

5.

Preliminary

model of I-Alcufe and I-AlPdmn.

The structure of the

'perfect'

and stable

quasicrystals [20, 21]

^with resolution-limited widths of the diffraction

peaks

attracts the

greatest

attention. Here _we shall discuss

only

the atomic

skeleton of the TCC based

model~

blind to the atomic

species.

Clearly,

the structure is of I-Almnsi type: atoms

occujy

with _some

probability vertices,

faces and

body diagonals

of the rhombohedral

tiling

and there _{are no}

mid-edge

sites. In

I-Almnsi,

it is still reasonable to describe the first shell of icosahedral clusters _{as a} small

Mackay

icosahedron and

assign

the atoms to the

edge midpoints

of

(some)

12-fold

vertices,

while

displacive

modulation is

expected

to accommodate stresses, ^introduced

by

too short

mid-edge/vertex

and

mid-edge/mid-edge

distances. However, this is

certainly

not the _case in I-Alcufe and

I-AlPdmn,

where the diffraction

analysis

leads to _a first

pseudc-icosahedral shells, represented by

_a

partially occupied pentagonal

dodecahedron

[22].

The second shell of the icosahedral clusters

(icosahedron

+

icosidodecahedron)

is the _one observed in I-Almnsi.

We _propose the

following

atomic _structure for this class of

quasicrystals:

(I)

the two types ^of

pseudc-icosahedral

clusters 'even'

(E)

and 'odd'

(O)

_are located _at the

even and odd TCC vertices.

(ii)

the

positions

of the

'glue'

atoms in C and D-cells interiors _are

strongly

constrained

by

the ico clusters and _are similar to those

proposed

for I-Almnsi [11].

E-type

cluster consists of the three shells:

(1)

8 atoms _on the vertices of

pentagonal

dodecahedron in _< it

I>-type

directions. There _are five

independent

choices of the

<ill>-type

directions with respect to the 20 ico 3-fold _axes.

(2) 'large'icosahedron

+ icosidodecahedron

(42 atoms)

(3)

r-inflated icosahedron ₊ triacontahedron truncated

along

3- fold _axes

(yielding

20

mutually nonsharing triangles)

12 + 60 atoms.

Starting

from the second

shell~

the

proposed

cluster is identical _to that observed

by

Fowler [10] ^{in a-Almnsi.} Note that the atomic sites of the r-inflated triacontahedron _are

projections

of the

body

center motif in 6D.

O-type

cluster is like

E,

but without the 3-rd shell.

Along

3-fold icosahedral

directions,

all inter-cluster

linkages

_are of E-O type ^{and the 3-rd} shell atoms of the E-clusters _are the icosidodecahedron _atoms of the O-cluster.

Along

ico 2-fold

directions,

O-O connection is

again perfectly compatible,

but in E-E _case the

pairs

of atoms of the truncated

triacontahedron, falling

_on the

'top'

and 'bottom'faces of rhombic

dodecahedron, decorating

this

bond,

_are too

closej

while in the I-Almnsi _case

they

_are shifted towards I and 2 thirds of the face

diagonal (from

the r~~ _and r~2

points),

_{we assume} there is

only

_one atom at these two sites and _occupy the _two 'canonical'

positions

with the

probability 1/2.

Another

degree

of freedom in the model is connected with the

possible

atomic

jumps

in the I-st and from the 2-nd to the I-st

(pseudo)-icosahedral

shell. This is due to the fact that

fully occupied

I-st shell _can accommodate without conflicts 14 atoms 8 _on 3-fold and 6 _on 2-fold _axes

(on

the vertices of the r-deflated

icosidodecahedron)

which is

nothing

but the usual bcc local environment.

We have

already

checked that the

density

of the structure

(when

the _{atoms are}

assigned

masses of the

constituting

elements

weighted by composition)

match with the

experimental

value

(see

^Tab.

I).

The

specification

of the chemical order and _a

precise comparison

of the

calculatedlexperimental

diffraction data is _on the way.

N°3 ICOSAHEDRAL

QUASICRYSTALS

691

6. Discussion.

All of the TCC-based models of icosahedral

quasicrystals

discussed here have their 6D _ana-

logues:

I-Almnsi _[12]~

[13],

^I-AlcuLi

[18], [23],

^I-Alcufe

[25].

^As a matter of the

fact,

the

frequent

_occurrence of the icosahedral clusters in the model _structure is

anticipated

in both

approaches, directly

in TCC-based _one and

through

the

procedure

of

tailoring appropriate

atomic surfaces in

6ll-type

^models. While in 6D

approach

^the

quasiperiodicity

is _a

priori guaranteed,

the

complexity

of the atomic surfaces increases when

adjusting

the

density,

the TCC guarantees

significantly higher density

of cluster-centers than _any known

quasiperiodic bc-network,

but the 6D motif of

quasiperiodic

TCC is still _not known and _can be

expected

to be

(very) complex.

In fact, recent results [26]

suggest

^that _an

important questions might

be raised: if the details of the 'atom~c surfaces'

(though

burried for _an

experimental

observation in

modulation

of the

low-intensity

diffraction

peaks)

_are

physically important,

the realistic

picture

of such

hypersurface might

include

(cylindrical)

fractal features around the lines in- tersections of the 2-fold

planes [26].

But such

hypersurface

would be _very

hardly,

if _at

all,

derivable

analytically.

At the _same

time~

there _are

quite

strong reasons to seek the densest

quasiperiodic packings (I)

^both two main classes of the known

truly quasiperiodic

icosahedral

sphere packings l'unit sphere'

and

'12-fold')

contain _a small amount of _{a very poor}

sphere

local

environments;

(it)

_as

suggested ^by

^{the TCC}

^example,

^{the densest}

packings might

_prove to be much _more

simple

and uniform [27] ⁱⁿ

real-space.

We

conjecture

^{that these} considerations

might

^be valid _on both inter-cluster _as well _as interatomic distance scales: the

perfection

of the short and medium range order

produces

fractal-like deterministic chaos around the

edges

of the 2-fold

planes

of the atomic surfaces in

perpendicular

_space.

References

[ii

HENLEY C.L., Phys. Rev. B ₄₃

(1991)

993.

[2] ^HENLEY C.L., Phys. Rev. B 34

(1986)

797.

[3j NEWMAN hi. and HENLEY C-L-, preprint.

[4) hiIHALKOVIC hi. and hiRAFKO P., accepted to Europhys. Lett.

[5] ^The coordinates of the TCC'S _up to

13/8

case are available _on request ^{from the authors} [6j ^CAHN J-W-, GRATIAS D. and MOZER B., J. Pllys. France 49

(1988)

1225.

[7] ^{ELSER V. and HENLEY} Cl., Phys. Rev. Lett. 55

(1985)

2883.

[8] ^{GUYOT P. and AUDIER} M., Philos. Mag. B 52

(1985)

L15.

[9] ^{MA Y. and STERN} E.A., Phys. Rev. B 38

(1988)

3754.

[10] ^FOWLER H.A., MOzER B. and SIMS J., Phys. Rev. B 37

(1988)

3906.

[11] ^{MIHALKOVIC M. and MRAFKO} P., to appear in J. Non-Cryst. Solids

(Proceedings

of the LAM-8 Conference, Wien

1992).

[12j DUNEAU hi. and OGUEY C., J. Pllys. France 50

(1989)

135.

[13) ^YAMAMOTO A., in Quasicrystals, Ed. T. Fujiwara and T. Ogawa

(Springer-Verlag 1990).

[14j ^GRATIAS D., CAHN J-W- and hiOZER B., Pllys. fleb. B 38

(1988)

1643.

[15] ^There might be _a confusion, what is _meant by cluster's shell: in _case of I-Almnsi, the 'three-shell' 126-atom cluster has 5 shells~ when _one considers _a shell _{as a} set of vertices of polyhedron, like in I-AlcuLi _case. We have followed the terminology of cited works.

[16j AUDIER hi., PANNETIER J., LEBLANC hi., JANOT C., LANG J.-M. and DUBOST B-j Physica B 153

(1988)

^136.

[17j ^{HENLEY C-L- and ELSER V., Pllilos.}

Mag.

B

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L59.

[18j^{de BOISSIEU} M., JANOT C., DUBOIS J-hi-, AUDIER M. and DUBOST B., J. Phys.: Cord. Matt.

3

(1991)1.

[19j MIHALKOVIC M. and MRAFKO P., submitted to J. PA_ys. Cond. Matt.

[20] ^TSAI A.P., INOUE A. and MASUMOTO T., Trans. JIM 29

(1988)

521.

[21j ^TSAI A-P-, INOUE A., YOKOYAMA Y. and MASUMOTO T., Pllilos. Mag. Lett. 61

(1990)

9.

[22j BOUDARD M., JANOT C., DUBOIS J-hi- and DONG C., Pllilos. Mag. Lett. 64

(1991)

197.

[23] ^YAMAMOTO A., Phys. Rev. B 45

(1992)

5217.

[24] ^{AUDIER M. and GUYOT} P., J. Phys. CoJloq. France 47

(1986)

405.

[25j CORNIER-QUIQUANDON M., QUIVY A., ^LEFEBVRE S., ELKAIM E., HEGER C., KATZ A., GRATIAS D., Phys. ^{Rev. B} 44

(1991)

2071.

[26j ^SMITH A-P-, preprint

[27] ^The uniformity of the rhombohedral tiling, inscribed into arbitrary TCC

(via

the classification of the vertex local environments defined in [2]) is surprisingly greater than in the perfect 3D

Penrose tiling, I.e, the number of allowed local environments is smaller.