Packing at Random in Curved Space and Frustration: a Numerical Study

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Packing at Random in Curved Space and Frustration: a Numerical Study

Rémi Jullien, Jean-François Sadoc, Rémy Mosseri

To cite this version:

Rémi Jullien, Jean-François Sadoc, Rémy Mosseri. Packing at Random in Curved Space and Frus- tration: a Numerical Study. Journal de Physique I, EDP Sciences, 1997, 7 (12), pp.1677-1692.

�10.1051/jp1:1997162�. �jpa-00247479�

Packing at Random in Curved Space and Frustration:

a

Numerical Study

R4mi Jullien

(~,*), Jean~franqois Sadoc (~)

^and

Rdmy

Mosseri

(~)

(~) Laboratoire des

Verres,

Universit6

Montpellier

II, Place

EugAne Bataillon,

34095

Montpellier

Cedex 5, France

(~) Laboratoire de

Physique

des

Solides,

Universit6 Paris~sud, Centre

d'orsay,

91405

Orsay

Cedex. France

(~)

Groupe

de

Physique

des

Solides,

Universit6 Paris VII, Place Jussieu.

75251 Paris Cedex 05, France

(Received

3

July

1997, received in final form 11

August

1997,

accepted

18

August J997)

PACS.61.43.Bn Structural

modeling:

serial-addition models, computer simulation PACS.61.20.Ja

Computer

simulation of

liquid

structure

PACS.61.72.~y

Defects and impurities in

crystals;

microstructure

Abstract. Random

packings

of discs _on the

sphere

52 and

spheres

_on the

(hyper~)sphere

53 have been built _{on a} computer using an extension of the

Jodrey~Tory algorithm.

Structural

quantities such _as the volume fraction, the pair correlation function and _{some mean} character~

istics of the Voronoi cells have been calculated for various

packings

containing up ^to N

= 8192

units. While the disc

packings

_on 52 _converge

continuously,

but _very

slowly,

to the

regular

trian~

gular lattice,

the

sphere packings

_on 53 _converge to the disordered frustrated Bemal's

packing,

of volume fraction _{c ~} 0.645, in the

(N

=

cc)

flat _space limit. In the 53 _case, the volume

fraction exhibits maxima for

particular

values of N, for which the

corresponding packings

have

a narrower

histogram

for the number of

edges

of Voronoi

polyhedra

faces.

1. Introduction

Random

packings

of identical hard

spheres

in three dimensions have been used

throughout

the last decades to

represent

^the structure of

liquids, amorphous

solids _or

glasses [1-6].

Since the

pioneering

work of Bernal

iii

it is known that the volume fraction of _a random

packing

of

spheres

cannot exceed _a limited value of _{cb "} 0.645

significantly

smaller than the

packing

fraction 0.7405 of the most

compact regular

structures

(face~centered~cubic

and

hexagonaL closed~packed)

in three

dimensions,

in contrast with the two~dimensional _case where the _sur~

face fraction of the

triangular

lattice of discs

(0.907)

can be reached

continuously.

It is _now believed that this

property

^is due to the so~called

geometrical

"frustration" associated with the

impossibility

to tile the d

= 3 _space with identical

perfect

tetrahedra

only [7-9]

while the

plane

can be tiled with identical

perfect triangles.

A way ^to vary, and sometimes

eliminate,

frustra~

tion is to consider curved spaces. The recent

investigations

of atomic

arrangements

in curved spaces has been proven ^to be _very

useful, mainly

for the

understanding

of disordered materials and

quasicrystals [8,9],

but also for the

study

of

systems

with

long

_range interactions

[10]-

(*)

Author for correspondence

(e~mail: remislldv.univ~montp2.fr)

@

Les

#ditions

_de

_Physique

₁₉₉₇

In this paper, we

investigate numerically

the

geometrical properties

^of random

packings

of

spheres

built in _a curved _space of constant

positive

curvature, I-e- the

"sphere 53".

Such _a

space is the extension to _one dimension _more than the standard

sphere,

the

sphere 52. Indeed,

discs

packings

_on the

sphere 52

have also been

investigated

for the purpose of

comparison.

It is worth

mentioning

that the

problem

of

packing

discs with the maximal volume fraction _{on a}

sphere

has

already

been

investigated ii Ii

and is very close to the old Thomson

problem

which

consists in

finding

the most stable

configuration

of _a set of

equal charges (by minimizing

the Coulomb

energy) _[12].

Our

packings

have been

analysed by calculating

their

pair

correlation functions and

by characterizing

the statistics of their "Voronoi cells". We recall that the Voronoi cell for _one

sphere

is the

region

of _space closer to that

sphere

center than _to all other

sphere

centers.

Knowledge

of the Voronoi cells has _proven to be very useful in the

analysis

of local

geometrical properties

of random

packings

_[2]- Not

only

have _we extended to curved _{space an} efficient

algorithm

introduced

by Jodrey

and

Tory _[13]

to generate ^the

packings

but also the methods of calculation for both the

pair

correlation function and the Voronoi tessellation. In that

framework the

present

work _can be considered _{as an} extension of _a recent paper in which

large sphere packings

_were studied in the Euclidian d

= 3 _space

[14, lsj.

Smaller

packings

have

previously

been built _on

53

but with other

algorithms dealing

either with soft

[16]

or hard

ii?]

spheres.

In Section

2,

_we describe the

principles

of the numerical

methods,

in Section 3 _we present and discuss the results and in Section 4 we conclude.

2.

Principles

of the Numerical Methods

2.I. GENERALITIES. Since

they

_{are very}

similar,

_we will

present together

the numerical methods that _we have used to build and

analyse

the discs

packings

_on

52

and the

sphere packings

_on

53.

For

simplicity

_we will

speak

in both _cases of

'~spheres"

and

"volume",

_even

if,

in the _case of

52,

this _means "discs" and "surface". When

working

_on

Sd-i,

_{we are}

considering

the

sphere

centers _{as a} set of N

points

M~

(I

₌

1, 2,

,

N),

in _a d~dimensional Euclidian _space

(d

= 3 _or

4),

such that their distance _to the

origin

is

equal

to a constant R which is the radius of curvature of the

spherical

_space.

Introducing

the coordinates _x~k

(k

₌

1, 2,.-, d)

^of the

vector r~ =

OM~,

this reads:

'~~'~

"

~

_X~k

~

~~. (~)

k

The

expression

^for the

"geodesic"

distance d~j ^{between two}

points

M~ and

Mj (I-e-

the shortest distance counted

along

the curved

space)

is

simply

related to the scalar

product

of _r~ and rj:

d~j =

Rcos~~

^~~

)~

₌

^Rcos~~ ~~

~(~~~ (2)

R R

The volume of _a

sphere

of radius _{r can} be

expressed

_{as a} function of _uJ

= r

/R,

u(uJ)

=

2~(1 cosuJ)R~ (52) (3a)

u(uJ)

=

2~(uJ "j")R~ (53) (3b)

as

depending

^whether one is

working

_on

52

_or

53.

As _a consequence, the volume

fraction,

also called

"packing fraction",

_c, of _a

packing

made of N

spheres

of diameter

do

is

equal

to the total volume

occupied by

^the

spheres,

which is

Nu(do/2R),

divided

by

the total volume of

Sd-1,

which is

u(~).

Therefore:

c =

~

(l cosuJo) (52) (4a)

2

~

~"° ~~~~~

~~~ ~~~~

with:

~

"o #

). ₍₅₎

It is worth

noticing

that _{one recovers} the well known

expressions

for the definition of the distance and for the

packing

fraction in the limit

R/do

_{~ cc} of _a flat _space.

2.2. EXTENSION _{oF THE} JT ALGORITHM. We recall that the

Jodrey-Tory (JT) _[13]

aI~

gorithm

is _an iterative method in which _a set of N

points

is modified

sequentially

in order

that two chwcifistic

(geodesic) di§fhncwdxi

and

dj~(dfim)~

WA

don#ergiq

tithe fibhl

sphere

diameter

do Considering

the N

points

as centers of

spheres

of diameter

dm,

or

dM,

one

gets packings

of

non~overlapping,

_or

overlapping, spheres, respectively,

with

packing

fractions cm < cM. The distance

dm always corresponds

to the lowest distance between

pairs

of

points

while the distance

dM

is set to an initial value

d[

and is

continuously

decreased

along

the iterations

according

to _a rule which will be

given

below. The initial set of

points

is chosen

randomly

and

uniformly

in the available space.

The method that _we have used to

generate

^{the initial} set

depends

_on the _space dimension.

In the _case of

52,

_we consider the standard

(polar coordinates) parametrization

of _a

sphere

in the Euclidian 3d _space:

xi #

Rsinbcos#;

_x2

# R sin b sin

#;

_x3

# Rcosb

(6a)

where b _runs from 0 to _~ while

#

_runs from 0 to 2~.

According

to the

expression

of the

elementary

solid

angle:

d~Q

= sin

bdbd#

=

-d(cos b)d# (7a)

one can build _a uniform random distribution of

points

on

52 by choosing,

for each

point,

_a value for _cos b

uniformly

at random between -I and +I and for

#

between 0 and 2~- In

practice

we

use:

fi

=

cos~~(1 2fi)I

_§i

~

2~f2 (8~)

where

fi

and

f2

are two

independent computer generated

random numbers

uniformly

dis- tributed between 0 and I. The _same kind of

procedure

has been used _on

53 except that,

for

convenience,

_we have used the

following parametrization (which

is not _a

straightforward

exten~

sion to 4d of the

spherical

coordinates of

equation (6a),

but rather "toroidal" coordinates

[9]):

xi " R _cos b _cos

#;

_x2

" R sin b _cos

#;

_z3

~ R _cos ifi sin

#;

_z4

~ R sin ifi sin

# (fib)

where the

angles #,

b and ifi ^run from 0 to

~/2,

2~ and

2~, respectively.

The

elementary

solid

angle

is then:

d~Q

= sin

#cos # d#dbdifi

=

~d(sin~ #)dbdifi. (7b)

2

Accordingly

_we

take,

for each

point:

#

₌

sin~~ fit

_b

=

2~f2i

~fi #

2~f3 (8b)

where the

f~'s

_are random variables between 0 and 1.

Then the JT

algorithm proceeds

as follows. At each step of the iteration

(p)

^the two

points Mi

and

M2

at shortest distance

dk~

_are identified.

They

_are then

pushed

apart

symmetrically

in the direction

MiM2

so that their _new distance is

equal

to

d$~.

^{To do} so, one first determines the coordinates of the mid

point

C of

MiM21

rc # R ^{~~ ~ ~~}

(9)

ri + _r2 The _new

position M[

of

Mi,

is

given by:

~~~~~

rc ri

di~

~ ~

(11)

R2 ^{' ~ ~°~} 2R

one can determine ~

by:

c~/(1 s2)(1 c2) ii s)(1 c2)

~

~

ii

_S)12C~ + S

1)

^~~~~

The _new

position M[

of

M2

is determined

similarly. Then,

before

going

to the _next

iteration,

the distance

dM

is decreased

according

to:

dj~~~~

= dj~~

~

(cj~~

c~f~)" (13)

where

c(~

and c~f~ are the

packing

^fractions associated with

d(~

and d~f~

through

formulae

(4, 5)

^and where the parameter K and the

exponent

a are two

input parameters

of the al-

gorithm.

Note that _we have used

here,

_as in reference

[14],

a

simpler

formula than the _one

initially

introduced

by

JT

[13j-

^The process stops at the iteration step

(n)

when _one gets

d~~

< d~0~- Then

db~

_{becomes the} actual

sphere

diameter

do

for the

resulting packing.

2.3. SOME TECHNICAL POINTS. The

input

parameters of the computer ^code are

R, N,

d(,

K and _a. Note that the value taken for the _space radius R does _not

play

_any ^role since it fixes

only

the unit of

length.

A _more

important parameter

is N which fixes the size of the

packing.

In the limit of

infinitely large N,

the

resulting sphere

diameter

do

tends to zero. In the

following,

all the

lengths

will be counted in units of the

sphere diameter,

and therefore R will stand for

R/do. Consequently

the limit N _{~ cc}

corresponds

to the limit R _{~ cc} of

an

infinitely large,

and

flat,

_space. The choice for the value of

d~~

is not very

important.

^It should be

sufficiently large

however to

guarantee

that

spheres

of diameter

d~~

^would

overlap.

In

practice,

we take:

d~J

₌

2Rcos~~(1 j) _{(52) (14a)}

d(~

₌

2R(

^~~

)i _{(53). (14b)}

This choice

corresponds

to

c(~

₌ l in the _case of

52,

as in the

original

_paper of JT

[13j,

^and

c~~

very close to I in the _case of

53.

The

remaining

parameters ^{K and} a are essential not

only

to fix the final

packing fraction,

but also _to determine the rate at which it is reached. For

given

a, the smaller

K,

the

larger

the

resulting packing fraction,

but the

longer

the

computation

time. Also the

larger

N

is,

the smaller K must be to

get

the _same accuracy on the

resulting

volume fraction. Here _we have taken _o

= 0.5 and 0.33 in the _case of

52

and

53, respectively

and in most of the calculations

reported

below _we have taken K

=

10~~

_{Note that the set} of parameters used _on

53 corresponds

to the _one used in _a recent calculation in the Euclidian

d ₌ 3 _{space were} the Bernal's volume fraction _was recovered within _more than 0.I per ^cent

accuracy for

packings containing

N

= 8192

spheres [14j-

Some tricks have been used to

optimize

the JT code and there is _no need to describe them in

great

details here. In

particular

the search for the

pair

of

points

at minimum distance is

optimized by updating

at each iteration _a list of distances in

increasing

order

together

with

a table _{to recover} the

particles

for _any

pair

on the list. Also the search for

points

in the

neighborhood

of another _one is accelerated

by using

an

underlying hypercubic

lattice in the d-dimensional space which defines

(hyper)boxes

used to locate and label the

sphere

centers.

To

give

_an

idea, generating

_a

packing

of 8192

particles

_on

53

with K

=

10~~

_takes

a few hours of IBM RISC

6000/380 computer

time.

2.4. PAIR CORRELATION FUNCTION AND VORONOI TESSELLATION. After _a

packing

has

been constructed it is characterized

by calculating

its

pair~correlation

function

g(r)

and

by performing

_a Voronoi tessellation. The

pair~correlation

function, which is

proportional

to the

probability

to find _a

point

at _a distance _r from _a

given point

^{and which is normalized} to

unity

for infinite _r, is calculated

by using

the formula:

gin)

=

(jlUo

_~~~~

where dN is the number of

points

located in the volume du

corresponding

to distances between

r and _{r +} dr from _a

given point

and _uo is the volume of _an individual

sphere.

The _presence of

c in the denominator _ensures that

g(r)

= I when

considering dN/du

=

N/u(~)- Introducing

uJo #

do/2R

and _uJ

=

r/R,

this formula becomes:

~~~~

~

c

i~~~

_~

~~~~ ~~~~~

~~~~

~

^~~

2c

~ii~~~~

^{~ ~~~~ ~~~~~}

In

practice

we have considered dr

= 0.05 in diameter units and dN has been

averaged

_over all the

points

of the set.

To

perform

the Voronoi

tessellation,

_we have extended to curved space an efficient

algorithm

which has been described in reference

[14].

We have first determined the

"simplicial cells",

which _are

triangles

_on

52 (resp.

tetrahedra _on

53),

which _are, among all the ensembles of three

(resp- four) points,

those such that _no other

point

of the set lies inside their circumscribed circle

(resp. sphere). By

circumscribed circle

(resp. sphere)

_{we mean} the set of

points

which

are at the _same

geodesic

distance from _a

given point

Q _as the three

(four) points.

^To do _so,

one should be able to find the _center Q of the circle

(resp- sphere)

circumscribed to _a

triangle MiM2M3 (resp.

tetrahedron

MiM2M3M4). Defining

the

~~'s by:

d

rn =

~j _{[r~ (17)}

~=1

they

must

satisfy

the _set of d

equations:

~j

_{~~r~ rj}

=

R~

_cos

^~~ (18)

R

Fig.

1- Two-dimensional projections ^of disc

packings

_on 52 with N values

ranging

from 4

(top left)

to 12

(bottom right)-

The

projections

of the discs centers _are indicated

by

white dots _or black dots

(in

the _case of

multiple projections)

and the bonds

(corresponding

to

simplicial

cell

edges)

are

represented by straight

line segments ^{instead of} ellipses _arcs. Back lines _are dashed.

where

Rn

is the unknown circle

(resp. sphere)

radius. This is done

by using

standard linear inversion

computer procedures.

After the

simplicial

cells have been determined the construction of the Voronoi cells becomes trivial if _one knows that the

simplicial

cells

edges correspond

to

Voronoi cell

edges (resp- faces)

and that the Q's _are Voronoi cell vertices. In

particular

^the number of

edges (resp, faces)

of _a Voronoi cell

surrounding

_a

given point

is

equal

to the number of

simplicial

cell

edges

connected to this

point. Moreover,

in the _case of

53,

the number of

edges

of _a face is

equal

to the number of

simplicial

tetrahedra

having

_{a common}

edge,

which is the

corresponding

nearest

neighbor

distance. We have written _a code that reads the list of

point

coordinates and

produces

the fractions

fe

of cells with

e~edges,

in the _case of

52,

and both the fractions

fF

of cells

having

F faces and the fractions

fe

of faces

having

_e

edges,

in the _case of

53.

3. Numerical Results

3.I. EXAMPLES _FOR SMALL N. It is

quite

_easy to vizualize the discs

packings

obtained

on

52

and _we

provide

_some

typical examples

obtained for values of N

ranging

from 4 to

12,

in

Figure

I. The

corresponding

values of the

packing fraction,

the radius of the _space in diameter

units,

the _mean coordination numbers _z, and the fractions

fe

of Voronoi cells

having

_e

edges,

have been

reported

in Table I- In the

figure

_we

represent

^{the horizontal}

projections

_on the

plane (xi, x2)

after

having performed,

for each

packing,

the

adequate

rotation of the

sphere 52

to get a

point

on the north

pole (x3

~

R)

whose

projection

is therefore

always

^located at the center of the

figures.

The

projections

of discs centers are

represented by

white dots

(or

black

Table I- Radius R

of

the

sphere 52 (in

disc diameter

unit), packing fraction

_c, coordination number _z, and

fraction fe of

Voronoi cells with _e

edges for packings of

discs _on

52

with values

of

N

ranging from $

to le.

N R _{c Z}

f3 f4 f5 f6

4 0.523 0.8453 3 1 0 0 0

5 0.637 0.7322 3.6 0.4 0-6 0 0

6 0.637 0.8787 4 0 1 0 0

7 0.736 0.7774 4.285 0.142 0.428 0.428 0

8 0.765 0.8236 4-5 0 0-5 0.5 0

9 0.812 0.8258 4.667 0 0.333 0.667 0

10 0.866 0.8101 4-8 0 0.2 0.8 0

II 0.903 0.8214 4.909 0 0,182 0.727 0-091

12 0.903 0.8960 5 0 0 1 0

dots when

they correspond

to two discs

centers)

and the bonds

connecting

nearest

neighbors (edges

of

simplicial triangles)

_are

represented by straightlines

instead of _arcs of

ellipses.

It is worth

noticing

that most of the

configurations depicted

here _are

quite symmetric

^and are not

truly

"random". In

fact,

for such small N values

(except

for N

=

5),

the

topology

of the final

configuration

does not

depend

_on the choice of the initial random

configuration.

This is no

longer

verified for

large

N values _as it will be discussed below.

Anyway

the most ordered and

symmetric

structures

correspond

to the

largest

values for the

packing

fractions. In

particular,

in the _cases N

= 4

(top left),

6

(top right),

12

(bottom right),

the

points

_are located at the vertices of the

only existing perfect polyhedra

made with

perfect triangular faces,

which _are

tetrahedron,

octahedron and

icosahedron,

with

integer

coordination numbers _z

=

3,4

and

5,

respectively,

and with

fe

= 0 for _e

#

_z and

fe

₌ I for _e

= z. The

corresponding packing

fractions _{are c}

=

0.845,

0.879 and 0.896.

They extrapolate nicely

to the

trianglar

lattice which

corresponds

to N

= cc, z = 6 and _c

=

0.907. All the other _cases exhibit much less

symmetry

and have _a broader distribution of

fe's.

For

example,

in both _cases N

= 5 and N

=

II,

the

structure-corresponds

to _a

perfect polyhedron

with _a vacancy. Note that in the _case N

= 5

(at

the top center of

Fig. I)

there is _a "continuous

degeneracy"

since the three

points (white dots),

which _are not _on the north and south

poles (black dot),

_are free to rotate around them

(within

hard _core

constrains).

The _case N

= 8

(at

the center of

Fig, I)

is

tricky

since the

points

are not located _on the vertices of _a

perfect

cube.

They

reach _{a more} compact structure

exhibiting only

two

parallel perfect

_square

configurations

of

neighbors,

twisted

by

_an

angle

of

~/8.

Note that these _square

configurations

_are

degenerate

in the _sense that there _{are two} different choices for the

simplicial triangles (our

code then

performs

_an

arbitrary

choice for the

bonds).

Nevertheless the

configuration

is

unique.

It is worth

noticing

that all _our smalLN

configurations

_very often

correspond

to the _ones

recently

obtained

numerically

for the

closely

related Thomson

problem [12j.

It is

obviously

less _easy to vizualize the

sphere packings

on

53.

Given _a

single plane projection

is

generally

not sufficient to

figure

out how the

points

are

organized. Anyway

the _same trends

are observed: the _more

compact

is the

packing,

the _more

sylhmetric

is the

configuration

and the less rich _are the

fF

and

fe

distributions. The

simplest

Voronoi

tessellations,

with identical

perfect polyhedral cells,

_are obtained for N

=

5,

8 and

120,

where the values of _{z are}

4,

6 and 12,

respectively.

The cells _are

perfect

tetrahedra

IF

= 4, _{e =}

3),

cubes

IF

₌

6,

_{e =}

4)

and

Fig. 2. Two-dimensional projections of sphere

packings

_on 53 with N

= 5

(top left),

8

(top right),

120

(center

and

bottom).

The

projections

of the

sphere

centers are indicated by white dots _or black dots

(in

the _case of

multiple projections)

and the bonds _are

represented by straight

line segments. ^In the _case N

= 120, three different

projections

_are shown.

Table II. Radius R

of

the

sphere 53 (in

disc diameter

unit), packing fraction

_c, coordina- tion number _z, number _e

of edges of

Voronoi cells

faces, for packings of spheres

_on

53

with N

=

5,

^{8, 120.}

N R _{c z e}

5 0.5484 0.6806 4 3

8 0.6366 0.7268 6 4

120 1.5915 0.7741 12 5

dodecahedra

(F

=

12,

e =

5)

and the

packing

fractions _are

0.6806, 0.7268, 0.7741, respectively.

They correspond surely

to

polytopes (3, 3, 3), (3, 3, 4)

and

(3, 3, 5) using

the notation of Coxeter

[18j.

Two~dimensional

projections

of these

packings

_are

represented

in

Figure

2 and the

corresponding

values of _c and R _are

reported

in Table II. For these

projections

_we have

performed

rotations of

53

in order that two different

points

are

projected

at the center of the

figure.

As in

Figure I,

when two, or more,

sphere

centers have the _same

projection, they

are

represented by

_a black dot. Therefore the center

point

^is

always

_a black dot. Note that for N

= 8

(top right)

it

corresponds

to fur

sphere

centers. For N

= 120 _{one recovers} the

so-called

polytope (3, 3, 5) [18,19j

^{which exhibits five~fold}

symmetry

and

corresponds

to the most compact

packing.

In

Figure

2, we have

represented

three different

possible projections

of

polytope (3, 3, 5)

which exhibit

symmetry 10,

6 and 4. Note that its volume fraction is

larger

than the one of face-centered~cubic and

hexagonaLclose-compact

structures in flat _space.

o.95

o.90

0.85

0.80

0.75

0.70

II .(j/

_'

0.65 _{II ;.}

.>I'/l~~ l~~',

_',"

..

" ....°

o.60

~ ~~ loo iso 2°° ~~° ~~~

N

Fig.

3. Numerical results for the packing fraction _{c as a} function of N _{up to} N

= 300 in both the

52

(upper curve)

and 53

(lower curve)

_cases. Five independent results have been recorded for each N value.

3.2. RESULTS FOR INTERMEDIATE N VALUES. The numerical results for the

packing

fraction _c for N values _up to 300 _are

depicted

in

Figure

3 for both the

52

and

53

_cases. We have _run the code five times for each value of N

using independent

seeds for the random

generator. Despite

_some noise which _can be attributed to both the lack of

efficiency

of the

algorithm (in

the

figure

_we have taken _a

= 0.33 and K

=

10~~,

_as

explained above)

and the randomness of the initial

configuration,

the "curves"

c(N)

turn out to be

qualitatively

^different

in the two cases, at least in the range of N values

reported

in the

figure.

In the

52

_case, the

dispersion

of the

points, already

found for the small N values discussed

above, disappears progressively

_as N increases while the _mean c~value increases _very

slowly

from _c cf 0.835 for N ₌ 100 to N _m 0.845 for N

=

300,

a value still

significantly

smaller than the

packing

fraction 0.907 of the

triangular

lattice.

In the

53

_case, the

dispersion

of

points

turns out to be such that _{one can} define

roughly

two different _{mean curves.} The lower _curve is almost flat and

corresponds

to _a c~value of about

0.655,

close _to

(and slightly larger than)

the Bernal's

packing fraction,

while the _{upper curve} exhibits _a series of maxima. We will comment _{more on} these two _curves later

(Sect. 3.4)

but

we would like first to focus _on the maxima of the _{upper curve.}

They

are

asymetric

in N since it is easier to create a vacancy than to add _an extra hard

sphere

to _a

compact

structure. In

particular,

all the _most

compact

structures obtained from N

= III to N

= l19 turn out to

be the

polytope (3, 3, 5)

with _a

given

^number of vacancies. As _a consequence the _upper

c(N)

curve is linear in that range of values of N.

Moreover,

as was

already

noticed _on

52

for small

values of

N,

_we observe that the maxima

always correspond

to the more

regular

structures with

narrower distributions for the

fe's.

This is

obviously

the _case of the

sharp peak

at N

= 120

.o

0.8 O 75

~ . 76

u 77 o

u 0.6

0A

O.2

~ O

O-O

2 e

Fig.

4. Fraction of Voronoi cells faces with

e~edges, fe,

_{as a} function of _e for

spheres packings

_on 53 with N

= 75

(open circles),

76

(filled circles)

and 77

(squares).

~

fiO

§

Q

fibCIJ~jJ

^o

~o ~O

O

~

~ ~~ ^jfj

~~j~~jj~~li~

llJ

o~

^°

cc ^O

Fig.

5. Two-dimensional projections of sphere

packings

_on 53 with N

= 20 and 76 with the _same

conventions _as in

Figure

2. The bonds _{are not} shown for N

= 76.

but this is also true for all the

secondary peaks.

To

give

_an

example,

we have

reported

the

fe histogram

for the most compact

packings

obtained with N

=

75,

76 and

77,

in

Figure

4. As

one can see, these

histograms

_{are more}

peaked

for N

= 76 which

corresponds

to _a maximum

of the

c(N)

_curve.

However,

for

larger

values of

N,

the

packings corresponding

to maxima

do not exhibit

special symmetries.

This is illustrated in

Figure

5 where _we have shown the two~dimensional

projection

of the

packings

obtained for N

= 20 and N

=

76,

both of which

corresponding

to maxima of the _curve. While the N

= 20

packing

exhibits _some symmetry the N

= 76

packing

looks random. In both _cases the Voronoi tessellation

only

involves two

topological

kinds of

cells, polyhedra

with 9

(fg

=

0.8)

and 10

(fir

₌

0.2)

faces in the _case

o.95

o.90

*

0.80

"

O.75 . lO"~

o.70

,

.65

O.60

~~

~~

~'~

l/N o.95

o.90

O.85

~ ^° _l

lo'~

o.75

o.70

,

O.65

.60

O.OO

.05 -lo .15 .20 .25

b)

Fig. 6. - Large N _numerical results for K =

The packing

and

53

curve)

cases.

Fig.

7.

Stereographic

projections of

packings

of discs _on 52 with N ₌ 4096 and N

= 8192.

Only

the 1024 discs closest to the

projection

center have been

represented.

The discs with _e

~

6 _are indicated by black dots.

their

regularity (geometric progression)

rather than for any technical _reasons. The results for the the

packing

fractions _are

reported

_{as a}

plot

of _{c versus} I

IN

in

Figure

6a and _versus

II log

N in

Figure

fib. In the

52

_case the convergence ^to the

triangular

lattice does not show up in the I

IN plot

of

Figure

6a _as well _as in any

plot

of the type I

IN".

As _seen in

Figure

fib the convergence to 0.907 is _more

likely

of the

type II log N,

^which is _an

extremely

slow convergence.

To illustrate this slow convergence we have shown in

Figure

7 _some

stereographic projections

of the

packings

obtained with N

= 4096 and N

= 8192. The

stereographic projection

is _a standard

procedure

used in

cartography. Imagine

a north

pole

N and _a south

pole

S. The

projection

^M' of any

point

M of

52

is obtained

by intersecting

NM and the

plane tangent

to

52

in S. Such a

projection

has the

advantage

to transform circles into circles. In the

figure only

the 1024 discs closest to the south

pole

have been

represented

in both _cases. Note that the effect of curvature, ^{which is} to increase the apparent diameter of the discs when

going

from the center to the

periphery

of the

projection,

is

hardly

visible on the

figures,

due to the

large

values of N. The discs centers

corresponding

to

defects,

I.e. to Voronoi cells with _e

# 6,

are

indicated

by

black dots. These defects form

lines,

_or

_"grain boundaries",

whose

density

does not decrease

significantly

when

increasing

N. The

necessity

to

keep

_an

homogeneous

_array of

grain

boundaries _as N increases is due to the constant curvature of the space.

In the

53

_case the convergence ^to the Bernal's value is _so

quick

that it _can

already

be _seen in the I

IN plot

of

Figure

6a.

Indeed,

there is _no way ^to

imagine

_a convergence to the fcc value.

The difference between the

52

and

53

_{cases can} also be stressed on the

g(r)

_curves

(Figs.

8a and

8b). While,

in the

52

_{case, one sees an} evolution towards

sharp peaks

_as N

increases,

as

it should be when

approaching

_a

crystal,

this

sharpening

is absent in the

53

case. It is worth

noticing

that the

g(r)

_curve obtained for N

= 8192 is

remarkably

close _to the _one

corresponding

to _a Bernal's

packing

of the _same number of

particles

^built in _a cubic box

[14j.

^All the well known characteristic features _are

present,

ⁱⁿ

particular

the twin

peaks

at _{r =}

v5

_and

r = 2.

One _sees in

Figure

8b that the

peak

at _r

=

vi

_{results from}

a shift to

larger

_r of the second

neihgbor peak

of the

polytope (3, 3, 5).

The statistics of the Voronoi tessellation _are also very similar to those for Bernal

packings.

In

Figure

9 _we

give

the results for the coordination number _z, which is the _mean number of faces of the Voronoi

cells,

_{as a} function of I

IN.

Note that there is _an exact

relationship

between N and _z in _two dimensions

(z

=

6(1- 2/N)) _[20j

which has been verified

by

_our

numerical results. In three dimensions _one

gets only

an

approximate

^{linear behavior} in I

IN.

It is worth

noticing

that the

extrapolation

to N infinite

gives

z = 14.22 _as the coordination number of the "ideal"

(infinite)

Bernal's

packing.

Such _a value is consistent with values of _z

5.o

4.O 256

8192

3.O

2.O

i .o

~'~0

2 3 4 5 6 7 8 9 lo

a)

r

3 0

I

I

t 128

2.5

II 256

Ii

8192

11

~ ~ /j

' l

# 5 '

1 ~

C'

/ ji

)

1.0 _'

'

~~ Ii

0.5 ^{' ,'}

O-o

0 2 3 4 5

b)

r

Fig.

8. Pair correlation functions

g(r)

for

packings

of discs _on 52 with N ₌ 256 and 8192

(a)

and for

packings

of spheres _on 53 with N ₌ 128, 256 and 8192

(b).

In

(b)

the delta

peaks

of the

g(r)

_curve

for the

polytope (3,

3,

5) (N

₌

120)

_are shown with arbitrary units.

obtained from finite Bernal

packings

built in _a cubic box

[14].

^{Note that other}

algorithms

based _on tetrahedra

packings (which

do not deal with hard

spheres)

lead to _z values of about 13.4 [9,

21, 23]

in close connection with the statistical

honeycombs

of Coxeter

[22].

^{It would} be

interesting

to understand whether these _two values

(14.2

and

13.4)

_are related.

14.5

14.0 13.5

~~'i.000 _.001

1/N

400

300 128

256 512

)

200

loo /,

/ ,

'

'~-~

'

/ '

~

' 0

0.64 0.65 0.66 0.67 0.68

c

Fig.

10. Histogram of the

c values obtained when running the

Jodrey~Tory

code

fifty

times for

packings

of

spheres

_on 53 with N

= 128, 256 and 512. All results have been obtained with K

=

10~~

converges

slowly

to the

triangular

lattice while the three~dimensional

sphere packing

_converges

to the frustrated Bernal

packing.

For _some

particular

values of the number of

spheres

the

packings

_on

53

_{are more compact} and exhibit _narrower

histograms

for the Voronoi cells char~

acteristics. Such _a

study presents

a way ^to understand the mechanisms of frustration in real disordered

systems.

It also

provides

an alternative _way to reach the

thermodynamic

limit in numerical methods

deeling

with

large

three dimensional systems. Instead of

working

in _a flat

space within _a cube

subject

to

periodic boundary

conditions

(which

is

topologically equivalent

to

working

_{on a}

torus),

and

trying

to reach the

thermodynamic

limit

by increasing

the

edge length

L of the

cube,

_{one can} work _on

53 by increasing

^the _space ^radius R. The geometry ^is

slightly

_more

complicated geometry

but there _are considerable

advantages.

In

particular

the space remains

isotropic

^{for all} values of R. This

might

be _an

advantage

in

studying glasses

for which it is well known that the

anisotropy

of the

boundary

conditions _can induce artefactual

crystallization.

In the future _we intend to

develop

numerical methods such _as Monte~carlo _or molecular

dynamics

to

study glasses

in curved _space.

Acknowledgments

Numerical calculations _were done at CNUSC

(Centre

National Universitaire Sud de

Calcul), Montpellier,

France. One of _us

(R.J.)

would like to thank discussions with Didier

Caprion

and

Philippe

Jund.

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