Packing fraction for hard disk random heaps

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Packing fraction for hard disk random heaps

R. Perkins

To cite this version:

R. Perkins. Packing fraction for hard disk random heaps. Journal de Physique I, EDP Sciences, 1994,

4 (3), pp.357-359. �10.1051/jp1:1994143�. �jpa-00246911�

J. Phys. 1FYa~lce 4

(1994)

357-359 MARCH 1994, PAGE 357

Classification Physics Abstracts

05.70 46.10

Short Communication

Packing fraction for hard disk random heaps

R-S- Perkins

University of Southwestern Louisiana, Lafayette, ^Louisiana 70504-4370, U-S-A-

(Received

23 September 1993, re~ised 29 December 1993, accepted 17

January1994)

Abstract. Structures for the ramdam distributions of disks _{in one} dimension _are used ta calculate trie packing density of two dimensional random heaps of disks. Trie calculated values

are compared with those obtained from computer simulations.

Studies bave been done on trie random

packing

^of

objects

in one, two, and three dimensions.

Such studies have been done

by

computer simulation _as well _as with

experiments using

actual

objects.

Summaries of studies in two

iii

and three _[2] dimensions _are available. A

long-

time

goal

has been _a mathematical

description

of random

packing.

A

complete

mathematical

description

has been done

only

for the _one dimensional case; for the random distribution of

equal

fine segments

along

_one

dimension,

the

problem

has been solved

mathematically by Renyi

[3]. ^Other work for the one dimensional _case is referenced in _a review

by

Solomon and

Weiner [4]. ^A mathematical

description

for _{a very} small two dimensional _case has been

given

[5]. ^{In the} present paper, an

approach

is used to calculate the

packing

fraction of _a

large

two dimensional random distribution. The

starting point

is the mathematical

description

of trie

one dimensional _case.

In _{a recent}

study, Acharyya

_[fil did _a computer ^{simulation of hard disks}

falling

in two

dimensions onto _a

randomly positioned

base

layer.

Trie positions of the horizontal diameters of the disks in trie base

layer

_are described

by Renyi's theory

_[3]. His

theory gives

the fraction of

one dimension

occupied by randomly packed

fine segments as 0.74760 1?i. ^For

randomly packed

segments ^{of unit}

length,

this

corresponds

to _an average gap between segments of 0.33761.

Therefore,

trie base

layer

in

Acharyya's

simulation _is made _up of disks with gaps between them. The gaps range in _size from _zero to almost _one disk

diameter,

but trie average gap

between disks is 0.33761 diameters.

In

figure

1, the bottom

layer

of disks is

positioned

non

by using

_a distribution of gap sizes but

by using

trie average gap size for _one dimension. This

representation,

which I call _an average random structure, will not give information about trie details of _a random _structure such _as distribution of gap sizes or radial distribution

function,

but it _cari

give

trie overall

property ^of

packing density.

In this _{paper, a} two dimensional structure _is obtained from trie

358 JOURNAL DE PHYSIQUE I N°3

Fig. 1. Average random structure for two dimensional heap of disks formed _{on a} randomly placed

base layer.

one dimensional _average random structure of trie base

layer

in

figure

1. Trie

packing density

of the

resulting

two dimensional collection of disks _is calculated and

compared

to results of

simulations and the _comparison used _as evidence that _a two dimensional _average random

structure con be constructed from _{a one} dimensional _average random structure. Trie Upper

layers

in

figure

1 show the arrangement ^{obtained when disks fall}

along randomly placed

vertical

trajectories

to stable

positions.

Trie pattern in

figure

_can be

represented by

varions unit cells _{or areas, one} of which is shown. Two sides of the unit _{area are} one disk diameter _m

length.

The other _two sides _are 1.33761 diameters in

length.

A

straightforward

calculation

gives

trie _area of this unit _{area as}

A = [1

(1.33761/2)~]

^~~~

^(1.33761)

^d~

where d is the disk diameter. The cell contains _a total of _one disk and the

packing density (Ad;sk/Ac~jj)

is found to be 0.78980. This _agrees with the value obtained

by Acharyya

of 0.795 + 0.01.

The _average random structure _can be found for

heaps

formed in another _way. Meakin

and Jullien _[8] bave also done _a computer simulation of

falling

hard disks which form _a two dimensional random

heap.

Their base

layer

_is different thon in the _case considered above. It is formed

by

disks

falling

to it at random locations but also

by

disks

falling

_{on an}

already

present disk and

by

rotation

coming

to rest _on the base

layer touching

the

neighboring

disk it first

contacted. The base

layer

in finis simulation bas _more

touching

disks thon

in trie first _case and trie _{average gap} size would be

expected

to be less.

Solomon has treated such a one dimensional arrangement borin

theoretically

and

through

simulations [4]. ^{He found} a theoretical value for the

packing density

of 0.8087.

Computer

simulations done for this

study using

about 4000 disks per simulation

give

_{an average}

packing density

for twenty simulations of 0.8091. A value of 0.8087 for

packing density corresponds

to _an average gap of 0.2366. Trie _average random structure for the simulation of Meakin and Jullien _[8] could be

represented by

_a

diagram

similar to

figure

1. Trie

only

difference would be that trie two

long

sides of trie unit _area would bave

lengths

of1.2366 diameters.

Using

trie _same

procedure

as above

gives

a two dimensional

packing density

of 0.8081. Meakin and Jullien did

large

and small scale simulations and found densities

ranging

from 0.8170 to 0.8210 [8].

They reported

values of 0.820 + 0.005 for trie small scale and 0.8180 + 0.0001 for the

large

scale

simulations. These values _are somewhat

langer

^{than those calculated here.}

It is

tempting

to use

figure

1 to obtain _a value of the

angle

of _repose, which is trie

angle

between the

idge

of the

heap

and the horizontal. The

angle

of _repose~

however,

is non an

overall property ^of the entire array of disks. It is _a property of

only

_part of the array and to calculate it, trie distribution of gap sizes must be used. It is

relatively

_easy to calculate trie doute

angle

^of trie unit _area in

figure

1.

Using

trie distribution of _{gap sizes} found

by

N°3 PACKING FRACTION FOR HARD DISK RANDOM HEAPS 359

Mackenzie [9], ^trie

probability density

_{as a} function of _gap size _was calculated. The

integrals

m Mackenzie's distribution _were

numerically

evaluated at intervals of _0.02 diameters

using

commercial computer ^software.

Trigonometry gives

the

angles corresponding

to the different gap sizes. The

product

of

probability density, angle,

and

interval,

summed _over all gap sizes

gives

the average

angle.

The value obtained is 53.6°. This value is close to that

given by Acharyya (r- 52°).

The calculated distribution of

probability

for different

angles

is very similar to that obtained

by Acharyya through

simulations. But the

angle

calculated here is not the

angle

^of _repose. The

angle

^of _repose

depends

not

only

_on this

angle involving

the first and second

layer

but also _on the

corresponding angles involving

^ail other

adjacent layers.

These

angles

involve the distribution of _{gap size} between the first _two disks _m the base

layer (as

in trie above

calculation)

but also trie distribution of _{gap sizes} in

succeeding

_gaps across trie base

layer.

Using

a two dimensional _average random _{structure as a} base

layer

to generate a three dimensional _average random structure _is

possible

but there _are several

problems

involved.

Using

trie _two dimensional _structures

generated

bene does _not lead _to

unique

three dimensional structures.

Also,

trie structures

generated

bene _are not the base

layer

of _a realistic three dimensional

heap.

A realistic _case would _{use a}

randomly deposited

two dimensional base

layer.

There is _no mathematical

description

of such _a

layer langer

than 4

partiales

_{[5] so} the

goal

of

mathematically describing

_a three dimensional random

heap

of

spheres

cannot at this time be achieved

by

_an extension of the method outlined in this _paper.

References

[ii

Hinrichsen E-L-, Feder J., Joessang T., Phys. Rev. A 41

(1990)

4199.

[2] ^{Nolan G-T- and} Kavanagh P-E-, Powder Technology 72

(1992)

149.

[3] Renyi A., Publ. Math. Inst. Hung. Acad. Sci. 3

(1958)109.

[4] ^{Salomon H, and Weiner} H., Commun. Star. Theor. Meth. 15 (1986) 2571.

[5] Zheng Y., J. Star. Comput. Simul. 29

(1988)

105.

[6] Acharyya M., J. Phys. I fronce 3

(1993)

905.

[7] ^Blaisdell B-E- and Solomon H.< ^J. Appt. Prob. 19

(1970)

382.

[8] Meakin P. and Jullien R., J. Phys. fFa~1ce 48

(1987)

1651.

[9] Mackenzie J-K-< J. Chem. Phys. 37

(1962)

723.