Configurational entropy of a 2-D continuous random network

Journal of Non-Crystalline Solids 75 (1985) 141-146 North-Holland, Amsterdam

CONFIGURATIONAL

ENTROPY

M.A.

K.

KLENIN

and

Department of Raieigh, N.C.

OF

141

A 2-D

CONTINUOUS

Physics, North 27695-8202

Carolina

State

The

calculation solids

experiment. based

on

of

is

the

for

Estimateslha~e

ring

structure

been or

on

made

a straightforward

describe in

random

here ~hich

the

network

dimensional

model

dimensional

analog

certain

two-,

discussed entropy imposed

(CRN). as

describes

fourfold

a

implicitly

on

The

model

the

four-particle

under

algorithm.

Thus

the

of

number

network provides history limits

it

a fixed of the

the

the

is

of

configurational

but

in

standard

a

quantitative CRN

the

to

relates

the

the

function

in

those

a continu-

a simple

in

been computed

algorithm.

discrete its

values

building

matter

to

enumerate

explicitly

available

at

each

in

the

its

entropy estimate

construction,

at

current

further.

CRN its

where

0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

each

role

in

of

step,

but

the

configuration steric

models effect

the

the

algorithm

spatial

This

of

step

building

possiblities

two-

three-

constraints

building only

for

order

geometrical

by

general.

whose

has

and

not

by

model, range

permits

In p r i n c i p l e

number

technique This

result

to

a simple

sample--i.e,

estimate

of

cases,

analysis

are

structures,

network

"moves"

possibilities

known~'

main way

correlation

valid

our

models with

special

described

intermediate

consideration

construction.

reducing

the

Our

well

case.

coordinated

qualitative

some

statistical techniques

is

apply

for

comparison

entropy

order We

for

such

a paradigm

else~here~ in

but

local

of

combined

We

systems

entropy

purposes

arguments,

in

eonfigurational entropy parameter distributions. to the geometrical

configurational

important

phenomenological

of

University,

INTRODUCTION

glassy

ous

NETWORK

PROMISLOW

A d i r e c t m e t h o d of c o m p u t a t i o n of t h e is u s e d f o r a C R N m o d e l w i t h d i s c r e t e Entropy maxima and minima are related properties of t h e m o d e l .

i.

RANDOM

four-

is is

hindrance well

difficult

particle

--

M.A. Klenin, K. Promislow / 2-D continuous random network

142

Lorrelation

is p e r m i t t e d

a b l e to c o r r e l a t e ~haracteristics at t h e

twofold

the entropy peaks

t o t a k e on a r b i t r a r y

entropy

estimates

of t h e n e t w o r k coordinate

site

as a f u n c t i o n

correspond

with

values.

the space

We are

filling

as t h e e f f e c t i v e

bonding

is v a r i e d .

are observed

of b o n d a n g l e .

to t h e e x i s t e n c e

Peaks

It a p p e a r s

of e s p e c i a l l y

angle

that

simple

in

these

tiling

patterns.

2. M O D E L I N G

PROCEDURE

We u s e a p l a n a r

two-,fourfold

comprised

of r e c t a n g l e s

specified

by t h e b o n d i n g

site.

The blocks

are strictly

was

found

between

attain

a constant

U s e of d i f f e r e n t

v a l u e at s i z e s randomization

kinetics

typical

large systems.

displayed valid

3.

is j u s t

moves

of

in our

is a l l o w e d . )

In n o c a s e

The

running

had

sample

kinetics

averages

less than about

procedures

the

Thus there

are discrete.

also

of

of the

appeared

to

1000 b l o c k s .

indicated

that

t a k e n on b e h a v i o r

The configurational

t h e s u m of

angles

of b l o c k s ,

these properties

samples

at e a c h b u i l d i n g

bond

for any pair

10,000 blocks.

. Most

is

coordinated

limiting. T y p i c a l

by maintaining

the growth of

sampled

to be self

quantities

ratio

so t h a t all

(Edge b o n d i n g

3000 and

were monitored

various bulk

structure

at t h e f o u r f o l d

configurations

of t h e d i s t r i b u t i o n s

sizes ranged growth

angle

maintained.

the algorithm

coordinated

length-to-width

are corner-connected

are only four possible a n d all

whose

logarithms

entropy

of

estimate

the number

of

step.

RESULTS Figure

shape o f

1 r e p r e s e n t s two t y p i c a l the

"bricks"

is

t h e same i n

c o r r e s p o n d s t o a bond a n g l e o f site.

t h e two cases,

60 ° a t

the fourfold

and coordinated

The bond a n g l e s a t t h e t w o f o l d c o o r d i n a t e d s i t e

120 ° and 1 0 7 . 5 ° , r e s p e c t i v e l y . the f o r m a t i o n of

triangular

f r e q u e n c y . There a r e i n formation--

i.e.,

rings,

this

are

The 120 ° bond a n g l e p e r m i t s and t h e s e appear w i t h h i g h

case no l a r g e r e g i o n s o f

"crystal"

no l a r g e r e g i o n s p o s s e s s i n g t r a n s l a t i o n a l

symmetry. The s t r u c t u r e smallest sides,

c o n f i g u r a t i o n s o b t a i n e d . The

allowed r i n g ,

and two l o n g .

corresponding t o a rectangle,

This

is

not a r i g i d

formed w i t h h i g h f r e q u e n c y a t a l l

I 0 7 . 5 ° has as i t s

composed o f

two s h o r t

structure

bond a n g l e s .

and i s

T h i s sample

M.A. Klenin, K. Promislow / 2-D continuous random network

appears the

to

be

Forced

rectangular

loosely

co~inected.

geometrical

te

ring

"crystal"

~orm

structure,

All

with

samples

parameters

had

constructed

a virtually

FIGURE 1 Portions of t y p i c a l 1 2 0 ° a n d 1 0 7 . 5 °, r e s p e c t i v e l y .

1.0

I

I

I

I

I

I

I

I

I

I

I

1

I

domains these

I

I

arrays.

with

The

consisting

domains

identical

bond

143

being

this

set

of only of

appearance.

angles

are

f

I

500 ---- I 0 0 0 ..... 9000 ----

0.9 0.8

O.Z o 0.6

0._ o

FIGURE 2 E n t r o p y v e r s u s bond angle at the cornerconnected s i t e .

0.5'

~- 0.4

5

0.3

0.2 0.I 0.0

I

I

I

I

I

120

I

I

I

I

I

I

I

140 BOND ANGLE

I

I

I

160

I

I

I

180

M.A. Klenin, K. Promislow / 2-D continuous random network

144

Figure of

the

shape for

2

shows

the

bond

angle

at

of

the

run

sizes

features

of

gradual

rise

maximum

at

than and

that its

Fig.

of the as

at

The

angle and

a sharp

the

are

120°,

(!)

variable

but

it

a point

the

near

this

is

such The

the

maximum;

minimum

and

less

are at

main

120°~

increased~

all

sharp

made The

at

is u n d e r s t o o d .

the

displayed

"bricks."

is

a function

with

estimates

maximum

in

as

site~

maximum

angle

appears

origin to

3000

bond

155°;

site

values

with

curve

1 corresponds to

constant.

bond

per

coordinated

500~I000~2500

(2) and

obvious

series

of

runs~

120 ° sample the

a (3)

of

107.5 °

value.

DISCUSSION gradual

rise

angle

The

increases

is

there

is n o

between The as

an

successive of

entropy

the

degeneracy

grid

builds

an

with pattern

is

The

grid most

the

structure

rectangular

strong

forced~

underlying

maximum.

of

the

and

angular domains

structure

striking

as

If

correlation are

formed.

manifests

case

grid.

underlying interfere

with

pronounced

In t h i s

lattice~ one

m~ximum

appearance

of

rings

sides~

respectively.

phases.

the

network

despite 155 ° and

of

Thus

ring four

one

domains

angle

dense

itself

occurs

of

tend

packing.

structure,

two

these

slightly

imposed

160 ° also

short

has

is

elements

between

of

bond

the

network

another

consisting

Formation of

the

a specific

square

values

case

and

sides

when

phases

on

the

not

to

The

less

namely and

four

the

on a

corresponds

coexisting

different

bond

rings.

a n g l e reaches 120°~ and t h e b l o c k s occupy p o s i t i o n s

hexagonal

the

in

associated

underlying

presence

bond

5.

held

approximately at

entropy

twofold

each

geometrical

sample,

4.

"bricks"

a single

sample

calculated the

to

a pair

of

long

"crystal"

is

observed

from

1 6 0 °.

at

CONCLUSION D i r e c t e n u m e r a t i o n used as an e n t r o p y measure

prediction

of

e x i s t e n c e of structure

regimes o f

"crystal"

regular tilings.

domain

E x i s t e n c e of

does n o t encourage domain

permits

f o r m a t i o n and t h e an u n d e r l y i n g g r i d

f o r m a t i o n , but r a t h e r

permits the f o r m a t i o n of a h i g h l y disordered s t r u c t u r a l by r e d u c t i o n o f

steric

hindrance.

state

a

M,A. Klenin, K. Promislow / 2-D con tinuous random network

ACKMOWLEDGMENT We grate~ully NOOO

acknowledge

support

oT O N R u n d e r

Contract

I zi-- 7 Q -C- ._-.1333.

,-,EF,= : , , _ N C E.S

t~ B e r n a l ~ J . D . ,

Proc. R . S o c . 2 8 0 A ~ 2 9 9 ( 1 9 6 1 )

2)

King~ S. V. , N a t u r e ~ Lond. 213, 1 1 1 2 ( 1 9 6 7 )

3)

BelI~R.J.~

a n d P.

D e a n ~ P h y s . Chem. G l a s s e s

9~125(1968)

4) T a d r o s , A . , M . A . K l e n i n ~ and G. L u c o v s k y ~ J. Solids64~215(1984)~also this conference.

Non-Crystalline

145