The effects of size polydispersity in nearly hard sphere colloids

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The effects of size polydispersity in nearly hard sphere colloids

Ichirou Moriguchi, Kyozi Kawasaki, Toshihiro Kawakatsu

To cite this version:

Ichirou Moriguchi, Kyozi Kawasaki, Toshihiro Kawakatsu. The effects of size polydispersity in nearly hard sphere colloids. Journal de Physique II, EDP Sciences, 1993, 3 (8), pp.1179-1184.

�10.1051/jp2:1993190�. �jpa-00247895�

Classification Physics Abstracts

64.70P 82.70D 81.30D

Short Communication

The eJllects of size polydispersity b1 nearly ^hard sphere ^colloids

Ichirou

Moriguchi, Kyozi

Kawasaki and Toshihiro Kawakatsu

Department of Physics, Kyushu University 33, Fukuoka 8121 Japan

(Received

13 April1993, accepted in final form 24 May

1993)

Abstract. We have investigated the effects of size polydispersity _on glass transition in nearly hard sphere colloidal systems in non-equilibrium condition after rapid compression by the Brownian dynamics simulation method. The interparticle interaction is chosen _as the Yukawa potential and the size distribution of colloidal particles is assumed to be Gaussian. We found that the critical standard deviation of the size distribution divided by the _mean size is about 6 _~- 7Sl above which _no crystallization takes place for arbitrary density, and also found that there exists _a disordered state _even with _a smaller size polydispersity than the critical value in the high density regioni _as is observed in the real colloidal suspensions. We also mention

the possibility that by performing Brownian dynamics simulations _on colloidal systems with _a

sufficiently reduced level of thermal noisei _{one can} observe _a sharp glass transition.

The

simplest

version of the mode

coupling theory

has

predicted

_a discontinuous

ergodic

to

non-ergodic

transition close _to the

glass

transition

point ill.

The transition of this type is called ideal

glass

transition. In real systems~ ^activated processes, e,g, the

jump

motions of

particles,

^often _occur and lead _to residual diffusion _even in the

non-ergodic

state, which _smears out the

sharp

transition. Such _a residual diffusion _can often be observed in the atomic

glassy

systems which

obey

the Newtonian

dynamics [2,3]. However,

in

glassy

colloidal

suspension

systems _[4] ^which

obey

Brownian

dynamics,

the noise level _can be

quite

small because of

large

sizes of colloidal

particles.

Therefore the activated _processes in colloidal systems are

expected

to be

extremely infrequent.

In this situation there is _a

possibility

that _{one can} observe _a

sharp glass

transition since the

ergodic

to

non-ergodic

transition coincides with the

glass

transition in absence of such activated _processes

ill.

In order to simulate such colloidal systems, one has to choose lower temperatures than those used in

ordinary

molecular

dynamics

simulations of atomic systems

[2,3].

^This

point

has not been

consciously

put forward in past simulations

[2,3].

As the temperature and the

density

_are not

independent

parameters ^{in the}

soft-sphere

systems often used to model

glassy

systems, ^such a

suppression

of activated _processes cannot be realized in such systems

[2,3]. However,

a Brownian

dynamics

simulation with _a realistic

interparticle

potential

should be able to

provide

_an idealized model of real

suspension

systems ^of

spherical

colloidal

particles

in _a solvent which _may

correrspond

to the colloidal suspension ^studied

by

1180 JOURNAL DE PHYSIQUE II N°8

Pusey

et al. [4].

Although Pusey

et al,

reported

that the size

polydispersity (SPD)

of their

sample

is small

enough

to be

regarded

_as

effectively monodisperse,

there has been _no successful

computer ^{simulation that}

reporduces

_a

glassy

state

using

_a

monodisperse

_system with _a

simple interparticle potential. Dzugutov

and

Dahlborg recently reported

^that a one component system

can form _a

glassy

state [5]. However

they

used _an

interparticle potential

which has _a barrier between _a minimum at short distance and _a

long

distance tail of the

potential.

Such _a

potential

leads to _an

effectively

two component system when the temperature is lower than the

potential

barrier and _may not be

regarded

_{as a}

monodisperse

system in its true sense. Moreover the

sample

used in the

experiment

^of

Pusey

et al, consists of

nearly

hard

sphere particles

with small

SPD,

which cannot be modelled

by

the soft

sphere

system. ^This motivates _us to

perform

a Brownian

dynamics

simulation

using nearly

hard

sphere potential

under

non-equilibrium

conditions and to compare the results with their

experiments.

In this

study

_we

investigate

the effects of

polydispersity by

Brownian

dynamics

simulations.

The effects of SPD and

charge polydispersity(CPD)

have been

previously investigated by

_com- puter simulations [6,

ii

and

theoretically

_[8]. Dickinson et al.

performed

_a computer simulation

using

_a ^small

sample

and

reported

that the

melting

transition

disappears

when the SPD is in- creased.

However,

this

investigation

was focused

only

_on the

equilibrium

states. Unlike their

simulations [6,

ii

and the

density

^functional

theory

[8], ⁱⁿ a real

experiment

_one has to _com- press or

quench

_a

liquid sample

in order to

produce

_a

glassy

state, ^which may be _a metastable state. So it is

important

to examine in detail the effects of SPD and the parameter _range where

glassy

state _can exist

by performing

simulations under

non-equilibrium

conditions.

We consider

spherical particles interacting through

the

following

Yukawa

potential;

#ij(r)

= e

~~~

exp

l~

~~~

(l)

r ajj

with aij ⁺

(az

+

aj) /2,

_ai

being

the diameter of

particle

I. We have assumed that the

particle

diameter _az

obeys

^{the Gaussian} distribution with _{a mean} size il and the standard deviation

~£l, ^where ~ is the standard deviation divided

by

the _mean

size,

and I is the effective

screening

parameter. If this parameter ^is

sufficiently large, equation (I)

_represents

nearly

hard

sphere potential

like

polymethylmethacrylate

used in the

experiment

of

Pusey

et al. In _our

simulation,

we chose 1

= 15 to represent a

harshly repulsive potential.

The

equation

of motion of colloidal

particles

is [9]

f@

=

Fz(t)

₊

Rz(t) (2)

where

f

is the friction

coefficient, ri(t)

is the

position

^of

particle

I,

Fi(t)

is the total interaction force _on

particle

I and

Ri(t)

denotes the

Langevin

random force

acting

_on

particle

I

obeying

the

following fluctuation-dissipation relationj

(Il~,~(t)Rj,p(t'))

=

2fkBT6i,j6~,p6(t t') (3)

where _a,

fl

_are the Cartesian coordinates and kB is the Boltzmann constant. Here the solvent- mediated

long

_range

hydrodynamic

interaction is

neglected

for

simplicity, anticipating

that this will not affect _occurrence of the

glass

transition

itself, although time-dependent

behavior may be altered _as is

reported

in reference [4].

For the

simulation,

_{we use a} cubic box

containing

N

= 254 colloidal

particles

with

periodic boundary

condition in each direction. In order to

integrate

equation

(2),

_we

adopt

the finite

difference method

(Euler method). Temperature

is chosen at the value T" _e

kBTle

= 0.1

[10] ^{which is consistent with the values used in the other} computer simulations

ii, 11-13].

Time mesh is At

/rB

_"

0.001,

where _rB

=

fa~ le

is the

microscopic

time scale of the Brownian

dynamics.

The

starting configuration

for each _run is chosen _{as a} dilute

liquid

state, ^because metastable

glassy

state cannot be obtained if

starting configuration

is _a

crystal

_{as was}

adopted by

Dickinson et al. _[6] and Tata et al.

iii.

The system ^{is then}

rapidly compressed during

^about 0,lrB, and after

reaching

_a

stationary

state, the

physical quantities

_are ^{calculated from the} data

during

2 x

10~steps(= 200rB).

For each parameter set, _we

performed

two

independent

runs from different initial

configurations,

in order to check the influence of the initial states.

For the _purpose of

checking

whether _a

crystallization

takes

place

or not, ^either the _ra- dial distribution function _or the time

averaged

radial distribution function _are

usually used,

where the time

averaged

radial distribution function _is the radial distribution function of the

time-averaged configuration

of

particles

where the effects of thermal noise _are reduced. Since

particles

have _a size

polydispersity,

the

peaks

of these radial distribution functions _are smoth- ered and

they

_are not

appropriate

to serve as a method _to

judge

the

crystallization.

As _a

more useful

method,

here _we

employ

the bond orientational order

ill,14].

The "bond vector"

bj

is defined _as the vector

connecting

the center of _a

particle

with the center of _one of its

nearest-neighbors,

where the

nearest-neighbor particles

_are defined _as those within _{a range} of 1.3 times the distance _at the first

peak

of the radial distribution function. The

position

of _a bond is defined _as the

midpoint

of the line

joining

the _two

neighboring particles.

The bond

orientational correlation functions _are defined _as

ill,14]

(fl (cos @jj)6(r rij))

~~~~~

Go(r)

^~~

^~'~'~'

^~~~

Go(r)

=

(Po(cos@z~)6(r rzj)) (5)

where @jj is the

angle

between bj and

bj,

^{and the P's} are the

Legendre polynomials.

The

average is taken _over all

pairs

of bonds with _common center

particles. Figure

I shows the

typical

behavior of

G6 IT),

which is the most sensitive to the

disordering

in the system, for both

an ordered

(crystalline)

state and _a disordered

(liquid

_or

glassy)

state.

Though

the behaviours of

G6(r)

for these two _{cases are}

qualitatively

the _same within _a short

distance, they

behave

differently

at

long

distances. There is _an apparent distinction between these two cases, I-e-

G6(r) decays rapidly

at

large

_r for the

glassy

state, while it remains finite _at

large

_r for the solid.

Therefore,

for the

judgment

of the

crystallization

this method _can be used

together

with the time

averaged

^radial distribution function.

Figure

2 shows the

stability diagram.

The closed _squares and the _open circles

correspond

to

crystalline

and disordered states,

respectively.

The _open

triangles

_are the points where the final state _was disordered _or

crystalline depending

_on the initial condition. Above _some

SPD, stationary

^disorderd states are realized

irrespective

^{of the}

density.

This threshold SPD is found _to have the _~ value denoted _{as ~~} in the _range 0.05 < ~~ < 0.08 from

figure

2. In the

high density region,

the

glassy

state _can exist _even below _~~. As the

particles

interact via _a

nearly

hard

sphere potential,

_a

particle

_can

perform

thermal motion

only

within _{a very} small

region

at

high density.

^Therefore even a small SPD _can

produce enough

structural frustration to prevent ^disordered state from

crystallizing.

We identified the

crystal-glass

transition

point

observed in the real

experiment

_[4] with that obtained

by

_our simulation and determined the effective random close

packing density

for _our simulation system

by using

the liner relation

between the volume fraction and the

p*

which is defined _as

p"

₌ ^N£l~

IV,

Such _an effective random close

packing density

is also indicated in

figure

2

by

_{an arrow.} In real

experiments,

the

shapes

of

particles

_may be distorted when the

density approaches

to the random close

packing density.

Thus

beyond

such _a

density

_our simulation may ^not

correspond

to _a real situation

[15].

^{We also} note that the

stability diagram

_may ^{somewhat be altered for smaller} T*. The

1182 JOURNAL DE PHYSIQUE II N°8

1.

$,

[(i' t'

$~ 7, ~~,,

,,

~/ l' ',

iO , / ', ,", /

~ I' '""1~""

'

l '

ill

Fig. 1. The bond orientational correlation functions

G6(r)

_as functions of the distance _are shown for _{a =} 0.05

(broken line)

and

a = 0.08

(solid line).

Dimensionless number density of particles is

p* ₌ Nfi~

IV

= 0.879 for both _cases, V being the volume of the system. The broken line is for the

case of the crystalline state, ^and the solid line is for the

case of disordered state.

o o o o

liquid glass

~

o

o o o o o o

o o o o

0

_{o . O o}

crystal

0

1-1

p* f

Fig. 2. Stability diagram _on the

p*-a

plane. The closed _squares show crystalline states and the open circles show disordered states. The _open triangles _are the points where the final state is either _a crystalline _{or a} disordered state depending _on the initial condition. The _arrow indicates the effective

random close packing density. The boundary between crystalline state and disordered state is shaded

as a guide to the _eye.

stability diagrani

_may also have _a

compression

rate

dependence. However,

_a

larger compression

rate than that used in the present simulation is not realistic because such _a

rapid compression

cannot be achieved in real

experiments.

With _a smaller

compression

rate it is

expected

that

the

crystal region

in

figure

2 may somewhat be extended. At least in _our

simulations, glassy

states could not be observed _up to the random close

packing density

for the

monodisperse

case

(~

₌

0).

On the other hand for _~

=

0.05, liquid-crystal-glassy

transitions _appear upon

increasing

the

density.

This _can

correspond

to the real

experimental

situation [4]. ^The

stability diagrani given

in

figure

2 is

qualitatively

similar to the

phase diagram predicted by

the

equilibrium density

functional

theory

_[8]. Note that the

phase diagram

in reference [8]

is for the hard

sphere

system with _a

triangular

distribution of

particle size,

and that _{as was} mentioned before the

stability diagram

and the

phase diagram

have different

meaning.

The

stability diagram

is for

non-equilibrium

states, ^whereas the

phase diagram

is for

equilibrium

states. The

outstanding

^unsolved

question

here is whether _a

truly monodisperse

system _can exhibit

glass

transition below the random close

packing density. Although

_so far _we failed to

give

_an affirmative _{answer to} this

question,

^further extensive simulations with different

interparticle potentials

and different temperatures _are needed to

give

_a ^definitive answer.

In summary, we found the

following.

^{The bond} orientational order is _a useful _{tool to} decide whether the system ^is

crystalline

_or disordered. We found that _our simulation for

~ = 0.05

shows _a behavior similar to the

experiment

of colloidal

glasses

with _a SPD of about _~ ci

0.05, which shows

liquid-solid-glass

type transitions when the

density

^is increased. This indicates that _{even a} small SPD _can have

important

effects _on colloidal

glasses.

Further

investigations

_on ^the

comparison

^{between the}

stabiity diagram

^{and the}

phase diagram

of _our model system, and also _on the temperature

dependence

of the

stability

and

phase diagrams,

are

indispensable,

^which _can

provide

_a basis for extensive

investigations

of the

dynamics

of the

sharp glass

transition.

We would like to thank

K.Fuchizaki, M.Itoh, Y.Hiwatari, T.Odagaki, J.Matsui,

dud J.- N-Roux for

helpful

comments and discussions. K-K also

acknowledges

useful conversations with Professor

P.N.Pusey.

This work is

partially supported by

Grant-in-Aid for Scientific Research from the

Ministry

of

Education,

^Science and

Culture, Japan,

and also

by

the Institute for

Molecular Science.

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_They

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1184 JOURNAL DE PHYSIQUE II N°8

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