End effects in adsorption of homopolymers and triblock copolymers

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End effects in adsorption of homopolymers and triblock copolymers

F. Aguilera-Granja, Ryoichi Kikuchi

To cite this version:

F. Aguilera-Granja, Ryoichi Kikuchi. End effects in adsorption of homopolymers and triblock copoly-

mers. Journal de Physique II, EDP Sciences, 1993, 3 (7), pp.1141-1159. �10.1051/jp2:1993188�. �jpa-

00247888�

Classification Physic-s ^Abstiacts

05.20 05.50 82.65D 61.25H 82.20W

End effects in adsorption of homopolymers and triblock

copolymers

F.

Aguilera-Granja (I)

and

Ryoichi

Kikuchi

(2)

(')

Instituto de Fisica

« Manuel Sandoval Vallarta _», Universidad Aut6noma de San Luis Potosi, San Luis Potosi, S-L-P. 78000, Mdxico

(~) Department of Materials Science and Engineering,

University

of Califomia, Los

Angeles,

CA 90024-1595, U-S-A-

(Received 29 Octobei 1992, revised 18

Februaiy

1993, accepted 23 Maich 1993)

Abstract. The effects of end segments on the conformation of

polymers

adsorbed _{on a} flat surface _are studied

theoretically

by the _use of _an

analytical

statistical mechanics (the Cluster

Variational Method, CVM) and a simulation based _on the CVM output. Based on the simulation,

we derive information _on the thickness of the adsorbed

layer,

the number of

loops,

the _area covered

by

_a polymers, fractions of the train, tail and

loop portions

of _an adsorbed

polymer,

the average number of tails, and related

geometrical quantities.

The conformations of the adsorbed

polymers

show _a strong

dependence

_on the end segment behavior. The result derived in the paper give us

guidelines

for obtaining _some specific conformation when triblock

copolymers

are used in

the

adsorption

studies.

1. Introduction.

The steric stabilization of colloid

suspension

is

becoming increasingly important.

Often the interaction between colloid

particles

in

suspension

is controlled

by coating

the

particles

with

polymers.

It is

generally

believed that the interaction

depends

_on the conformations that the

polymers adopt

_on the

particle

surface. Therefore the

important question

is how _{we can} control the

configuration

of the adsorbed

polymers.

With the purpose of

controlling

the

configuration

of the adsorbed

polymers,

different types ^of

homopolymers

have been tried.

However,

it

seems difficult _to control the

polymer configurations only by

the _use of different

homopoly-

mers. For this _reason, the _use of

polymers

with _some

special

architecture like diblock

copolymers

^and triblock

copolymers

has been started

[1-7].

In this paper, we report a

study

^{of the}

configurations

of the triblock

copolymers.

The

study

of triblock behai,ion of

polymers

_are classified in intrinsic and extrinsic _cases. The intrinsic

case is _a

homopolymer

in which the end segments

(the

first and the last

polymer segments)

have different interaction behavior than the intemal segments. ^{The difference} may come as a

consequence of the different arrangements ^{in the chemical bonds} at the ends of the

polymer.

1142 JOURNAL DE

PHYSIQUE

II N° 7

Under such

conditions,

it is

expected

that _a

homopolymer

behaves like _a triblock

copolymers

(ABA), where the ends of the

polymer belongs

to one kind of block

(A)

and the main

body

^of the

polymer belongs

to the other type ^{of block}

(B).

The way in which the blocks A and B _are

adsorbed _on the surfaces may be

completely

different. The tiiblock behavio/. of

homopolymers

has been

pointed

out also

by Scheutjens

and coworkers

[8].

The

explicit

case is when the triblock

copolymer

is indeed _a

polymer

built

(synthesized)

from three differents blocks

(ABC).

However, in this paper we limit ourselves _to the _case in which the blocks A and C _are

equal

and

they

are much shorter than the central block. The

particular polymer

concentration in the solution studied here is I

fb,

the parameter ^{for the bulk}

length

is L

=

39

[9, 10],

and the size of the system ^{used in the} simulation is _a box of 50 _x 50 _x 20 lattice _constants. The

length

of the

polymers generated by

the simulation process is denoted

by ^I [10].

The unit of the energy is

kT,

T

being

the temperature.

2. Model and method.

Before _{we start} the

simulation,

the

equilibrium

state of the

polymers

close _to the surface is determined

using

the Cluster Variational Method

(CVM) [9, 10].

The

analytical

results _are used _as the

input

for the simulation

[9].

Since the method and all the details about the way in which the

analytical

calculation and the simulation _are done _can be found elsewhere

[9, 10],

we

simply highlight

the

important points

of this method.

ii)

ANALYTICAL _INPUT. In the present case, we use the

pair approximation

of the CVM in

the

simple

cubic lattice

[9-12].

The CVM is _a statistical mechanical

technique

of

calculating

the

equilibrium

_state

by minimizing

the free energy ^F with respect to the variables chosen _to describe the state of the system

[13].

It _was

applied

for the first time to

polymer problems by

Kurata and coworkers

[14],

which showed

good

results in the entropy ^evaluation for

polymers

in the bulk. The CVM is _a herarchical system of

approximations.

Each

approximation

is

defined

by

the basic cluster

(made

of

neighboring

lattice

points)

and subclusters. The _most

rudimentary approach

is the

point approximation

which is constructed

using

the

probability

variables for the

species occupying

_one lattice

point.

This

approach

is

usually

called the _mean-

field

approximation

_or the

Bragg-Williams approximation [15].

We _can mention that the

classical

theory by Flory [16]

is

essentially

a mean-field like

approximation.

The

physics

contained in the

pair approximation

of the CVM which is used in this work is

equivalent

to the well known

Quasi-Chemical

Method

[17], although

the details _are different. The

advantage

of the CVM

approximation

used here is that the entropy

expression

can be

improved

systematically

when we choose _a

larger

cluster _as the basis of the

formulation,

and the CVM

expression

is the _most efficient

(for practical purposes)

for the chosen cluster.

In this work the

thermodynamic

behavior of the system ^{is described}

by

the

point

variables and the

pair

variables. The

point variable,

_x~, is the

probability

of

finding

_a

species

I on a

point,

and the

pair variable,

_y~~, is the

probability

of the I

j pair

_on

nearest-neighboring

lattice

points.

The energy E and the entropy

S,

and then the free energy F

=

E

TS,

_are written _{as a} function of the x, s and

y~~ ^s. The

equilibrium

state is derived

by minimizing

F with respect to these variables. In the present

application

Of the

CVM,

_we work _on a

layer

above _a

substrate,

and hence the system ^is

inhomogeneous.

This makes _{us use more}

subscripts

and

superscripts

to

specify

_{x s} and _{y s.} Also, because of the

length

^and the

density constraints,

_we minimize the

grand potential

fl rather than the free energy F. We describe the details in the

following.

The lattice

plains parallel

to the substrate _are numbered

by

_{n, n}

= I

being

the first

plane

closest _to the solid substrate

(surface).

The

configurational probabilities

of the lattice

points (the single

site

probabilities)

in the

plane

_{n are} written _as x~.,, for _a solvent

(I

= 0

),

for _an end

polymer

segment or end _monomer

ii

=1,

2, 3)

and for _an intemal

polymer

_{segment or}

internal _monomer

ii

=

4,

5,

6, 7).

A

polymer

_segment has

connecting bonds,

an end segment

being

bonded _{to an}

adjacent segment

^and an internal segment

being

bonded _to its _two

neighboring

segments. ^{This is} illustrated in table

I,

where all

species

and their statistical

weights (w,

^{in the} case of _a

simple

cubic lattice _are shown

[9].

A

pair

of

adjacent

_are either

connecting

_or

non-connecting.

^The

pair

is called connected when the

segments

on these _two

points

_are

adjoining

segment ^of a

polymer

and is called

non-connecting

when both

points

are not

adjoining

in the _same

polymer

_or when the two segments

belong

to different

polymers.

When

they

_are in the _same

plane

_n, their

probabilities

_are written _as

yj~~,~

^and y)(~~_ for

connecting

and

non-connecting, respectively.

When in the

pair

the segment ^{I is} on the

plane

_n and

j

on the _{n +}

I,

their

probabilities

are written _as

?j~/~

and

?)~/j~.,,~ ^(with

h

m 1/2, and _{n +} h

being

the middle

point

of the _two

planes).

The

probabilities

_on the

plane

n ₌ I need

special

attention because _a chemical bond cannot

point

downward toward the solid

surface,

the

probabilities

with bonds

pointing

towards the surface _{are zero.} The number of

equivalent configurations

due _to the different type of

pairs (connecting

and

non-connecting)

are listed in tables II and III. Those

pairs

_not shown in the tables _{are zero.} In the simulation however, each of

equivalent configurations

^of the

bonding

arms is treated _as

distinguishable [10, 12].

The energy of the system ^{is made of} two terms. For the sake of

simplicity

we consider that the interaction between the surface and the

polymer

_segment is extended

only

to the segments

on the _n

=

plane

as is illustrated in

figure

I. The first contribution is the

surface-polymer

attraction

only

for intemal

polymer

segments. This is _{a van} der Waals type ^of interaction and is

proportional

_to the number of

polymer

segments on the _n

=

I

plane

:

7

Ej

=

Ls

jj

_{w~ a-j}

,

(l)

,=5

where _~ is the parameter to describe the

strength

of the

interaction,

_.ij,

(i

=

4,

5,

6, 7)

is the

probability

to find _an intemal

polymer

_{segment on} the

plane

_n

= I, and L is the number of lattice

points

within _a

plane parallel

to the solid surface.

Table I.

Definition of

the

subscript

I and the statistical

wieight factor

_(w~

) for

the

single

site

probability

used in _our model.

D»finition of the subscript I _{and the} weights tU;

Species and Connections I td;

Solvent molPcule @~ 0

End _monomer '

@

2 4

3

Internal mono>Twr 4 4

*

₅

6

6

7 4

l144 JOURNAL DE PHYSIQUE II N° 7

Table II.

a)

The statistical

weight factors for

the

connecting pairs

within the _same

plane (w)i~,,~ ).

^{For the}

first layer

_n

=

I,

the statistical

weight

vanishes when I _or

j

is 4,

b)

The statistical

weight factors for

the

connecting pail-s

between

planes _(w~~(,

~

).

For the

first layer,

n =

I,

the statistical

weight

vanishes when I is 6. Those

pail-s (I, j

not shown in the tables _are

=eio.

la)

I j ² 4,7 5

2 0 I 3

4, 7 I I 3

5 3 3 9

16)

I _{on n} ion

(n+I)

4 6

3 0 4 I

6 ^I 4 I

7 4 16 4

Table III. In

(a)

the statistic-al

weight factors for

the

non-connecting pairs

within the _same

plane

_(w>),f(

~

).

For the

first layer

n = I the statistical

weight

vanishes when I or

j

is 1, 4 _or 6.

In

b)

the statistical

weight factors for

the

non-connecting pairs

^between

planes (w~f( ).

^For the

first layer

n ₌ the statistical

weight

vanishes when I is I _or 4. Those

pairs (I, j

not shown in the tables _{are zero.}

ja)

I j 0, I, 3, 6 2, 4, 5, 7

0,1, 3, 6 3

2, 4, 5, ^{7 3 9}

jb)

I _{on n} ion (n+1) 0,3 2,7 5

0, 4 6

2, 4 4 16 24

5 6 24 36

n= 3 n=2

n=

/~

_. «

-i

' surface

Fig.

I. Illustration of the interaction energies ^{used in} this model. e is the energy of interaction felt by the B block (internal segments) _{on n} I, ^and A is the energy felt by the A block lend segments), on

n = 1.

The second contribution is for end

polymer

_segments

(the

first and the last segments ^of

polymers)

and may be

attractive, repulsive

_or the _{same as} the _case of intemal

polymer

segments. ^This energy is also

proportional

to the number of

polymer

segments on the

n = I

plane:

3

E2

₌ ^LA

£

_{W, xi,,}

,

(2)

,=~

where A is _a parameter ^{that describes the} interaction of the end

polymer

_segment and the solid

surface,

and xi,,

(I

=

1, 2, 3)

is the

probability

_to find _an end

polymer

_{segments on} the

plane

n =

I. The total energy E is the _sum of the two contributions. In this model the solvent condition is assumed _«

good

» so that _we do _not consider any interaction between the solvent and the

polymer _segments.

Note that xj,~ ^and xi, are not allowed _on the

plane

n = I, and thus _are

missing

in

(I)

and

(2).

Also _note that _our convention for the energy parameters ^{is that}

they

_represent attraction when

they

_are

positive

and

repulsion

when

negative.

In _our energy

formulation,

_we

prefer

_{to use} the

pair

_energy interactions _e and A instead of the

respective

_{Jfs Parameters} which _are the _same

pair

interaction energy divided

by k~

^T.

The entropy of the

system

^{is written in the}

pair approximation

of the CVM for the

simple

cubic structure _as

[9-13]

S=k~Ljj ~7

2+5

jjw,£(x,,,)-

n ,=o

2

I _{lW)( (,}

j

£(Y)(~,,

j

)

₊

W)( (,

j

£

(Y)I

^,,

j)1

,,,

i

_[W)~~,

j

£(Z~~

h _<,j + W~~~,_j £(Z~~~ _h,

<,j

)1)

,

(3)

<,j

where w,,

w)[(

~,

w)[

), ^are the statistical

weight

factors

(number

of

equivalent configurations)

of the

single

site

probabilities (x,,

,

), pair probabilities

within the _same

plane ~y)1(

and

pair Probabilities

between

plains (z)flh., _j)

_as shown in tables I, II and III,

respectively.

The

function

£(x)

^{is defined} as x In _{x x.} This

expression

reduces _to the _case of the bulk formula when the variables do _not

depend

on n

[I I].

The

equilibrium

distribution of the

system

^{is calculated from the} minimization of the

grand potential

D with respect to the

pair probabilities

^for

given

values of the interaction

energies,

^the

temperature T,

the bulk

polymer length

_parameter

L,

and the chemical

potential

values. The

grand potential

D can be written _as follows _:

D=E-TS-L

I

_p,p~,,

(4)

,,,,

where E is the intemal energy defined in

equations

^and

(2),

S is the entropy as in

equation (3),

_p, is the chemical

potential

_{and p,~}

,

is the

density

of the I-th

species

_on the

plane

_n. After all the

equilibrium

cluster

distributioni ix,,,,), _(y)[(,~ ), _{(z)fl~., ~)}

have been solved for the

equilibrium

state, we can

proceed

to the simulation.

(ii)

SIMULATION _PROCEDURE. When all the cluster

probabilities

of the system are known

we are

ready

to

begin

the simulation process

using

the

Crystal

Growth

Probability

Method

(CGPM) [18].

The

advantage

^{of the simulation} when it is done _{as an} addition to the

analytical

statistical method is that the simulation

gives

_us information that otherwise would be

inaccessible. The

advantages

of the simulation based _on the CVM method _over the traditional

l146 JOURNAL DE PHYSIQUE II N° 7

Monte Carlo

(MCS) [19]

simulation _are first the

speed,

and second that _our simulation is based

on the distributions that

satisfy equilibrium

conditions, _so that _no further relaxation processes

are needed.

The simulation process is _as follows. We construct the system ^{from bottom} to top

beginning

next to the

substrate,

from left _to

right,

and from back to

front, by placing

a

species

(a

polymer

segment or a solvent

molecule)

_{at a} lattice

point

_one at a time. In order _to

place

_a

probability

in

a

given point,

_{we use a} random number _to choose the

probability

in such _a way that the

pair probability

distribution

(coming

from the

CVM)

is satisfied

[10, 12, 18].

The

procedure

is

repeated

at each lattice

point,

and when the entire lattice is

filled,

the simulation is done.

However,

with this type ^of

simulation,

it is

required

to repeat ^the

procedure

_many times _to avoid bias in the different statistical patterns

[10,

12,

18].

In the present

polymer

simulation,

we need _a

special

constraint _to take _care of the direction of chemical bonds

[12],

and in order _to avoid conflict with the chemical bonds _we need to use a

pseudo six-point

cluster

[10, 12].

An

important

feature of this multi-chain simulations is that the

sample

of the

polymers generated by

this method is

polydispersed [10, 12].

In the

analytical

results the

polymer length

in the bulk is

specified by just

one

parameter

^L

[9-12].

The parameter L is defined _as the total

number of

polymer

segments ^divided

by

^{half of} the end

polymer

segments.

Although,

the

average bulk

length

L is

given

in the

analytical

results _{as we}

quoted above,

it does _not guarantee a

monodispersed polymer

system ^{in the} case of the simulation

[10, 12].

For all the results

presented

in this paper we use an average bulk

length

parameter ^of L

=

39. In this simulation the average

length

in the bulk measured

by

the direct

counting

in the simulation field

(f )

is

expected

to be

roughly

the _{same as}

L,

but

actually

is

f~~~

₌ ^{22. The}

polydispersity

of the

generated sample

^{is understood due} to the fact that the L parameter ^that controls the

length

in the bulk is fixed

only

in the bulk for the

analytical

CVM

calculations,

and _we let the

polymer length freely

evolve towards the surface. The free evolution of the

polymer length

in the

sample originated

_a distribution _on the

length (f _).

_{From the simulation results,}

we know

that in the bulk

(far

_away from the

surface)

the

length

distribution of the

polymers

is

negative exponentially [12],

with

polydispersity

index N

=

1.82

(the

ratio of the

weight-average

molar

mass

M~

to the number average mass

M~ [12].

Our

polymer

system ^is an open system ^which is in

equilibrium

with _a bath

(reservoir)

of

polymers

of different

lengths,

and in the

equilibrium

situation the

polymer

system ^{and the reservoir}

exchange polymers freely

between them.

The size of the system used in this simulation is 50 _x 50 _x 20 lattice constants, and the system ^{is truncated in five of the six faces. Because of the truncation of the} system, some

polymers

_{near a} truncation surface have

only

_one

ending point

inside the simulation field. For the sake of

simplicity,

when the statistical count is done these

polymers

near the

edge

are

disregarded.

Another

anomaly

^of this simulation is the existence of closed

loops

^of

polymers,

which are also

disregarded

in the statistical

counting

since the main _concern of this paper are

the linear

polymers.

The

probability

of closed

loop polymers

in this type ^{of simulation base} on

the CVM decreases when

bigger

clusters _are used

[12].

In the present paper, we work with the type of triblock

copolymers

in which the central part of the

polymers

_are adsorbed with the energy

e(~

0

),

and for the end segments we consider

both attractive A _~ 0

(ATT)

and

repulsive

A

~ 0

(REP)

_cases. The main purpose of this work is to

study

how the conformations of the adsorbed

polymers

vary

depending

_on the REP and ATT

conditions.

It is

important

to

emphasize

that the CVM calculation describe the

equilibrium

state of the system, ^{and that the simulation is} a new extension of the CVM

developed by

one of the authors in 1980

[18].

The simulation is _not needed

[9,

11, 13,

18]

if _one is _content with the

analytical

results calculated

by

the CVM _{as were}

generally accepted

^before this _new type ^{of simulation} had been devised. However, the simulation is _a very useful and

powerful

tool that

displays

in _a

visual form the information which is contained in the

analytical

distribution obtained in the CVM.

By doing

^the

simulation,

_{we can get a}

physical representation

of the system, ^from which _{we can} obtain _a

large

amount of information about the

shape

of individual

polymers

adsorbed _on the surface. We may say that the simulation is the

key

^that allows _{us to} show the information that is hidden in the CVM

analytical

^results.

3. Results.

Before

going

to the simulation

results,

_we first show in

figure

2 _some of the

analytical

results obtained

by

the CVM. These results _{are some} of the _{ones on} which _our simulation is based. In

(a),

_we show the

density profile

_as a function of the distance away from the surface. Both the

Density (a)

o.oi

lo zo

% En d

_X

loo

owomooo

',,, ~o--°~

"',,,, ~o~° ^(b)

,~~

o.oi

Distance

^{'° ~°}

Fig. 2. Analytical results used _as the input in this simulation _: (a) the density profile and (b) the percentage ^{of end} segments as functions of the distance away from the surface. The circles (REP) and the squares (ATT) correspond to the calculated points.

lI48 JOURNAL DE PHYSIQUE II N° 7

ATT

(e

= 1.0 and A

=

3.0),

and the REP

(e

=

1.0 and A

=

1.0)

cases have

practically

the

same curve. The two values used here for the A parameter are chosen

just

to illustrate the

dependence

of the conformation _on this parameter. ^In this paper the ATT _case is

represented by

a square, and the REP _case

by

a circle unless otherwise stated. The concentration of the end segments ^is so small in these _cases that their

displacement

towards _or away from the surface does not make

changes

in the total

density

of

polymers

close _to the surface.

However,

_even when the

global

_property like _the

density profile

does not

change,

the conformations

adopted

by

the _{two cases} may be

completely

different. To find the differences of the conformations of

the _{two cases} in detail is the main purpose of this paper.

Although

the

density profile

^does not

change,

the percentage of end segments shows _some difference in the ATT and the REP _{cases, as are} shown in

(b).

The dashed line

(together

^with the

square)

is for the ATT, and the solid line

(together

with the

circle)

^is for the REP _case.

After the second

plane,

the _{two curves are} almost

equal.

This is understandable since

only

the segments on the _n

=

plane

_are

interacting

with the surface in this calculation. In the _same

figure,

_we include the dotted line _as the reference which is when all segments ^{feel the} same

attraction from the surface

(F

=

1.0 ). We call this _case HOM

(homopolymer),

and represent ^it

by

a

triangle

unless otherwise stated.

For all the results

presented

in this paper the coverage 0

(concentration

of

polymers

on the

surface)

and the adsorbed _amount

r~~~

_are

kept fixed, being

their values 0.58 and

0.89, respectively.

It is

important

to mention that the definition of the adsorbed _amount used here is the _as the _{same one} used

by

Roe

[20].

For the _case of

homopolymers,

it is well established

theoretically [10, 21, 22]

and

experimentally [23]

that in the

adsorption

of

polydispersed polymer samples,

there is _a

preferential adsorption

for

longer

chains than the average ^{in the bulk}

(solution). Now,

_{we are}

concemed how the

properties

of the

homopolymers

_are modified because of the triblock

copolymer

behavior

(intrinsic case)

of the

homopolymers

_or what

properties

_are

expected

in the _case of extrinsic triblock

copolymers.

One of the results of this simulation is shown in

figure 3,

^which shows the

probability

of adsorbed

polymers

_{as a} function of the

polymer length

~

~

25 ~~

o

',,U

~ ,

O

£~

^O

8

_

/

'~,

Q ~° ~'~,

~'~b~~

~~ ~

o,,,~

>~~

° ~~ '°

Le/~th

^{~°° ~~~ ~~°}

Fig.

3. Simulation results of the distribution of the adsorbed

polymers,

_or

adsorption probability,

_on

the surface. The circles _are for REP and the squares for ATT.

(f).

_{Note that}

_although

_the

_analytical

_{CVM results}

are for the

polydispersed

_system, the simulation

technique

can select different

lengths separately.

There is _a

preference

of

adsorption

for short

polymer

^{chains in} the ATT _case, while in the REP _case the

preference

is for

long

chains. The average

polymer length

of the adsorbed

polymers

_on the surface for the ATT is

f~~~

=

40,

and for the REP is

f~~p

=

80,

_to be

compared

with the average bulk

length

in both

cases which is

f~~~~=22.

The

polydisperse

index N

[12]

of the distributions is

N~~~

=

1.63 and

N~~p

=

1.34, respectively.

It is

interesting

to compare the results with the HOM _case

[10],

for which the average

length

of the adsorbed

polymers

on the surface is

f~J~~

₌

72,

the

f~~~~

₌

22,

and the

polydisperse

index is

N~J~~

=

1.41. The

comparison

of

HOM _case with the ATT and REP indicates that in the _case of the triblock

copolymers

(intrinsic

_or

extrinsic)

the distribution of the adsorbed

polymers

_as well _as the

polydisperse

index _can be controlled

by

the behavior of the end segments ^{of the}

polymers.

In the _case of the intrinsic triblock

copolymers,

this

change

in the distribution of the adsorbed

polymers depends

on the

change

of the chemical and

physical properties

of the

ends,

_{as a} consequence of the different arrangement ^{of the bonds} at the end segments ^{of the}

polymer,

and therefore _{we cannot} control this type ^of

properties,

because

they

_are intrinsic in this

type

^of

polymers.

On the other hand, for the _case of extrinsic triblock

copolymers,

the

properties

of the end may be controled

by changing

the type ^{of end}

blocks,

and therefore in this way we can control the distribution of the adsorbed

polymers.

We _can obtain _more information about the conformation

adopted by

the adsorbed

polymers

when _we

study

the end-to-end distance. This

quantity

in this kind of simulations for _a

polymer

in the solution

obeys

a power law

[12],

_as

rj r~ =

cf

"

,

(5)

where _c is _a constant that

depends

_on the model and _v is _an exponent

depending

_on the

dimensionality

of the system ^and

independent

of the lattice model

[10,

24,

25].

The results for

this simulation _are shown in

figure

4. The exponents ^obtained here _are v~~r =

0.65 and

v

i

Length

Fig. 4.

- The

average^nd-to-enddistance for

adsorbed polymers on the surface as _a

function of the

l150 JOURNAL DE PHYSIQUE II N° 7

v~~p = 0.66. The _two lines in

figure

4 _are almost

parallel.

The exponents v are almost the _same

as that in the HOM _case

(vHoM

_~

0.62) [10].

The small difference in the exponent v does _not

necessarily

_mean that the conformations _are similar it

simply

means that the difference in the conformations cannot be differentiated

by

the measurement of the end-to-end distance in the

case of the

type

of triblock

copolymers

studied here.

In order _to

study

the difference in the conformation, _we examine the

layer

thickness in the

REP and ATT _cases for _a fixed value of the coverage and the adsorbed amount. The definition

of the

layer

^{thickness used here is} as follows. We record the

largest

_n

(plane number)

of each chain adsorbed _on the surface. The

largest

_{n may} be at _an end of the adsorbed

polymer

_{or on an}

intemal segment. ^{We call} this

quantity

_Z~~~ and the results _are shown in

figure

5a in _a

log-log

scale. The Z~~~

approximately

^follows a power law in the

polymer length (cfY ).

As

expected,

the Z~~~ ^is

bigger

^{in the REP} than in the ATT for short

polymers.

This is

understandable,

because for short

polymers

the tails _are the main contribution _to

Z~~~.

^{As the}

polymer length

increases, the difference in the _two

Z~~~'s

^decreases. This is because _as the central part ^{of the}

polymer (B block)

_grows, the difference between the REP and ATT decreases. This behavior

makes _sense, because the end size

oust

the first and the last segments ⁱⁿ the

polymer

_or the block

A)

_are

proportionally decreasing.

The exponent _y ^is

bigger

in the ATT

(0.61)

than REP

(0.47)

until

approximately

50 segments,

beyond

which Z~~~ ^{follows the} exponent ^of the REP,

which is in agreement ^with _{y =} 0.5 of the HOM _case

[10].

There is _a second way of

characterizing

the adsorbed

layer,

which _we call the thickness T.

Using

the

polymer length f,

its fraction in

loops _fj~~~

and its average number of

loops (Nj~~~),

^{the thickness T for} a

polymers

of

length ^f

^is defined _as

~

2

~~fl)

^~~~

The factor two in the denominator appears because in every

loop

about _a half of the segments

are used to go up and _a half to go down. The results _are shown in _a

log-log

scale in

figure

5b.

The thickness T _can be

interpreted

_as the average

height

of the

loop.

The thickness T also

obeys

a power ^law

(cf~ _).

_The

_loops

in the ATT are

higher

than those in the REP because of the

preference

of the REP _to have

longer

tails _{as a} result of the end

repulsion.

The exponents are

p~~r = 0.48 and p~~p =

0.56. The

bigger

_{p exponent} in the REP _case

implies

that for very

long polymer

^{chains the} difference in the thickness T will

disappear.

As _a

reference,

we may quote ^{that the}

exponent

p in the HOM _case is 0.51

[10].

The number of

loops

of the adsorbed chains also

gives

_us information of the conformations

adopted by

the

polymers.

The average number of

loops (Nj~~~)

^{for the ATT}

(squares)

and the

REP

(circles)

_are shown in _a

log-log

scale in

figure

5c. The number of

loops

is

larger

in the ATT than in the REP _{case ;} what is

responsible

^{for this} behavior is

basically

the

preference

^of

the ends in the ATT _to be close to the surface. The difference

persists

for chains of _as

long

as

90

segments, showing

that the conformations of the adsorbed

polymers

_are

strongly

modified

by

the end

behavior,

_even when the size of the ends

(A block)

_are

relatively

small. The number of

loops obeys approximately

a power law

(cf~ _).

The exponents K are I. II and 1.89 in the

ATT and the

REP, respectively.

The exponents ^{in both} cases are different from the HOM _case 1.72

[10].

The fact that the exponent K is modified _more

strongly

in the ATT than in the REP indicates that in the HOM _case there is _a natural

tendency

of the ends to be away from the surface due to the entropy ^effect ; this effect is shown

by

the

preferential adsorption

for

longer

chains.

Before

continuing

with _our simulation

results,

it is worthwhile to comment that in many of the

experimental

data the thickness of the adsorbed

polymer layer depends

on the

technique

used in the measurement as has been

pointed

_out

by

_many authors

[3, 8, 26, 27].

The methods

(a)

Z

max

o o

,,6

a

,d

~

i lo loo zoo

(b)

a

Thickness

o

o

lo loo zoo

(c)

Number of loops

,,' a ,,'

,,' a

,,'

,"" _o

,,"

~°

Lenqth

^{~°° "°}

Fig. 5. -Some of the geometrical properties of the adsorbed REP (circles) ^and ATT

(square)

_as

functions of the length (a) Z~~~, (b) thickness T, and (c) the average number of loops

(N

_j~~~). Marks _are the simulation output.

1152 JOURNAL DE PHYSIQUE II N° 7

used for the measurement can be

mainly split

in

dynamics

and statics _as Cohen Stuart and coworkers

suggested [3],

and it _seems that different

techniques give

information about different aspects of the

layer

thickness.

They give hydrodynamic layer

thickness and static

layer

thickness. The

experimental

data indicate that the

hydrodynamic

thickness is

always larger

than the thickness determined

by

the static

techniques

_:

ellipsometry

and neutron

scattering [3, 8, 26, 27].

A very

interesting

observation _was made

by Cosgrove

et a/.

[27]

who found that the

profile

measured

by

^small

angle

neutron

scattering

vanished

completely

at a

distance which is about half of the

hydrodynamics

thickness.

Cosgrove's

observation

strongly

supports ^{the idea that the adsorbed}

polymer layer

has _two parts a dense part ^close to the solid surface and _a sparse part

(sponge like)

at the top ^{of the dense} part. ^{The dense} part ^{of the}

layer

thickness may be related with the

ellipsometry

and _neutron

scattering

measurements while the

outermost

(exterior)

_may be related with the

hydrodynamic

measurements. Since in _our

simulation results the thickness T is related with the average

height

of the

loops,

we associate this with the dense part ^{of the} adsorbed

layer

while the

Z~~,

which is related with the farthest

polymer

segment is

interpreted

as the _«

hydrodynamic

_»

layer

^thickness. For the purpose of

comparing

our results with the

experimental

observation _we calculate the

following

ratio _:

~max

The results for C _are shown in

figure

6. Since _most of the

experimental

results _are for very

long polymers,

we are

just commenting

_on the calculations for

polymers longer

than 50 segments

(although

^the entire range of the

length

^is shown in the

figure).

For this range the C value increases very

slowly

as a function of the

polymer length.

Its values in the REP and HOM

cases are in the same order of

magnitude

as

Cosgrove's experimental

observation in which he

points

out that the dense part ^{of the adsorbed}

layer

thickness is about _a half of the

hydrodynamic

thickness

[8, 27].

In the ATT _case the dense part ^{of the adsorbed}

layer

is

larger

than that in the REP and HOM _cases due to the fact that the end segments are

strongly

attracted

to the

surface, encouraging loop

formation and

resulting

in the

larger

values of T and hence the

larger

values of C in the ATT _case. A

question

that may arise is whether _{or not} the C ratio

keeps

_on

growing

_or levels off _as the

polymer length

increases. We _{cannot answer} this

question

because _our simulation results _are [imitated to a finite

length

_range.

~

).

~Gl

_'Gt

_~

u ; ._ t- ~ ~_~ ~-~ ^{Q -tH a ~}

'A.,

~_ ~_

i

0 25 50 75 100 125 150

Length

Fig.

6. Simulation results for the C

=

~ ratio _{as a} function of the

polymer length

^{for the} cases

~max HOM (triangle), ^ATT

(square)

and REP (circle).

The _area covered

by

the adsorbed

polymers

is also sensitive _to the triblock behavior _or the end effect. In order _{to see} this, we calculate the _{area cover}

by

an adsorbed

polymer,

which is

the square area that circumscribes those

polymer

_{segments on} the _n

=

I

plane

for every adsorbed

polymer

chain. We take the average for all chains with the _same

length.

We call this

area the

A~~~.

^{The results of this} calculation in units of square lattice _{constant are} shown _on the

log-log

scale in

figure

7. The results show

clearly

that the

A~~~

^follows a power law in the ATT

and REP _cases

(cf~

_). The

exponents

bare 1.2 and 1.6 in the ATT and REP,

respectively.

The

exponents are

approximately equally

deviated from the HOM _case for which the exponent ^is 1.42

[10].

Amax

~b ,6'

_~

u

~

,,'

~r#~ ~~

~p

a

~

,u'~

,,d' ,'~

io loo zoo

Length

Fig. 7. Simulation results of the A~~~ or the maximum _area covered by the adsorbed polymers for REP

(circle) and ATT

(square)

_{as a} function of the

length.

We _can get more information about the conformation and its variation with the

polymer length using

A~~~ ^and

Z~~~.

^{In order} to find _more information about the conformation _we

calculate the

following

ratio

~

fi(8)

~max

This R

gives

_us information about the lateral view of the adsorbed

polymer

_; there _are three

cases as is illustrated in

figure

8a. The _{cases are} the

followings, I)

When R

~ l, _a small part ^of the adsorbed

polymer

touches the surface and the _rest of the

body

is

mainly

extended into the

solution. The

polymer

can be contained in _a vertical

rectangle,

and the entropy ^{effect is} more

important

than the energy effect in this _case,

ii)

When R

- I the adsorbed

polymer

can be

contained in _a square, and in this _case there is _a balance between the entropy ^and the _energy contribution and both effects _are

important. iii)

When R _~ l the adsorbed

polymer

is

mainly

kept

close to the surface and the

polymer

can be contained in _a horizontal

rectangle,

so that in this _case the energy effect is _more

important

than the entropy ^{effect. In}

figure

8b _{we can see}

l154 JOURNAL DE PHYSIQUE II N° 7

la)

( I R=I R>1

~'~ ~

tl~

~

~'~~ ~

mm nT~

~'~

ar _{~_} ^w

o.m w. a.

.a.'~'

o.50

(~)

O.30

O 25 50 75 loo 125 150

Length

Fig. ^8. Simulation results for the R

=

~

ratio _{as a} function of the polymer

length.

This ratio

gives

ax

us information about which is the leading factor in the conformation of the adsorbed

polymers

the energy-dominated when R

~ l and the entropy-dominated when R

~ l. The marks correspond to the simulation

points,

the

triangles

_are for HOM, the circles _are for REP and the squares for ATT.

that in the ATT _case the conformation

adopted by

the adsorbed

polymers

_are

mainly

dominated

by

the energy process in the entire range of the

length,

while in the REP _case there is _a

continuous evolution from conformations in which the entropy dominates the

adsorption

for

short

polymers

to conformations in which the energy dominates the

adsorption

for

long

polymers.

In the _same

figure

also shown _are the HOM data _{as a} reference and _{we can see} that its behavior is _more similar to the REP _case than _to the ATT _case. In the _case of very

long polymers,

the results _{are as}

expected,

the end effects

becoming

less

important,

and the

ATT,

REP and HOM show the _same behavior.

We counted the fractions in the

train,

tail and

loop portions

of the

polymers

adsorbed _on the surface. The fractions _are shown in

figures

9a and b for the REP and

ATT, respectively.

The tail fraction is enhanced in the

REP,

while it is reduced in the ATT. The train fraction in the REP shows a weak

dependence

on the

length

and later it tends _to level

off,

while for the ATT there is _a clear

dependence

on the

length.

The

loop

fraction increases in both cases as the

polymer length

increases, but also the

levelling

off should be present ^for very

long polymers.

The

loop

fraction is

bigger

in the ATT than in the REP since the

preferential adsorption

^of the end segments enhances the

loop

formation. In both cases the

loop

fraction becomes the

largest

fraction for very

long polymer

chains. The

loop

fraction is

bigger

in the ATT than in the REP in

agreement

^with the

larger layer

thickness in the ATT than in the REP _{as was} shown in

~

Toii (a)

lJ> _,..6.

..."~"'K"~.&.A,~

~

b ___f.."~'~

~~~

4 ..."""'

I

°

o

T~0i~

~

LL-

o

a

Loop

lo loo zoo

~,~ ^o

Troin ^~~~

#~ O~

^~

Toil

b

G b~

O

~o~

~

a~~

~a~

ad

~.A

o~o

,a..~

~ ~

...~.""~

nooa~~~"

a

Loop

..:

~~

' '°

Le _n gth

^{'°° ~°°}

Fig.

9. Simulation results of the fractions in the trains, tails and loops ^{of the adsorbed}

polymers

_: (a) for the REP and (b) for the ATT.

figure

5b. Our calculation for the train, tail and

loop

in the ATT _case is in

qualitatively good

agreement ^with some recent Monte Carlo simulation

(MCS)

results for

amphiphilic

triblock

copolymers [28]

as is illustrated in

figure

10 where the MCS _are

presented. However, slight

numerical differences _are present ^because in the MCS the central block is not adsorbed while in

our case the central block is

adsorbed, although

_more

weakly

than the ends. This

qualitative

agreement

^{indicates that both} the MCS and the CVM simulations

predict mainly

the _same behavior for similar type ^of

polymers.