Geometrical properties of chemically adsorbed copolymers at surfaces

(1)

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Geometrical properties of chemically adsorbed copolymers at surfaces

P. Haronska

To cite this version:

P. Haronska. Geometrical properties of chemically adsorbed copolymers at surfaces. Journal de

Physique II, EDP Sciences, 1993, 3 (3), pp.357-365. �10.1051/jp2:1993137�. �jpa-00247838�

(2)

Classificafion

Physics

Abstracts

05.40 36.20 61.40K

Geometrical properties of chemically adsorbed copolymers at

surfaces

P. Haronska

Max-Planck-Institut flit Kolloid- und

Grenzflichenforschung,

DO-1530 Teltow-Seehof, Kantstra6e 55, Gennany

(Received

26 November 1991, revised 4 November 1992,

accepted

24 November

1992)

Abstract. In this _paper _we calculate the mean-square end-to-end distance and the mean-square

perpendicular

distance of

chemically

adsorbed

copolymers

ôf type Â-B. ^Tile âdsorbed monomers are assumed to be _a

quenched

system. ^The calculation of the mean-square end-to-end distance shows that tllis distance is

fully

determined

by

the frozen-in disorder of the adsorbed _monomers.

The result obtained for mean-square

perpendicular

distance

predicts

_a

dependence

of the fraction of adsorbed _monomers. In

comparison

with

homopolymers

there is _a difference within the _power law behaviour, but _no difference _occurs for the

physical adsorption

of

copolymers.

1. Introduction.

During

the last years the

adsorption

of

polymers

at surfaces has found _a considerable interest

[1-8]. Although

most studies _were restricted _to

homopolymers only,

the

adsorption

of

copolymers

at surfaces

plays

_an

important

role for colloid

stabilization, wetting,

and microemulsion formation.

Recently, Marques

and

Joanny [9]

and Garel _et al.

[10]

studied the behaviour of random

copolymers

at a

liquid-liquid

interface. The

adsorption

of block

copolymers

at

liquid-liquid

interfaces _was considered in

[11,12]

and the variation from random to block

copolymers

was studied

by Yeung

et al.

[13].

In the paper we calculate the size of

chemically

adsorbed ideal

copolymers,

I.e. chains without excluded volume. The macromolecules _are

copolymers

of type

A-B,

where

only

_one type, ^A _say, can

chemically

adsorb _at the surface.

By varying

the fraction of the

chemically

adsorbed _monomers the size of the

copolymers

will be

changed,

_so that the

geometrical properties

_are functions of the

degree

of

polymerization

and of the fraction.

While

physical adsorption

has been

intensively

studied

theoretically, chemisorption

^of

polymers

has not received such attention. The

chemically

adsorbed _monomers _are frozen into their

position

at the surface.

Experiments

_on

polymer

systems are not

performed

_upon _a

single

isolated chain. Even when the

polymer

concentration is

sufficiently

low that chains _are isolated from _one

another,

all measurements involve contributions from _a

large

number of

polymers.

(3)

358 JOURNAL DE

PHYSIQUE

II N° 3

The measurement process involves therefore _an average over an ensemble of

polymer

molecules. Such _a circumstance is _a characteristic feature of

quenched

systems

[14, 15].

The paper is

organized

as follows. In section 2 _we recall the formulas of interest in the framework of the continuous model

proposed by

Edwards

[16].

In section 3 the

replica

trick is used for

calculating

the

quenched

_average. Section 4 contains the calculation of the _mean-

square end-to-end distance and section 5 is devoted to the calculation of the mean-square

perpendicular

distance from the surface. Both

quantities

_are used for

characterizing geometrical properties

of

chemically

adsorbed

copolymers.

Section 6 is devoted _to _a

comparison

between

chemisorption, physical adsorption

and other theoretical

approaches.

Furthermore _we

give

some

concluding

remarks. Technical details of the calculations _are

given

in the

appendix.

2. Continuous model.

A suitable

description

for continuous

polymer

chains is the Edwards model

[16].

The

conformation of _a chain is then

given by

a vector

position

function

r(s) (0

w s w N

),

where N is the number of

links,

each of

length ^I.

The

spatial position r(s)

of _a

polymer

_segment is

represented

as a vector in the 3-dimensional

half-space

p ~

R~,

z ~

[0,

_co

],

_r

=

(p, z) (2.I)

If

repulsive short-range

interactions

(excluded volume)

between _monomers _are

neglected,

the Hamiltonian reads

where fl denotes

(kB T)~ '.

_In _order _to _take _into _account _that _M

monomers of each

polymer

_are

chemically

adsorbed at the

surface,

_a constraint of 3-functions is used for

calculating

the

partition

function Z. Then the

following path integral representation

holds

[17, 18]

M-I

z

" Dr

fl 31(Z(S~)) 32(P(Sa) Pa)eXp(- pHirj). (2.3)

a=0

Here p~ is a random variable which determines the

position

of _a chemical adsorbed _monomer at the

surface,

^and _s~ denotes the

corresponding _segment.

We _assume that consecutive

adsorbed _monomers are

separated by

the _same number of segments. ^Then

s~ = a

~

(2.4)

where _a _runs from I to M I. Note that the choice

given

in

equation (2.4) implies

that both ends _are attached to the wall.

The size of adsorbed

polymers

_may be characterized

by

the mean-square

perpendicular

distance from the surface and the mean-square end-to-end distance. For the former _one _we _use here

lM

ⁱ

h~

= z~ Dr

fl 31(z(s~ )) 32(P(s~

_p~

)

_x

a o

x

~j~

z~

(k

₊ 1/2

)

^~ _exp

(- flH [r] ). (2.5)

'~

k o

'~

(4)

An

equivalent expression

is obtained

by using

the

following generating

function

M- I

l~(~

i'

~

)

"

j

i~Z

j

i~P fl fir (Z(S«)) fi2(P(S« )

_Pa

)

_X

x exp

A

~ ~

i

~jj~

z2[ ^(k

⁺

ⁱ¹²⁾ ~N

+

Aj p2(N ₎ ~H jr i (2.6)

~ o

then the mean-square

perpendicular

distance from the surface becomes

h~

= lim

~

ln

Z(A~, Aj

=

0) (2.7)

A~ ~o

°Ai

and the

corresponding

mean-square end-to-end distance reads

Ri

=

lim ^~ ln

Z(A

~ ⁼

0,

A

) (2.8)

A, ~o

°A1

3.

Replica

trick.

Mean values defined

by equations (2.7)

and

(2.8)

_are those of _a fixed distribution of adsorbed

monomers. This circumstance

requires

_an additional

averaging

_over all distributions. One has

therefore _to calculate the free energy

flf(A~,

_Ajj

)

=

lnZ(A~, Aj ).

However, due to the

logarithm

this average ^cannot be

performed directly.

To _overcome this

difficulty

let _us

apply

the

replica

trick

[14, 15]

lnZ(A~, Aj)

₌ lim ^~

Z~(A~, Aj). (3,1)

n~0 °~

For any

positive integer

_n _we _can _express

Z(A~, Aj )

in terms of _n identical

replicas

of the system. ^Then

Z~(A~, Aj )

reads

n

lM-1

2~(A

j,

Aj )

_#

fl ^DZ«> j _Dp'> _fl

31(Z«>(S~ )) 32(p~~~(Sa ) pa)

_X

. I a =0

~ ~~

l~

ⁱ

j ^j~ ;>~j

~~ ~

i/2)

^N ^~

(

j~ ^sj ^~~°~ ^~

P

~M-1,~~~~~

_M-1

_2f~,=1

0 ~

+

Aj j _(p«>(N ₎₎₂ j ^j _j~

_ds

(

j~l. ^(3.2)

. =i ,=i o

The

quantity N/(M

I

)

which enters in

equation (3.2)

is the number of links between two adsorbed _monomers.

The

positions

of the adsorbed _monomers _are assumed _to be distributed

randomly.

Let p

(p ) represent

^this

distribution,

then for _a

large enough

value of

N/M,

_p

(p

is

govemed by

the

(5)

360 JOURNAL DE PHYSIQUE II N° 3

central limit theorem

[19].

Hence the

following

Gaussian distRbution holds

P

(P«

+ i Pa

)

=

(2 arb2 )-

_exp

_'~ jp~

~ i Pa l~

(3.3)

b

where b is the _mean distance of consecutive adsorbed _monomers. Then

equation (3.2)

becomes

n M-I

2~(A

j,

At )

#

fl fl

DZ~~>

31(Z~~>(Sa))

X

>=I a=0

«exPlA.t _lids _lfll~l~+Ai i,tltlzi>~i(k+1/2) _/~~1ilx

X

lDp~> ^d~pa

^P

^(pa

⁺¹ ^pa

⁾ ^32(p°~(S~ ⁾

^pa

⁾

^X

x exP

A

ii

.I

~P~>~N

))~ i,t, _II

_dS

_~il~ ^~l

_~3

_.4)

From

equation (3.4)

it follows that the calculation of

geometrical properties

_can be done

independently

of those of the z-component. ^Such a circumstance is _a characteristic feature of Gaussian chains

[20].

In order _to

parameterize

the 3-function _a

length

_a _can be introduced such that

a « b

(3.5)

Then the

following approximation

is valid for

length

scales

larger

than _a

32(P)

~

("a~) _eXp(- _p~la~) _(3.6)

Inserting equation (3.6)

into

equation (3.4) yields

z~(Ai, Ai )

₌

zn(A~

=

o, Aj

^~

j j"~ _j~ j

lDp«>

^exp

Aj j

(p«> (N))2

,=o ,=1

(3.7)

We have

replaced

in

equation (3.7)

the _sum in the

exponent resulting

from

approxi-

mation

(3.6) by

_an

integral,

I.e.

M

I

-1

P~(~~

_~

p

~ ^N ^d~

P~(~l' _(3.8)

a 0 0

Furthermore the

following replacement

has been used

j ^Z~ _lP«

+ i Pal~

=

@f _l~

^ds

_~()~

^~

(3.9)

~ =o o

where

P~°~

(s)

= p~

(3. lo)

(6)

The _use of this effective

replica requires

_an additional normalization in

equation (3.7).

Finally

the

(n

₊ I

)

_x

(n

+ I

)

matrix

w;j

^is

given by

In

^-I ^-I

D= ~' '

o (3.ll)

-1

0

4.

Mean-square

end-to-end distance.

The location of the _monomers _near the surface is described

by

_a harmonic

potential.

This is

quite

different _to the usual treatment of

physical adsorption

where _a 3-function is used

[2-8].

For technical _reasons it is necessary ^to introduce normal coordinates. Then _a Hamiltonian is obtained which is _a _sum of

independent

harmonic oscillators. For such _a transformation

Dp

is _an invariant

[18].

In the

appendix

the

following

relation is derived

Z

P~'~~ ₌

Z

_'~~~^~

~ i

(t~°~

₊ n~'~

'~~

_~)~

(4. ')

. i _, o

where the

(~'~'s

_are normal coordinates. The _use of the nornlal coordinates

yields

~~~~ °'

~

~~~~~~~~ (~ l~ i1

_~~~~~

_~ ^~~~~~~l

~

x

~ ~~ ~~~ ^~

D(~°~

exp ^Aj

(1 ^{(~°~~ (N )} ^flJco ^[(~°~]

^x

3

b~

n + I

x

fi _Dil')

exp (A

ii

iii

⁾²

(N fl

_3C,

[i~')] (4.2)

where

°~~

n l ^~ _n l

~~

~~~ l~

^~~

^~~~

^~ ^~~ ^~~

p

_~c~

j~ii

_>j

=

~

i

~

+

~ _~

12

~'( _j~

_ds

^(~~

^~

2

i~

n + I _n ₊ 3 b _o S

+

~~~ (n

+ I

j~

^ds ^(~~~~

^(s) ^(4.4)

2 _a

o

and for I _m 2

PJC

i~~

_>i

=

A _II

_dS

~ll'~ ~

+

Ill _II

_dS

_'~'~~

~S~

~4.5~

With the

exception

of

replica

I

= 0 the

partition

function

(4.2)

is that of

harmonically

localized random walks. In order to calculate the

partition

function it is convenient to _use the

general

structure of

equations (4.3)-(4.5)

Jti

₌ A

II1

~~~~ ₎

_~

) _j)

_~~

^~

~

~) _~

~~ ~~~~~ ~~ ~~

(7)

Then the evaluation of the

path integrals

3~~ ⁼

lDj exp(- Jt~) (4.7)

is standard and _can be found in the literature

[18].

Because _we are interested in the limit

bla ~ oJ, we can assume

ground

state dominance.

Choosing

furthermore

j(0)

to be the

origin,

and

integrating

over

((N)

one gets ^for ^I m

lq

^I

3$I

(

A

jj, ⁼

(I)li~ ~ (q;li ₎ j

A

j eXp q~ N/2

(4.

8

2 11 and for I

= 0 it follows

3&o(Ajjo)

=

(illi~f (2/3 Ni()~ _~~( jjo)

(4.9)

Nfo

Using

Ri

=

lim ^~ ^~

Z~(A~

=

0, Aj (4.10)

n, A, ^~o °n

°Ai

one finds

Ri

=

rim ~

~ )j~ )( _lfl &;(Aj,))

(4.ii)

n,A,~o

~ ~

I

~o

After _a

straightforward

but tedious calculation the

following

result is obtained

ql12

~

b~ Jf ~

i~

_N~~

(4. '~)

i~ qb~

where

q =

$$. _(4.13)

In the limit bla

~ oJ,

R(

becomes

Ri

=

b~

M.

(4.14)

Equation (4.14) corresponds

to the well known result for the mean-square end-to-end distance of random walks

[20].

The appearance ^of ^the ^effective

step length

b in the last

equation implies

that the mean-square end-to-end distance of

chemically

adsorbed

polymers

is

fully

^determined

by

the frozen-in disorder of the adsorbed _monomers.

5.

Mean-square perpendicular

distance.

In order to calculate the mean-square

perpendicular

distance from the surface let _us start from

equation (2.7).

In section 3

(compare Eq. (3.4))

we have _seen that

averaging

_over the

quenched

disorder has _no effect _on the

geometrical properties

of the

z-component.

^Therefore we may

(8)

write

il

= lim

)

_In

~fl~

_Dz

_31(z(s~ ₎₎

exp

~- ^~ j~

^ds

~

^~

+

A~ ~0 1

a =o 2 ₀

+

A~ '~j~

z~[ ^(k

⁺

^1/2)

~

1. ^{(5. I)}

k=o

An altemative

expression

is

given

_as follows

~ ^~

jz(s,)=0

_~

js,

^~~ ²

~

A~~0

^~~ _l

~~

z(0) 0

~~ ~~~

2

i~

0

~ ~~ ^~

1 ^M ⁱ

+

A~

^z~

(1/2 si)) ^(5.2)

M

Would _we

neglect

the restriction to the

half-space geometry

^the ^evaluation ^of ^the

path integral

in

equation (5.2)

_can be

performed directly. Writing Dz(s)

as a

product

of

D; _z(s)

for each of the intervals 0

~ s _< 1/2 _s

i and 1/2 si ~ s _~ si the

following expression

holds for the mean-square

perpendicular

distance

where

[7, 8]

G(z

_; _s

₎

=

~ ^~~~

~~p

(_

³ ^z~

2

wsi~

₂

~f2 ^(5.4)

To obtain _a formula which

belongs

_to the

half-space

_geometry _we _must add the

image

ternl to

equation (5.4) [7].

Then the calculation of

h~

is

perforated easily by integrating

_over the half-

space. The result is

There is

no

ependence on istribution of the _adsorbed

monomers. This

is, on

the one hand, due to the

used Gaussian bond probability and,

on

the

other

_hand,

model of chains

without

xcluded volume. A more _realistic model

should give a _dependence

6.

Comparison

between

chemisorption

and

physical adsorption.

Physical adsorption

^is induced

by

an attractive wall without any chemical reactions between

wall and _monomers. In the _case of _an ideal

homopolymer

the

following

_power laws hold

[4]

_:

(z~ )

~

(N/M

_)~

(6. 1)

and

(R()

~

N

(6.2)

(9)

A

comparison

between

equations (5.5)

and

(6. I)

_as well _as

equations (4.14)

and

(6.2)

shows

clearly

that there is _a difference between

chemisorption

and

physical adsorption

with

respect

to the power laws.

Marques

and

Joanny [9] investigated

the

physical adsorption

of random

copolymers. They

found that the thickness is

given by

h N/M

(6.3)

when N _» M.

Equation (6.3)

_agrees with the

corresponding expression (5.5)

derived in this paper.

To calculate the distances of interest _we have used _a very

simple

model. The

advantage

of this model lies in the fact that _an exact

analytical

solution of the

replica

_symmetry

breaking problem

has been found

[14, 21].

Such _a circumstance is very

helpful

for the

perturbative

treatment of _a _more realistic model. This will be the

topic

of _a further paper.

Appendix.

In this

appendix

_we shall derive

identity (4. I).

At first the

knowledge

of the

eigenvalues

of the matrix

(3.ll)

is needed. The

eigenvalues

read

wo = 0

(Al)

wi = n +1

(A2)

and

w~ =1.

(A3)

Note that the

degeneracy

^of ^the latter is _n I. The nornlalized

eigenfunctions

are

e1°~

=

(n +1)~~'~ _(l, _I,.

,

I) (A4)

ell~

=

(n/n

₊ I

)l'~ (I,

I/n,

,

I/n) (A5)

Owing

to the

degeneracy

of the

eigenvalue

_w~ the

corresponding eigenfunctions belong

to a

sub-space orthogonal

to e1°~ and

ell~

_The

general

structure is

e°~

=

(0, _e~i,

,

e~~) (A6)

The nornlal coordinates

(l'~

_are connected with the

original

_ones

pl'~ by

the

following

relation

jj10) jIn)jT ^~ j ~(0) pin)jT ~~)

where the

orthogonal (n

+ I _x

(n

+ I

)

matrix

b

_is

_given _{by [22]}

b

~

j~l0iT ^~l')T ^~12)T

~

~ln)Tj (A8)

, , ,

Then

i

pi°>2

=

jji°>,

,

fin>) b 0 bT (ji°>,

,

tin>)~ (A9)

0

(10)

where

1/(n

+ I

), nl'~/(n

₊

_1), 0,

,

0

b 0 b~

=

~~~~~~~ + '

~' ~~~~

+

~' °' '°

(A10)

~ i, _~i

,

I,

,

I

Inserting (A10)

in

(A9) yields

p1°)2

=

(~1°)

+

nl'2 (l~~f _{(Al I)}

n +

Using

f _P~'~~(N)

=

f _il')2(N) _p1°)2 _(A12)

identity (4.I)

follows

immediately.

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