Adsorption of long polymers dissolved in a semi-dilute solution of shorter chains

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Adsorption of long polymers dissolved in a semi-dilute solution of shorter chains

J.-U. Sommer, M. Daoud

To cite this version:

J.-U. Sommer, M. Daoud. Adsorption of long polymers dissolved in a semi-dilute solution of shorter chains. Journal de Physique II, EDP Sciences, 1994, 4 (12), pp.2257-2278. �10.1051/jp2:1994260�.

�jpa-00248130�

Classification Physics Abstracts

61.40K 82.65D

Adsorption of long polymers dissolved in

_a

send-dilute solution of shorter chains

J.-u. Sommer and M. Daoud

Laboratoire L40n Brillouin, ^CEN Saclay, 91191 Gif-Sur-Yvette Cedex, Fkance

(Received

24 March 1994, re~ised 15 June 1994, accepted 20 July

1994)

Abstract. The adsorption of dilute long polymers of length N dissolved in _a semi-dilute

solution of shorter chains of length P is considered using _a scaling approach. The short chains themselves _are considered _as non-adsorbable- Two qualitatively different phase diagrams in the

concentration adsorption-strength diagram _are predicted depending _on the value of

N/P~.

An

overall concentration and chain length controlled adsorption _{cross-over may} be found. An ideal adsorption regime is possible for N _< P~. _The scaling of the radius of gyration of the adsorbed chains _as well _as the concentration profile _are derived in the various adsorption regimes. The

surface concentration effects of the adsorbed long ^chains _are ^calculated from the condition of chemical equilibrium between the surface and the bulk. The plateau adsorption density~ which may be considered

as the maximum amount of adsorbed polymers, _may be controlled by the

overall bulk concentration of the bimodal system. This provides _{an easy way} to control the

polymer surface concentration.

1. Introduction.

Adsorption

of

polymers

from solution _on solid surfaces

plays

_an

important

role in many in- dustrial _processes like colloidal

stabilisation,

adhesion _or lubrication. It is also _an

interesting problem

from the theoretical point of view. Because of these _{reasons, a} lot of work has been devoted to this

subject involving

different mathematical methods _as well _as computer ^simula- tions

(see

^for instance references

[1-6]).

In most _cases the

polymer

solution _was assumed to be

monodisperse,

_as usual in other fields of

polymer

science.

However,

actual

polymer

solutions exhibit

polydispersity,

and the

investigation

of its influence _on the

adsorption

behavior _appears to be _a serious

problem.

For

example

it is well-known that

already

adsorbed short chains _are

replaced by longer

ones and that

longer

chains _are

preferentially

adsorbed onto the surface when

polydispersity

is present [7, 8]. ^This

implies

that the distribution of chain

lengths

_on the

surface is different from that of the bulk from which the chains _are adsorbed. For _a detailed

investigation,

it is useful to consider the

simplest

_case of

polydispersity, namely

^the case where

only

two different chain

lengths

_are present.

Beyond

this

simplification,

_a bimodal

polymer

g Les Editions de Physique 1994 _IOURNAL

DE pHysiouE,< T 4 N.12 DECEMBER19q4

solution exhibits _an

interesting phase diagram

_even in the bulk _as

previously pointed

out

by

de Gennes [17] ^and

by Joanny

et al. [10]. ^The reason for this

interesting

behavior is due to the fact that the shorter chains control the excluded volume behavior of the

larger

_ones.

In this paper, we discuss the

problem

where

only

^the

long

chains _may be adsorbed but their

configuration

statistics is influenced

by

the presence of the short chains. Furthermore _we restrict the discussion to the _case of semi-dilute solutions of short chains. This allows _us to use a

general

treatment of the overall concentration effects

by mapping

the

problem

onto _a lattice of concentration blobs.

The _{paper is}

organized

_as follows: in section 2 _we remind

briefly

the

configurational

_prop-

erties of

long

chains dissolved in _a semi-dilute solution of shorter chains. In the section 3 _we discuss the

configuration

of _a

long

adsorbed chain. The section 4 is devoted _to the _concentra- tion effects in the surface

layer,

I,e. the chemical

equilibrium

between the surface and the bulk.

The dilute _as well _as the bidimensional semidilute and the

plateau regimes

_are

investigated

in detail. The concentration

profile

of the

long

chains _near the surface is considered in section 5.

2.

Configuration

of

long

^{chains in} _a semi-dilute solution of short chains.

In this section _we

give

a brief review of the

configuration

of

long

chains made of N _monomers which _are dissolved in _a semi-dilute solution of short chains with P units in _a space of dimension

d. We _assume that the solvent is

good

both for the short and for the

long

chains. This avoids

further

complications

due to the interaction effects

recently

discussed

by

Schifer et al.

[ll].

We first consider the _case of _an overall concentration 4l

equal

to

unity,

that

is,

_{we assume} the

long

chains to be dissolved in _a melt of short chains. It is known that for P ₌ I the

long

chain is swollen because it is in _a

good

solvent. In the

opposite

limit where P

=

N,

_we have _a

melt,

where the attains _are ideal. Therefore

by changing P,

_we expect _{a crossover} between these two limits. In other

words,

the effect of the short chains is _a renormalization of the excluded

volume parameter _uo of the

long

chains into _an effective _one denoted

by

_u in the

following

_way

U =

p

,

(I)

which _was obtained

by

_a random

phase approximation [15, 17]. Minimizing

the

Flory

free energy for _a

single long

chain leads _to the

following

radius of

gyration

R for the

long

chain

R

+~

N"P~~/~+~

,

(2)

where _{v =}

)

_{is the usual}

_Flory

_{exponent. We}

now introduce _{a crossover} chain

length

gjd "

P",

below which the radius of

gyration

^{becomes ideal.} For shorter

probe chains,

the

excluded volume interaction is screened R

+~

N~/~ _Comparing

the latter-relation with

equation (2),

the exponent _{p is} ^found to be

» =

) ₍₃₎

It is worth

noting

that for d

= 2, _{p is}

equal

to one and hence full

screening

of the excluded volume is

present only

when the chains _are of the _same

length (N

₌

P).

In this _{case a very}

large length

effect is present: as soon as there is _a

slight

^difference in the _masses of the

probe

and matrix

chains,

excluded volume interactions _are present.

Using

these results, it is

possible

to introduce ideal blobs when the

probe

chain is

sufficiently

long: as we just saw, its overall behavior is swollen.

Still,

if _we consider _a part ^of it, the interactions _may not be

sufficiently

_strong to swell it. The

largest

parts ^{of the}

probe chain,

made of _{gid monomers,} that _are not swollen _are called ideal blobs. Let

(id

be their

length

scale.

Thus _we have

gid ~ P" and

(jd

_~

P"/~ (4)

It may be checked that

equation (2)

is

equivalent

to

R/f;d

+~

(N/gid)" (5)

Thus the

probe

chain is _a

self-avoiding

walk of ideal blobs. The

previous

results _are valid in _a dilute

solution,

when the

large

chains _are far from each other. For

larger

concentrations, in the semi-dilute _range, the

long

chains

overlap.

Another distance _appears~

namely

the mesh size

(

of the

overlapping long

chains. This

corresponds

to a concentration

blob,

with _mass g,

along

the

probe

chains. Because of the

overlapping,

excluded volume interactions _are

screened,

and the radius of

gyration

varies _as for _an ideal chain. More

precisely,

_we have

R/1

~

(N/g)1/2 (6)

( and g are related

by

4lN +~

,

( (7)

where 4lN is the relative concentration of the N-chains and the relation between

(

and _{g is} similar to

equations (2).

The various results _are summarized in table II for d ₌ 2 and 3,

respectively. They correspond

to the SDS

region

in

figure

8. Note that R and

(

_merge at the

overlap

concentration 4l

[

_+~

N/R~.

Thus for _a semi-dilute solution in _a

good solvent,

the structure of _a

probe

chain is ideal for distances smaller than the size

(;d

of the ideal blob. It is swollen for distances between

(jd

and the concentration blob

(. Finally,

it is screened and thus ideal

again

^{for distances}

larger

than

(.

A second _crossover concentration 4l

p,

_occurs when the

correlation

length

of the

probe

N-attains is of the _same order _as the ideal blob size, I.e.

(

+~

(;d.

Using equations (4), (6)

and

(7)

_{we get}

Alp

+~

P~% ₍₈₎

For concentrations

larger

than this _cross-over the excluded volume is screened for all dis- tances, and we are left with ideal

long chains,

_see SDI in

figure

8. It is worth

noting

that for d

= 2 this concentrated region does not exist. This is due to the fact that in this _{case p}

= I,

see

equation (3)

above.

Let _{us now} consider the _case where solvent is added _to this system. ^{This situation is sketched} for

overlapping

N-chains in

figure

I: Let 4l be the overall _monomer concentration. The

following

two restrictions will be

applied throughout

this _paper:

. The concentration Alp of the short chains is assumed to be in the semi-dilute _range

. When the

long

chains _are also in the semi-dilute _{range, we} will consider

only

the _case 4lN _< Alp.

This allows _{us to} take the overall concentration

dependence

into account

by simply rescaling

all

physical quantities

discussed _so far. The first step ^{in this} direction is to introduce overall concentration blobs with

length to

and _mass go, such that

to

°~

4l~*h

_3lld

go °~

4l~Z~~ ₍₉₎

Fig.

I. Sketch of the various scales appearing in the

region

of overlapping long chains for

N/go

>

(P/go)~.

The various regions are the circles. For increasing distances, _we find the

(swollen)

concentration blob fo~ the ideal blob f;d and the mesh size f of the long chains. For distances larger than ii the configuration is again ideal. The inset shows the _coarse grained situation.

Note that

to

<

(,

_see

figure

I. It is _now

possible

to renormalize all chain _masses and

spatial

distances

by expressing

them in units of _go and

to

The latter defines _a size of _{a new coarse}

grained

lattice which _we call the blob-lattice. Since the concentration blobs _are

closed-packed

the renormalized system is

effectively

dense. In units of the concentration blobs the semi-dilute solution _may be considered _{as a} melt. For _more detail _see the

appendix.

For the renormalized

quantities

_on the blob lattice

(see Eqs. (52-54)),

the above results for the melt _case may be

applied.

Here _we will

only

discuss the _most

important

_new

feature, namely

the existence of _a

crossover concentration 4l+ for d ₌ 3. The _cross-over condition

f;d

~ R has to be reconsidered.

If R is much smaller than

f;d

the whole

long

chain will be screened _as discussed above for the

melt _case. In the

opposite

limit the

long

chain is swollen.

Using equation (4)

the _cross-over

condition in d

= 3 is written in the

following

form

N/go

+~

(P/go)~ (N

+~

g;d). (lo)

This

yields, together

with

equation (9)

and the

Flory

exponent, 4l+

+~

(N/P~)~/~ (ll)

Hence if N is smaller than P~ the _cross-over between the swollen and the screened state of the

long

^chain _may ^{be controlled}

by

^{the overall} concentration. This is the

qualitative

_new

effect in the semi~dilute state

compared

to the melt state.

The effect of the semi-dilute _state of short chains for the behavior of

long

chains is _{a concen-} tration

dependence

of all

configurational quantities.

In the

following

sections _we will _see that this has

interesting

consequences for the

adsorption

of the

long

chains.

3.

Adsorption

of _a

single long

chain.

Because of the _presence of _many

lengths

_even in the

single

chain

problem,

_we will first look at the conformation of _a

single long

^chain in the _presence of _an attractive surface. This chain is assumed to be

grafted

in order to ensure its presence in the

vicinity

of the surface. Next

section deals with _a dilute solution.

In the

following,

all

energies

^{will be measured in} units of

kBT.

Let 6 be the

adsorption

free _{energy per monomer} of _a

long

chain. This is the free _energy

gain

when _{one monomer} of _a

probe

chain is _on the surface. The aim of this section is to derive the

(6,4l)-diagram

for _a

single long

chain in _a semi-dilute solution of short

polymers.

We _saw in the

previous

section that three different distance scales _are present: the overall concentration swollen

blobs,

with _{go monomers,} the ideal

blobs,

with g;d

units,

and the chain

length

N. As _we will _see

below,

the size

Rp

of the short attain is also _an

important

parameter ^{for the} adsorbed state.

In addition _to these

lengths,

the adsorbed chain has _a thickness D. Because of the _presence of these

scales,

different kinds of surface behavior _are

expected: depending

_on the value of

the interaction

6,

D may be

positioned anywhere

in this distance scale.

Figure

3a illustrates the situation. This defines _one non-adsorbed and three different adsorbed _states which will be

discussed

separately.

The

corresponding regimes

_are called

A,

Bi and 82 In order _to

proceed,

it is _necessary to know the number of _monomers Na which _are in direct contact with the surface if the latter is

effectively neutral,

I.e, neither

adsorbing

_nor

repelling.

This number is

assume(

to be

proportional

to

N~/~

_for

a

self-avoiding

walk _{near an}

impenetrable

surface [2, 12~14]. ^For ideal Gaussian chains

Na

is

proportional

to

N~/~

The first _{case we} will consider is the

adsorption

of _a

long

chain when 4l _< 4l+ Thus the number of _{monomers on} the surface _per swollen blob is

+~

g(/~

The number of elements of

a swollen chain _on the surface

obeys

_a similar relation. The number of units of _an ideal blob made of _n;d

"

g;d/go

elements _on the surface is

proportional

to

n)j~.

As _a consequence,

Na

may be written in the

following

form

N~ _~

(N/g;~)3/5(g;~/go)1/2gl/~ (12)

Using

the results of the

previous

section and the

appendix

_we get

Na

+~

N~/~P~~/~4l~~M (13)

In order to define _a dimensionless

adsorption

_energy, it is necessary to compare the

adsorption

energy

Na6

with the free _energy loss when the chain sticks _to the surface. This is of the order of the translational _energy of the chain in the

bulk,

which is of order unity in _our convention.

The

scaling

parameter ^{A is} then

~

~ ~

~3/s p-i /s~-1/4

_~i~~

Thus the

adsorption

threshold da

corresponds

to A

+~ I:

d~ ~

N-3/s pi /s~i/4

~15~

This separates ^the nonadsorbed from the adsorbed _states

(see Figs.

2a and

b). Therefore,

whereas for interactions smaller than da the

probe

attain is

basically isotropic,

^for

large

^surface

interactions it

adopts

_a

pancake shape

with width D. This is summarized in the

following

j c '° §~j _B~ ^'P

,~

~jj~

B~

L;

~.,

A _i~

N'~'P^'~~

Ii

₁

a)

j c

Lo

~

~ip

~ _{_,n}

P'"

~

iii

n~~~

6j

';d

N"~

A

'a

_{N"'P "'}

N ^~~

ii _i~

i I

b)

Fig.

2.

a)

Adsorption phase diagram for N _> P~. Regions Bi and 82 differ in the relative size D of the adsorption blob and Rp of the short chains: D » Rp in region Bl; D < Rp in region 82.

This is displayed by the insets in this region where _an adsorption blob

(circle)

and _a short chain _are shown,

b)

Adsorption phase diagram for N < P~. _{There is}

an additional ideal adsorption _crossover between the region of nonadsorbed screened long chains and region Bi Two concentration controlled

adsorption-desorption _{crossovers are} possible, shown by the _arrows.

~2 ~i

^~

>

l~ R~

1,~

R~

a)

B~ ~~

^B

>

l~ R~ R~

_1;~

b)

Fig.

^3.

a)

Sketch of the length scales appearing for the _case ~ _< ~+ _The

arrows show the possible

positions of the proximal distance D.

b)

Sarne _as in figure 3a but for the _case ~ > ~+ _{The ideal} blob size is only virtual since it is larger than R. The whole chain is ideal

on all scales larger than to- Hence, the adsorption region A cannot appear.

scaled form for the dimension of the

polymer

_[5]

D +~

Rf(A) (16)

Assuming

_{a power} law behavior for

f(A)

when A » I and

using

relation

(14),

_we find the width of the

polymer

in the direction normal _to the surface when it is adsorbed. Because the conformation has the

shape

of _a

pancake

in the adsorbed state, we

impose

the condition that D be

independent

of N

D

+~

6~~ (17)

When the

long

chain is adsorbed, and A _» I, _{we may} ^define

adsorption

_or

isotropic

blobs [5]

that _are still in _an

isotropic configuration.

These _are parts ^{of the chain made of} gD monomers

that _{are on} the verge of

being

^adsorbed.

Generalizing

^relation

(14),

_{one may}

get

_gD ^{and clieck}

that the radius of these blobs is D. Note that D and _{gD are} related

by

D +~

g§/~P~~/~4l~~/~ (18)

according

to the

corresponding

entry in table II since the blobs consist of swollen chain parts.

Taking

^the

isotropic

blobs _as

units,

the adsorbed chain _may be considered _{as a} two-dimensional walk. Therefore its radius Rjj ^{of the adsorbed}

long

^chain

parallel

to the surface is

Rll

ID

'~

(N/gD)~~~

,

(ig)

where _we used the excluded volume exponent v in _two dimensions.

Using equations (18)

and

(17)

_we

finally

get:

j~

~

113/4 p-1/4 fit/4§~-5/16 (~~)

ii

The

resulting

structure of the adsorbed

chain,

valid in

region A,

includes several

lengths

that _we summarize: for distances smaller than the concentration blob

to,

the chain is swollen.

Screening by

short chains _occurs for distances smaller than

f;d.

For

larger distances,

short chains _may be considered _as a

good solvent,

and the

probe

chain is swollen

again.

All the

previous

kinds of behavior _are

isotropic,

and the surface effects _are present

only

for distances

larger

than the size D of the

adsorption

blob. For

larger distances,

the behavior becomes

two-dimensional because the chain is adsorbed.

As the surface interaction is

increased,

the width D of the adsorbed

polymer decreases,

and

a cross-over occurs when it becomes of the _same order _as the size of the ideal blob. This

implies

that for

larger

than _{&;, to} be defined

below,

the

probe

chain is adsorbed _{as an} ideal

polymer

and the

properties

of the two-dimensional adsorbed chain

change.

The _crossover

corresponds

to

f;d

_'~ D.

Using equations (A.6)

and

(17),

_we get 6; +~

P~l~~~/~ (21)

Thus for surface interactions

larger

than

6;, probe

chains _are ideal for distances of the order of the

adsorption

blob D. For _a

given

value of the surface interaction

6,

this _{cross-over occurs} for

overall concentrations

larger

than 4l; that may be evaluated from the _same

equation.

Because the adsorbed blob is ideal in the present _case,

equation (18)

is

replaced by

D

+~

gi~4~-~/~ (22)

Using equations(19)

and

(22),

_we get ^the

parallel

radius of the chain in this

regime

R~ ~

N3/4 &1/2~-3/16 (~~)

This

corresponds

to

region Bi

in

figures

2 and 3

respectively.

Note that in this _case Rjj ^is

independent

of P because the

adsorption

blobs _are ideal

(Eq. (22)). Finally,

let _us note

again

that in this

regime

^the

probe

chain is still

locally swollen,

for distances smaller than the size

to

of the concentration blob.

When the interaction 6 is still

increased,

the size D of the

adsorption

blob becomes smaller than the radius Rp of the short solvent chains. The

corresponding

_{crossover occurs} for

6p

+~

P~~/~4ll/~ (24)

where _we used

again

relation

(17),

valid in all

regimes

of the adsorbed chain.

For D <

Rp,

_or 6 _>

6p,

_we enter

region 82.

The

important

^difference with

region Bi

lies in the fact _that the

screening by

the short chains _now has _a two-dimensional character: _as discussed

above,

for distances

larger

than the size D of _an

adsorption blob,

the

probe

chain is adsorbed and lies flat _on the surface. Because Rp is

larger

than

D,

the

configuration

of the

large

chain that is

being

screened is two-dimensional. As discussed in section 2 the important

implication

is that the size of the

corresponding (ideal)

^blobs is of the order of Rp and that

the adsorbed chain has swollen conformation _at

larger

distances [16]. ^Thus

equation (22)

has to be

replaced by

R[[/RP

_'~

(N/P)~~~ (25)

and

Rjj +~

N~MP~~M4I~~/~ (26)

The differences between the situations Bi and

82

_are shown in

figures

4a and 4b. In

figure

4b the short chains _are smaller than the

adsorption

^blobs of size D.

Hence,

outside these blobs the

long

chain behaves _{as a}

self-avoiding

walk. In contrast,

figure

4a presents ^{the situation} when the short chains _are much

larger

than the

adsorption

blobs. Since the

problem

is two-

dimensional the

screening length

is

given by

the size Rp of the short chains Rp

+~

P~/~4l~~/~

(see Eq. (3)).

If _we still increase the monomer-surface

interaction,

^there is _{a cross-over} to _a

regime

^where segments

along

the chain of the _same size _as the concentration blobs _are adsorbed. This

happens

when D

+~

to Using equations (9)

and

(17),

_{we get} for the _cross-over value do do +~

4l~/~

(27)

The

regime

⁶ > do will not be considered here for several _reasons.

First,

in this _case the

adsorption

blobs _are smaller than

to

and and the

simple mapping

^of the

problem

to lattice of concentration blobs does not work because _{we are now}

considering

distances smaller than

to

The

simple approximation

for the effective excluded volume constant _u

given

in

equation

(I)

is not

longer

valid. Note that for 4l

= this

corresponds

to the strong

adsorption

limit

(6

₌

1)

^which is also _not considered here. A second _reason is that inside the

pancake-shaped long

chain, ^the local -concentration in

probe

_monomers becomes

higher

than that of short chain

monomers. Thus there is an

interesting wetting problem

that

arises,

but is outside _{our scope.}

For 4l >

4l+,

_{we saw} in section 2 that the structure of the

probe

is _no

longer swollen,

but is ideal for

large

distances. The

possible

relation between the different

length

scales _on the

probe

chain and the

adsorption layer

^width is illustrated in

figure

3b. This

implies

that the

equation

for the

scaling

parameter

(14)

has _to be

changed

A

+~

6N~/~4l~~/~

,

(28)

where _we have used the fact that two

length

scales _are present: the swollen concentration blobs

to,

and the ideal chain size R. This leads _to

6;d _'~

N~l/~4ll/~ (29)

Note that this

adsorption

of _an ideal chain takes

place only

for N < P~

(see Fig. 2b).

For

» &;d, the conformation of the adsorbed chain is identical that in

region Bi

that _we discussed above.

The

resulting phase diagrams

_are ^shown in

figures

2a and 2b for the _case N _> P~ and N _< P~

respectively.

It is worth

noting

that for _a

given

value of _&,

by changing

the overall concentration, ^the

probe

chain _{crosses over} from _an

isotropic-non

adsorbed-to _a

flat-adsorbed-configuration

when

one goes

through

the various _cross-over lines that _we considered above.

R~

D

a)

D

b)

Fig.

4. Illustration of _a part of the structure at the surface. When the short chains

are

larger

^than

the adsorption blob size D, they _screen the excluded volume interaction of the adsorbed long chains within _a distance Rp. For the two-dimensional _case this also implies _a local compactification of the chain

on the surface.

b)

Opposite ^situation _as in 4a. Outside the adsorption ^blobs the adsorbed long chain behaves _{as a} two-dimensional excluded volume chain.

4. Concentration effects.

Having

^studied the different

possible configurations

of _a

single probe

chain at the surface, _we

now consider the _case of _a dilute solution of

long

^chains. We recall that

they

_are dissolved in

a semi-dilute solution of shorter

polymers

in _a

good

solvent. We will consider the

adsorption

isotherm and the surface _saturation

regime

for _a finite bulk concentration

4lN

in

long

chains.

To this end _we consider the free energy F _per chain within _a

Flory approximation.

This _was done

by

de Gennes in the

simple

solvent _case, and for _a surface of defects [17, 18]. ^{The free}

energy per chain consists of the three

following

terms

F # Fad +

Ftrans

+

Frep (30)

The first _term is surface attraction contribution. The second _term is the translational free energy of _a chain _on the

surface,

and the last _one is the contribution from the excluded volume

interaction of the chains in the semi-dilute surface

region.

We evaluate these contributions.

Each

adsorption

blob contributes _an

adsorption

_energy of the order kBT since it is

defined

to be at the _verge of

being

adsorbed. Thus _we get a

simple scaling expression

^for the free

adsorption

_energy

Fad '~

-N/gD (31)

Regions

A and B have _to be considered

separately.

This is done

by taking

the

appropriate

values of _gD

using equations (18)and (22).

The results _are

displayed

in table I.

Region

82 needs further discussion because the size Rp of the screened ideal domains is

larger

than that of the

adsorption

blob D

(see Fig. 4b).

Thus the ideal blobs _are

two-dimensional,

as mentioned earlier. Because _a two-dimensional ideal chain is compact, these domains _cover the

adsorbing plane.

This is in _contrast with

region A,

^{where the surface} is covered

by

the

adsorption

blobs. Because of

this,

_one of _our

previous

arguments ^{fails: the} translational _energy of the

adsorption

blob is _no

longer

of the order of thermal _energy. More

precisely,

_every ideal

domain has _{an energy} of this order. This

implies

that the free _energy of _an

adsorption

blob is smaller than the thermal _energy.

Thus,

in this _case, the

adsorption

_energy is _no

longer kBT

_per

adsorption blob,

but rather _per ideal blob. The total

adsorption

_{energy per} chain is therefore

Fad '~

-NIP

for

Rp

» D

(32)

This is the number of ideal domains _per chain. One _can understand this _crossover in the

following

_way: in

region

Bi the

adsorption

_process for _a

single

chain consists in

placing

ad-

sorption

blobs of size D _on the surface. Each blob has _an

adsorption

_energy

kBT (unity

in _our

conventions).

This _energy is necessary ^to overcome the translational free _energy of that blob in the bulk and hence adsorb it. In

region 82,

because of the

screening by

the short

chains,

the adsorbed blobs _are in _an ideal

configuration

_on the surface. To this

screening corresponds

an ideal

blob,

that is,

larger

than the

adsorption blob,

and has _{an energy} of order kBT. This

implies

that the

adsorption

blobs _are

feeling

the

repulsion

_energy

(or

surface

pressure)

and that their translational _energy is smaller than

kBT.

As _a result the

adsorption

_energy is reduced.

In all

regions,

the translation entropy Firana is

nran~

+~ In

(£

,

(33)

where r is the surface fraction. More

precisely

r

ID

is the _average concentration in the adsorbed

layer

[5].

I

=

I £°° ^~N(z)dz

^~

l _£~

_~N

_{(z)dz (34)}

where

4lN(z)

is the surface fraction of

probe

_monomers at _a distance _z from the surface.

Finally,

_we have to consider the

repulsive

contribution

Frep

^{within the}

adsorption layer

for the semi-dilute surface

regimes.

^{Once the chains}

overlap

in the

surface,

_{we may use}

previous

Table I. The various

physical quantities

in the various

adsorption regions,

_gD denotes the

n umber of _{monomers per}

adsorption

^blob. _Rjj is the extension of the

long

chain _on the surface for the dilute surface

regime.

r* is the surface concentration where the

long

^{adsorbed chains}

start

overlapping.

roar

gives

the

plateau

surface concentration which is

independent

of the

amount of

long

chains in the bulk.

Fad

is the

adsorption

free _{energy per} adsorbed

long

^chain.

Fre~

^{is the}

repulsive

free energy contribution for the

overlapping long

chains at the surface.

§-5/3

pi

/3~5/12 §-2~l/4

_p

113/4p-1/4§1/4~-5/16 113/4§1/2~-3/16 113/4p-1/4~-1/8

r~

~-i/2~-1/2pi/2~s/8 ~-i/2~-1~3/8 ~-i/2pi/2~i/4

r ~

~i/3pi/3~s/12 ~i/4 ~i/4

Fad

-N&~/~P-1/~4l-~/l~ -N&~4l~~/~ -NIP Frep Nr~&P~~4l-~R Nr~&?4l-~R Nr~P-14l-1/~

results for semi-dilute solutions

[19, 20]:

_every ^surface concentration blob has _a contribution of kBT

(unity

in _our

conventions).

Thus

Fre~

is the number of surface concentration blobs _per

probe

chain. In order to obtain this

number,

_we have _to

investigate

the semi-dilute surface

regime

more

carefully.

The

overlap

concentration r* is:

r*

~w

N/Rj (35)

Using equations (20),(23)

and

(26)

_{one can} obtain the

explicit

results summarized in table I.

The number of _monomers per two-dimensional concentration blob _g2d is obtained

by using

the

scaling

assumption

g2d ~ N'

f(r/r~) (36)

In the semi-dilute range, for r _»

r*,

_{we assume}

f(x

_»

I)

_+~ x~. The exponent a is determined

by

the condition that _g2d is

independent

of molecular

weight.

This

implies

that _a

= -2 in all

cases.

Using

the fact that F~e~ +~

N/g2d

_one gets the

repulsive

part of the free _{energy per} chain

in the different surface

regions.

The

explicit

results _are

given

in table 1.

4.I THE DILUTE SURFACE REGIME. When the solution is

dilute,

_{Fr~p may} be

neglected

and the free _energy has

only

two contributions:

F

+~ In

(£)

+

Fad. (37)

We may assume this free _{energy to} be the surface chemical

potential

_per chain _pa, that

is,

the free energy necessary for

adding

_one

polymer

_on the surface. The surface concentration

r

(al

r~

I~

Fig.

5. The adsorption isotherm. For _very low bulk concentrations ~, the surface concentration is proportional to ~. The slope is proportional to

exp(-Fad).

tsar is the surface concentration

at saturation. Here the adsorbed _amount varies very slowly with bulk concentration and may be

considered _as

a constant.

r is evaluated

by equating

_pa with the chemical

potential

_pb of the

long

^cliains in the bulk.

Because the latter is assumed to be _a dilute

solution,

_we have

pb '~ In

l~'~

_N

⁽³⁸⁾

This

gives

^the

equilibrium

surface concentration r

r =

D~Ne~~~~ (39)

As for

adsorption

from _a solution in _a

simple

solvent [5] ^the surface concentration is propor- tional to the bulk concentration for _very low bulk concentrations. In the present _case, there is

an additional

dependence

in the overall concentration that is hidden in Fad

(see

Tab.

I).

Note that because of the exponential factor the value of r is much

larger

than 4lN. ^Thus

although

the bulk is

dilute,

the surface _may be

concentrated,

_{as we now} discuss.

4.2 THE BIDIMENSIONAL SEMI-DILUTE AND THE PLATEAU REGIME. For concentrations

larger

than

r*,

in the semi-dilute

regime

the

repulsive-excluded volume-part

of the free _energy

F~e~ comes into

play.

Two _cases have to be

considered, depending

_on the value of the ratio

F~e~/Fad.

When this ratio is much smaller than

unity,

relation

(39)

remains valid and there is

no

qualitative change

in the

adsorption

isotherm sketched in

figure

5.

For

sufficiently large

concentrations, the

repulsive

interaction becomes of the _same order _as the surface attraction, and the

plateau regime

is reached. The surface concentration saturates at a value

independent

of the bulk concentration of

long

chains. The _reason is that in this _case the attractive and the

repulsive

free

energies

_per chain _are much

larger

than the translation free

energy. In

fact,

n~ana in the surface

layer

_may be

neglected

and the bulk chemical

potential

is

a small

perturbation

of order I

IN.

For _more

detail,

the reader is referred to [5].

JOURNAL DE PHYSIQUE II T 4, N' 12 DECEMBER 1994 83

In order _to get ^{the saturation} concentration roar _one has to minimize the free _energy F ₌ F~e~ ⁺

Fad.

For the various

regions

in

figures

2a and

2b,

this

gives:

I~$~~ ^+~ d~~~P~~~4~~~~~

(40)

~nBi ~B2

~l/4 (~i)

where the

corresponding

entries for the free

energies Fre~

^and Fad of table I _are used.

The

geometrical meaning

of the saturated surface

regime

is _a close

packing

of

adsorption

blobs _on the surface.

By comparing

the situations sketched in

figures

4a and

4b,

_{one can} understand the

meaning

of

equation (41)

in both _cases, the saturated state is _a dense ensemble of ideal

adsorption

blobs of _same size D. Another _reason is that for all distances

larger

than D the excluded volume interaction is screened in both _cases in the saturated

regime.

This

argumentation

may also be viewed _{as an a}

posteriori justification

^of

equation (32).

Figures

6a and 6b

display

the surface concentration in the

plateau regime

_{as a} function of

adsorption

_energy and concentration for the various _cases in the

phase diagram

of

figure

2.

5. The concentration

profiles.

Finally,

_we derive the concentration

profiles 4lN(z)

of the

long

chains in the

plateau regimes.

z is the distance to the wall.

Following

de Gennes _one has _to

distinguish

between the

proximal region

z <

D,

the central _or self-similar

region

D < z < R and the distal

region

z > R for the _case of _a dilute bulk

regime

of N-chains. When the bulk is

semi~dilute,

^the

profile

does not

exiend

_{to the radius} of

gyration R,

but

only

to the overall

screening length (.

In the distal

region,

^the concentration

profile decays exponentially

with _z and will not be discussed

further. We will consider the

adsorption regimes

A and B

separately. Regimes

Bi and 82 do not show _any differences _as far _as the concentration

profile

is concerned. Because the surface is saturated

by adsorption blobs,

the _monomer concentration 4lN~ ^at the surface is the number

of _{monomers ga per} such blobs _at the surface divided

by

the _area of _a blob:

4lN~ +~

9s/D~ (42)

Since the definition of the

adsorption

blob

implies

_gad

+~ I, _we

immediately

find 4lN~ +~

(43)

The latter result is valid for all saturated

regimes.

5.I THE _{SWOLLEN cAsE:} ADSORPTION FROM REGION A. Let _us first consider the _case

where the

long

chains in the bulk _are swollen. We start the discussion with the

proximal region,

for _z < D. Since the size D of the

adsorption

^blob is

larger

than that of the ideal blob

(;d,

three different kinds of behavior have _to be considered _{as z}

increases, namely:

_z <

to, to

< z <

(;d

and

(;d

< z < D. In order to derive the

profile 4lN(z)

in the various

regions,

_we

assume,

following

de

Gennes,

_{a power} law

decay

with the

proximal

_{exponent m:}

4lN(z)

m 4lN~ z~'~

(44)

The exponent m was shown to be

equal

to

1/3

for _a swollen chain and to 0 for _an ideal chain

near an

impenetrable

surface. The three different

regions

mentioned above

correspond

to ideal

and excluded volume behavior. Thus _we get

4lN(z)

oJ

&z~l/~ (a

< _z <

to (45)

Note that

4lN(z

=

to)

'~

&4llR

_{For distances} in the _range

to

< z <

(;d,

the behavior is

ideal,

_so that _{m =} 0. Thus the

density profile

remains constant.

Continuity

with the

previous

relation

implies:

I~N(Z)

~ dl~~~~

((0

_{< Z <}

(id) (46)

For _z

larger

than the

screening length, (;d

the chain is swollen

again,

so that _m

=

1/3.

Continuity

of the

profile

for

(;d

leads _to

~ -i /3

4lN(z)

+~

&4llR ((;d

< z <

D) (47)

(id

Note that

4lN(z

=

D)

+~

&~/~Pl/~4l~/l~ (48)

For distances

larger

than D, we enter the central

region.

This is _a

special region

between the

(surface) proximal

^{and distal}

regions

^{where the adsorbed chains build} up a self-similar

profile,

_as discussed

by

^{de Gennes. The} _reason ^{for such} a behavior is that for every distance

z to the

surface,

the local concentration is in the semi-dilute _range. This

implies

that there is

a local

screening length ((z).

The latter is

given by equation (A.5)

with 4lN

being

the local concentration _at distance _z.

Accepting self-similarity

of this

layer implies

^{that this}

length

is

proportional

to z

z +~

( (49)

In other

words,

the mesh size in the central

region

is

proportional

to the distance _z to the wall.

Using equation (A.5),

_we get:

4lN(z)

+~

z~~/~Pl/~4l~/l~ (D

_{< z <}

R) (50)

Note that the latter

expression

_{crosses over}

smoothly

to

equation (48)

for _z

= D. The

resulting

concentration

profile

for

region

^A is

given

in

figure

7a.

5.2 THE IDEAL cAsEs: ADSORPTION FROM REGIONS B. The main dilserence between the

present and previous cases is that the size of the

adsorption

blob D is _now smaller than that of the ideal blob

(;d. Therefore,

the last

region

in the proximal

regime

does not exist anymore.

The first _{two ones are} unaffected. Thus

equation (45)

and

equation (46)

_are valid in the ideal

adsorption

cases.The

only

difference is that the latter extends

only

to

D,

rather than to

(;d.

This is shown in

figure

7b. For distances

larger

than

D,

_we enter the central

regime.

The central part ^{of the} concentration

profile

is different from that in

region

A for distances

z <

(;d.

This is because the behavior of _a chain in this distance smaller _range is ideal

(see Eq. (A.7)).

Two _{cases may} be met,

depending

_on the overall concentration 4l: for 4l »

4l+,

we saw in section 2

(see

also

Figs. 3b)

that

(;d

< R in the bulk. This

implies

that the

probe

chains _are

completely

screened in the bulk.

Taking

^this into account, _we find [21]

4lN(z)

+~

z~~4l~M (D

_{< z <} R;4l » 16+)

(51)

I'" I>I+

I'" I<j+

~

~

sat

$r~(j~~(~~i't'°n

non ideal adsolptian

crossover

Lo

_L~j L;d

a)

~al

, _~p3/5

concentration controlled

adsorption _cross-over

6'~

1)

1 _(L~) I _(L,,) 4

b)

Fig.

6.

a)

The plateau surface concentration _{as a} function of 6. The figure shows the result

given in table I for two different overall concentrations. For N < P~

(Fig. 2b)

both ideal and _non- ideal adsorption _are possible depending _on the value of the overall concentration. At the adsorption

cross-over the surface concentration varies rapidly with the adsorption strength _as displayed in figure

5. These transitions _are indicated by the _arrows.

b)

The plateau surface concentration mat _{as a} function of the overall concentration of the binary system. ^The figure shows the result given in table I.

Concentration controlled adsorption cross,overs can occur for 6 < N~~/~ when N <

P~,

_see

figure

2b and when 6 _<

N~~/~P~/~

_{for N >}

P~j

_{see figure} 2a. At the overlap threshold of the short chains ~)

the result for simple solvent is recovered, mat

~

6"~ _{[5]. At} the adsorption crass-avers the function

r(~)

varies exponentially.