Finite extensibility and density saturation effects in the polymer brush

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Submitted on 1 Jan 1989

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Finite extensibility and density saturation effects in the

polymer brush

D.F.K. Shim, M. E. Cates

To cite this version:

Finite

_{extensibility}

and

_density

saturation effects in

the

_polymer

brush

D. F. K. Shim and M. E. Cates

Theory

of Condensed Matter, Cavendish

_{Laboratory, Madingley}

Road,

Cambridge

CB3 OHE, G.B.

(Reçu

le 9

_juin

1989, révisé le 18 août 1989,

accepté

le 31 août

₁₉₈₉₎

Résumé. 2014 Nous

avons étudié le

_problème

des chaînes de

polymères

neutres attachées par un

bout à une interface

_plane

_(«

brosses » _de

_polymères),

à

_{l’équilibre statistique.}

Nous traitons le

problème

de l’extensibilité finie des

_polymères,

en utilisant une forme de

_l’énergie

libre d’une

chaîne étirée,

qui diverge

à

_l’approche

de l’extension maximum. En

_adoptant

aussi une

_équation

d’état pour le

potentiel chimique

local du type de

_{Flory-Huggins,}

nous obtenons des

_profils

de densité de monomères et des distributions de la densité des extrémités de chaînes pour diverses conditions du solvant. Nous utilisons une

_{approximation}

de

_champ

self-consistent pour les chaînes

très étirées,

proposée

par Milner, Witten et Cates

_{(Macromolécules}

21

_{(1988) 2610), qui}

trouvent

un

_profil

de densité

_parabolique

pour des brosses de « densité modérée » dans un bon solvant. Notre travail

_généralise

cette

_approche

en l’étendant aux densités élevées et/ou aux mauvais solvants ; dans tous les cas, le

profil

de densité est

_{beaucoup plus plat qu’une parabole.}

Abstract. 2014 We

study

the

_equilibrium

statistics of flexible neutral

_polymer

chains attached

_by

one

end onto a flat interface

_(the

polymer

« brush

»).

We account for the finite

extensibility

of the

polymers, adopting

a form for the free energy of a stretched chain that

_diverges

as its

fully-extended

length

is

_{approached. Adopting}

also a

_{Flory-Huggins equation}

of state for the local chemical

_potential,

we obtain monomer

_{density profiles}

and chain-end

_density

distributions under various solvent conditions. We use an

_analytic

self-consistent field

approximation

for

long, highly

stretched chains, as

_{developed by}

Milner, Witten and Cates

_{(Macromolecules}

21

_{(1988) 2610),}

who found a

_{parabolic density profile}

for brushes at « moderate densities » in a

_good

solvent. Our

work

_generalizes

their

_approach

to _{allow very}

_high

_{coverages and/or poor solvents} to be considered ; in each case the

_{density profile}

is much flatter than a

_parabola.

Classification

Physics

Abstracts 36.20 - 61.40K

1. Introduction.

The uniform

_grafting

of

_monodisperse

flexible neutral

_polymers

onto a

_{non-adsorbing}

substrate,

immersed in solvents of different

_quality,

is of both theoretical

_[1-20]

and

experimental

[21-23]

interest. An

_important

use of the

_{resulting polymer}

« brush » is in the

steric stabilization of colloidal

_suspensions,

with

_applications

in waste-water treatment, in

foodstuffs,

pharmaceuticals, paints,

etc. ;

polymer coatings

are also

_important

in

lubrication,

adhesion,

and related areas. It is

_apparent

that

_optimal

use of

_polymeric

stabilizers entails

detailed

_{investigation}

into the

_equilibrium

of

_polymer

_brushes,

since these

_play

a direct role in

determining

the interactions between coated

_particles

in

_suspension.

Experimentally, grafted

chains may be realised in several ways [21-22]. One method is to

use chains end-functionalised with

(say)

a zwitterionic _group at one end which then attaches

itself to the substrate surface

_[22].

Another common

_procedure

is to use an

_asymmetric

diblock

_{copolymer AB,}

chosen so that the shorter A block adsorbs

_favourably

onto the substrate. Theoretical studies have been made for

_{block-copolymers}

in selective and

non-selective solvents

_[19].

In

_previous

theoretical treatments of

_terminally

anchored chains

_[1-20],

lattice models energy balance

arguments,

scaling

arguments

and self-consistent field

_(SCF)

methods were

employed.

In _{this paper} we shall use the SCF

_approach,

as initiated

_by

Dolan and

Edwards

_[3],

later

_{augmented by}

Helfand and Wasserman

_[24],

and

_Scheutjens

and Fleer

_[6].

This reduces the many chain

problem

to a

_single

chain one

_involving

a

_{position dependent}

chemical

_potential,

which must be found

_{self-consistently}

_[25].

We

_exploit

an

_analogy

between the brush

_problem

and one

_involving

the motion of

fictitious classical-mechanical

_{particles moving}

in an external field related to the chemical

potential.

The

_{correspondence}

between

_highly

stretched chains and classical

_trajectories

was

first

_{exploited by}

Semenov

_[26]

in the context of molten block

_copolymers.

This

_analogy

was

used

_extensively

in recent papers

by

Milner et al.

_[15-17].

In reference

_[15],

Milner,

Witten and Cates

_(MWC)

showed that for a

_typical

chain in the

_brush,

the

_{only configurations}

contributing significantly

are those that follow the « classical

_path

»

(of

least « action » or free

energy)

between the

_end-points

of the chain. This holds because chains in a brush are

_highly

extended,

at least in the

_long

chain limit : the

_{height h}

of the brush scales

_linearly

with the chain

_length

N.

Adopting

the usual Gaussian

_elasticity

form for the

_stretching

_{energy of} a

_chain,

MWC

_[15]

showed

_{quite generally}

that for

_monodisperse

chains the

_only

stable form for the

_spatially

varying

monomer

_potential

is a

_parabola

_{(A -}

Bz ,

with z the distance from the

_grafting

surface).

MWC restricted attention to the « moderate

_{density » regime,}

defined as that in

which the «

equation

of state »

relating

the local chemical

_potential

to the monomer

_density

reduces to a linear

_{dependence. Clearly}

in this case the

_{density profile}

is itself a

_parabola.

At low coverages, their results are in

_agreement

with numerical calculations

_{[27, 28]}

based on the

lattice SCF theories of

_Scheutjens

and Fleer. The

_scaling

laws

_relating

brush

_height

to surface coverage and chain

length, predicted by

the full SCF treatment, are consistent with the energy

balance

_arguments

of Alexander

_[5]

who did not calculate the

_profile,

instead

_making

a

_simple

step

function ansatz.

Departures

are seen at

_high

_{coverages, which is}

_expected

since the

_{corresponding}

densities

are not

_moderate,

but close to saturation

_(at

volume fraction of

_{polymer 0}

--+

1).

At these

densities

_{(and in any}

case to _{describe poor solvent conditions) it is necessary} to use a more

realistic

_equation

of state. For sensible results at

_high

_{coverage, it is also essential} to correct

for the

_{finite extensibility}

of the chains.

Indeed,

it is easy to confirm that at

_{high enough}

coverage, retention of a Gaussian elastic term will

_always

lead to

_predictions

of the brush

height

that exceed the

_{fully-extended length}

of the chains.

In the

_present

_paper we

_rectify

both of these

_shortcomings

of the MWC

_theory.

We

_adopt

for

_simplicity

the

_{Flory-Huggins equation}

of state, which allows _{for saturation and poor} solvent

effects ;

and we choose a

_suitably

modified elastic term that

_prevents

_unphysical

chain extension. Thus we are able to

_{interpolate smoothly}

between the

_{parabolic profile}

of MWC and a

_flat,

_{step-function profile}

which must

_obviously

be recovered at _coverage

_unity

_(when

each chain has no choice but to stretch to its

_fully

extended

_length).

Since we are

_dealing

with

density 0 =

1),

we

_expect

the SCF

_approach

to be valid in all cases we consider. In section 2

we set out the

_theory

to account for saturation and the finite extension of chains in the

brush,

and

present

the results obtained. Some final comments and conclusions are

_given

in section 3.

2. Self-consistent mean field

_theory.

In this section we set _{up the} 1-dimensional self-consistent

_{equations describing}

the

brush,

appropriate

for a flat

_geometry

with translational invariance

_parallel

to the

_grafting

surface. As usual we

_adopt

a structureless model

[29]

for the

_polymer

chains since we are interested

only

in the

_macroscopic

statistical

_properties.

We

_adopt

the same

_Lagrangian

formalism as

used

_by

MWC

_[15].

The

_{configurational}

sum f2 of all

_polymer

chains,

characterised

_by

the

positions zi

(n ),

with

_{n = 0, 1, ... N, of}

the ends of the N monomer

_segments

of the i-th chain

is

_{given by}

_(working

in units of

_kB

T =

1)

or in the continuum

_language

where

_{F (K )}

is the free energy for K = O"A chains with cr as the surface coverage

(o-

= 1 is

saturation)

and A the

grafted

surface _{area ;} T is the free energy of

stretching

of the

chains and W is their _{interaction energy.}

T is the sum of

_{independent stretching energies Ti.}

Our model for

_Ti

which accounts for the finite chain

_{extensibility}

is defined as

where

_ii

=

dzi/dn

and

y is

a

_{dimension-dependent}

_parameter

which we shall discuss later in detail.

_(MWC

took y = 1

[15]).

We have chosen this form for

Ti

in _an ad hoc _manner

subject

to the

_requirements

that

_(i)

it reduces to the usual Gaussian

_stretching

_{form 2}

(

dz 2

when 2

dn )

1 Z

|

-+ 0 ;

and

_(ii)

it has a

_divergent

derivative

_as

_{| z}

_[

- 1 . The second of these

_requirements

enforces an

_{extensibility}

limit

_at

_{| f}

₁

=

1;

correspondingly

_a classical mechanical

particle

with

T;

as its

_Lagrangian

would have a

_{limiting speed}

of

_unity.

To have a

_stronger

_analogy

with classical

_mechanics,

it would be attractive to choose instead of

₍₂₎

the

_Lagrangian

of a relativistic

_particle.

This choice was first

_proposed

_by

Vilgis

and Kilian in the context of rubber

_{elasticity theory}

_{[30] ;}

it leads to more

_complicated

calculations than

_equation

_(2),

as shown in

_appendix

A. Various other functional forms can

be obtained from

_specific

local models of the chain at the monomer level

_{[31] ;}

however,

we

emphasize

that there is no _{reason, a}

_priori,

for

_choosing

one form over another

(subject

to

requirements

(i)

and

_{(ii) above).}

The corrections to Gaussian

_elasticity

for

_{polymers, though}

universally

present,

do not take a universal

_form,

but should vary from one

_species

of

_polymer

to another

_[31].

With this in

_mind,

the choice in

₍₂₎

is motivated

_{mainly by analytical}

convenience : it

_happens

to lead to

_{simple Laplace}

transform

_properties

which

we

_exploit

below. ;

The

_remaining

term in

₍₁₎

is

W,

the interaction energy of the chains. It includes the

Flory-Huggins

lattice

_expression

_[29]

_{for the free energy of}

_{mixing F mix}

_per

_site,

which for

_long

chains reduces to

where 0

is the local monomer

_density

and

_{x is}

as usual

_Flory’s

interaction

_parameter.

In

_(3),

0 (z)

is defined in a self-consistent way as

From

₍₃₎

we have

The

_change

in the

_{configuration}

sum

₍₁₎

on addition of a test

_{chain j}

_{say, is}

_{given by}

where the «

_{action » Sj}

is

and we introduce an « effective

potential

» V

(Zj)

which is the

negative

of the space

dependent

monomer chemical

_potential.

(The

_change

in

_sign

is introduced so that

₍₇₎

looks like the

Lagrangian

of a

_particle

in the

_potential

V.)

2.1 THE CLASSICAL LIMIT TO THE SELF-CONSISTENT EQUATIONS. - When the brush is

sufficiently

dense that the chains are stretched far

_beyond

their

_{unperturbed gyration}

_radii,

the

_only

_path

which contributes

_{significantly}

to

₍₆₎

for a

_{chain j}

is the one which minimizes the

action Sj

at fixed

_positions

for the chain ends. In the

_long-chain

limit,

perturbations

about this

stationary path

can be

_ignored

in

_{(7). (These}

results of reference

_[15]

are unaffected

_by

our

adoption

of more

_general

_expressions

for the

_equation

of state and the

_stretching

_energy).

From

_{(1), (2)}

and

_(5),

_{the total free energy}

_F(K)

of the brush is

where each of the chains

_separately

minimizes

_(7).

_Suppose

that in

_equilibrium

the free ends of the brush are at

_{height Ç ;}

for

i = 1, 2, - .., K.

Then the

_change

_{in free energy}

AFi

on

_adding

a

_single

chain i while

_keeping

all others fixed in space is

The minimisation condition of

₍₈₎

is

_simply

the set of

_{Euler-Lagrange equations}

of the

Lagrangian

of the chains

_{given by}

which is a sum over

_single

chain

_{Lagrangians L;.}

Hence each chain i satisfies

(dropping

the

suffix

_i)

where we have

_(for

future

_{convçnience)}

_decomposed

the effective

_potential

_V(z)

as

In this

_expression,

_{A (h )}

is a constant _{for fixed coverage u (hence} fixed

_{equilibrium height}

_h)

and is chosen so that

and

2.2 THE EQUAL TIME REQUIREMENT. - The results of the

_preceeding

section

_specifically

relate to our choice of

₍₂₎

for the modified

_stretching

_{energy but} are

_independent

of the form

of the

_equation

of state. As was found

_by

MWC for the case of a Gaussian

_stretching

_energy, the

_problem

of

_calculating

V

_{(z )}

_{simplifies dramatically}

if one assumes that the

_density

of free chain ends is nonzero

_throughout

the brush. Their

_proof

_[15]

that this is indeed true for a

Gaussian

_stretching

_{energy carries} over almost

_unchanged

to the

_present

case

_(see

_Appendix

B).

Thus the

_limiting

case of a

_step

function

_profile

at _coverage

_unity

_(for

which the all chain ends are at the front of the

_brush)

is

_approached

via a _{sequence of}

_end-density

distributions

that remain nonzero

_everywhere.

Bearing

in mind that the

_trajectory

of a chain is found

_{by minimizing}

the action

Sj

(7),

we note that

_finding

the

_{configuration}

of a chain whose end

_happens

to lie at

_{height C}

above the

_grafting

surface is the same as that of

_finding

the

_trajectory

of a

_particle,

with

(7)

as

its

_{Lagrangian, starting}

from rest at this

_position.

_(The

initial

_velocity

is zero since there is no

external force on the chain

_end).

Since all chains have the same

_length

_N,

our task is to find the form of the

_potential

_{V (z )}

such that a

_{particle starting anywhere}

within the brush will reach the wall in

_equal

time,

N.

For a Gaussian

_stretching

_{energy, the}

_equal

time

_potential

is

_simply

a

_parabola

(V

= - A +

BZ 2)

_[15].

The

_problem

of

_finding

the

_potential

for a

_particle

with

_Lagrangian

given by

(7)

requires

a more detailed

_analysis,

which we now

_give.

Since the

_single

chain

_Lagrangian

₍₁₁₎

does not

_{explicitly depend}

on « time _{», n,} there exists

a constant of the motion H

_[32]

such that

for the

_velocity

i of the

_particle

at z, where we have defined

_{à U =- U (e) - U (z ).}

With these

definitions the

_equal

time constraint becomes

To calculate our

_potential

_U(z),

it is convenient to use U itself as the

_integration

variable

[17] ;

equation

(17a) may then be written

where

_{f (U’ )}

=

dz/dU’

is now the unknown function to be determined. For DU small

₍₁₇₎

reduces to

which is the case of a _{Gaussian stretch energy ; the}

_analysis

of MWC then follows in this limit and the

_{potential approaches}

a

_parabola.

_(Note

also from

₍₁₆₎

that as àU- 00,

Ii

1

-

1,

which

imposes

the

required

maximum

extensibility

fo the

chains).

2.3 FORM OF THE EFFECTIVE POTENTIAL. -

Taking

the

_Laplace

transform of

_(17b)

and

inverting yields

[33]

where a =

B/2/7ry.

Using

the

boundary

conditions

(14)

gives

The inverse of this function

_(which

can

_easily

be found

numerically)

is the

_equal

time

_potential

U(z)

for a

_particle

with the

_Lagrangian

of

_(7).

_Figure

1 shows a

_plot

of

U(z)

versus distance in

units

_z/N.

With

₍₁₃₎

this

_specifies

the effective monomer

_potential

_{V (z).}

For

_{z « N,}

the

parabolic

form is recovered as

_expected,

whereas

_U(z)

_diverges

at z = N.

To

get

from the monomer

_potential

to a

_{density profile,}

one must invoke an

_equation

of

state ; in the

_{Flory-Huggins}

_{approximation}

_{(using (3), (10)}

and

₍₁₃₎₎

we obtain

By equating

this to

_U(z)

_obeying

_(19),

the

_{density profile}

_{cp (z) may}

now be constructed for

various values of the

_parameter

A

_{(h ).}

This

_generates

a

_family

of

_{profiles 0}

(z )

_{corresponding}

to different brush

_heights.

The

_{corresponding}

values of the coverage cr may be found

afterwards

_{by integrating}

the

_{density profile.}

These

_procedures

are

_{implemented numerically,}

but this is

_{fairly straightforward}

in view of our closed-form result for

_{z(U). (We}

should note that the existence of a

_{relatively simple}

expression

for this function is what motivated our

_particular

choice for the

_stretching

term

Ti

in

_(2).

The

_{corresponding}

result for a « relativistic

particle

»

Lagrangian

is discussed in

Fig.

1. -

Equal

time

_potential

_{U(z )}

from

_equation

₍₁₉₎

versus

_height

in units

_z/N.

2.4 DENSITY PROFILES : ATHERMAL

_(X

=

0)

_CASE. - The athermal _case has _a

particularly

simple

form

_{for 0 (z).}

From

₍₂₀₎

_{putting X}

= 0

gives

for the local _monomer

density

The constant

_{A (h )}

is fixed

by

the conservation of

_material,

which

_requires

We now _{argue that the}

_equilibrium

brush

_corresponds

to one in which the monomer

_density

vanishes

_smoothly,

rather than

_{discontinuously,}

at the front of the brush. This was shown for

the

_{parabolic profile by}

MWC

_{[15] ;}

their

_argument

_generalizes

in a

_{straightforward}

manner so

_long

as

_{F,,,i,, (0 )10}

is a

_{monotonically}

_decreasing

function of

_0.

This holds

_throughout

the

good

solvent

_regime

_{(X _ 1/2) but fails in poor solvents}

as we will discuss in section 2.7.

Since

_{for y}

= 0 there is no

_step

_{discontinuity}

at the front of the

_brush,

we _may set

0 (h)

= 0 and thus determine

A (h )

to

obey

Equations

(21)

and

₍₂₃₎

determines the brush

_{profile :}

₍₂₁₎

_gives

the

_profile

as a function of

the

_parameter

A

_{(h ),}

and

₍₂₃₎

fixes this

_parameter

in terms of

_{coverage o- corresponding}

to an

equilibrium height

h.

The

_dependence

of h

and 0

(0)

for y = 1 on u is shown in

figure

2. At small these show

characteristic u 1/3 and u 2/3

behaviours

_{respectively,}

which were

_predicted

in

_previous

treatments

_[4,

7,

15].

As coverage

approaches

saturation,

h becomes almost linear

in 0-;

in this

limit,

the brush can

_crudely

be

_thought

of as a slab of material at maximum

_density,

whose

height

therefore increases

_linearly

with coverage.

Figure

3a shows normalised

_{density profiles}

Fig.

2. - Monomer

density 0 (0)

and

_equilibrium

_height

_{hl N}

versus _{coverage a-}

_{represented by}

dotted and solid curves

_{respectively.}

’

Fig.

3.

_{- (a)}

Normalised monomer

_{density 0 (z)l 0 (0)}

_profiles

versus normalised

_height

_z/h

of the

present

theory

for _{various coverages r} = 0.025, 0.25, 0.50 and 0.70

corresponding

_to solid, dotted,

dashed and dotted-dashed curves

_{respectively.}

_(b)

End

_{probability density}

_e(z)

_profiles

versus

normalised

height

z/h

for coverages cr = 0.025, 0.25 and 0.50

represented by

solid, dotted and dashed

curves

_{respectively.}

The curves are scaled

by

Emax, the maximum value

of -- (z).

Corresponding

values of Emax for the three coverages are 6.7, 5.8 and 8.5

_{respectively.}

recovered at _{rather low coverages ( a} «

0.025 ) ;

this is not

surprising

since the corrections to

Gaussian

_elasticity

enter whenever

_h/N,

which varies

as 0’113,

is a

_significant

fraction of

_unity.

For

_higher

a- the

_profile

becomes

_gradually

flatter,

as

_expected.

Note that since the

elasticity

corrections are

_largest

for the most extended

chains,

the first

_{significant departure}

from the

2.5 DETERMINATION OF THE END DISTRIBUTION. - In this section

we consider how the chain ends are distributed

_through

the brush. As discussed

_already

in section 2.2 and

Appendix B,

we

_expect

a continuous

end-density,

which should however

_approach

a

delta-function at the front of the brush in the limit a - 1

_(when

all chains are

_fully

_stretched).

Define the

_probability

that a chain end in a brush of

_{height h}

lies

between ,

and

e

+ d’as

Now the monomer

_{density 4>}

_{(z )}

is

_simply

the sum of

_{contributions 1 dn/dzi}

₁

that each chain i

starting

at

_Ci

makes at

_position

z i.e.

Note

that,

as shown in reference

_[15],

chains

_starting

at £

- z do not contribute to

_(25).

In the

particle analogue

the term inside the modulus is

_simply

the

_{velocity. Using}

₍₁₆₎

and

transforming

to

_potential

U as the

_integration

variable

₍₂₅₎

_becomes,

with

_Uo

=

U(h)

where 0

(U) _

0

(z(U))

and

_{e (U ; Uo)}

denotes the end

_density

in

_{« potential}

_{space » [17].}

Inverting

(26)

for

_{Z (U ; Uo )}

_yields

the end

_probability

distribution as

To obtain the

_end-density

_{in real space} we _{may write}

_{£(z ; h) = ë(U; Uo)(dU /dz)}

where

dz /d U

obeys

(18).

The

_{resulting end-density}

distributions for the athermal brush are shown in

figure

3b for various _coverages. As with the monomer

_{density profile,}

the

_asymptotic

form

e (z ;

A) =

(31h 3)

z (h 2 -

Z 2)1/2

predicted by

MWC is recovered

_only

at _very low _coverages.

For o- > 0.1 the curves show

_{pronounced upward}

curvature and an

_{emphatic peak}

near the front of the brush. For

_example,

with .-, 0.5

roughly

45 % of the ends lie in the front 10 % of

the

brush ;

the

_{corresponding figure}

at _{low coverages, u = 0.025 is 12 %.}

In

_figures

4a-d we _{compare the results of this and the}

_previous

section with those found from a numerical mean-field

_approach

based on

_{Scheutjens-Fleer theory, using}

chains of 100

steps

on a

_simple

cubic lattice in three dimensions and a

_{Flory-Huggins}

_equation

of state.

_(The

lattice data was

_kindly

_{provided by}

Milner

_[34]).

To make a

_quantitative

_comparison,

we must

select a value of

y in

equation

(2). This may be done

by

comparing

the

_equilibrium

end-to-end

distance of a chain with its

_fully

stretched

_length.

The latter is

_equal

to N on a unit cubic lattice

in any

dimension,

presuming

the direction of extension to coincide with a cubic axis. Thus to

reproduce

the well known result

_{(R2) =}

N for the end-to-end distance we must set

y ==

d,

the dimension of space. In

figure

4 we

_present

curves for y = 3 and also for

y = 1

(as

would be

appropriate

for a

_strictly

one dimensional

theory).

We see that for a _{low coverage, cr} =

0.05,

the _y = 3

profile

_{compares well with the}

numerical result whereas for

_higher

_coverages our results based in

equation

(2)

give

a

Fig.

4. -

(a)-(b) Monomer density 0 (z)

and end

_{probability density c(z)}

_profiles

versus

_height

z/N

respectively

for a = 0.05. The solid and dashed _curves

correspond

to

_{dimension-dependent}

parameter y = 3 and

y = 1

_{respectively.}

The dotted curve is the lattice mean field result based on the

Scheutjens-Fleer theory.

(Data

for the lattice mean field are for 100 monomers _per chain

kindly

provided by

Milner). (c)-(d) Monomer density 0 (z)

and end

probability

density c(z)

profiles

versus

height

z/N

respectively

for o- = 0.25.

agreement

in this case. These differences

give

an indication of the

_degree

of

_{nonuniversality}

for

_{strong-stretching}

corrections to the Gaussian elastic limit. There is no reason to

_prefer

one

type

of calculation over the

other,

since the use of a lattice is at least as

_arbitrary

as the introduction of

_equation

_{(2). (Indeed,}

_{large changes}

in the

_profiles

at

_high

_{coverages would}

coverages close to

_unity,

where the

_profile

_approaches

a

_{step-function}

(see e.g.,

_Fig.

3).

The

corresponding

calculations

_using

the lattice SCF

_approach

are

_extremely

slow to _converge

[34],

because of the

_high

_{osmotic pressures} near the

wall,

and the

_{sharply peaked}

end distribution

_{[35]. (The}

_{practical limit}

for these calculations seems to be near cr = 1/2

[34]).

2.6 FREE ENERGY OF FREE AND COMPRESSED BRUSHES. - It is shown in

appendix

B that in

equilibrium

the

_{action Si}

₍₇₎

_takes,

_{for fixed coverage, the} same numerical value for all chains

in the brush. The free energy of the brush may then be obtained as follows.

Selecting

a chain near the

surface,

we

_get

Constructing

the brush

_{by progressively adding}

_chains,

the _{free energy,}

_F,,

for some fixed

a,

_{corresponding}

to an

_{equilibrium height}

h is

where

_{A (h)}

satisfies

_{(20). For illustrative purposes,}

we restrict attention to the athermal case

x = 0.

Using

(23)

and

(29)

_we obtain

h

where

_{g (h )}

=

dz exp

U (z).

The non-athermal case _{may be treated}

_using

similar argu-0

ments.

We now consider the case of a brush

_{compressed against}

a hard

_wall,

or

_{equivalently,}

another brush

_{[15]. Clearly}

we can

_expect

_repulsive

forces on

_compression

to a

_height

z -

h,

h

being

the

equilibrium height

for some _coverage,

_{o, (h).}

The

_integration

in

₍₃₀₎

is now

performed

in two

_steps.

_Firstly

_{the brush is built up until the}

_equilibrium

_height

is

z(/ï), say,

corresponding

to

_{some o (z) ;}

from

₍₃₀₎

we have

The second

_step

of the

_integration

is to fix the

_height

z and _{increase the coverage from}

u(z)

to

_u(h).

Therefore we have

The free energy of the

compressed

brush is

We obtain the

_compression

force for z -- h in the athermal _{case X} = 0 as

Fig.

5. - Force

77 (z) required

to _compress an athermal brush to a

_height

_z/h

«-- 1 for coverages

a = 0.025, 0.25 and 0.50

represented by

the solid, dotted and dashed curves

_{respectively.}

2.7 OTHER SOLVENT CONDITIONS. - The case

_{of X}

finite and less than 1/2 defines the

_good

solvent

_regime.

The mean field

_approach

is valid for

sufficiently high

o- and the non-Gaussian

stretching

energy is

important.

The results for

the y

= 0 _case, as

_given

_above,

are

representative.

More

_generally,

from

₍₂₀₎

we

_get

Equation

(33) may be solved

numerically by fixing

h « N.

_Using

_(19),

DU is known for all z,

hence the

_{density profile}

_4>

_(z)

can be found

_by

_solving

the transcendental

_equation

(33).

The

surface coverage a can be inferred from

(22).

One

_special

case,

_namely

behaviour in a theta solvent

_(X

=

1/2)

at low coverage may be

obtained

_{by replacing}

the

_{Flory-Huggins}

_equation

of state with a three

_{body potential}

(V

oc

02) .

This

_gives

an

_{elliptical profile}

rather than a

_parabolic

one with vertical

_asymptote

at the front of the brush.

The bad solvent case

_(X

>

1/2)

is more subtle. In this

_regime

there is a

_{discontinuity}

at the

front of the

brush,

which means that

₍₂₃₎

is no

_longer

valid. To see

_why

this

arises,

we recall

[29]

that the

_equation

of state

_{F mix ( cf> )}

_{given by}

₍₃₎

for such a solvent exhibits an

_instability.

There is an

_optimal

volume

fraction 0

* _{for which}

F.àl 0

is minimized :

A bulk

_polymer

solution

at 0 -- 0

* _will

phase

separate

into a state

_{of 4> = 0}

*

coexisting

with

a state

_{of 0 =}

0.

_(This

holds in the limit N --+ oo, which has

already

been taken in

_(3)).

Clearly

then,

a brush

_profile

which vanishes

_continuously

at the front

_edge

is

_locally

unstable in a bad solvent

(X :::. 1/2 ) :

both the osmotic term and the

_stretching

_{energy favour}

which

_corresponds

to a local

_{density 0}

=

cP *.

(This may be checked

by

considering

the force on a

_confining

_plate

_{pressed against}

the brush : this is

_equal

to the osmotic pressure at the front of the brush

_[15]

and must vanish in

_{equilibrium).}

Using

(20)

this redefines the

_boundary

condition for

_{A (h )}

as

which

_yields

for the brush

_profile

Fig.

6. -

(a)-(c) Effects

of solvent

_quality

on the brush

_density

_{profile cf> (z)}

versus

_height

z/N

for coverages u = 0.025, 0.25 and 0.50

respectively.

The solid, dashed and dotted _curves

correspond

to

_Flory

interaction _{parameter X} = 0.75, 0.50 and 0.10

The effects of

_varying

solvent

_quality

for various fixed are shown in

_figure

6a-c. In

particular

we note that

_{for y =}

_î/2

_(theta

solvent

_case)

the

_profile

_goes

_continuously

to zero

with vertical

_slope

at the front _{of the brush for the three coverages. At the} lowest _coverage, an

elliptical profile

is

_approached

as

_predicted

above.

_{For X}

1/2 the brush as

_expected

is

relatively

compact

(since

chains attract each

_other)

_although

the brush

_height

remains

proportional

to the chain

_length

N

_[36].

This ensures that the chains are

_strongly

stretched

even in a _poor

_solvent,

so that results based on the classical limit of the SCF

_equations

remain

reliable.

3. Conclusions.

In this paper we have

_generalized

and extended the

_analytical

SCF

_approach

to

_polymer

brushes as introduced

_by

Milner et al. in reference

_[15].

The more

_complicated

chain

Lagrangian

studied here lead to a somewhat less tractable

_analytical

structure, which means

that some numerical work is

_required

to obtain

_profiles

from the

_theory.

Nonetheless our

approach,

which starts from the

_strong

_stretching

_(«

classical

_»)

limit of the SCF

_equations,

is

quite

different in

_spirit

from one

_involving

the full numerical solution of the self-consistent

problem

as

_{typified by}

the

_{Scheutjens-Fleer approach}

[6,

_27,

34].

The main aim of our work is

to allow

_higher

_{surface coverages and chain densities} to be described within the MWC formalism. Our results reduce to those of reference

_[15]

at low

_enough

_{surface coverages, and}

interpolate

all _{the way} to the

_{step-function profile}

of a

_completely

saturated surface. One

might

expect

that in this

limit,

some of our results for force

_profiles

could be obtained

_by

a

suitable modification to _{the energy balance}

_arguments

of Alexander

_[5],

which were based on a

_{step-function profile.}

_However,

this turns out to be difficult since the

_{Flory-Huggins}

interaction becomes a _very

_strong

function of

_density,

and small

_departures

from a flat

_profile

become very

important energetically.

By

decreasing

solvent

_quality

we _{may also}

_approach

the

limit of a brush whose

_density

is constrained to be constant, as arises in surfactant

_bilayers,

for

example

[11,

36,

37].

To obtain reasonable results in

_high

_coverage

_regimes

it is essential to include not

_only

a

realistic

_equation

of state, but also corrections to the Gaussian

_stretching

energy used

by

MWC. A drawback of our work is that the nonlinear

_stretching

_{energy chosen}

_{(Eq. (2))}

and the

_{Flory-Huggins equation}

of state both involve ad-hoc

_{approximations}

of restricted

universality.

In

_principle

it would be

_possible

to include more

_precise

forms in each case,

although

the mathematics

_rapidly

becomes more

_complicated

_(see,

_e.g.,

_{Appendix A).}

A

measure of the « error bar » inherent in our work is seen in

_figure

4 where there are

_significant

departures

between our

_predictions

and those of

_{Scheutjens/Fleer}

_type

calculations

_[34]

at

coverages of order o- = 0.25. We

_expect

the main difference between the lattice calculations

and our own to relate to the choice of

_stretching

term in

_equation

_(2),

and while it would

certainly

be

_interesting

to

_implement

our SCF calculations for several different choices of this

term

_{[30, 31],}

that lies

_beyond

_{the scope of the}

_present

work. We

_emphasize

_{in any} case that

lattice-based models have no more claim to fundamental

_validity

than our own

_equation

_(2),

and indeed that the results for different lattices vary

significantly

[28, 34].

A

_{major advantage}

of our

_approach

is that results can

_easily

be obtained for coverages

_right

_up to

_unity,

in

contrast to _{the convergence} difficulties encountered

_by

lattice calculations for coverages

larger

than about 1/2 in the athermal case

_{[34, 35].}

Even

greater

advantages

are

_expected

in

poor solvent conditions.

Finally

we

_point

out that the

_original

MWC

_theory,

which involves a Gaussian

_stretching

term and a

_{quadratic equation}

of state

_(F

_{(0 ) - 0 2)@}

_yields

brush

_heights

that are a

significant

fraction of the

_fully

extended

_height

even at _{very low coverages. (For}

_{example u}

= 0.05

yields

0-1/3

for the brush

_height),

there is at best a limited

_regime

in which the

density

is

high enough

for a mean-field

_approach

to be

_suitable,

but the brush

_height

is small

_enough

for

non-Gaussian corrections to the

_stretching

term to be

_negligible.

In this sense our extension of the

theory

to «

high

» _{coverages is relevant} to _{coverages that would}

_normally

be considered rather small. Note also that the main influence of the non-Gaussian corrections at low o- is to make the

_{density profile}

vanish more

_steeply

at the front of the brush than would otherwise be the

case, which

_give

a relative

_{« hardening »}

of the force/distance

_profiles

for

_compressed

brushes. For force/distance curves, the earlier MWC

theory

is at the

_stage

of

_(almost)

parameter-free

fits to

_experiment

in certain

_systems

at _{very low coverage [22, 38]. We may}

therefore

_hope

for

_{quantitative experimental}

tests for some of our new results in the near

future.

Acknowledgments.

We are

_{especially grateful}

to Scott Milner for

_providing

the lattice SCF data

_[34]

shown in

figure

4. We also thank Scott

Milner,

Jan

_Scheutjens,

Tom Witten and Matthew Turner for

helpful

discussions. DFKS is

_grateful

to

_{Trinity College, Cambridge}

for financial

_support.

Appendix

A.

It is

_interesting

to _{compare the form of the kemel (16) (which} results from

_adopting

equation

(2)

as the

_stretching

energy)

to that of a relativistic

_particle.

The

_Lagrangian

for such a

_particle

_(with

mc2 =

₁₎

in one dimension acted on

_by

conservative forces

_independent

of

i is the familiar

_[31]

from which the total energy of the

particle

is

Following

(17)

we

_get

Taking Laplace

Transforms

_gives

an

_{algebraic equation}

which on inversion

_yields

where C is the

_Laplace

_operator

and

_K,

is the modified Bessels function of the second kind of order one.

This form of

_{f (U)}

is less tractable than

₍₁₈₎

which is our main reason for

_adopting

equation

(2)

for the

_stretching

energy rather than T

= - .J 1 - i2

as

_might

be

_{suggested by}

Appendix

B.

In this

appendix

we

_give

_arguments,

_directly

_generalizing

those of reference

_[15],

which establish that the end

_density

in a brush is

_everywhere

nonzero and that the action of every

chain in the brush is the same.

We start

_{by establishing}

the local

_stability

of a brush whose chains

_obey

the classical

equations

of motion

_arising

from the action

_(7).

For the brush to be

stable,

the minimum of the

_{action Sj}

from

₍₇₎

should be

_stationary

with

_respect

to

_changes

to the end

_{position :}

Using

the

_single

chain

_Lagrangian

₍₇₎

and

₍₁₂₎

we

_get

(dropping

the

_suffix)

At the lower

_limit,

aL

₍₀₎

= 0 since

i(O)

= 0

(there

is _no force on a free chain

end),

and at

ai

the upper

limit,

az

(N )

= 0 since all

particles

terminate at the wall. Thus

S(e)

is indeed

96

stationary

with

_respect

to small

_changes

in the

_position

of a chain end.

This

_implies

that

_{S (C )}

is constant _{in any interval}

_{of (where}

chain-ends are found. The

action S is thus the same _{for every chain}

_(as

assumed in Sect.

_2.6)

so

_long

as the

_end-density

is

everywhere

nonzero. To establish

this,

let us consider a

_hypothetical

« dead zone »

[15]

in

which no ends are

_present.

From

_equation

_(25),

and

_using

the _{conservation of energy}

(Eq. 15)),

we find that a self-consistent solution for the

_density

in such a dead zone is one in

which

the 0

is a constant

_throughout

the zone. Consider now a chain whose free end is at the

outer

_edge

front of the zone. If we

_bring

this free end

_slightly

towards the wall

(so

that it lies

within the dead

_zone),

then S for that chain

_decreases,

since

_stretching

_{energy is released} at no

cost in _{osmotic free energy.}

_{Alternatively,}

we may observe that such a chain must have an

initial

_velocity

toward the wall so as to arrive there in

_equal

time to a

_{particle starting}

from the inner

_edge

of the zone. From the

_expression

for

as

derived above

_(and

_using

also

₍₁₂₎₎

this

os

inward initial

_velocity

_(negative)

_implies

a

_positive

value of

as ,

so that

_moving

the chain end towards the wall lowers the free energy of the

system.

This shows that dead zones are

_locally

unstable,

and it follows that the end

_density

is nonzero

_throughout

the brush.

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