Forces between asymmetric polymer brushes

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Forces between asymmetric polymer brushes

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

To cite this version:

Short Communication

Forces between

_{asymmetric polymer}

brushes

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

Theory

of Condensed

Matter,

Cavendish

_Laboratory,

_Madingley

Road,

Cambridge

CB3

OHE,

G.B.

(Reçu

le

_{25 janvier}

1990,

accepté

le

_{6 février}

₁₉₉₀₎

Abstract. 2014 We

study

the

_{equilibrium compression}

of

_{asymmetric polymer}

brushes

_grafted

on flat

plates,

under athermal and theta solvent

conditions,

using

a lattice self-consistent field

_(SCF)

ap-proach.

We find that the

_{separation d}

between two

_plates

coated

_{asymmetrically}

with brushes of type

1 and

2,

as a function of the force

_{F, obeys}

the "bisection

_rule",

_{d(F) = (d1(F)}

+

_{d2(F)) /2}

where

d1(F)and d2(F)

are the

_{corresponding separations}

for the

_symmetric

brushes of _type 1 and 2

re-spectively.The

bisection rule is

_{easily esplained}

if one assumes no

_{interpenetration}

of the two brushes.

However, a

_study

of the

_polymer

volume fraction

_profiles

_suggests that the bisection rule is well

_obeyed

even when there is _strong

_{interpenetration.}

The relevance of this work to recent

_experiments

on

asym-metric brushes is discussed. Classification

Physics

Abstracts 36.20 - 61.40 k

1. Introduction.

The

_physics

of end-absorbed

_{polymer layers (grafted polymers)}

finds

_application

in fields such as

tribology,

adhesion,

pharmaceuticals,

waste-water treatments etc.. Grafted

_polymers

on colloidal

particles provide

an efficient _way of

_controlling

the

_stabi2ity

and other

_{physical properties}

of col-loidal

_suspensions

_[1].

_Knowledge

of the interaction between such

_particles

coated with

_polymer

chains is of interest in the context of

_coagulation

and structure formation in

_disperse

_systems

[2 - 3].

In this short communication we

_give

the first detailed theoretical

_analysis

of the force

between

_asymmetric

_{grafted polymer layers.}

We

_study

both the force

_profiles

and the

_degree

of

interpenetration

of the two brushes.

In the so-called "classical" limit

_[4]

of

_{infinitely long}

chains,

_polymers

in a brush are

_strongly

stretched and even when forced

_together

do not

_{interpenetrate}

_{[4 - 5].}

This

_gives

rise to the

"bisection rule":

which states that the

_separation

d of

_plates

coated with

_asymmetric

brushes

_{corresponding}

to

somme force F is the _average of

_separations

of

_symmetric

brushes of

_type

1 and 2 with the same

force. the results

_presented

below of our lattice self-consistent field

_(SCF)

calculation show that this rule is well

_obeyed,

not

_only

for

_long

chains,

but also for

_relatively

short chains and when there is

_{interpenetration}

between the

brushes;

however in this case the

_physical

_{interpretation}

of

702

the rule is different.

_{Preliminary experimental}

data of watanabe and Tirrell

_[8]

_appear

to indicate

departures

from the bisection rule for certain _{combinations of surface coverage} and chain

_lengths

in the two

_opposing

brushes. This contrasts with the calculations

_presented

here and the

_origin

of

any

experimental departure

therefore remains

_{unexplained by}

the

_present

_theory.

2. Self-consistent field

_theory.

Numerical methods based on the lattice self-consistent field

_{equations developed by}

_Scheutjens

and Fleer

_(SF)

have

_previously

been

_applied

to

_{grafted polymer}

brushes

_{[6 - 7].}

We refer to these

papers

and references therein for détails of the method.

Essentially

the

_object

is to determine the Boltzmann

_segment

_weighting

factors in a self-consistent

potential

field. Thèse

_weights

are then combined to form the statistical

_weights

of chains in their

various conformations

_subject

to the

_boundary

condition that each chain is attached to a wall. The

Flory-Huggins

approximation

is

_applied,

which

_averages

out chain concentration fluctuations in each

_layer.

The result is a cheminal

_potential

field

_{perpendicular}

to the

_grafting

surface. The

volume fractions of the various

_segment

_types

are determined

_by

this

_potential

field which

de-pends self-consistently

on local concentrations

_according

to the

_{Flory-Huggins equation}

of state

[1,6- ].

In order to obtain

_{force/séparation profiles}

we

_require

an

_expression

for the free _energy of a

_system

of two

_{opposed plates}

coated with

_polymers.

We used the

_expression

obtained

_by

Van Lent et al.

[7]for

the excess free _energy

_per

site in units of

kBTfor

a

_system

of

_grafted

and free

_components

in

_equilibrium,

where

kB

is the Boltzmann constant and T the absolute

_temperature.

3. Results.

All the results below are for

_{non-adsorbing polymers}

of the same

_{species grafted}

to ftat surfaces in two

_opposed

_layers.

_Figures

la and b are

_{log-log plots}

of

_{force/separation profiles}

for the

compression

of

_{grafted symmetric}

and

_asymmetric

brushes in an athermal solvent. The notation

N(O")

indicates chains

_having

N

_segments

and

_coverage

0". From

_figure

la we observé that the bisection rule is well

_obeyed

for the

_asymmetric

brushes

_having

_{Nlong/ Nshort}

= 2 and

uiong lashon

=

1

_{(case A).}

The

_{force/separation}

_profile

lies

_midway

between the

short/short

and

_{long/long profiles.}

The dots are bisection rule

_values,

i.e.

_{d(F) = (dl(F)}

+

_d2(F))

/2,

with

_dl(F)

and

_d2(F)

those found for the

short/short

and

_long/long

cases.

Figure

Ib has

_{Nlong/ Nshort =}

2 and

_aionglashon

=

1/2

for the

asymmetric

brushes and hence

the same

_grafted

amount on each

_{plate (case B).}

The

_{force/separation profile}

for the

_asymmetric

brushes

_again

follow the bisected values

_(dots).

At small

_enough

_{separations, roughly}

_speakipg

less than

Ils,

where

_Rs

is the sum of the radü of

_gyration

of a free chain from each side of the

_plate,

the three curves for case B

_converge.

This is

_expected

since the force is then

_purely

osmotic in

origin

and

_{depends only}

on the total

_grafted

amount.

_Figures

_lc,d

are for the same sets of brushes

(case

A and case

_B)

_respectively

but in a theta solvent. The brushes are now less extended from the

surface,

but the bisection rule is still

_obeyed

for both cases. The above

calculations

were

repeated

for a

_range

of different values of N and u and we observed no

_significant

deviation from

the

_predictions

based on the bisection rule.

Figures

2a ,

c show the

_polymer

volume fraction

_profiles

for

_symmetric

brushes

_{20(0.05)/20(0.05)}

Fig.

1. -

(a)

Log-log plot

of force of

_compression

_F(d)

vs.

_separation

d of

_plates

for brushes

_20(0.05)/20

(0.05), 20(0.05)/40(0.05)

and

_{40(0.05)/40(0.05)}

in athermal

_solvent.(b)

_log(F(d))

vs.

_log(d)

for brushes

20(0.05) /20(0.05), 20(0.05)/40(0.025)

and

_{40(0.025)/40(0.025)}

in athermal

_{solvent.(c) Log{F(d))vs. log{d)}

for brushes

_{20(0.05)/20(0.05), 20(0.05)/40(0.05)}

and

_{40(0.05)/40(0.05)}

in theta solvent.

_(d)

_Log(F(d))

vs.

log(d)

for brushes

_{20(0.05)/20(0.05), 20(0.05)/40(0.025)}

and

_{40(0.025)/40(0.025)}

in theta solvent.

40(0.025)/40(0.025)

respectively

at

_separation

d = 8. We make the

following

observations. As d

decreases,

the

_degree

of

_{interpenetration}

increases. All of these brushes are

_quite

far from the

classical limit and

_{interpenetration}

effects are

_obviously

_strong.

When the

_separation

is less than

RS,

the total volume fraction of the

_polymer

become almost

_homogeneous;

from

_figures

_2a,d

and e there is for d = 8 a near-uniform total

_density

_Ot.t.,

N 0.275 in each _case. In this

region

we

expect

that the force becomes insensitive to chain

_length

and _coverage, and even to the fact the chains are

_{grafted, depending}

instead

_only

on the total amount of

_polymers

between the

_plates.

For

_{larger separations (d}

>

_Rs),

the

_dependence

of the

_profile

on the

_{separate parameters}

704

Fig.

2. -

(a,b,c)

Volume fraction

_profiles

for

_symmetric

brushes

_{20(0.05)/20(0.05)}

in athermal

_{(solid curves)}

solvent and theta

_{(dashed curves)}

solvent conditions for

_{separations d}

=

8,

12 and 16

respectively.

(d,e)

7b

_quantify

the

_{interpenetration}

between

_opposing

brushes,

we introduce the

_parameter

_{3

at

sep-aration

_d,

defined as

where

_4>l(Z)

is the volume fraction in

_{layer z}

of brush 1 and

₀₁

=

Ez d

1 (Z)

its total

grafted

amount;

_similarly

for brush 2. We motivate this definition of

₍₃

_by

_imagining

a situation with the

two

_components

1 and 2

_uniformly

distributed

_throughout

the volume

_{(irrespective}

of whether this

_corresponds

to the minimum of free

_energy).

This

_gives

an _average volume fractions

_()1/d

and

_{02/d in}

each

_layer

and #

is defined to be

_unity

for this limit of total

_{interpenetration.}

Figures

3a,b

show

_plots

of

_#(d)

vs.d for cases A and B

_respectively

under athermal

_{(solid curves)}

and theta solvent

_{(dashed curves)}

conditions. We note that the

_#

_parameter

increases

mono-tonically

with

_{decreasing separation}

d

_indicating

a

_greater

_degree

of

_{interpenetration.}

For

_good

solvents the behaviour of

_,0

at onset

_{(large d)}

is

_roughly

_cubic,

as would be

_predicted

from the

classical limit

_picture

_[4]

were one to

_{simply overlay}

the

_profiles

from each brush

_(since

in that limit the

_density

of the brush rises

_linearly

from sero at the

_tip).

Of course this is not consistent with the fact that brushes in this limit should

_hardly

_{interpenetrate}

at all

_[4],

but the observation

may be useful for

_estimating

_{qualitatively}

the behaviour of

_relatively

short-chain brushes such as

the ones considered here. For small

_enough

_{separations (d}

_Rs)the

_#

curves for the athermal and theta brushes _converge, i.e. the athermal and theta volume fraction

_{profiles collapse}

onto

each other as the effect of solvent

_quality

become less

_{significant. (This}

is

_already

evident from

figures

2a,d

and e for

_example

at

_separation

d =

_8.)

Fig.

3. -

(a)

Plot of

_{interpenetration}

_parameter

_(3(d)

_vs.

_separation

d of

_plates

for brushes

_20(0.05)/20

(0.05), 20(0.05)/40(0.05)

and

_{40(0.05)/40(0.05)}

in

athermal’

_{(solid curves)}

solvent and thêta

_{(dashed curves)}

solvent conditions.

_(b)

Plot of

_{interpenetration}

_parameter

_0(d)

vs.

_{separation d}

of

_plates

for brushes

_20(0.05)

/20(0.05),

20(0.05)/40(0.025)

and

_{40(0.025)/40(0.025)}

in athermal

_{(solid curves)}

solvent and theta

_(dashed

706

The effects of

_{interpenetration}

are elucidated further in

_figure

_4,

which is a

_{log-log plot}

of the

force/separation

profile

for the

_compression

of

_symmetric

brushes

_{20(0.05)/20(0.05).}

The solid

curve is the

_profile

obtained when there is

_{interpenetration}

between the brushes while the dashed curve show the

_profile

if

_{interpenetration}

were not allowed to occur.

_(This

is the same as the force

required

to

_compress

a

_single

brush

_against

a hard

_wall.)

We see that for moderate

_separations

the force of

_compression

is

_considerably

smaller when

_{interpenetration}

is

_allowed,

_confirming

that in the non-classical

_regime

of short chains and low _coverages we cannot

_ignore

the effects of

interpenetration.

,

Fig.

4. -

Log(F(d))

vs.

_log(d)

for

_symmetric

brushes

_{20(0.05)/20(0.05)}

with

_{(solid curves)}

and without

(dashed

curves)

interpenetration.

4. Discussion.

As mentioned earlier the bisection-rule is

_{easily explained}

for the "classical limit" case of N oo, since in the

_asymmetric

brush the force F of

_compression

is determined

_by

_matching

the

discon-tinuity

in osmotic

_pressure

at the front of each brush

_[4];

the brush

_heights

_d/2

for each side of

the

_system

_may

be added to

_give

the total. In the other extreme of

_strong

_compression,

the force

between the

_plates

_may

be

_thought

of as that due to a

_{near-homogeneous}

concentration of

poly-mers, whether or not

_{interpenetration}

occurs. The fact that the force

_{depends only}

on the mean

density §

between the

_plates

leads

_directly

the bisection-rule for small

_enough

_separations

d >

_Rs,

as

_may

be seen from the fact that d

_{(§) =}

_(dl

_(§)

+

d2

_(§))

/2.

Our numerical results indicate

that,

between these two

_limits,

there is little

_scope

for

_{significant departures}

from the rule.

Because of the

_relatively

small number of

_segments

and low

_coverages

used in our numerical .

iterations,

the size of the brush is

_only

a few times the radius of

_gyration

of free

_polymer

in the

same solvent but still a

_significant

fraction of the

_fully

extended

_length.

This of course

_implies

that we are well within the non-classical

_régime

and so we should

_expect

the

_{interpenetration}

zone

(overlap regime)

to be a

_significant

fraction of the

_separation

of the

_plates

whenever d

_Rs.

This

is indeed observed. One would

_expect

these conditions to be

_optimal

for

_{observing departures}

from the bisection rule but all our results

_suggest

that even under these

_fairly

extreme conditions

the

_departures

are not

_significant.

’

Part of the motivation for this work comes from force-balance

_experiments

on

_{polymers grafted}

deviations in the

_{G/R profile}

for the

_asymmetric

brushes of case B towards the

short/short

sym-metric brush

_profile.

Here G is the force between the curved

_plates

and R is the radius of

curva-ture of the

_cylindrical

surface. Under the

_Derjaguin

_{approximation}

_[1]

the measured force

_G(d)

at

_separation

d is related to the

_energy

of

_{compression V (d)}

in a flat

_plate

_geometry

_according

to:

’Ib check for the bisection rule

_{experimentally}

one would in

_principle

have to do a numerical differentiation of

_{G( d)/27r R}

to obtain the

_{corresponding}

flat

_plate

_geometry

force

_F(d),

and then

apply

the bisection rule.

_{Alternatively}

for theoretical

_{predictions (such}

as those found

_analytically

using

the classical limit

_equations)

_[4]

one must first

_apply

the rule and then

_integrate

the

_resulting

F(d)

curve to

_get

_G(d).

In terms of

_fitting

_experimental

data,

either method is very cumbersome and it would be

_helpful

to find a

_simpler

_approach.

One

_suggestion

_[8]

is to construct

_{G(d) =}

_(GI

_(2di)

+

G2

(2d2))

/2,

and minimize over

_di

and

d2

subject

to constraint

dl + d2 =

d. For

_{non-interpenetrating (classical)}

brushes on curved

surfaces,

this

_corresponds

to a variational

_procedure

in which the

_boundary

of the

_opposing

_layers

is

pre-sumed

flat,

i.e.

_parallel

to the horizontal axis between the curved

surfaces,

and at a

_position

chosen so as to minimize the _{total energy. Consideration of} the same minimization

_{applied directly}

to the

potential V(d)

in the

_{corresponding}

flat

_{geometry suggests}

that this is exact in the classical limit. For the numerical curves we have

_generated,

we have found that this

_{procedure gives}

results for

V (d)

very close to those

_{actually computed,}

even in cases

_{of relatively}

_strong

_{interpenetration (for}

which the

_physical

_{interpretation}

is much less

_clear).

Thus the above construction for

_G(d)

does

appear

to

_provide

a reliable basis for the

_analysis

of

_experimental

data.

Acknowledgements.

Most of this work was carried out at

_{Wageningen University,}

The

Netherlands,

under the

_support

of E.S.R

_(A.A.

in

_Chemistry

and

_Physics

of

_Polymer

Surfaces and

_Interfaces)

and

_{’Ii-inity College,}

Cambridge.

We are

_grateful

to R.

_Israels,

B. van

_Lents

Barneveld and C.

_Wijmans

for use of their

original

computer

program

and assistance in its modifications. We also thank Jan

Scheutjens,

’Ibm

Witten,

Matt Tirrell and Pete Barker for

_helpful

discussions. We thank Matt Tirrell for

_showing

to us the

_experimental

results of reference

_{[8] prior}

to

_publication.

DFKS

_{gratefully acknowledges}

financial

_support

from

_{TMnity College, Cambridge.}

Part of this work was funded

_by

N.S.E

_(USA)

under

_grant

number PHY82-17853 at the

_University

of California in Santa Barbara.

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K.,

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_W.R.,

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