Polymer Mushrooms Compressed Under Curved Surfaces

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Polymer Mushrooms Compressed Under Curved Surfaces

D. Williams, F. Mackintosh

To cite this version:

D. Williams, F. Mackintosh. Polymer Mushrooms Compressed Under Curved Surfaces. Journal de

Physique II, EDP Sciences, 1995, 5 (9), pp.1407-1417. �10.1051/jp2:1995190�. �jpa-00248241�

Classification Physjcs Abstracts

36 20Cw 87 15-v 46 30Lx

Polymer Mushrooms Compressed Under Curved Surfaces

D.R.M. Williams _(~~~<*) and F.C. MacKintosh _(~~~)

(~)Institute

_of Advanced Studies, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, Australia

(~) Department of Physics, University of Mich~gan, Ann Arbor MI 48109-l120, USA

(~)Institute

_for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, USA

(Received

6 January 1995~ received in final form 6 May 1995, accepted 17 May 1995)

Abstract. We study the problem of _a single surface-tethered polymer chain _or "mushroom"

compressed by _a curved circularly symmetric obstacle, such _{as an} atoJn1c force _microscope tip,

m the presence of _a good solvent. In response ^to the compression the chain breaks _into

a series

of blobs For _an obstacle modeled _{as a} finite disk _we find _an escape transition and hysteresis,

as predicted in _previous studies. We also _examine compressions under _concave and _convex power-law shaped surfaces defined by h

+~ r", where h _is the distance of the obstacle from the

grafting surface and _{r is} the radial coordinate For _a <

1/2

the chain is effectively confined

to two dimensions For

o > I (i _e., all

convex

surfaces)

the chain is effectively unconfined.

Between these two limits the chain radius obeys _a scaling relation Rcha,n _'~ N~/~~"+~~. Finally,

we examine compression under

a surface with sinusoidal roughness In this _case there _can be

a

large

number of "escape" transitions.

1. Introduction

The deformation of end-tethered

polymer

coils is important in _a number of

applications,

_par-

ticularly

in steric stabilisation of colloids and

liposomes

_{[1, 2].}

Polymers grafted

to surfaces

can extend

short-range repulsions

between

particles

^and prevent

aggregation.

There _are, in

general,

two limits of

grafting.

If the

grafting

is

dense,

then _a "brush" forms [3, 4]. ^{In this} paper we are interested in the

opposite limit,

where the

grafting

is less dense and the

polymers

form individual "mushrooms" attached to the surface

[5-7]

The

compressional

behaviour of these mushrooms was first studied

by

Daoud and de Gennes [6], ^who

argued

that the system forms _a series of blobs.

They

considered

compression by

a flat surface of infinite _area where the blob size is the distance between the

grafting

surface and the

compressing

^obstacle. More recent studies have focussed _on

compression by

_a flat disk-like

particle

of finite size

[8,9].

In that _{case a} novel

"escape"

transition _was

predicted (Fig. I), whereby

part of the chain _escapes

(*) Author to whom correspondence should be sent. E-mail:

cuisses@calvino.physics.lsa.umlch.edu

© Les Editions de Physique 1995

<al

ibi

(cl

h(r)

~

Fig. 1.

a)

A grafted chain compressed under

a flat obstacle The chain breaks _up into _{a series} of blobs;

b)

A chain compressed _even further than _m

a)

_escapes. In doing _so it lowers the free energy; c)

A chain compressed under

a curved surface The blob size varies gradually _{m space}

from under the obstacle to avoid compression. In many

practical

situations the obstacle _com-

pressing

_a mushroom will _not be

planar.

This _can for instance _occur when the obstacle is _a curved tip of _an atomic force

miscroscope.

^Our aim here is to

generalise

the planar ^studies to the _case of

compression by

_a curved obstacle. This

problem

is more

complicated

than that of

a flat

obstacle,

since here the blob size varies in space

(Fig. 1c).

In order to

study

this system,

we write down _a

generalised Flory-type

free _energy and then

locally

minimise this _energy for

a number of different surfaces. In

doing this,

_{we are} able to

study

_any surface with

cylindrical

symmetry,

including

the

planar

surface considered earlier. In the next section _we

briefly

review

some of the results for

planar

surfaces and compression beneath finite disks. In Section 3 _we derive the free _energy for _a chain under _a curved surface. In Section 4 this is

analysed

for _a chain

compressed

under _a

power-law

surface.

Finally,

in Section 5 _we present some numerical results for compression under _a wide variety of surfaces: infinite

planes,

finite

disks,

witch's hats and sinusoids.

2.

Compression

Under

a Flat Disk

The

problem

of

permanently grafted

fixed mushrooms

compressed by

_{a very}

large

^flat obstacle

was first discussed _{many years ago}

is,

6]. ^{Let the chain have N} monomers with _monomer size

a. The natural radius of the mushroom in _a

good

solvent is about the _{same as} its

Flory

radius

RF3

_*

aN~/~.

_{If the}

compressing

obstacle is _a distance D above the

grafting surface,

then there is

only

_a minimal effect for D > RF3

However,

for D <

RF3,

the chain breaks _up into

a series of blobs

[5,10]

of radius D. Within each blob the _monomers behave _as

they

would inside _a

large Flory-type

excluded-volume chain. Thus if each blob has g monomers we have D ₌

ag~/~.

The total number of blobs is

given by Nb

=

N/g

₌

N(a/D)~/~,

and each blob is associated with _a free _energy cost of order kT. The blobs

repel

each other like hard

spheres

and form _a two-dimensional

self-avoiding

random walk of radius R2F ₌

DNll~

The situation is different if the

compressing

disk has _some finite radius Rd;sk. This _was the

case studied in reference

[8,9],

and _we review _soIne of the features

predicted

there. Provided Rd,sk »

R2F

the Daoud de Gennes

analysis

is valid

(Fig. la). However,

in the

regime

where Rd,sk * R2F the chain _can relieve _some of the compression

by partially escaping

from under the disk

Complete

_escape is

impossible

since the chain is

grafted

to _one

point

_on the surface

(Fig. lb).

The chain thus forms _a tether [8,

9,11],

which stretches from the

grafting

site to outside the disk. The _{monomers not} used in the tether form _a

large,

^unconfined blob at the disk

perimeter

whose energy is

ignored

in the

analysis

in reference [8]. The free _energy of this

configuration

_can be estimated _as follows. Let the tether have _nt blobs. The intrinsic energy associated with the tether is thus kTnt.

However,

the natural radius of _a chain of _nt blobs in two dimensions is Rto =

Dn(/~,

and if Rd;sk >

Rto

it is necessary ^to pay a

stretching penalty

to stretch the chain. Since the tether will be

strongly~stretched,

the

stretching penalty

is

given by

_a blob argument as

kT(Rd,sk/Rto)~ i12].

This

implies

that the total free _energy is

Ft =

kT(nt

₊

(Rd,sk/D)4n/~). (l)

The minimum of this free _energy lies at nt "

(Rd,sk/D)

and is F

=

kTRd,sk/D. Equating

^this

to the free _energy of _an

imprisoned

chain

yields

the critical value of D at which the _escape transition takes

place.

This is Dc

=

Do

where

DO % N~~~a~~~ /Rd>sk~~~ "

Rd>sk(RF3/Rd>sk)~~~.

~2) Note that Do is the value of D at which the chain _can make _a tether just

long enough

to reach the disk

edge.

Thus the chain _escapes at

roughly

the point where it is

squeezed

from under the disk. Now let _{us compare} the end-to-end distance of the

chain,

Rcha>n, just ^before and just

after the escape. Before the escape it has _a radius

given by

the _two dimensional

Flory

form

RF2(D=Dc)

"

Dc(N(a/Dc)~~~)~~~

"

Rd>sk(RF3/Rd>sk)~~~ (3)

After the escape the chain has end-to-end distance

slightly larger

than Rd,sk.

Assuming

Rd,sk >

RF3,

I.e, ^{the disk} is

actually larger

than the

original

mushroom size, _we find _a "first-

order"

jump

in the chain radius at the _escape transition.

As in all first-order transitions there is _an energy barrier to be _overcome in

escaping.

^This can

be estimated _as follows. Just prior to escape the radius of the chain is

RF~(D=D~) (3).

In

forming

a tether this chain must be stretched first _to the

edge

of the disk. Once this is done _some blobs

can escape. This initial

stretching

costs _an energy

kT(Rd~sk/RF2(Dc))4

=

kT(Rd,~k/RF3)~/~.

Since Rd,sk > RF3, the energy barrier is

larger

than kT. As shown

previously [8,9],

it is of order NkT. At the critical compression Dc, the chain must overcome this energy barrier. However, for certain stronger

compressions,

the

imprisoned

chain becomes

absolutely

unstable to escape.

This defines _a

spinodal line,

which _can be calculated

by

_a

simple geometrical

argument. ^The

imprisoned

^{chain becomes unstable when} its radius RF2 ₌

D(N(a ID

_)~/~

)~R

is

equal

to the disk radius. Once this _occurs there is _no

longer

_any

penalty

associated with

forming

_or

stretching

a tether and the escape transition _must take

place.

This _occurs at D~- =

N~a~Rd,sk~~

"

Rd,sk(RF3/Rd,sk)~ (4)

A second

spinodal

^line may occur for _an

escaped

chain that is

being decompressed.

It becomes unstable _to

imprisonment

at D

= Dc. However, there is _an energy barrier to be

imprisoned

that vanishes at _a second

spinodal

_at D ₌

D~+.

This of _course is the _same

barrier,

of order

kT(Rd,~k/RF3 )~/~,

calculated

previously.

The

spinodal

_occurs when the chain of blobs _can

just

stretch _to the outside of the

disk,

I.e., ^{when the} external

escaped

blob has _zero size. This

°~~~~~ ~~

D~+

₌ Do ₌

aN3/2(a/R~,~~)3/2

=

R~,s~(RF~/Rd,sk)~/~ (5)

Note that _to within numerical

prefactors

^this is the _{same as} that for the critical line Dc.

The _energy barrier needed to

undergo

the transition _can lead _{to super}

compression lexpansion

and to

hysteresis. Super-compression

can take

place by compressing

_an

initially

free chain.

The _escape transition takes

place

not at D

=

Dc (2)

^but at _a smaller value of

D, given by

D ₌ D~-

(4).

Between these _two values the chain is

super-compressed. Similarly,

the chain

can be

super-expanded

between D

= Dc and D ₌

D~+.

^The second effect of the barrier

is

hysteresis

in any measurement of the chain

properties

_versus D. Consider for instance the end-to-end distance of the chain. Under

compression

a

jump

in the chain size, Rcha;n

occurs at D

= D~-. Just prior to the

jump

it is

Rd;sk,

whilst

just

after the jump it is Rafter _" Rd,sk +

al~$/~~~~j

^{where Nextemai} " N

(Rd;sk/D~-)(D~-la)~/~

is the number of

monomers in the

escaped

blob. This leads to a

jump

of

RF3(1- (RF3/Rd,sk)~/~)~/~.

A similar

jump

can occur when the chain is

decompressed

at D

=

D~+

The chain radius

jumps

from Rd,sk to

D~+(N(a/D~+)~/~)~R

=

Rd,sk(RF3/Rd,sk)~/~.

In summary, the main conclusions of references _[8] and _{[9] are} that the chain will

undergo

_escape and

imprisonment

transitions and will show

hysterisis

under

compression

^and

decompression.

3.

Compression

Under _a Curved Surface

The previous

analysis [8,9]

was based upon a

simple

free _energy and the

postulate

of _a tether.

For _a curved obstacle this kind of

approach

is _no

longer appropriate

because the blob size varies in space

[13].

^{The free} _energy can be lowered

by reducing

the _monomer

density

in

regions

where the

height

h

(and

therefore blob

size)

is

small,

and

increasing

the

density

in

regions

of

large

h.

In this section _we introduce _a

generalised Flory

free energy based _on the blob

picture.

We

neglect

all numerical

prefactors

m

deriving

this free energy. There _are three terms: a birth

term

Fb,rth

caused

by

creation of the

blobs,

and thus loss of chain entropy, ^which contributes kT _per

blob;

_an excluded-volume term

l~nt

caused

by hard-sphere

interactions between the

blobs;

and _a

stretching

term F~tretch caused

by

the chain of blobs

being

stretched

beyond

its

Gaussian radius.

Consider _a mushroom _{on a} flat surface centred at the

origin

_r

= 0. A curved

plate

is _now

placed

above the mushroom with

plate-surface

distance

h(r).

We _assume

cylindrical

symmetry,

so that

h(r)

is

enough

to describe the system. ^{The mushroom should divide into} a number of blobs of size

h(r).

Let

p(r)

be the surface

density

of blobs. The number of blobs between _r and _r + dr is dnb

"

rp(r)dr.

The birth term is just the total number of blobs:

Rchmn

Fb>rth _" ^kT

drrp, (6)

where the chain is assumed _to extend _{out to some} radius llcha>n. Within the mean-field _approx- imation, the excluded volume interaction for the

region

between _r and r+dr is

dn(h~ /(hrdr)

₌

p2rh~dr.

Thus the total excluded volume interaction contribution _is

Rcha,n

l~~t

₌ kT

drp~rh~. (7)

The

stretching

_energy contains two different contributions. If the

stretching

is weak then _a Gaussian

approximation

is

appropriate

_[5],

I.e.,

to stretch dnb blobs

requires kT(h~~dr /dnb )~dnb However,

at stronger

stretching

the Pincus expression for two dimensions is

required

_[12]. This is

kT(h~~dr/dnb)~dnb.

There _{are many ways} of

interpolating

between these two expressions.

The _{one we use} here is

simply

to add them. This

provides

accurate results in the _two limits of weak and strong

stretching.

The total

stretching

_energy is then

Rcha,n

Fstretch _" kT

drTp((prh)~~

+

(pTh)~~). (8)

It is useful to introduce the variable _n

= pr. The total free _energy then reads Rcha,n

F/kT

₌

/ _drn~~h~~

+

n~~h~~

₊

n~h~ /r

_{+ n.}

(9)

0

At present ^{the chain}

radius, ll~ha;n,

is unknown. This _can be eliminated

by changing

variables

from _{r to} the _monomer index _{n as} follows. We let _monomer number _m lie at radius _x.

Then,

by counting

the _monomers from the _centre

m(x)

=

/ ~ drn(r)(hla)~/~. (10)

Thus

dm/dx

₌

n(x)(h(x)la)~/~

_or

n(r)

=

(a/h)~/~l/r',

where the prime _means differentiation with respect to monomer number. We _can then insert this into the free _energy and

integrate

over monomer number rather than radius. This

gives

_a ^free _energy ^of

N

F/kT

₌

/ dmh~~(r')~(hla)~/~

₊

h~~(r')~(hla)~

₊

(a/h)~°/~h~(r')~~r~~

₊

(a/h)~/~. (ll)

o

In order to obtain the chain size, _we need _to minimise

(11)

with respect to

r(m), subject

to

r(0)

= 0 and free-end

boundary

conditions _{at m}

= N.

Taking

the functional derivative with respect to

r(m), bF/br(m)

= 0,

yields

_a

highly

non-linear differential equation for

r(m)

that has _no obvious

general

solution. In the next section _we present _a

scaling analysis

valid for

power-law surfaces,

which avoids

directly solving

the differential

equation.

4.

Compression

Under _a Power-Law Surface

The first kind of non-trivial surface _we

analyse

is a

power-law surface,

where the distance of the obstacle from the surface is

given by h(r)

=

a(r It)" (a,

I constants, a >

0).

We

postulate

that the _monomer

position

also

obeys

_{some power} law

r(m)

= wm"

(w

_a

constant).

Each term in the free energy

(before integration)

_can be written _as

FGauss _'~

m~"

~ ~"~~, Fp;ncus ~'m""~~" ~, F;nt

'~ m ~"

~""~~, Fblrth

'~ m

~"". ₍₁₂₎

For

large

_m, and hence

large N,

_{we can see} which of these terms is the most

important.

To do this _we choose _a > 0 and equate two of the _{terms to} get _u. ^{We then} compare the free energy for these _two

against

the free energy of the other terms.

The results _are:

I)

When Fp;r~cus ₌

l§nt,

u =

15/(7a +18)

and the exponents ^of m in the

integrand

are ip,ncus " 1>nt "

~(12 +13a)/(7a +18),

_ibirth

"

~25a/(7a +18)

and _iGauss

"

-(6 +19a)/(7a

+

18).

^This

comparison

is

self-consistent, provided

_a > I, I.e., in that

Case ib>rth < 1Pincus and _{iGauss < iPincus.}

2)

When Fbirth _"

F,nt,

_{~ "}

~3/(tY 6),

_ibirth

" lint "

5a/(O 6),

_{ip,ncus "}

-(7a 12)/(a 6)

^and _{iGauss =} -I. This

comparison

is self-consistent

provided

_a < 1.

3)

When

Fp,ncus

_" Fb>rth, U "

3/(2a

+

3),

_{ip,ncus " 1b>rth " 1Gauss "}

~5a/(2a

+

3)

and

i,r~t = -I. This

comparison

is self-consistent

provided

_a < I. Note however that the

Gaussian _{term cannot} be

ignored

here since it is of the _same order _as the Pincus and birth _terms.

4)

When FGau~~ = Fb,rth we have the _same situation _as

regime (3).

5)

When FGauss _"

l~nt

_we have _U

#

3/(4

₊

a),

_iGauss

" 1>nt "

~(2

+

3a)/(4

+

a),

_ip,ncus

"

-I and _ib,rth

"

-5a/(4

₊

a)

To have this

regime

hold

self-consistently,

_we

require

~Gauss > ip,r~cus, which

requires

a < 1. However, _we also

require

_~Gauss > ib,rth which

implies

o > I. Thus this

regime

never occurs, except in the _case where _a

= 0, for which the birth term is constant and _can be

ignored.

In that

particular

_{case we} obtain the 2D

Flory

exponent of _u

=

3/4.

We thus find several different

regimes. However,

there _are

important

additional constraints

on this

analysis.

The first is that the radius cannot increase _more

slowly

than for _a free chain.

Thus _{u >}

3/5.

If _we find _{u <}

3/5,

then _{we can assume} that the chain is

asymptotically

unconfined. This is _true for _{o >} I.

Also,

_we find that _case

(2)

above results in _u <

3/5.

Thus

we are limited _to

regimes (3)

and

(4),

where _{u =}

3/(2a

+

3). Secondly,

since _{o >} 0 _we cannot have the chain radius

increasing asymptotically

_more

rapidly

than for _a chain under _an infinite flat disk. Thus _{u <}

3/4. Hence,

_we require that _{a >}

1/2.

Thus _{we are} led to the

following

conclusions about the asymptotic ^{behaviour of} the

radius,

R

rw N". For _o <

1/2,

_we have _u

=

3/4.

That is,

asymptotically

the chain behaves _as

though

it _were confined to two dimensions

by

_a flat

plate.

For

1/2

< a < I, we have _u

=

3/(20

₊

3).

Physically,

this

corresponds

to _a

regime

in which the "birth"

penalty

for blobs _is balanced

by

chain

stretching

_energy. On the _one

hand,

the "birth"

penalty

favours the

expansion

of the chain to

regions

of less _severe

confinement,

and hence fewer blobs. On the other

hand,

such

an

expansion

costs

stretching

_energy. In this

regime,

we find that the Pincus and Gaussian

stretching

terms are of

equal

size, and that the excluded volume interaction between blobs is

asymptotically unimportant.

The chain

configuration

_is

essentially

a balance between two

different

entropic

terms. For _{a >} 1, _we have _u

=

3/5.

This

corresponds

to chains that _are

asymptotically

unconfined The

regime 1/2

< _a < I is the _crossover between confined 2D behaviour and unconfined 3D behaviour. It also

happens

to be the _crossover

region

between weak

(Gaussian)

and strong

(Pincus) stretching.

In the _next section _we present some numerical

results with which _our

predictions

_can be

compared.

5. Numerical Results

For _a

general

surface the

scaling approach

used in the

previous

section cannot

apply

and _we

are forced to resort to _a numerical solution of the

problem.

This _we do

by

first

discretising

^the

free energy

(11)

into segments ^of one monomer or

larger.

We then search for local minima in the free energy either

by solving

an

equation

of motion for each

segment dr~/dt

=

-dF/dr~

or

by locally searching using

Newton's method. Both methods _can be _very

costly

in computer time, which scales

roughly

_{as N]~~~~~~~,} where N~egme~ts ^{is the number of} segments ^{each chain} is broken _up into. We have to limit _our

searching techniques

to those which make small local

jumps,

since otherwise

hysterisis

effects

might

_go undetected. Sometimes in _our

examples

_we

use small numbers of _monomers and

compressions

close to _{one monomer.} The reader should

la) j~j

1000 10~

w xx

w Z iQ2

x~

_fl IQQ _,, ',,

~ ,, /~ ^',

/~ ^,, ",

,,, ',

(

lo

"'

~ 1°~

fi

°

~~o

~10 _{100 1000} lo'~ 10'~ _lo" lo° lo~ lo~

Number of Monomers Obstacle Distance

Fig

2

a)

A log-log plot of the chain radius _versus the number of _monomers for

a chain compressed under

a flat infinitely

large

disk with h

= 1. The dashed line has the slope of the Daoud de Gennes scaling prediction, Rcha,n

'~

N~/~ _{The points}

(lower

line) _are from _a numerical mimmisation of the free _energy

(11), b)

A log-log plot of the chain radius

versus the compression ^distance. The chain _is

compressed under _a flat, infinitely-large disk. The points _are for the numerical mimmisation of (11) for N ₌ 20

(open

circles) and N

= 30 (bullets) and N

= 1000

(crosses).

The dashed fine has the slope

-1/4

predicted by Daoud and de Gennes The solid line _is that predicted by _comparing the Pmcus

stretching _energy to the interaction energy, and has slope

-7/18

For strong compressions the Daoud de Gennes prediction is verified, but for weaker compressions the Pmcus term in the free _energy modifies the results.

keep

in mind that in these limits the blob concept breaks

down,

and _so the results for strong compressions _can

only

be considered _as

semi-quantitative.

Throughout

the remainder of the numerical

analysis

_we set _a

= I, and all

lengths

_are

measured in units of _a. We _can test _our program

by

first

examining

the _case of _a

flat, infinitely large

disk. In this _case the Daoud de Gennes

scaling prediction

is that the radius of the

chain is

Rcha,n

~

a~MN~Mh~~M.

_{This may be obtained from}

(11) by balancing

the Gaussian

stretching

_energy

against

the interaction term.

By

varying h and N _{we can measure} the two

exponents

(Figs. 2a,2b)

Good agreement ^{is found} for the

scaling

with N The agreement ^for the

scaling

with h is less

satisfactory.

This is because of the presence of the Pincus term in the free energy

(11).

This term creates corrections to the

scaling

results. Note that in

(11)

the Gaussian term has _a

prefactor h~~/~

while the Pincus term has _a

prefactor

h. The numerical results _are thus Inost affected at

large

h. If _we balance the Pincus

stretching

terIn

against

the

interaction term _we obtain Rcha,n

~

h~7/~8

_{which agrees with the weak} compression results shown in

Figure

2b. Of _course, under _a flat

plate

the chain is not

strongly-stretched

and the Pincus term is an artifact.

However,

it must be included in _our free energy ^to allow for

strong-stretching

under other

geometries.

We _now examine the _case of _a finite disk, of _some radius Rd,sk

(Fig. 3).

This is the _case considered reference [8j. ^In fact, for

computational

_{reasons, our} disk does not have

perfectly

vertical

edges,

but _ones which

slope slightly

from the vertical at _an

angle

9 _< 1

(Fig. la).

Various disk sizes and chain

lengths

_were

studied,

and in all _cases there _{were many common}

features,

_many of them

predicted

in [8].

Thus,

at strong

compressions

_an

initially imprisoned

chain escapes. As the system ^is

decompressed,

the chain

again

becomes

imprisoned.

The

escape transition involved _a small

jump

of the chain radius while the

imprisonment

transition

(al

~~

lb)

~~

80

I

II

~§ .I

^~~

~

/~ _~~

)

₆₀

~ 9

~ ₍

50

~ ~

~

~ 8 U 40

~0

_{5 IO 15 20}

3°o

5 10 15 20 25 30 35 40

Obstacle Distance Obstacle Distance

Fig. 3. Compression and decompression of chains with different N under disks of finite radius Rd,sk.

a)

N ₌ 50 and Rd,sk ₌ 10 The chain radius _is here taken to be the position of the

(N

_1)~~'

monomer The _arrows indicated the direction of motion Under compression the attain _is initially fully

under the disk. At about h

= 9 it undergoes _{an escape} transition, and then escapes further _as it _is

compressed _more. On decompression the chain remains escaped and looses _{monomers to} the interior At about h

= 12 the chain undergoes _an imprisonment transition The bullets _are the points of lowest free energy, while the circles _are metastable states. Note that upon decompression the escaped states

are almost always of higher free energy. The hysterisis under

compression/decompression

_is clear;

b)

N = 500 and Rd,sk ₌ 70.

usually

involved _a much

larger

jump. The systems ^exhibited

hysterisis, I.e.,

the behaviour _on expansion is different from that _on compression. In

particular,

the _escape and

imprisonment

transitions do not _occur at the _same compression. One

major point

of difference between the numerical results and the

predictions

_{of [8]} is _as follows. In [8] ^{it is assumed that} that _once the chain escapes

slightly

it escapes

fully,

I-e-, _once the chain has reached the

edge

of the disk it then

suddenly

puts _a

large

^{number of} monomers outside the disk and reaches _some

equilibrium escaped configuration.

The numerical results show

something

^different.

They

show that there

are two

escaped configurations

at _a

given compression,

_one stable, the other metastable. This is made _very clear in

Figure

3a where for h < 9 there _are two

escaped

states. Often the state obtained under

compression

is the stable _onq, and the _one obtained under

expansion

is

metastable. In other

words,

under

compression

the chain _{escapes to some energy} minimum.

Under expansion ^it becomes

trapped

in _a

high

_energy

escaped

state and remains there for _some time, even

though

this

escaped

state has energy

higher

than other

escaped

states. The two

states _are caused

by

the free _energy of the external blob _or

blobs,

which _was

neglected

in [8].

We _can also test the

predictions

of the

previous

section for

power-law

surfaces. To test these _{we use a}

power-law

surface

touching

the

grafting

surface and vary N. The radius

always obeyed

_a

power-law

in

N,

Rcha,n

~

N", (Fig. 4a).

The measurements of the exponents are

plotted

in

Figure

4b _as is the

scaling prediction

of

the previous section. The numerical results show the _same trends _as the

prediction, although

it is

fairly

clear that the

correspondence

is not

precise.

This is

presumably

because the

scaling

predictions

only

becoIne valid for _very

long

chains.

It is of interest to consider1nore exotic surfaces. For instance, _{we can coInpress} the chain under _a witch's hat

h(r)

= H AT. This surface has _a

height

which decreases with

increasing

radius. We thus expect ^{that for small}

compressions

the radius

might

increase.

However,

for

(al ₁₆₎

40

j

³⁰ _

i

~~ ~

.

( (

j

IO

~

~i

U g

f

~10

_{100 0 0.2} O.4 0.6 0.8 1

Number of Monomers Obstacle Exponent

Fig. 4. Chains compressed under power-law surfaces a) A log-log plot of the chain radius _uersus

the number _monomers

m the chain for

a chain compressed under three different power-law surfaces,

h = 1.0r". The values of _{a are a}

= 0.1

(squares),

_a

= 0.6

(open

circles) and

a = 1

(bullets).

Note that the radius is well described by _a power-law _{r rw} N~ _over the _range of _monomer numbers plotted;

b)

the chain radius exponent v uersus the power-law exponent _o for the compressing obstacle, such that Rcha,n

'~

N~ under _an obstacle of shape h

= Ar". The top line is the analytic scaling prediction

v =

3/(20

+ 3) ^The points _are from

a numerical Ininimisation of the free _energy

(11),

_one for A ₌ I

(bullets),

the other for A

= 0 1

(circles),

measured from data taken between N

= 10 and N

= 100 In

reality the scaling expressions only apply for

1/2

< _a < I

(see text).

large compressions

the chain becomes

strongly

^confined in the vertical direction and _{can remove}

some of this confinement

by shrinking

in radius. This is in fact what is _seen

numerically (Fig. 5a). Finally,

_we consider _a sinusoidal surface

h(r)

₌ H +

Asin(kr) (Fig. 5b).

This

(al (b)

5.5 io,5

IO

fl

⁵ _{.~ 9.5}

%

/~4.5

9

,(

8.5

~ 4

f

~

~~

~°~4

_{6 8 IO 12 14 16 18 20 ° 5 IO l~ ~° ~~}

obstacle Distance

~

°bS~~C~~ ~~~~~'~~~

Fig. 5.

a)

Chain radius _uersus compression and decompression for N

= 25 under _a witch's hat h

jr)

= H-r. The obstacle distance _is the value of H, _e

,

the distance of the hat apex from the grafting

surface. Note that under compression the radius first increases and then decreases;

b)

chain radius

uersus compression and decompression ^{for N} = 50 under the sinusoidal surface h(r) ₌ sin(12 lr). The

obstacle distance is the distance of the grafting surface from the _mean height of the obstacle. Thus at D = I the obstacle _is touching the surface Note that several transitions _occur

m this system.

might

^{model what could be}

expected

for confinement under _a

rough

surface. What is found is _a sequence of _escape transitions under

compression

_as the chain gets

squeezed

under the sinusoidal

bumps.

Under

decompression

^there is _a similar _sequence of

imprisonment

transitions and _some

hysterisis.

6. Conclusion

In this _{paper we} have studied

compression

of _a

single grafted polymer

chain _or mushroom under _a curved surface. A number of candidates for such surfaces exist. The most obvious

one is the tip of _an atomic force microscope Another

might

be _a

protein

molecule _or other

foreign particle

in _a

biological

system

impinging

_upon a

polymer-clad liposome.

We have been able to

study

this system

by writing

down _a

generalised Flory

free energy. This allowed _{us to} extend the earlier studies of flat and finite surfaces

[6,8,9].

For chains

compressed

under finite disks the _escape transitions and

hysterisis predicted

_{in [8] have} been

substantially

confirmed

by numerically minimising

the free _energy.

However,

_we note one

major

difference.

Numerically

we find _two

escaped configurations

for all compressions.

We have studied

compression

under

power-law surfaces, h(r)

m~

r",

both

analytically

and

numerically.

We have

argued

that the asymptotic

scaling

^{of the} chain radius Rcha~n ~ N" _as

a function of N

obeys

the

following

rules. For _o <

1/2

_we find

scaling

identical to that of _a chain confined under _a flat

plate,

I.e., u =

3/4.

For _{a >} I the chain is

essentially unconfined,

I.e., Rena;r~ _rw

N~/~.

_{The regime}

1/2

_{< a <} I is _{a crossover}

regime

between two and three

dimensions,

where _{u m}

3/(2a

+

3).

Note in

particular

that _an atomic force microscope tip is often modelled _as

having

^either _a

spherical

_or

parabolic

form. In this _{case o}

rw 2 and the chain is

asymptotically

unconfined.

We have also studied compression ^under more exotic surfaces. Surfaces which have _a de-

creasing height

as a function of radius _can have _a

decreasing

^{chain radius} _{as a} function of compression, I.e., the chain gets "smaller" _as it _is

compressed. Rough

_or ^sinusoidal surfaces

can show _a number of _escape and

imprisonment

transitions and

hystersis,

_{just as} is _seen in _a finite disk.

We conclude

by noting

some limitations to this study. We have assumed

throughout

that the blob size _at any radius is

given by

the distance of the obstacle from the

grafting

^surface at

that radius. In _{some cases} the chain

might

be able _to lower its free _energy

by choosing

smaller blobs and

stretching

out of _an undesirable

region

[14]. ^{We have}

neglected

this

possibility.

Our

analysis

is of the

Flory

mean-field type. ^This means we have

neglected

fluctuations.

Noise is of

particular

relevance in _our

study

of

hysterisis. Hysterisis

_occurs because there _are energy barriers between local and

global

minima in the free _energy. Noise allows the chain to

overcome such barriers

provided

_one waits

long enough.

Thus _our calculations _assume that

any experiments are done

rapidly

_so that noise is unimportant. ^{As shown} in [8] ^the barriers

can in fact be many kT.

Acknowledgments

P.

Pincus,

G.

Subramanain,

Claus

Jeppesen,

Jacob Israelachvili and

Tonya

Khul _are thanked for _many interesting

suggestions.

D.R.M.W _was funded

by

_a

QEII

research

fellowship.

The

authors also

acknowledge partial

_support from the Donors of the Petroleum Research

Fund,

administered

by

the American Chemical

Society,

from the Exxon Education

Fund,

and from the NSF under Grants No.

PHYB9-04035, DMR91-17249,

and DMR-92-57544.

References

Iii

^Kuhl T.L

,

Leckband D.E

,

Lasic D D and Israelachvili J N., Biophys. J 66

(1994)

1479.

[2] Woodle M-C and Lasic D-D-, Biochim Biophys. Acta ll13

(1992)

171.

[3] Halpenn A

,

Tirrell M and Lodge T P, Ado. Polymer. So loo

(1992)

33

[4] MiIner S-T, Science 905

(1991)

251.

[5] de Gennes P -G

,

Scaling Concepts _m Polymer Physics

(Cornell

University Press, Ithaca, NY-, 1979)

[6] Daoud M. and de Gennes P-G.~ J. Phys.

(Pans)

38

(1977)

85 [7] de Genne P-G, Ado. Cottoid interface Sm. 27

(1987)

189

[8] Subramaman S., Williams D R-M and Pincus PA-, Europhys. Lett 29

(1995)

285.

[9] Subramaman S

,

Williams _D R M. and Pincus P A

,

submitted to Macromolec [10] ^Pincus P-A, Macromotec 9 (1976) 386

[iii

Williams D-R-M, J Phys. ii France 3

(1993)

1313 [12] ^Pmcus P-A-, Macromolec. 9

(1976)

387.

[13] ^{Note that} there _are other systems where spatially-varying blobs _{sizes occur.} See for instance Brochard-Wyart F, Europhys. Lent. 23

(1993)

105.

[14] ^Detailed Monte-Carlo simulations of compressed chains _are currently being undertaken by C.

Jeppesen. In principle these should avoid this limitation