Branched polymers in restricted geometry : Flory theory, scaling and blobs

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Branched polymers in restricted geometry : Flory theory, scaling and blobs

T. Vilgis, P. Haronska, M. Benhamou

To cite this version:

T. Vilgis, P. Haronska, M. Benhamou. Branched polymers in restricted geometry : Flory the- ory, scaling and blobs. Journal de Physique II, EDP Sciences, 1994, 4 (12), pp.2187-2196.

�10.1051/jp2:1994255�. �jpa-00248125�

J.

Phys.

II Franc-e 4 (1994) 2187-2196 _DECEMBER 1994, _PAGE 2187

Classification Physics Abstracts

36.20 61.40 5.40

Branched polymers in restricted geometry

_:

Flory theory, scaling and blobs

T. A.

Vilgis ('),

P. Haronska

(2)

and M. Benhamou

(3)

(')

Max-Planck-Institut ffir

Polymerforschung,

Postfach 3148, 55021 Mainz,

Germany

(2) Max-Planck-Institut for Kolloid- und

Grenzflichenforschung,

Kantstrasse 55, 14513 Teltow-

Seehof,

Germany

(3) Laboratoire de

Physique

des

Polymbres

et Phdnom~nes

Critiques,

Facultd des Sciences Benm'sick, Casablanca, Marocco

lRec.erred 3 Maich J994, iei,ised 26 Ju/j, J994, accepted J8 August J994)

Abstract. This paper discusses the behavior of

polymers

with arbitrary

connectivity

in restricted

geometries,

such _as pores and slaps. The _use of

Flory

theories, blob models and

scaling

theories for linear chains is well-known and does not lead _to any

problems,

I-e- all three

approaches

agree with each other. In the _case of branched molecules this is _not the _case and e-g- no blob model exists. Indeed Flory ^free

energies

and

scaling

theories may lead _to contradictions, when applied to

branched

objects

^and

polymeric

^{fractals without} further information. In this paper we will suggest a

strategy, ^how to use both in combined form. The such obtained results _are sensible scaling forms for the radius of

gyration

and the

filling

fraction. It turns out that _a blob model _can be constructed for branched polymers. This will be demonstrated in the _case of randomly branched polymers. It is also shown that the _new results for

arbitrary connectivity extrapolate

to the well-known _case of linear chains, I-e-

polymers

with one-dimensional

connectivity

and

predicts

new scaling ^{laws for} the _case of two-dimensional tethered surfaces.

1. Introduction.

Scaling theories, Flory

theories and blob models for linear

polymers

introduced

by Flory

and de Gennes _a

long

time ago are very successful tools _to find the

asymptotic

behavior of

polymer

chains in different situations

[I].

There _are many

examples

where it _was demonstrated that these

approaches

can be used _to

provide

the main

insight

and _{answers to}

problems

in

physics.

This becomes most obvious when dilute

polymer

solutions or

polymer

melts in restricted

geometries

are studied. This

approach

to

polymer problems

has been used many times and is also the main part ^{of de} Gennes'

monograph.

The

major point

to be made is that in most cases

the _{answers can} be

guessed intuitively.

This _can be _{seen at} the

following

instructive

example

_:

consider _a

polymer

in _a small

cylindrical

_pore. The _use of _a

cylindrical

_pore is _not essential but it is used here _as

special example.

This

problem

can be solved in three ways. The first method of choice is to _{use a}

simple Flory argument,

the second is the blob

model,

and the third is

scaling theory

[1].

@Les

Editions de

Physique

1994

In

Flory

type arguments ^{the elastic} part ^{and the confined excluded volume} interaction of the free energy are balanced and the result is _a

basically

stretched

configuration

of the chain in the

cylindrical

_pore.

The blob model introduces a new scale in the

problem,

I-e- the blob

size,

which is

naturally given by

the diameter of the pore.

Together

with the natural size of the chain these two

length

scales _are the

only

_ones

present

^{in the}

problem.

The

portion

of the chain inside _one blob in the pore takes its natural dimensions and

shows, however, good

solvent behavior. The size of the blobs must be

entirely

determined

by

the diameter of the pore. These

assumptions

_are

enough

to find the dimensions of the chain in the pore.

The third method _uses the

scaling hypothesis,

which is well-known from the

theory

of

phase

transitions

[2].

In this

approach

it is assumed that for

large

pores the natural radius of

gyration

of the

polymer

the chain is not

perturbed.

Whenever the diameter of the pore becomes smaller than the natural chain size the

polymer

extends in direction of the pore. Such results for linear chains _are reasonable and well-established

[1, 3].

It is not too

surprising

that the results agree from the three methods for linear chains when

they

_are

compared

to each other.

So far _our remarks

apply only

for linear

polymers.

Another

important problem

is when the

polymers

_are branched. Branched

polymers

do have several restrictions due _to their

large connectivity [4].

Therefore it is

tempting

to

study

the behavior of branched

polymers

ⁱⁿ small pores ^to model the behavior in _a

simple

_porous media. In fact in

[4]

it has been shown that small pores could be used

experimentally

to select between linear and branched

polymers,

I-e- in _a

chromatograph.

In this paper we would like _to

study

the behavior of branched

polymers

in

such small pores. In these _cases the restricted geometry ⁱⁿ space competes ^{with the}

connectivity

of the

polymers.

It _{turns out} below that the

theory

_can be formulated for

general polymeric manifolds,

such _as linear

polymers,

branched

polymers

_or tethered surfaces.

Further

problems

occur if

higher dimensionally

connected

polymers,

such _as branched

polymers,

_{or more}

generally

D-dimensional

polymeric

manifolds in restricted

geometries

_are

investigated.

Such

higher

connected

polymers

_are not able to fit into small

enough

_pores. In _a

previous publication

this has been shown before

[4].

In this paper it _was

simply

assumed that the

D-dimensionally

connected

polymer

in

good

solvent does _not fit into pores where the

diameter is less than 5~

=

aN ^{l~ ' Y2} where _a is the size of the _monomer and N the number of

monomers in _a

given

direction. That _means the total number of _monomers

(proportional

to the

total

mass)

is M

=

N~

_{in the branched}

polymer.

The latter statement is easy ^to

verify.

For linear

polymers,

D

=

I this is trivial.

Imagine

a two-dimensional connected

polymer sheet,

I-e- a tethered network. N is the number of _monomers

along

one side. The total _amount of

monomers in the whole manifold is then

N~.

_{This relation}

can be

analytically

continued for

non-integer

values of the

spectral

dimension D very

simply

and will be used later in the paper.

A

general

_{strategy was} not

given

for the combined _use of

Flory theories, scaling

and blob

pictures

for D-dimensional manifolds in this

publication,

^since

only

^the

Flory theory

^has been used. Nevertheless the

approach

is able to

predict

the minimum of the pore size _{as a} function of

the molecular

weight

which the branched

polymer

is

just

able _to pass

through.

A

general

solution and the _use of blobs and

scaling

for manifolds in restricted

geometries

is _so far not

present.

^This is _one of the aims of this paper.

A proper

description

of

higher

dimensional

polymers

^is needed and will be achieved

by

_a

simple generalization

of what is known in the literature. Linear

polymers

are one-dimensional

objects,

surfaces _are of two-dimensional

connectivity,

and

randomly

branched

polymers

can

be

approximated

as

polymeric

fractals with _a

spectral

dimension D of about

4/3,

I.e. the _mean field limit of

percolation

clusters. This is not exact in d

~

6,

but

reasonably

close to the real values obtained

numerically [5].

Nevertheless for the purpose of _our

study

based _{on a}

Flory

estimate

only

the _mean field values _are of relevance.

N° 12 BRANCHED POLYMERS IN RESTRICTED GEOMETRY 2189

In the

following

_we demonstrate that in _cases of branched molecules

actually

both

approaches,

I.e.

Flory theory

and the

scaling

method has _to be used to find sensible _answers to the

problem.

Another

problem

that is

obviously present is,

^that at first

sight

no

appropriate

blob

description

of D _~ l

-polymers

is

possible,

and _one has it is natural to

rely

on

Flory

arguments.

Nevertheless it is difficult to

rely

on such arguments ^{and it will be shown how the additional}

use of

scaling arguments

^{will rule} out

insignificant

results. The basic

difficulty

in the

swelling

and conformational behavior of branched

polymers

_was

already pointed previously [6, 7]

and has been used earlier

by

the

principal

author of this paper

[4]

_: branched chains _are in

contrast to linear chains _never be able to stretch _{out to a} size

ROOM,

where

M is the total molecular

weight. Only

_a chemical

path

which contains N _{monomers can} be

stretched _out

completely,

but this does _{not mean} that the size of the

polymer

follows the

scaling ^lai

R _cc M. From this statement follows

immediately

that _two limits exist for the size of

higher

connected

polymers (D-dimensional

in

connectivity) first,

the branched

polymer

can extend _as far _as R _cc N

(stretched

_out

completely

in any

direction).

^This limit

yields immediately

_a minimum fractal dimension of d~ = D, I-e- _a maximum size

R$~~

cc M and

corresponds

to the maximal

possible

extension for all

physical

_processes such _as

swelling

_or

mechanical deformation. The other obvious limitation is

given by

the condensed _or saturated

phase,

I-e- the minimum size

given by

the maximum

density R$,~

cc M, ^where d is the space dimension. Therefore it _can be concluded that D _w d~ w d.

These

limiting

cases will _not be very

important

^{in the} often studied solution _cases, but _are essential for the behavior of

melts, polymers

in restricted

geometries,

_or melts in restricted

geometries. Polymer

melts of linear chain molecules

(D

= I _are

relatively simple

to treat within the

scaling approach.

As the chains _are of one-dimensional

connectivity they

are able _to

interpenetrate

in the dense melt state. When the melt consists of

higher

connected

polymers

(D

_~ l this cannot be

generally

the _case.

Polymerized

membranes

(D

=

2

),

for

example,

_can

only interpenetrate

for

special

_{cases, e.g.} in the flat

phase [8],

but in the crumbled

phase [8]

they

cannot. Melts of such

highly

connected

polymeric objects

_saturate, I-e- the individual

polymers

form dense balls

and, however, macroscopically

is the fractal nature is _no

longer important [9, 10].

^{Such melts}

placed

in restricted

geometries,

for

example cylindrical

_pores,

will show _a very subtle behavior _as the conformations of the individual molecules _are sensitive to the geometry of the space.

The paper is

organized

_as follows. In the

following

section the basic definitions will be

repeated briefly

for the _case of

arbitrarily

connected flexible molecules in

good

solvent.

Then,

a manifold in

good

solvent between two-dimensional

plates

is

investigated.

Three methods _are used

Flory theory, scaling

and _a blob model for such systems ^is introduced. In the next section it is shown that these methods break down when the manifolds _are

put

^into a

cylindrical

pore. For all manifolds whose

connectivity

is

larger

than _{one a} minimum pore size has _to be defined. This pore size is

entirely

determined

by

^the

connectivity,

I-e- the

spectral

dimension.

In the last section

polymer

melts in pores are considered.

2.

lXdimensionally

connected

polymers

in

good

solvent.

The arguments ^{from above} can be made _more familiar if _{a common}

generalization

of the Edwards Hamiltonian for linear

polymers [8, 11]

is introduced for

polymeric

^{fractals and D-}

dimensional manifolds in the

following

standard way

N N N

H

=

0 d~x(V~R)~

v

o d~x 0 d~x' &~(R(x) _R(x')). ( la)

D is the

spectral

dimension _as noted earlier. Well-known

special examples

are D

= I, that _are

linear

polymers

and D

=

2,

I.e. random tethered surfaces. The

analytic

continuation of D _{to non}

integer

values, I-e- I ~D

~ 2

correspond

to any

polymeric

fractal of

arbitrary connectivity.

_x is _a D-dimensional _vector of the manifold embedded in d space dimensions

described

by

the vector R and N is the linear dimension in _one

arbitrary

direction. The first term is the Gaussian

connectivity

and the second _term the usual excluded volume interaction.

In this paper

only objects

with D

~ 2 _are considered for convenience. The Hamiltonian

(I)

does _not make _sense for fractional values of D, without

defining

fractional differentials and

integrals properly.

In the

scaling

limit it _can be used without

problems.

We restrict ourselves _to this latter _case.

Although

we do _{not use} the full Hamiltonian in this paper, its introduction is

helpful

_to derive its

scaling properties (see [2, 8]), especially

for readers who _are not familiar with the notation used below. The standard estimation uses the

replacement R(x)

- R and

(x(

^{-N. All}

integrations yield trivially

_a factor of

N~.

_{It is remembered}

again

that

N is not the entire number of _monomers but

only

^{the number of} monomers in _one

given

direction in

spectral

vector space. Then Hamiltonians such _as

given by equation (I)

_can be transformed

easily

into a

Flory

free

energie by

dimensional

analysis [12].

F

=

R~/(a~ N~

^~ ₊

a~ N~ ~/R~ (16)

By

the substitution M

=

N~

_{it is easy to} show that

equation (16)

transforms into the well-

known

Flory

form of

polymeric

fractals

[4-10]

with the Gaussian fractal dimension

d~ =

2D/(2

-D),

which _{recovers cases} of linear

polymers (D

=

), randomly

branched

polymers (D

=

4/3)

and tethered membranes

(D

=

2).

The standard result is obtained

by

minimization of

equation (16),

that

yields

^the usual

Flory

_exponent for the size of the

polymer

in the swollen

(crumbled)

state is found to be

[8]

v=~~~

(2)

To avoid

misunderstanding

at this

early

_stage we mention

again

that these exponent accounts for the linear

(chemical)

size N and _not for the total _mass M.

By simple

_arguments it is

easily

found that the

corresponding

fractal dimension is

given by

_d~

=

DIV. In the

following

we

restrict ourselves _to the three dimensional

(d

=

3)

embedding

_space.

3.

lXdimensionally

connected

polymers

between two

parallel plates

in

good

solvent.

As _a first

example

the case of _a D-dimensional

polymeric

manifold in

good

solvent between two

plates

is studied. The first attempt to solve the

problem

is the _use of

Flory's theory.

This is very

simple

and the

Flory

free energy

(see Eq. (lb))

for this

problem

is

given by

F

=R(la~N~~~ +a~N~~/(%R().

_{d~ is the distance between the two}

_{parallel plates}

_and

Rj

_measures the size of the of

polymer parallel

to the

plates. Minimizing

^the free energy with

respect to the size Rjj

yields

the desired result

a 1/4

~~ _~ ~~~

~~ ~

j~

~ ~~~

Note that the N

dependence

in

equation (3) corresponds

to a two-dimensional

polymer

with

spectral

dimension D. For D

=

I

(and

^N

=

M)

^the correct

exponent

v = 3/4 is recovered

[1, 13].

For

polymer sheets,

such _as

flexibly polymerized

membranes

(D

=

2 the reasonable

exponent

v = I is obtained. The latter

corresponds

to the case where the tethered membrane is flat between to very narrow

parallel plates

with undulation fluctuations of the order of the

N° 12 BRANCHED POLYMERS IN RESTRICTED GEOMETRY 2191

distance between the _two

parallel plates. Although

reasonable limits _are

predicted by

this

Flory

model it is difficult _to be _sure about the

validity

of this result if it is not derived

by

_a different

method,

such _as, e-g- ^the

scaling theory.

This will be

provided

in the next

paragraph.

The

scaling analysis

can be done in close

analogy

to the _case of linear chains. The radius for the chain between _two

plates

can be written _as

R~

RI

"

RF f ~ ⁽⁴⁾

where

Rj

is

again

the extension of the chain

parallel

to the

plate

and 5) the distance between the two

plates. R~

is the

geometrically

unconstrained

Flory

radius in

good

solvent and is defined

by

the

Flory

_exponent

given

above

by equation (2).

The

scaling

function

f(.<)

has _two limits. The

R~

first is when _x

= ~ tends _{to zero,} I.e. when the

plates

are

separated

_very ^{far from} each other J)

and thus

f (.;

- I. The

opposite

limit, when the

parallel plates

_are

placed

_{very narrow} the _two- dimensional

configuration

_appears, ^{which determines in usual} manner the exponent

x of the

scaling

^{function. This}

corresponds

to the two-dimensional

configuration

Of the D-

polymer

is calculated

by

the

Flory

model above. The usual standard argumentation

provides

the _{same answer as} derived in

equation (3).

A more

appropriate

form _is given

by

R11 ⁼ 5) ^~ N ^{l~+ ~~"}

(5)

5)

~~~

It is

tempting

to

generalize

de Gennes' blob

picture

to such

self-similar

branched D- dimensional

polymers.

This has _not been done yet,

although

it is

simple.

To do

this,

_assume the

D-polymer

between two

plates

behaves _{as a} fractal made out of blobs of size 9). Thus it is reasonable _{to assume} that the size of the

object

is

given by

_Rjj

= 5)n ^{~~+ ~ ~~~} where

n is the total number of blobs. Note that the fractal dimension of the

effectively

_two-

dimensional

object

_d~

=

(2

₊ D

)/4

D has been used _to account for the

full

_mass in the fractal.

The number of blobs _{n can} be calculated

by

determination of the _{mass m}

(number

of

monomers)

inside the blob. Inside the blob the branched structure shows

good

solvent behavior that

yields

_m

= (5)la )~~~l~

+~~.

Therefore the number of such blobs is

given by

n = M/m, where M is the total _mass

=

N~

as before.

Following

de Gennes'

argumentation

^the result

Rjj = 9)

~

)~~~Ml~+~'"~

^is

^obtained,

^{which is identical to those obtained}

^by

^the

^Flory

J)

theory

and

scaling approach.

This almost trivial

example

shows that the blob

picture

_can be

used to construct the _same

physically

reasonable results for branched chains similar _to the _case

of linear chains. The

point

^is to use the information also from the

scaling

in _terms of the number of _monomers in the chemical

path

N

additionally

to the _mass

scaling

in the blobs.

The _cross check of all results is to consider the

filling

fraction i

=

a~ N~/ (S~R~) ii.

It turns out later that this

quantity

is useful with _more respects. ^{For the}

polymeric

^manifold

(or

a 7/4

~~ ~~~~

polymeric fractal)

in the

slap

i is

given by

i

=

N This result makes

5~

physically

sense and for D

=

I the classical

polymer

^behavior is recovered

[1, 3].

For D

~ 2 the

filling

fraction becomes

unphysically large. Trivially

_a

polymeric

membrane _can be

pressed completely

between _to very narrow

plates,

I-e- 5~

=

O(a).

In this

special

case the

filling

fraction becomes

independent

of the molecular

weight,

_as it is

intuitively

clear

(lower

critical

dimension).

The

example

of the

arbitrarily self-similarly

branched

polymer

between two

plates

has been

explicitly

discussed in _more detail to demonstrate how the blob

picture

and the

scaling

_{arguments can} be

generalized

to branched

polymers

or

arbitrary higher

connected

polymeric objects.

4.

lXdimensionally

connected

polymers

in _a

cylindrical

_pore

(good

solvent).

Severe

problems

occur when such self

similarly

branched, non-linear

objects

_{are put} into

cylindrical

_pores when in other words the space available for the

polymer

^{is further} restricted.

The

simple

dimensional

analysis

from above has _to be modified in the usual _sense, that the d- dimensional Dirac function becomes

anisotropic

and the lateral dimensions _are

given by

the

pore size. Thus _we estimate the relevant excluded volume term from

equation (I)

to be

(R(x) R(x'))

_cc

~

where 5~ is _now the pore diameter and Rjj is

again

the chain size 5~

RI

parallel

to the pore. Now

begin

with the consideration of the

Flory

free energy for the manifold in the pore

F

=

R(/N~

^~ ₊

vN~ ~/

_(5~~

Rj ) (6)

Minimization

yields immediately

the result for the

parallel

exponent

vj =

~

)

^~

(7)

which agrees for D

= I with the standard exponent

[I],

I.e. _vii

=

I

corresponding

to the

stretching

of the linear chain

along

^the pore.

Obviously

this exponent ^{is ill defined whenever} D ~ l, that is whenever the

polymers

are of

higher connectivity

as the linear _ones. The linear

(chemical)

dimension

through

the fractal _or the membrane must not be

larger

than N itself that

corresponds

to the

entirely

stretched limit. Such

phenomena

^{have been found}

previously [6].

It is _now easy to

understand,

that

scaling

and _a blob model _as it _was

presented

for the

slap

cannot work anymore for _case of the

cylindrical

_pore. For

example,

if _a

scaling

argument

^{is considered which} assumes a

good

solvent behavior for

large cylindrical

_pores and

a linear

(entirely stretched)

branched

polymer

for _narrow pores, contradictions will show up such that the

filling

fraction is

unphysically large

for all values for the

spectral

dimension D ~ l. One way out of this

difficulty

is the

postulation

of _a minimum pore size

through

which

the branched

polymer

is able _to pass

through.

Thus the minimum pore size _can be found such that for the

parallel

direction the value Rjj ^cc N is taken. It is

given by

~~m'n ~~° ~ ^{~~ '~~}

" M ^{~~ ' '/~~} ~~~

This result makes

physically

_sense. The minimum pore size for linear

polymers

the pore size is

independent

of the molecular

weight.

Thus linear

polymers (D

=

I find their way even

through

_{a very} small pore, if 5~~,~ ^{is of the order of the Kuhn}

length,

but with _a

extremely

low

probability

and in _a very

long

time. The time the linear

polymer

^will need to pass

through

the pore will be

exponentially large,

I.e. its diffusion constant is

exponentially

small

[ii.

In branched

polymers

as D

~ l another limitation is

important

the

connectivity.

The

larger

the

connectivity,

the

larger

is the minimum pore size. It should be therefore

possible

to construct a porous medium that is able _to separate a mixture of branched and linear molecules

on a basis of their

connectivity.

This _can be done

by

an

appropriate

^minimum pore size

through

which linear

polymers

can pass, whereas branched

polymers

cannot. For the construction of such _a

chromatographer dynamical

aspects have to be taken into account, since the other

selection constraint is the finite time to pass

through

a pore, as mentioned in

equation (8).

Such aspects are left for the moment, since an

analogous equation

^{for the diffusion} coefficient in

terms of blobs is still

lacking.

It can be

expected, however,

that the diffusion constant shows _a similar behavior _as

given

in the D

=

I _case for pores

larger

^{than the minimum} pore size.

N° 12 BRANCHED POLYMERS IN RESTRICTED GEOMETRY 2193

The essential

point

to be made is that the minimum pore size is

entirely

defined

by

the

spectral

dimension and the molecular

weight.

Therefore the pore is able _to select

objects

with respect to their

connectivity,

I.e. their

spectral

dimension. This

possibility

has been called

spectral chromatography

in the

previous

_paper to

distinguish

to classical

chromatography,

which selects

only

with respect to molecular

weight. When,

for

example,

a membrane is put into _a small

enough

_pore it cannot flatten _out in the

remaining

_space but has either to crumble in

a

specific

direction, if the pore is

large enough,

_or to saturate

(collapse)

in smaller pores in the lateral direction. Whether _one finds the crumbled _or

collapsed

_case is determined

by

the value of the minimum pore size

5J~,~.

It could be

guessed

that the

geometric

restriction of the pore becomes less

important

if _a Theta

(9

solvent is used for

transporting

the

polymer through

the pore

[14].

For _a linear

polymer

the

9-exponent

is _v

=

1/2 that is less than the swollen exponent

I].

Thus the total size

of the chain in 9-solvent is smaller,

compared

to the

good

solvent _case. For

polymeric

manifolds of

larger connectivity

this is also the _case and it could be concluded that the limitation defined

by

the pore is less _severe. This is indeed _not the _case. It _can be

immediately

estimated

by using

_a

Flory

_argument of the similar type as above but

including

the third virial coefficient, whereas the excluded volume is

effectively

zero

[I].

It _can be shown that the minimum pore size is

given by

the _same value

5)~,~

cc N ^l~ ~~~ as above. We will _not elaborate in _more detail _on this since it is well-known that the

Flory argumentation yields

_very bad values for the exponents _v ⁱⁿ two

dimensions,

but _exact values at the lower critical dimension for

three

body

interactions d

= I and the upper critical dimension d

=

3

[13].

Little is known for the

validity

of

Flory

values of branched _structures and manifolds under

o-conditions,

and we

do not discuss it further. The value for

5J~,~

^under o-conditions

is, however,

the indication that

the minimum pore size is determined

by

the

connectivity only

and _not

by thermodynamic

conditions.

It has been mentioned earlier that _an

interesting

check about the

consistency

of the results is to calculate the internal

concentration,

_or

filling factor,

~F

=

N~/5~~Rj.

The

filling fraction

becomes

independent of

the molecular

weight

M

=

N~ of

the

manifold

at the

point

when the pore takes its minimum size. This indicates that the pore size is still _a natural scale for the D-

dimensional

polymers.

5. Melts of fractals in restricted

geometries.

The _case of melts of fractals and branched

polymers

is also of interest. The _case of melts of linear chains in small

cylindrical

_pores has been studied

by

Brochard and de Gennes

[15].

Again

this _case of D

=

I-linear

polymer

melts in restricted

geometries

is

simple.

The

generalization

to branched

polymers

and

polymers

with

higher connectivity

is not _as

simple

_as

it

might

look _at first

sight.

In _an earlier

publication

it has been shown

[10]

that melts of

branched

polymers

with _a

spectral

dimension D

~ l must be divided into _two classes.

Whenever the

connectivity

is

larger

than _a threshold value

D~

^D

~D~

=

2d/(d

₊

2)

the fractals do _not

interpenetrate

in melts _as do linear chains. In such systems ^the

connectivity

and space

filling

are too

high.

Instead of

interpenetration

the

polymers

saturate and form

separated

balls of their natural

density,

I.e. R _cc

N~~~.

The short argument ^{will be}

repeated

_to understand better the

following argumentation.

The easiest argument can be

performed by using

_a

Flory

argument ^{for melts of such fractals. In} contrast to self similar

polymers

in

good solvent,

where the excluded volume parameter ^is

given by

v cc

a~,

_{in melts v is screened to a much lower}

value, I.e. _{v cc}

a~ N~~ [12].

The

Flory

free energy for melts

F

=

RilN~

^~ ₊

^{a~ N~/R~} (9)

JOURNAL DE PHYS<DUE U T 4,N' 12 DECEMBER 1994 82

leads

formally

to the D

independent

melt

exponent

v =

2/

(d

₊

2).

This result is of _no

physical

sense since it

yields

a fractal dimension

larger

than the space

dimension,

I.e. d~ = D

(d

₊ 2

~/2,

which is

larger

that the space dimension d whenever the

spectral

dimensions D is

larger

than

D~

=

2 d/

(2

+

d).

For smaller connectivities the situation is very different and the result is that the

polymers

take their

unperturbed size,

that is

given by

R _cc N^{l~ ~}_~~. This

happens

since for

all _cases the system ^{is above} the upper critical dimension. The upper critical dimension in the

melt is

given by

d~~ =

2

D/(2

D )

(10)

which is of factor 2 smaller _as in the _case of

good

solvent. The latter

point

is due _to excluded volume

screening.

The _two different _{cases are now} discussed in detail. For

simplicity only

the

case of the

cylindrical

_pore is considered. The _case of the

slap

is very easy and _can be done

straightforward.

The

following

subsection consider the two different _cases, when the

spectral

dimension is

larger

or smaller

compared

to the critical value

D~.

5. I D _~ D

~.

In this

regime

the melt of fractals is saturated. That _means

that,

unlike in linear

polymers, polymers

of

higher connectivity

do _not

interpenetrate

each other since their

connectivity

is _so

large

^{that this} cannot

happen.

One limitation for such

polymers

in the pore is that

they

form _{a row} of balls each of size R~ =

aN~~~

_The

filling

fraction for this situation is

easily

calculated and it is

given by

J

=

a~ N~/5l~

_R~

=

(a/5l)~ N~

^°~~

(l1)

The

filling

fraction _cannot be

larger

^than one

and, therefore,

the pore diameter is limited _to values D*

=

aN~~~,

which is the saturation radius of the

polymer

itself. This situation is sketched in

figure

I. When the pore is smaller each of the individual saturated fractals _can become

compressed

further _as their saturated radius of

gyration

_can be

elongated

to forrn

ellipses. Again

the maximum

parallel

radius of the

polymer

is

given by _R~~

=

aN,

and for this

case the

filling

fraction is

given by

i

=

a~ N~/5)~ R~~

₌ (a/5) )~

N~

^'

₍

_l

₂₎

and the

limiting

_pore size is

given by

5~~,~ cc

Nl~

'~~~ which is

naturally

identical _to that of

good

solutions and o-solutions.

Again

we find that the pore size does _not

depend

on the

thermodynamic

state of the manifold. 5)~~~ is

only

determined

by

the molecular

weight

and the

connectivity.

To pass

through

this minimum pore size each of the saturated fractals has _to be

Fig.

I. A melt of

highly

connected

polymers

or manifolds in _a small

cylindrical

_pore. ^The

connectivity

is

larger

than D

= 6/5. The manifolds _cannot penetrate ^{each other.} They form

separated

balls. The pore diameter is identical to the radius of

gyration

in the saturated _case.

N° 12 BRANCHED POLYMERS IN RESTRICTED GEOMETRY 2195

Fig.

2. Same _as in

figure

I but with smaller pore diameter. The individual fractals _can still be

compressed

until they _are stretched _to their maximum radius of gyration R~

cc N.

compressed

_a factor of A

= N~ l~~~~~~, which is for tethered membranes A

=

N'~~

and for

randomly

branched

polymers

A

=N~~~

as both _cases

belong

to the class

D~D~.

A

visualization of this situation is

schematically

shown in

figure

2.

5.2 D

~

D~.

In this _case the

physical picture

is very similar _to the _case of linear

polymers.

As

long

as the

connectivity

is less than the critical

D~

₌ ^{6/5 in three}

dimensions,

the fractal takes its ideal Gaussian

dimensions,

I.e.

Ro

₌

aNl~~~~~

_{This is because} the upper critical dimension for the melt is

always

less than the dimension itself, I.e. in the present case

d

= 3. Therefore excluded volume forces _are screened

completely

and the manifold behaves

ideal. In addition this _means that the manifolds _can

interpenetrate

each other _as the

connectivity

is very small. There exist

again

two basic

length

^{scales for the} pore size. The first

one is

given by

^the limitation where the manifolds pass

through

without

changing

their

shape.

This _can be read off

again

from the

filling

fraction.

The

limiting

value for the pore size is

given by

_s~o

=

aNl~~~~~

_For

larger

_pore size the

polymers

in the melt are able to pass

through

without

changing

their

shape.

Smaller pore sizes

are still

possible,

when the manifolds in the melt stretch out. The maximum

stretching

is

given by R~

₌ aN, and the

limiting filling

fraction

predicts

5~~,~

~~j, ⁼

aNl~ '~~,

which is

again

identical to the _same

limiting

_pore size _as in all other _cases. In the

regime (b)

the results _can be

extrapolated

to the _case of linear

polymers (D

=

I without

problems.

6. Discussion.

In this paper a

scaling theory

for

arbitrarily

connected

polymeric

manifold in

simple

restricted has been

presented.

The main

point

is that

larger

connectivities

give

rise to _severe restrictions for the conformation and the behavior of such molecules in restricted

geometries

such _as

parallel plates

or pores. The radius of

gyration

_can be calculated for the _case of two

parallel plates using Flory theory,

blob model, and

scaling theory.

The blob model has been constructed

using arbitrary spectral

and fractal dimension. This has _not been done yet. ^To our

knowledge

the blob model has

only

^{be used for}

regularly

^branched

^molecules,

^such as for

example

star branched

polymers [16, 17].

These treatments can break down when the D-dimensional manifold is studied in _a small

cylindrical

_pore. Indeed, the

study

of the behavior of the manifold in the

cylindrical

_pore

yields

the _most serious restriction. An observation that is in contrast to linear

polymers.

Manifolds with

larger spectral

dimension D

=

I do _not pass

through

small pores without

problems.

A minimum pore size has _to be assumed _to find consistent results. Another main result is that this minimum pore size does _not

depend

_on the

thermodynamic

conditions of the

manifold,

such _as

good solvent,

theta solvent, or melt conditions. The minimum pore size is determined

by

the

connectivity,

I-e- the

spectral

dimension

only.

The considerations

given

in this paper can be

generalized

_very

simply

_to

adsorption

problems

and _to

charged systems.

^Dense systems are also under present consideration and will be

reported separately.

Another

important generalization

is _a proper mathematical formulation

[18]

of such systems ^{and their behavior in}

arbitrary geometries.

^Indeed it has been

shown,

how

path integral

methods and field theoretic methods _can be

generalized

to the _case D

~ l. Linear

polymers

_are in these theories

only

_a

special

case.

Experimental

evidence of these results _are

given by

similar considerations _as for linear

polymers.

We have

already

mentioned the

possibility

of

spectral chromatography. Polymers

of

arbitrary shape

in porous media is another

example.

Realistic porous media _cannot be treated

by simple cylindrical

_pores, but fall into the _same class of

problems provided by

de Gennes and

others in

previous

work _on linear chains.

Acknowledgements.

T.A.V. thanks Mr. B. Evans for the _«

push

_».

References

[Ii de Gennes P. G.,

Scaling

_concepts in

polymer physics

(Cornell

University

Press, Ithaca, 1979).

[2] Zinn-Justin J., Quantum field theory and critical

phenomena

(Oxford, Clarendon press, 1993).

[3] Daoud M., de Gennes P. G., J. Phys. Franc-e 38 (1977) 85.

[4]

Vilgis

T. A., J. Phys. II France 2 j1992) 2097.

[5] _{see e-g-,} Stauffer D., Aharony A., Introduction _to

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[6] Vilgis T. A., J.

Phys.

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[7] Lhuillier D., J. Phys. Franc-e 49 (1988) 705.

[8] See for _a general reference: Nelson D. R., Piran T.,

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1989).

[9] Cates M. E., J.

Phys.

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Vilgis

T. A., Physica A 153 (1988) 341.

ill] Doi M., Edwards S. F.,

Theory

of

polymer dynamics

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[12]

Vilgis

T. A., J.

Phys.

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[13] des Cloizeaux J., Jannink G., Macromolecules in solution (Oxford, Clarendon press, 1990).

[14] Raphael E., Pincus P., J. Phys. II Franc-e 2 (1992) 1341.

[15] Brochard F., de Gennes P. G., J. Phys. Franc-e 40 j1979) L399.

[16] Daoud M., Cotton J., J. Phys. Franc-e 43 j1982) 531.

[17]

Halperin

A., Alexander S., Macromolecules 20 (1987) l146.

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