Dynamics of polymer chains trapped in a slit

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Dynamics of polymer chains trapped in a slit

F. Brochard

To cite this version:

F. Brochard. Dynamics of polymer chains trapped in a slit. Journal de Physique, 1977, 38 (10),

pp.1285-1291. �10.1051/jphys:0197700380100128500�. �jpa-00208698�

DYNAMICS OF POLYMER CHAINS TRAPPED IN A SLIT

F. BROCHARD

Laboratoire de

Physique

des Solides

(*),

Université

Paris-Sud,

^Centre

d’Orsay,

⁹¹⁴⁰⁵

Orsay,

^France

(Reçu

^{le 3 mai}

1977, accepté

^le

14 juin 1977)

Résumé. ^- On étudie les

propriétés

dynamiques des macromolécules en solution et confinées dans des lamelles

d’épaisseur d microscopique

(20-200

Å).

On utilise la technique ^des lois d’échelle pour inclure les effets de volume exclu et les interactions

hydrodynamiques

^entre monomères. Pour certaines valeurs de la concentration et de la masse moléculaire, ^{on montre} que les macromolécules

se comportent ^{comme des} chaines de Rouse à deux dimensions, l’unité étant le blob de taille d groupant

un grand nombre de monomères. On

distingue

^deux régimes ^{« 2d » :} 1) ^{dans le} régime ^dilué (chaînes séparées), ^{on attend un} coefficient de diffusion inversement

proportionnel

^{à la masse molé-}

culaire. Pour les modes internes, ^on obtient la loi de dispersion

039403C9q =T/~s d1/3 q10/3;

2) ^{dans le} régime semi-dilué (caractérisé par ^une longueur de cohérence 03BE2 supérieure ^à d), ^on

a des modes de type

gel

pour

$$ q03BE2 1 03C4-1q

⁼

Td1/3 q2/03BE4/32

^{et la structure} des modes internes pour

q03BE2>1.

En augmentant ^la concentration, ^on passe continûment du comportement « ^{2d » au} comportement dynamique ^{« 3d »} des solutions massives.

Abstract. ^- We investigate ^the dynamics ^of polymer solutions confined in ultra-thin slits (20-200

Å),

including both excluded volume effects and hydrodynamic interactions through ^a scaling analysis.

1) ^In the dilute regime, for chain extension R larger ^{than the} slit thickness d, the overall translational diffusion coefficient Dt ^is predicted ^{to scale like}

$$

which is much larger than the value expected ^{from the} Debye-Bueche approximation

$$

(where

RF2

^{is the chain} size measured in the slit plane). ^For the internal modes, ^{we find a structure of} the Rouse type ^at

wavelength

larger than d. This is due to screening ^of

hydrodynamic

interactions, ^an intrinsic feature of slit systems. ^The eigenmode frequency scales like

$$ 039403C9q = Td1/3

q10/3/~s

2) ^{In the « 2d »} semi-dilute regime characterized by ^a monomer-monomer correlation length 03BE2 larger

than d, ^we find modes reminiscent of a gel ^for q03BE2 1

(with a

relaxation rate $$ ⁾

and

internal mode structure of a confined chain for q03BE > 1. The self-diffusion coefficient of a chain is

predicted ^to scale like Ds ~ N-2 c-2 d-1/3. 3) ^At higher concentration, ^we reach the ^« 3d ^» semi- dilute regime (03BE d) ^{and we recover the} dynamics of the bulk polymer ^solution

(Ds ~

^N-2

c-1.75).

Classification Physics ^Abstracts 56.60 ^- 62.10 ^- 66.10

1. Introduction. ^- We consider flexible

polymer

chains

trapped

in ultra-thin slits. There are

already

measurements on

partitioning

and diffusion of macro- (*) Laboratoire associe au C.N.R.S.

molecules _{in pores}

having

^the

geometry

^{of a slit}

[1].

Such _pores can be realized in various _ways, for instance in porous

glass prepared by leaching

^{or with}

cleavage

^{fractures in mica. However such} systems

give

little information on

dynamical

^behaviour.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197700380100128500

1286

Lamellar

phases

^of

lipid-water

systems

]2[

^would

seem at first

sight

^{a more}

promising

system ^{to test the}

complete dynamical

behaviour of confined

polymer

solutions

by using light scattering techniques.

^We expect ^that

hydrophilic

^flexible

polymers

^{would be}

trapped

^{in the water sheet of the lamellar}

phase,

^of

thickness

varying

^{from a few to 200}

^A.

The statics of

polymer

chains confined in a tube and in a slit has been discussed

by

^a

scaling approach [3].

We have

recently

studied the

dynamics

^{of a}

single

chain

trapped

^{in a tube}

[8].

^We

investigate

^{here the}

statistical motions of

polymer

solutions confined in a

slit

using

^the

dynamical scaling

^method

[4]

^{which is}

known to

give reasonably good

^{results for bulk}

poly-

mer solutions. Our

assumptions

follows reference

[4] :

we

postulate

that friction between monomer and solvent is

dominant;

^we

ignore monomer/monomer

friction and also all internal

viscosity

effects. All our

discussion is restricted to chains dissolved in

good

solvents where monomers

repel

each other.

2.

Summary

of static behaviour : the

concept

^of

« blobs ». ^- The

configuration

of the chains under the

same conditions has been discussed

by

^{Daoud and}

De Gennes

[3]. They

^{find five}

regimes depending

upon the monomer concentration _c, the

polymerization

index N and the slit thickness d which are

represented

on the

diagram

^of

figure

^{1 .}

2.1 SINGLE-CHAIN REGIMES. ^- In domain

A,

^we

have a « 3d » conventional dilute solution of separate

coils of radius

RF, (RF3 ^d),

^where

RF3

^{is the}

^Flory

radius for bulk solution

In domain C

(RF3

>

d),

^the coil becomes a flat

pancake

of thickness d and radius

RF2.

^{We reach}

a « 2d »

behaviour,

characterized

by

^a

Flory

^radius

RF2 varying

^{like N3/4. The chain can be}

represented

^{as a}

real two-dimensional chain : the static units are not the monomers of size _a, but the blobs of size d contain-

ing

gd ^monomers

(Fig. 1) :

^{in a} scale smaller than d we

have a « 3d » behaviour and the blob size d is related

to gd by eq. (1)

on a

larger scale,

^{we have a o 2d »} behaviour and the ^« 2d » chain

extension, according

^to

Flory,

^is

This blob

picture

is very useful in

understanding

^the

dynamical

behaviour. If we represent ^{the chain}

by

^a

pancake

^{with a uniform monomer} concentration

c ⁼

N/ Rl d

^{as in the}

Debye-Bueche approximation,

we shall see that it would lead to too

large

^friction

effects. Because the monomer concentration is _very

inhomogeneous,

the solvent can flow

easily through

the chain.

To describe the

dynamical behaviour,

^{we need to}

know the elastic

restoring

force which _opposes chain

FIG. 1. ^- The five regimes of flexible polymer ^chains trapped ^{in a}

slit. We investigate the statistical motions of chains exhibiting ^a

« 2d » behaviour : 1) ^{in the dilute « 2d »} regime where the isolated chains can be represented by ^a 2-dimensional arrangements ^of blobs of size d (the ^{hatched zone} corresponds ^{to the} Debye-Bueche

model where the concentration is supposed uniform); 2) ^{in the}

« 2d » semi-dilute regime, ^{where « 2d » chains} overlap. ^{The blobs}

are statically correlated (on ^a length’ 2) ^but hydrodynamically independent ^{because of slit wall} proximity effects which screen the backflows on a scale d.

deformation. From the

scaling analysis

of reference

[3],

the elastic free _energy associated with a chain extension

2.2 OVERLAPPING ^{CHAINS. -} If

starting

^from

region

^{C we} increase the

concentration,

separate

pancakes overlap

^{and we reach the « 2d »} semi-dilute

regime represented by

^{domain D.} The coherence

length C2

^which

represents

the distance between chain crosslinks is

larger

than d. The chain can be seen as a

succession of

non-interacting pancakes containing g monomers. 03B62 is

^related to g

through

eq.

(3)

with the

relation g

⁼

cç 2 d,

^{it leads to}

At scales

larger

^than

ç 2,

^{the chain is} ideal and its extension R is

For the

dynamical behaviour,

^{the SD} solution behaves like a

gel

^at

frequencies higher

^{than the}

entanglement

relaxation rate. The

gel

^{modulus E is}

propostional

to the number of crosslinks :

If we go ^on

increasing

^the

concentration, (2

gets smaller and we reach the threshold line SN defined

by C2

^{= d.} Above SN all correlation

lengths

^are

smaller than d and local correlations are identical to those of bulk solutions. The chain extension is R ⁼

(Nlg) 112 (3’

^where

C3

^{is the} «3d» coherence

length

^defined

by :

In domain

E,

^R > d and each chain

occupies

^a

region

^{with the}

shape

^{of a}

pancake.

In domain

B,

^R d and the chain

occupies

^a

spherical region.

For both domains the

gel

^{modulus is}

The aim of our

investigation

^{is to} derive the

dyna-

mics of confined chains in the two-dimensional dilute

(o C »)

^and semi-dilute

(o

^D

») regimes.

^We

follow the

analysis

of reference

[4]

^for bulk solutions which

apply

^{to domains}

^{A, B,}

^{E. We} expect ^{a smooth}

crossover between the 2d-3d

dynamical properties

^on

the threshold lines

separating

^domains

A,

^{C and}

E,

^D.

3. The ^« Rouse ^» behaviour of confined chains. ^- If friction between monomers and solvent is

dominant,

the

equation

of motion for the nth monomer

including hydrodynamic

interactions can be written as

where

Bo

^{is the}

mobility

^{of a}

single

^monomer,

Tnm

^the

blackflow tensor and _(p. the total force

acting

^on

monomer m. (pm includes both external

cpm

^{‘ and}

internal

(pint

^t contributions

(cpm

^‘

represents

^{the local} elastic force due to

neighbouring

units and excluded volume effects between distant

monomers).

In an infinite medium

Tnm

^is

long

range and decreases like the inverse distance r

= I r n - r m I

The essential

point

^{for the}

dynamics

of confined chains is that in a slit the _range

of

^the

backflow

^kernel

is d. The

screening

^{of the} backflow is due to the

boundary

conditions on the walls. At distance r much

larger than d,

^the dominant factor in

T(r)

^is exp -

nr/d.

This result can be understood

qualitatively by treating

the wall effect

through

^{a method of}

images ] 5[

^{as shown}

on

figure

2. Thus the sum qJm in eq.

(11)

^involves

only

^a

finite number of monomers which are within a dis- tance d from n.

If we are interested _{in space} variations which are

slow

(qd 1)

^{we may}

replace

qJm

by

Tn in eq.

(11)

^and

we arrive at an

equation

^{of the Rouse form}

FIG. 2. ^- The method of images ^{to treat} boundary conditions at the walls of the pore : ^at distances r > d, the flow induced in a slit

by ^the point ^force T ^can be treated as the flow produced by (p ^{and its} images [5] (represented ^{as dotted} sources) ^{in an} infinite fluid. The

flow becomes very small at distances larger ^{than d.}

1288

where B

is a renormalized monomer

mobility :

We can

estimate B by

^a

scaling

argument. ^{Because the} range of the kernel is

just equal

^{to the} blob size d of chain units in the 2d

regime (domains C-D)

^we

have _gd terms in the sum

giving

^a contribution d -1.

Thus we have

The

mobility

^{of one blob is}

Successive blobs are

independent dynamically

^and

behave like

impenetrable spheres.

Remark. ^- We have taken into account

only

^the

geometrical screening (d)

^{due to}

boundary

^effects.

There is also a backflow

screening (K-1)

^{due to}

neighbouring

^{monomers discussed} in reference

[4].

As shown in reference

[4],

^the

screening length

^{K -1}

is identical to the monomer-monomer correlation

length.

^{It leads to}

^K-1

= din domains

C, D, K -1 == 03B63

in domains

B,

E, ^and K - 1 =

RF3

^{in domain A. The}

screening by surrounding

^{monomers has therefore to}

be taken into account

only

^{in the 3d}

regime

^where

^K-1

becomes smaller than d.

Conclusion. ^- In the two-dimensional dilute and semi-dilute

regimes,

^{the chains can be}

represented by

^{« 2d » Rouse} chains made of a succession of

impenetrable

blobs of size d. The blobs are

statically

correlated

(excluded

^volume

effects)

^{but are}

hydro- dynamically

uncorrelated.

4. « 2d » dilute

regime.

^{- 4.1 CHAIN MOBILITY.}

COMPARISON WITH DEBYE-BUECHE MODEL. ^- As

explained

ⁱⁿ

part 3,

^the

equation

of motion for the nth monomer of a

single

^{chain can} be written as :

Summing

^{over the index n, all the} internal forces must add _up to zero and we get ^an

equation

^{for the centre}

of mass

which defines the overall

mobility

of the chain

By

Einstein’s

relation,

^the

corresponding

^diffusion

coefficient is

Comparison

^{with the}

Debye-Bueche

^{model. - In}

this

paragraph, following

^the

Debye-Bueche approxi- mation,

^we

neglect

^the fluctuations of the monomer

concentration. We represent ^{the chain}

by

^a

pancake

of size

RF2

^and thickness d

containing

^a concentration of monomers

(see

^{hatched zone of}

Fig. 1).

As shown

by Debye [6],

^the

pancake

^{as a} porous medium excludes the solvent flows except ^{on a} thickness

KDBI

^defined

by KD

⁼

cBo/r¡; 1.

^Dimen-

sionally Bo - 6 7rt7, a

^and

KDBI

is very small

compared

to RF2 (KDS RF2 ^’ .Na/d > ^1).

^{Thus the}

pancake

^moves

as a

rigid body

in the slit.

We calculate the friction force

acting

^{on the}

pancake moving

ⁱⁿ the slit with the

velocity

^{U. A slit can be}

considered as a Hele Shaw cell

[7]

^{and the} flow induced at

large

^distance

by

the motion of the

pancake

^is

governed by

^the

equation

plus

^the

boundary

^conditions

In ^a porous medium there is no

boundary

^condition

for the

tangential velocity Vo (vo

^{falls to zero in a}

distance d for a

rigid pancake

^or

KDB

^{in our}

case).

Eqs. (20-21) imply V2 p

^{= 0.}

The correct solution

forp

^is

It leads to a viscous

dissipation :

^.

The friction force

acting

^{on the}

pancake

^is

In the

Debye-Bueche model,

^we expect ^{a chain}

mobility

Comparing

this result with _eq.

(18)

We find that the chain is much more mobile than

expected

^{from the}

Debye-Bueche approximation.

Conclusion. ^- A confined chain is much more

permeable

^than

expected

^{from a uniform concentra-}

tion

picture.

^{Because the} concentration fluctuates

largely

^{on a scale d}

(as represented by

^the

blobs),

^the

solvent can

easily

^flow

through

^{the chain. It leads to a}

much

larger

^chain

mobility.

4. 2 INTERNAL MODE STRUCTURE. ^- 4. 2.1 Funda- mental mode

qRF2

^{= 1. -} The elastic

restoring

^force

which _opposes the chain extension

(RF2 --+ RF2

⁺

^6R) is, according

^to eq.

(4)

If the time rate of

change

^{of R is}

bR,

^the

velocity

^{of all}

blobs is of order

03B4R

and the friction force

using

^the

result

(17)

^{is :}

Equating

^{these two} forces leads to the relaxation rate of the first mode :

4.2.2 Internal modes

qRF2

> 1. ^- We derive the

dispersion

^law of internal modes

by

^a

scaling

assump-

tion, following

^reference

[4].

^We determine the struc- ture of internal modes

by using

^the

product

^{of two-}

dimensionless functions

f (qR2) g(qd)

a)

The function

f,

which describes the crossover

between the diffusion mode of the whole confined chain

(qR2 1)

and the internal modes ^must have the

following properties :

- for

qR2 1,

^{we must recover}

f1Wq

⁼

^D, q 2

Thus

f (x --+ 0) _+ X 2.

^We

verify

^that

D,

⁼

R 2/0 1

^is

consistent with result

(14) ;

- for

qR2

⁼

1, f (1)

⁼

1 ;

- for

qR2 > 1, f(x) -

^{xP. The}

exponent

p is derived from the condition that

Ao-)q

^is

independent

of

RF2 in

^{the limit}

qRF2 >

^{1. It leads}

^{to q}

⁼

^10/3

and to a characteristic

frequency

b)

^The

function 9

^{describes the crossover between}

the two-dimensional

(qd 1)

and the three-dimen- sional

(qd

>

1) regimes :

To

satisfy

the condition

Acoq (qd > 1) independent of d,

^{we must}

have ’q

^{= -}

1/3.

^{It leads to}

in agreement ^{with the Zimm}

eigenfrequency [4]

ⁱⁿ

3 dimensions.

Remark. ^-

A(o 17

is the characteristic

frequency

associated with the monomer concentration fluctua- tions measured

by

^inelastic

light scattering.

^{We can}

derive from

ð.Wq

^{the structure of the N}

eigenmodes

_Tp

(with

p ⁼

N,

^...,

N/J, ..., ...1)

^{which come into}

play

in acoustic measurements. With a wave vector q,

we

explore

^a

region containing n

^{monomers such as}

q-1 - (n/gd)3/4

^{d in the limit}

qd

^{1. The corres-}

pondence

^between p ⁼

N/n and q is q - p3/4.

^We

expect

^{then for the}

eigenmodes

^{of a} confined chain

In the limit _p >

N/gd,

^the

correspondence is q - p3/5

and

1/Tp - p3X3/5 (= ^p1.8).

5. ^« 2d » semi-dilute

regime.

^{- 5 .1 GEL BEHAVIOUR}

(q’2 1).

^{- At}

frequencies higher

^{than the relaxa-}

tion time

Tr

^for

complete disentanglement

^{of one}

chain, overlapping

chains behave like a

gel,

^{of elastic}

modulus

ED

^defined

by

eq.

(8).

^{In the limit}

qC2 1,

the

polymer

^{solution can be described}

by

^{a continuum}

theory

^{and we}

specify by

^{r the}

displacement

^{of the}

gel

^and

c(r)

^the

polymer

concentration.

For

longitudinal

^{modes r =} ro

e"7x e-t/ ,

the elastic

restoring

^force per unit volume on the

gel

^is

where

ED = kB TldC’ (eq. (8)).

We

again

^{use the blob}

concept

^to write the viscous

11

force. The blobs’ frictions

(Bb- 1 = 6 nfls d )

^{are addi-}

tive and

Balancing

^{these two}

forces,

^{we are led to a relaxation} time

1290

On the threshold line

SN,

^defined

by (2 = ’3 ^{= d,}

we have a smooth cross-over with the o 3d »

gel ,

relaxation time

[4]

5.2 SINGLE CHAIN BEHAVIOUR

(qC2 > 1).

^{- For}

q(2

>

1,

^we expect ^{that the} characteristic

frequency LBWq

will become identical to the

frequency

^measured

for one isolated confined chain

(eq. (31), (32), (33)).

The cross-over between the

gel

^behaviour

(q(2 1)

and the

single

chain behaviour

(q03B62

>

1)

^{can be}

described

by

^a

scaling

^law

hypothesis :

where the characteristic

frequency

^d

separating

^the

two

regimes

^is

given by (30)

and

f(x) -

xP,

g(x) - xq

^for x >> 1.

p is determined

by

the condition that

Ao-)q

^should

be

independent

^of

03B62,

which leads to p ⁼

10/3

^and

the result

(32) ; q by

^{the condition that}

Aa)q

^{should be}

independent

^of

d,

^hence to q ^{= -}

1/3

^{and the result}

is eq.

(33).

5. 3 REPTATION TIME AND SELF-DIFFUSION COEFFI- CIENT. ^- We calculate the time

Tr

^to

disantengle

^one

chain of total

length

^{L =}

(N/g) C2 following

^refe-

rence

[4].

^{The friction to}

pull

^{one chain in} the network of the others is

proportional

^{to the} blob number

(N/gd).

The diffusion coefficient involved in this _process is then identical to the diffusion coefficient

Dt

^{of one}

isolated confined chain

(19)

^and

Tr,

^defined

by

^the

diffusion

law,

^is

given by

The

macroscopic

self-diSusion coefficient

D.

^{of one}

labelled chain is related to the chain extension

R(c) by

i.e.

6.

Concluding

^{remarks. -}

Statistically

^{a chain}

confined in a slit behaves as a two-dimensional self-

avoiding

^{walk if we take as units not the monomers}

(size a)

^{but the blobs}

(size d).

^The main result of the

dynamical analysis

^{is that these « 2d » chains of blobs}

behave as Rouse

chains,

^{i.e. the units are}

hydrodyna- mically

uncorrelated.

(In particular

this leads to a

diffusion coefficient for a

single

^chain

inversely

pro-

portional

^to the molecular

weight.)

Experimentally

^{the main} directions of research appear ^to be the

following :

1)

In porous media the

cooperative

diffusion coef- ficient of a semi-dilute solution should be measurable rather

simply.

^{A not too dilute}

polymer

^solution

penetrates

easily

inside the _porous medium

(1).

^If

the porous

sample

^{is then}

placed

^{in contact with a}

solution of

slightly

lower concentration some

polymer

will diffuse back from _{the pores} to the bulk solution : the time

required

^for

equilibrium

^{will lead to a deter-}

mination of D.

2)

^We

expect

^{that the}

study

^of

hydrophilic

^chains

dissolved in the lamellar

phase

^{of a}

lipid-water

system will allow for more detailed

investigations

^{on the}

dynamics

of confined chains. The use of

light

^scat-

tering [9] (2)

should detect both

cooperative gel

^modes

(qÇ2 1)

^and

single

chain modes

(qÇ2

>

1)

^{in the SD}

regime.

Self-diffusion coefficients in the dilute

regime

are also accessible

(2).

The Rouse behaviour may also be tested more

directly

^{in the dilute}

regime by

^a

simple viscosity

measurement in shear flow. If the flow direction is

parallel

^to the sheets and the flow

gradient

^{is normal}

to

them,

^the

viscosity il

is related to the monomer

concentration c in the

hydrodynamically independent

blobs

picture by

where _Tb ⁼

cd 3/gd

^{is the} volume fraction of the water

phase occupied by

^{the blobs. If} the chains had behaved

as

impenetrable pancakes,

^{as in the}

Debye-Bueche approximation,

^{we would} expect

where qJp

= - RF2 d

is the volume fraction

occupied

N ^F2

by

^the

chains,

^Thus

simple viscosity

measurements in

polymer-lipid-water

^mixtures may allow ^{a test} of the 2d Rouse behaviour of confined chains.

Acknowledgments.

^- The author has benefited from many discussions with Professor P. G. de Gennes and with F. Rondelez.

(1) Audebert, R., private communication.

(2) Rondelez, F., private communication.

References [1] COLTON, ^C. K., SATTERFIELD, ^C. M., LAI, ^C. S., AICHE Journal

21 (1975) ^289.

[2] GULIK-KRZYWICKI, T., TARDIEU, ^{A. and} LUZZATI, V., ^Mol.

Cryst. Liq. Cryst. ⁸ (1969) ^285.

DE GENNES, ^P. G., Solid State Phys. Suppl., ^{to be} published.

[3] DAOUD, M., ^DE GENNES, ^P. G., ^J. Physique ³⁸ (1977) ^85.

[4] ^DE GENNES, ^P. G., Macromolecules

9 (1976)

^587.

[5] SONSHINE, R. M., BRENNER, H., Appl. Sci. Res. 16 (1966) ^273.

[6] DEBYE, P., BUECHE, A., J. Chem. Phys. 16 (1948) ^573.

[7] ^{CHIA-SHUN YIH,} Fluid Mechanics (Mc ^Graw Hill) 1969, p. 382.

[8] BROCHARD, F., ^DE GENNES, P. G. (submitted ^for publication).

[9] ADAM, M., DELSANTI, M., JANNINK, G., ^J. Physique ^{Lett. 37} (1976) L ^53.