Non-linear behaviour of gels under strong deformations

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Non-linear behaviour of gels under strong deformations

S. Daoudi

To cite this version:

S. Daoudi. Non-linear behaviour of gels under strong deformations. Journal de Physique, 1977, 38

(10), pp.1301-1305. �10.1051/jphys:0197700380100130100�. �jpa-00208700�

NON-LINEAR BEHAVIOUR OF GELS UNDER STRONG DEFORMATIONS

S. DAOUDI

Collège

^de France, 75231 Paris Cedex

05,

^France

(Reçu

^{le 18 mai}

1977, accepté

^le

28 juin 1977)

Résumé. ²⁰¹⁴ Nous considérons un

gel

de chaînes de

polymères

^{flexibles en} bon solvant : une forte contrainte constante 03C3 est

appliquée

^au temps ^{0. Nous} étudions la réponse

élastique

prompte ^{à des} temps t tels que la fraction chaînes/solvent n’a pas changé. ^{La théorie est basée sur} les lois d’échelle établies par divers ^auteurs pour ^une seule chaîne en forte déformation. On prédit ^{une relation}

contrainte-déformation à grande déformation 03B1 de la forme 03C3 ~ 03B15/2 en traction et 03C3 ~ 03B13/2 en

cisaillement simple. ^Nous présentons ^aussi quelques remarques ^sur les diagrammes de diffusion de neutrons, la birefringence ^et les coefficients de sédimentation _que l’on attend pour ^un échantillon déformé. Ces considérations ^ne

s’appliquent pas

^aux caoutchoucs purs.

Abstract. ²⁰¹⁴ We consider a gel ^{of flexible}

polymer

^{chains in a} good ^{solvent : a} strong ^constant

stress 03C3 is

applied

^at time 0. We study ^the prompt ^elastic response ^at times t such that the chain/solvent

fraction has not changed. ^The theory ^{is based on the}

scaling

^laws established by various authors for a

single chain under strong deformation. The stress-strain relation at large ^{strain 03B1 is}

predicted

^to

be ^{03C3 ~} 03B15/2 in

longitudinal

deformation and 03C3 ~ 03B13/2 in

simple

^{shear. We also} present ^{some tentative} remarks on the neutron scattering patterns, ^the

birefringence

and the sedimentation coefficients, expected ^{for the distorted}

sample.

^These considerations do not apply ^to pure rubbers.

Classification Physics ^Abstracts

36.20

1.

Principles.

^{- Our}

present understanding

^{of the}

organisation

^{in an}

entangled polymer

^{solution is}

based on

scaling

concepts

[1, 2].

^{The basic} parameter for a solution of concentration c is the correlation

length 03BE(c).

^For

good

^{solvents this}

length

behaves like

where

RF

^{is the}

single

chain radius in the same

solvent,

and c* is the critical

overlap

concentration

(N being

^the

polymerization index).

^{At c = c* the}

different coils are

just

^{in contact.}

A

polymer gel

^{can be} described in _very similar terms : at

equilibrium,

^{in a}

good solvent,

^{the chains} tend to

separate

^{from each}

other,

^{but the cross links} force them to remain in contact : thus the

equilibrium

concentration c is

essentially equal

^{to c*}

(apart

^{from a}

numerical

constant).

^{The whole}

Flory theory

^of

gel swelling [3]

^is summarized

by

^this sentence : all the

dependences

^{on solvent}

quantity

^{are included in the}

single

chain radius

RF,

for which the

Flory analysis

of the

single

^chain

problem gives

^us very

good

^nume-

rical values.

For small deformations

(characterized by

^{a strain}

a)

the stress in the

gel

^{is a} linear function of the strain

where the elastic modulus scales like

[3] :

(Tis

^the

temperature :

^{we use} units where Boltzmann’s constant is

unity).

Eq. (1. 4)

holds for both

longitudinal

^{and transverse}

shear

(the only

difference

being

in the numerical

prefactors

^{- which we}

ignore systematically

^{in the} present

discussion).

Our aim here is to extend _eq.

(1.3)

^to

large

^defor-

mations

(a

>

1).

1.1 MACROSCOPIC VIEWPOINT. ^- For a

single chain,

we know from the work of Fisher et al.

[4]

^and

Pincus

[5]

that the free _energy at strong deformations is of the form

where r is the end to end

distance,

^and

RF --

^aNv

[For

^our purposes ^{v =}

3/5

^and

(1 - v)-1 - 5/2.]

Eq. (1. 5) ignores

^{some minor}

(logarithmic)

corrections which are

negligible

^at strong deformations.

Consider first the case of a

longitudinal

^deforma-

tion,

^{where the}

sample changes

^its

length

^from

Lo

to

aLo.

^Each

chain,

between its

junction points,

^{has an}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197700380100130100

1302

FIG. 1. ^- Schematic representation ^{of a} strongly elongated ^net- work, showing strings ^between junction points. ^Each string ^{is a}

collection of Pincus blobs. The amount of wiggling ^{of an individual} string ^{has been} purposely underestimated in this drawing, ^{to show}

the structure most clearly.

extension r

= ç

II ⁼

aRF

^{and a size at rest}

RF,

^{as shown}

on

figure

^{1. Then the} energy per

cm3

is

where

c/N

^{is the chain} concentration. We assume that

during

^{the time of the}

experiment,

^the solvent does not have

enough

^{time to}

permeate

^{in or out of the solu-} tion : the concentration c is then

unchanged by defor-

mation. In

fact,

^{for a}

sample

^{of width}

W,

^the permea- tion time would be of order Tp ⁼

W’lDc

^{where the}

cooperative. diffusion

coefficient

Dc

^{is known both}

from

scaling

theories and from

experiments using

^the

inelastic

scattering

^of

light [6].

^With

typical

^values

W ⁼ 1 mm and

Dc

⁼

^10-’ cm2/s

the time rp ^is of order one

day.

From eq.

(1. 5)

it is then _{easy to} find the stress

where S ⁼

So

^{a-1 is the cross sectional area.}

Inserting

eq.

( 1. 5)

^we get

In this formula one factor a is

trivial,

^{and is due to the}

reduction of cross-section. But the

remaining

^factor

rx3/2

is

interesting.

^{A similar}

argument

^{for shear} deformation

gives

the difference between

(1. 8)

^and

(1.9)

^{is related to the}

area on which forces are

applied.

^{This area is inde-}

pendent

^{of a for}

simple shear,

and varies like a-1 in

longitudinal

^stretch.

1. 2 THE LOCAL PICTURE : STRINGS AND PINCUS BLOBS.

-

Following

^Pincus

[5]

^{we can}

give

^{a much more}

physical picture

of the distorted system

(Fig. 1).

^Each

chain is connected to the network

by

^{its two extre-}

mities,

^{and feels a}

stretching

force ± ^{0. We call} the stretched chain a

string

^{with a}

length jjj.

^{At sizes}

r>03BE (larger

^{than the mesh of the}

^network)

^{we expect}

an affine deformation : this

implies

We can relate the force (P to the

macroscopic

stress a in

longitudinal

deformation. We have

c/N strings

per

cm3,

^each

carrying

^{a tension 0 over a}

length ç II.

^Thus

Consider now the inner structure of one

string :

^it

can be visualised as a

nearly-linear

sequence of smaller units

(Pincus blobs),

^each

containing

gp ^monomers and

having

^{a size}

çp.

^At distance r

çp

^{the force 0 is a}

weak

perturbation,

^{and we} find the correlations of a

swollen chain at rest. This

implies

that the relation between gp and

çp

still has the excluded volume form

At size r >

çp

^the force 0 becomes a strong per- turbation. At the crossover

(r

⁼

çp)

^{we must have}

Finally

^{we know that the whole}

string

^{is a}

nearly-

linear _sequence of

N/gp

^Pincus

^blobs,

^{and thus}

Combining

eqs.

(1.10)-(1.14)

^we get

and

This with _eq.

(1.11)

^{then restores the stress strain}

relation

(1.8).

2. Correlations in the distorted state :

longitudinal

case. ^- It is

important

^to observe first that the fraction of the volume

occupied by

^the Pincus blobs is small at

large

^{a. This} fraction is indeed

as can be seen from _eqs.

(1.10), (1.15).

Thus for _many

properties

^{we may} consider the

strings

^as uncorrelated. The average distance d

(measured

^{normal to the} direction of

stretch)

^between

strings

is such that

d 2 ç

^{II =}

^RF,

^{and thus}

Each

string

^{is not}

quite straight,

^{but has} fluctuations

(transverse

^{to its own}

axis)

^{of order}

çl.,

^where

ç

^BI

corresponds

^{to a random walk of}

N/gp

^{steps, each} step

being

^{of order}

03BEp.

^Thus

giving ç.l 1"’.1 RF ^{a-1/4 .}

^Hence

çl./ ç II 1"’.1

^{rx- 5/4 is}

^small,

but

039E./d

⁼

^rxl/4

^is

larger

^than

unity.

These remarks lead us to the

following conjecture :

the coherent neutron diffraction

pattern

^{should be}

qualitatively

^{similar to the}

scattering

^{due to}

indepen-

dent

(strongly aligned)

^Pincus

strings.

For a

given

^wave vector q, the coherent

scattering intensity I(q)

is related to the

pair

correlation function

by

^a Fourier transform. Our

conjecture gives

^the

following

features for

I(q).

a)

^{For low}

angle scattering I(q)

^--+

1(0)

^measures

the number of monomers in one correlated

region

b)

^At

intermediate q

^values

I(q)

^becomes

^smaller,

and

anisotropic. For q

⁼ qll II

(along

the stretch

axis)

the

intensity drops

^when

q - 03BE - ’ = ot - ’ ^{RF 1.}

^For

q ⁼ ql

(transverse)

^the

drop

^is

at q ~ 11 = rxl/4 Ri 1.

c) For q

>

03BEp 1

^{we recover the}

isotropic scattering

from a

single

^{chain in}

good

^solvents

[7]

These features are summarized in

figure

^{2. It must}

be

emphasized

^that

figure

^{2 should}

apply only

^{if the}

crosslinks do not contribute

strongly

^to the coherent

scattering.

^{In the}

opposite ^limit,

^a correlation

peak might

appear in

I(qll)

for well-calibrated

gels [8] :

^this

would reflect a

quasi-periodic repetition

^{of nodes}

along

the stretch direction.

FIG. 2. ^- Conjectured pattern for coherent neutron scattering by

a strongly elongated ^network. q is the wave vector

(qll

^{refers to the} projection along the direction of stretch).

3. Stress-induced

birefringence.

^{- In a}

strongly

stretched

gel,

^our

description

^{in terms of}

strings (of length jjj

^{II and width}

^{ç 1.) suggests}

^{that this inhomo-}

geneous distribution of monomers will

give

^{rise to an}

anisotropic

dielectric constant tensor E at a scale

larger

than

çp.

^{In the present section we discuss E}

^using

^the

approach by

Landau and Lifshitz

[9]

for hetero- geneous media. We define electric induction fields D and E related

by :

where 8,

^{is taken to be}

isotropic

^and

proportional

^to

the local concentration :

Thus the present calculation does not include _any effect due to an intrinsic

anisotropy

^{of the monomers.}

Separating

the small deviations from the volume average, ^we may write

and

We return with these relations to

(3.1)

and take the average

Taking

^the

divergence

^of

(3.1)

^we

obtain,

^{to first} order

If we

interpret

^{this as a Poisson}

equation

^{we find}

Then,

^from

(3. 5)

and the definition of

E(D

⁼

EE)

^we

have

Introducing

the Fourier transform and

using (3.2)

leads to

For in

longitudinal

deformation we assume a

simple Lorentzian

form

(at q ç; 1) :

where qll ^{and ql}

^{are the}

projections

^of q

parallel

^and

perpendicular

^to the stretch direction

respectively.

1304

The dielectric constant

anisotropy

^{is then}

given by

From a

straightforward integration

^limited

by

qçp

^{1 and after}

simplifications (a

>

1)

^we

finally

arrive at :

We see that the

anisotropy

^of

the

dielectric tensor is not

proportional

^{to the} stress tensor : this is rather

exceptional,

^{and is}

specific

of the non-linear behaviour discussed here.

4. Sedimentation and

cooperative

diffusion. ^-

Expe-

riments

measuring

the relative friction of

polymer/sol-

vent should also

give

^{us some} information on the

pair

correlation function

y(r),

^if

they

^{can be}

performed

^at

large

^{a and in the}

required

time scale.

1)

Consider first a

permeation

^or sedimentation

experiment.

This would be

extremely

^{hard to}

perform

in

practice,

^{but is}

conceptually simple.

^{In a suitable}

frame,

^{each monomer is}

subjected

^{to a force}

f,

^the solvent

being

maintained immobile on the _average.

We define a sedimentation coefficient s for this case as

the ratio of the average ^monomer

velocity

^{v to the}

force ^.

In terms of the Oseen

hydrodynamic

^tensor

(where

p, v ⁼ x, y, z, ns

being

^{the solvent}

viscosity),

the drift

velocity

^of one monomer due to the force

applied

^{at its}

surroundings

^is

[10] :

Inserting

^for

y(r)

^{the form}

appropriate

^for

fully aligned,

uncorrelated rods

(i.e. following

^the

conjec-

ture of section

2)

^{we arrive at}

Eq. (4.4)

may also be derived

directly

^{from the}

friction law for a viscous fluid

flowing

^{around a}

cylinder

^{- the} argument ^{of the In term}

possibly being slightly

different. In

practice, using

eq.

(1.10), (1.15)

we may

simplify

eq.

(4.4)

^to

where so is the sedimentation coefficient of the undis- torted

sample

2)

^Inelastic

light scattering experiments might

^be

easier to

perform. They

^{measure a}

cooperative diffusion coefficient [6, 8]

Again

^{this should} vary

essentially

^{like rx - 1 : our}

conjecture

^{leads to a}

slowing

^{down of} fluctuations

under stress. ^’

5. Conclusions.

-1)

^The

scaling

^{ideas of}

Fisher, Pincus,

and others on a

single

chain under strong deformations could

possibly

be checked on

gels,

^if

some

gels

^{can stand the} strong values of a which are of interest

(e.g.

^{a N}

10).

2)

^{It is}

tempting

^to

apply

^{the same ideas to solutions}

of

entangled

^{chains :}

provided

^{that the}

experiments

^are

performed

^{in a} time scale shorter than the

disentangle-

ment time

Tr,

^the

entanglements

^are fixed and the behaviour should be

gel-like.

^In

fact,

^{in a} first draft of the present paper, ^{we were}

putting

^the

emphasis

very much on

solutions,

since here we . are not limited in the a value

by

rupture

problems.

But there is a serious

difficulty :

^{in an}

entangled melt,

^for

instance,

^{the dis-}

tance between consecutive

entanglement points

^{on one}

chain _may be of order k - 100 monomer units or more

[11].

^{Thus we} may define

strings

^{of v units.}

But,

in this _case, the

strings

^are

strongly overlapping

^{and are}

not

independent

^{of each} other. The net result is that the

strings

^are

essentially ideal,

and the Fisher-Pincus nonlinearities are not

expected

^{to occur here. A similar}

effect _may well take

place

ⁱⁿ semi-dilute solutions : as

explained

in reference

[1, 2]

^{we then have}

independent

blobs

of g

monomers,

where g - (ca3)-5/4.

^The

number of units between successive

entanglements

Ve scales like _g, but _may contain a

large

^{factor k :}

ve(c)

⁼

kg.

^{Thus the}

strings

of interest _may contain

a

large

^{number of}

^blobs,

and should then behave

ideally.

’

3)

^{We have not found many}

experiments

^{which can}

be

meaningfully compared

^{to our} model : mechanical tests on

gels

^are

generally

restricted to small or

just

moderate deformations. Data on uncrosslinked _sys- tems have been often taken on

liquid

^filaments

extruded from a

spinneret,

^{but our}

requirement

^of

rather low

(semi-dilute)

concentrations is

rarely

^met.

However the data taken

by

Hudson and

Ferguson [12]

on

polybutadiene

^{in dekalin are relevant. In their}

experiments,

^the transit time of one chain

through

^the

probed region

^is

probably

^{of order}

1/100

^{of the disen-}

tanglement

^or

reptation time [6].

Thus it is indeed the

prompt response which is measured.

Also,

^{the authors}

have

plotted

their results

(for

concentrations 6-12

%)

in terms of stress a versus the

integrated

deformation a.

The

plot

^{shows a}

plateau

for intermediate values of a

(which

^{are not} taken into account

here)

^{and then a}

sharp

^{rise for a} >

3,

^{with a -} a3. This is to be _compar-

ed with the

prediction

^of eq.

(1. 8) a - ^rx5/2.

^{But the}

number of data

points

^{on the}

high

^{a side is not}

quite

sufficient at

present.

Acknowledgments.

^{- We} have benefited from _many related discussions with G. Jannink and M. Adam.

References [1] DAOUD, ^{M. et} al., Macromolecules 8 (1975) ^804.

[2] ^DE GENNES, P. G., ^{Israel J. Chem. 14} (1975) ^154.

[3] FLORY, P., Principles ^of polymer chemistry, Cornell Univ.

Press, Ithaca NY, 1967.

[4] FISHER, M., ^{J. Chem.} Phys. ⁴⁴ (1966) ^616.

FISHER, M., BURFORD, R. J., Phys. ^{Rev. 156} (1967) ^183.

Mc KENZIE, D., MOORE, M., ^J. Phys. ⁴ (1971) ^{L 82.}

[5] PINCUS, P., Macromolecules 9 (1976) ^386.

[6] ^{DE GENNES, P.} G., Macromolecules 9 (1976) 587, 594.

ADAM, M., DELSANTI, G., JANNINK, G., ^J. Physique ^Lett.

37 (1976) ^L-53.

CANDAU, S., ^{to be} published.

[7] EDWARDS, ^S. P., ^Proc. Phys. ^{Soc. 85} (1965) ^613.

[8] ^For undeformed gels, ^{a broad} peak shows up in I(q) : ^{it des-}

cribes short range correlations between blobs : see Du-

PLESSIX, R., Th. Strasbourg ^{1975. We} expect ^this peak ^to

be removed (at ^least for q ⁼

q)

^{when 03B1 ~ 1,} because the

strings ^{are less} closely packed ^{in this case.}

[9] LANDAU, L. D., LIFSHITZ, E. M., Electrodynamics of ^continu-

ous media (Pergamon Press, Oxford) ^1960.

[10] ^This approach ^{to friction} coefficients is described in more detail in BROCHARD, F., ^DE GENNES, ^P. G., ^{J. Chem.} Phys.

(Aug. 1977).

[11] FERRY, J., Viscoelastic Properties of Polymers (Wiley, N.Y.)

1970.

GRAESSLEY, W., ^{Adv. Pol. Sci. 16} (Springer 1974), ^{and to be} published.

[12] HUDSON, ^N. E., FERGUSON, J., ^Trans. Soc. Rheol. 20 (1976)

265.