The static and dynamic structure factor of polymer-like lecithin reverse micelles

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The static and dynamic structure factor of polymer-like lecithin reverse micelles

Peter Schurtenberger, Carolina Cavaco

To cite this version:

Peter Schurtenberger, Carolina Cavaco. The static and dynamic structure factor of polymer- like lecithin reverse micelles. Journal de Physique II, EDP Sciences, 1994, 4 (2), pp.305-317.

�10.1051/jp2:1994130�. �jpa-00247963�

Classification

Physics

Abstracts

31.20 61.25H 82.70

The static and dynamic structure factor of polymer-like lecithin

reverse

micelles

Peter

Schurtenberger

and Carolina Cavaco

Institut for Polymere, ETH Zfirich, CH-8092 Zfirich, Switzerland

(Received 2J September J993, accepted 3 November J993)

Abstract. We repon a detailed analysis of the static and

dynamic

structure factor of polymer- like lecithin _reverse micelles. We

previously

demonstrated that _{one can} directly apply the results from conformation space renormalization group

theory

for semi-dilute

polymer

solutions _to

« equilibrium polymers _» such _as worm-like micelles _or microemulsions, if _one takes into _account the fact that their size distribution is concentration

dependent

ii. The present study confirms these findings and shows that this approach _can be extended to the Q- and c-dependence of the static (S(Q II and

dynamic

(S(Q, i)) structure factor.

Introduction.

Surfactant systems ^{in which}

giant

^worm-like micelles _are formed have

frequently

been used _as

model systems ^for an

experimental investigation

of the static and

dynamic propenies

of

«

equilibrium polymers

_», where the term

equilibrium (or

«

living ») polymer

is used for linear

macromolecules that _can break and recombine

[2-9].

Therefore

equilibrium polymers

are

transient _structures with _a

relatively

short live time r~, and

they

exhibit _a wealth of

interesting dynamic propenies

_on time scales both

long

or short

compared

to r~. Due to the finite live time of the

particles,

the molecular

weight

distribution of

equilibrium polymers

^such as micelles is

not constant but in thermal

equilibrium

and thus

depends strongly

on surfactant concentration

and temperature

[2].

This is

quite

different from _« classical _»

polymers

such _as

polystyrene,

where the molecular

weight

distribution is

quenched

after the

preparation procedure.

It has

previously

been shown for _a number of aqueous surfactant solutions that it is

possible

to find conditions where the micelles grow

dramatically

with

increasing

surfactant _concen- tration into

giant

and flexible rodlike aggregates. ^{These worm-like micelles} can then

entangle

and form _a transient network above a cross-over concentration _c ^* with static

propenies

that _are

comparable

to those of semidilute solutions of flexible

polymers.

It _was demonstrated that

scaling theory

for

polymers

could

successfully

be

applied

to these semidilute solutions, and that _a number of

experimentally

accessible

quantities

such _as the osmotic

compressibility,

(dH/dc

)~~, or the static correlation

length, f~, directly obey

the _same

simple

universal

scaling

laws _as do classical

polymers.

It _was also

clearly

shown that the

understanding

of the

concentration

dependence

of

dynamic propenies

was much _more

complicated

due to the

transient nature of the aggregates. ^It was found that

dynamic quantities

such _as the

hydrodynamic

correlation

length, f~,

which is measured _{on a} time scale much shoner than r~,

again closely

follow the behavior of classical

polymers. However, dynamic propenies

_on

long

characteristic time scales such _as the micellar

long-time

self diffusion

coefficient, D~,

or the _zero shear

viscosity,

_~~, exhibit _a behavior which is

dramatically

different from

those found for classical

polymers,

and _are not yet

fully

and

quantitatively

understood

[2, 8,

lo,

lll.

One of the

major

difficulties in _a

quantitative understanding

of the

dynamic properties

of

equilibrium polymers

comes from the

explicit dependence

of

D~

or ~

~

on the molecular

weight

distribution. Since micelles have _a

concentration-dependent equilibrium

molecular

weight

distribution

n(N)

with _an average N which is believed to follow _a power ^law form

N c~, this leads _{to an} additional contribution to the

c-dependence

of

D~

^and _~~

[2].

A

quantitative interpretation

of the

experimental

data would thus

require

_an

independent

determination of _a, which is difficult to achieve.

Experimental techniques

for molecular

weight

determinations in

polymer

solutions such _as

light scattering

_or osmotic pressure

measurements

always

contain contributions both from the molecular

weight

distribution _as

well _as from intermolecular interaction

effects,

I.e., from the static _structure factor

S(0 ).

In classical

polymer

solutions N _can be obtained from _a concentration

dependence

at low values of _c,

I.e.,

in the virial

regime.

However,

unambiguous

information _on N is much _more

difficult to obtain for

equilibrium polymers,

since most systems ^form

giant

worm-like micelles

even at concentrations close _to the critical micelle concentration, the cmc. This makes _a deconvolution of contributions from

n(N

and

S(0)

_very

difficult, especially

because

simple

virial

expansions

of S

(0

_{are not} valid anymore ^at the values of c/c ^* which _can be achieved for most model systems

investigated

_so far.

We have

recently

shown that _{one can}

successfully

attack this

problem by directly applying

the results from conformation space renormalization group

theory

for semi-dilute

polymer

solutions to

equilibrium polymers

^such as worm-like micelles _or microemulsions if _one takes

into account the fact that their size distribution is concentration

dependent [I ].

When

compared

to

scaling theory,

renormalization group

theory

has the clear

advantage

that

prefactors

are

derived for the power laws in the limit _c »

c*,

and that the transition from the dilute to the semi-dilute

regime

is described _as well. We have tested this

approach using

lecithin water-in- oil

(w/o)

microemulsions _{as a} model system ^for

equilibrium polymers.

^This system ^is

quite unique

in the _sense that it shows _a characteristic

sphere-to-rod

transition

normally

observed in aqueous solutions

only.

The addition of _{trace amounts} of water to lecithin _reverse micellar

solutions induces one-dimensional

growth

and the formation of

giant

^flexible

cylindrical aggregates.

^For

lecithin/cyclohexane

solutions _{at a}

given

surfactant concentration, the micellar size increases with

increasing

_{water to} surfactant molar ratio, _wo, and reaches _a

plateau

at

wow14.

^At wow

lo,

the

resulting

micellar

particles

_are

extremely large

even at very low

surfactant concentrations. Based _{on a}

systematic

static

light scattering study

^{of leci-}

thin/cyclohexane solutions,

_{we were} able to obtain detailed and

quantitative

information _on both the micellar size distribution and the intermicellar interaction effects, I-e-, on the

weight

average molecular

weight, M~,

and _on the static structure factor,

S(0).

Here _{we now} extend _our

investigations

to

Q

# 0 and

present

a detailed

study

of the static

(S(Q))

and

dynamic (S(Q, t))

structure factor as a function of the surfactant concentration

using

static

(SLS)

and

dynamic (DLS) light scattering.

We _{now use our new} data _{as a} funher

quantitative

test of _our

previous findings,

and _we shall show that within this theoretical framework _a self consistent

interpretation

of the SLS and DLS results is

possible

and that _we

can obtain information _on the micellar

size,

_structure and intermicellar interactions _{over a} wide range of concentrations.

Materials and methods.

Soybean

lecithin _was obtained from Lucas

Meyer (Epikuron 200)

and used without further

purification. Cyclohexane

and iso-octane

(spectroscopic grade)

were

purchased

^from Fluka.

Samples

_were

prepared

_as described

previously [12, 131.

Static and

dynamic light scattering

measurements ~vere

performed

at a temperature ^of

25.0 _± 0. I °C.

Approximately

I ml of solution _was transferred into the

cylindrical scattering

cell lo _mm inner

diameter).

The

scattering

cell was then sealed and

centrifuged

between 20 and 420 minutes _at

approximately

5 000 g and 25 °C in order to remove dust

panicles

from the

scattering

volume. Measurements _were made with a Malvern 4 700 PS/AlW spectrometer,

equipped

with _an argon ion laser

(Coherent,

Innova 200- lo,

o ⁼ 488 nm), _a

digital

correlator

(128 points)

and _a computer ^{controlled and}

stepping

motor driven variable

angle

detection

system.

Static

light scattering (SLS) experiments

were

performed

_at 37 different

angles

(15°

_w 0 _w 150° for low concentrations and at 13 different

angles (30°

_w 0 _w 150°

)

for

high

concentrations, and 90 individual measurements were taken and

averaged

^for each

angle.

The

data _was then corrected for

background

(cell and solvent)

scattering

and convened into

absolute

scattering

intensities

AR(0) (I.e.

_{« excess}

Rayleigh

ratios

») using

toluene _{as a} reference standard. The _excess

Rayleigh

ratio of the

sample

was calculated

using

where A

(1(0 ))

and

(I~~~(0 ))

_are the average excess

scattering intensity

of the solution and the average

scattering intensity

^of the reference solvent toluene, R~~~(0 ₌ ^39.6 x lo ^~

m~'

_is the

Rayleigh

^{ratio of} toluene, and _n and _{n~~~ are} the index of refraction of the solution and the

reference solvent,

respectively [141.

Plots of

Kc/AR(0)

_1>ersus

Q~/3

_were

extrapolated

to

Q

~

0 _to

give intercepts Kc/AR(0),

where K

=

4

w~n~(dn/dc')~/(N~. Al),

dn/dc is the refractive index increment

(values

of dn/dc for

soybean

lecithin in

cyclohexane

as a function of W'o ^are

given

in reference

[121), Q

=

(4

wit/A

o sin 0/2 ) is the

scattering

vector, and _c is the concentration of surfactant

plus

water,

respectively.

The static correlation

length f~

was

determined from the

intercept

and

slope

of the

plot

Kc/AR

(0

_versus

Q~/3 using

a Lorentzian

scattering

law of the form

£j~

=

£ _ii

+

Q~ ill ₍₂₎

For

samples

^with

high

^{values of} _w-o ^{and low values of} c, I.e., for very

large

micelles in dilute solutions where

f~

^is

large, only

the linear part ^of the data at low

Q

_was included in the fit.

Dynamic light scattering (DLS) experiments

were

performed

at 37

scattering angles

(15°

_w 0 _w 150° ) at low concentrations and 7

scattering angles (30°

_w 0 _w 150° _at

high

concentrations,

and between 4 _to 20

intensity

autocorrelation functions _were

individually analysed using

a second order cumulant

analysis [lsl.

The results _were then

averaged

for each

value of

Q,

and _an apparent

cooperative

diffusion coefficient

D~~~ ^{= lim}

(r)/Q~

_was

Q -o

determined. From D~~~ ^a

hydrodynamic

correlation

length f~

was calculated

using

k~

^T

~~ ~ _{" ~}

0l~app ^~~~

where ~o is the

viscosity

of the solvent

[16].

Results and discussion.

Typical examples

^for the

Q-dependence

of the

scattering intensity I(Q)

and the apparent diffusion coefficient

(r)/Q~

as obtained from

lecithin/cyclohexane

reverse micellar solutions

at c<c* and c'~c* _are

given

in

figures

I and 2 for

w>o =14 and _c

=

5.ll

~/ml

^and

c =

30.7

mg/ml, respectively.

In the limit of

c-0, f~ corresponds

to

R~/

^{3, where}

R~ (R()('~

^and

(R()~

^{is the} z-average ^mean square radius of

gyration

of the

particles.

At

c<c*, scattering experiments

at low

Q

^values thus

primarily

reflect the _water- and concentration-induced

growth

and the

giant

^{size of} the micellar aggregates. The

resulting large

static correlation

length

results in _a very

pronounced angular dependence

of the

intensity,

and deviations from _a

simple

Lorentzian

scattering

law

(Eq. (2))

are

clearly

^visible at

higher

values of

Q.

An

analysis

of the low

Q

part of the

intensity

data shown in

figure

I for

c = 5.ll

mg/ml yields f~

₌ ⁶⁵⁹

I.

_{This value of}

_f~

_{is much}

_larger

_{than the value of the}

persistence length i~

=

lo _± 30

I previously

^{measured with small}

angle

neutron

scattering.

.1

0.000 0.001 0.002 0.003 0.004

Q jA-~j

Fig. I. -J(Q III (0) _{as a} junction of Q for

lecithin/cyclohe~ane

w./o microemulsions _at wo ¹⁴ and

c 5.I I

mg/ml

(o) and _c. 30.7

mg/ml

(.). Also shown _are theoretical

curves for f,

6591

_and

a

Lorentzian

scattering

law I(Ql/1(01=1/[1+ Q~

ill _(Eq.

_{j2)) (dotted line),} polydisperse worm-like chains with

f~=

I lo

I

_{(dashed line)}

or

polydisperse

semi-flexible chain~ with excluded volume interactions [19] (solid line). See _text for details.

At

Q. f~

m I,

scattering experiments probe

the interior of the micelles

(or

the _« blob

» if

cm

c*)

and the static

light scattering experiment provides

_us thus with information _on the

chain conformation. For

Q.f,»

I and

Q.i~

ml, where

i~

^{is the}

persistence length,

I

(Q

should reach _an

asymptotic

_power law of the form I

(Q ) Q~~

for Gaussian chains and

I

(Q Q~

^{~~~ for} «

perturbed

_» chains with excluded volume effects.

However,

the

asymptotic

limit _{can not} be reached with

light scattering

for _our system ^due to the restricted

Q-range.

Nevertheless _we

clearly

_see from

figure

I that the data at is in clear

disagreement

with the form

factor for

polydisperse (Gaussian)

worm-like chains shown _as the dashed line, but in

quantitative

agreement ^with theoretical calculations based _on chain statistics

appropriate

for excluded volume effects

(Fig.

I, solid line)

[17-191.

Figure

2 shows the

Q-dependence

of

(r)/Q~,

where

(r)

is the first cumulant obtained from _a second order cumulant

analysis

of the

intensity

autocorrelation

function,

for the _same

sample [201.

We find _two characteristic

regimes

for the

Q-dependence

^of

(r)/Q~.

At

Q.

SW I, the

dynamics

of concentration fluctuations examined

by

DLS _are dominated

by

translational diffusion of the

polymer

^{coil in the dilute}

regime (c

_<

c*),

and the relevant

relaxation process for concentration fluctuations is diffusive, I.e., characterized

by

(r(Q )) Q~.

We can define _an apparent ^diffusion coefficient D~~~ =

lim

(r(Q )) /Q~

,

from Q-o

which _{we can} then calculate _a

hydrodynamic

correlation

length f~ using equation (3).

For

c _- 0,

f~ corresponds

to the

hydrodynamic

radius

R~

of the

polymer

coil, which increases with

increasing

molecular

weight according

to a power law of the form

R~~M~,

where

v = 0.5 for theta solvents and _v

=

0.6 for

good

solvents,

respectively.

However, it is

important

to

point

out that for

polymers

the

experimentally

observed excluded volume exponent ^is

generally

^{smaller than the} theoretical

prediction [21, 221.

For

polydisperse solutions, R~

₌

(I/R~) j

The

dynamic light scattering experiments

also confirms the _water-

induced formation of

giant

_aggregates for wo =14 and _c

=

5.ll

mg/ml.

An

extrapolation

Q-0

leads _{to a} value for the

hydrodynamic

correlation

length

^of

f~=605i.

_At

Q, f~

^{ml, the}

relatively

weak

Q-dependence primarily

reflects the

polydispersity

of the

samples.

_

1.5'l 0 ^?

E

~.~'

~

~~o

*'~'

U

i o

io"7 ~<

/

C4 ~P°

o _oOO°

' P'

/S

f~o

.000 0.001 0.002

Q IA-11

Fig. 2.

(r(Q))/Q~

as a function of Q for lecithin/cyclohexane w/0 microemulsions at wo =

14 and

c = 5.I I

mg/ml

(o) and _c. 30.7

mg/ml

(.).

For

Q

f » I _we

primarily

see the intemal motion of the

polymer

chains. The

dynamic

structure factor is then characterized

by

a

decay

rate

(r(Q)) Q~

ⁱⁿ the so-called Zimm

model. As shown in

figure16,

for

Q. fh*0.6

_we do observe _a very

pronounced Q-

dependence,

and the

plot

of

(r)/Q~

_versus

Q

_appears to be linear. This indicates that

(r) Q~,

in agreement ^with the

predictions

for the contribution of internal chain

dynamics

to the

intensity

autocorrelation function

[221.

A similar behavior has

already

been observed for

aqueous ionic micellar solutions

[61.

At

higher

concentrations _c »

c*, f~

^and

f~

do not

depend

_on the individual micellar size anymore, but _are related _to the _« mesh size _» of the

entanglement

network which decreases

strongly

with

increasing

concentration. Under these

conditions,

I

(Q

is _now well described

by

equation (2)

_over the entire

Q-range

accessible in _our

experiments

(4.8 x10~~i~~

w

Q

_S ^3.5 x

10~~ i~~),

_and

(r)/Q~

is almost

independent

of

Q

over a

much wider range of

Q-values

as

expected

^for an

entanglement

network. An

example

for the

scattering

data obtained for c»c* is shown in

figures

I and 2 for

wo=14

and

c =

30.7

mg/ml.

For this

sample

_{we now} find

f~

₌ ^{2 lo}

I

_and

_f~

=

230

I,

which _can be

compared

to

f~

=

659 and

f~

=

605 for wo =

14.0 and _c

=

5.II

mg/ml.

The results from the SLS and DLS measurements are summarized in

figures

3a and

b,

where

f~

and

f~

are

plotted

as a function of wo and _c. We _see that the data at low concentrations

primarily

reflect the strong water-induced micellar

growth

and the concentration

dependence

of the micellar size distribution. At

w>o ^m

lo,

the water-induced micellar

growth

is

particularly

pronounced,

and the

resulting

microemulsion

panicles

_are

extremely large

_even at very low values of _c. Both

f~

^and

f~

^{then reach} a maximum at _c

w c ^* and decrease at

higher

values of _c

with _a power law of the form

f c~~,

where _x

=

0.65 _± 0.05 for

f~

^and x =

0.7 _± 0. I for

f~ [121.

^For c _~

c*,

both

f~

^and

f~

^become

independent

of wo.

We _{can now} try to use the data shown in

figure

3 for _a

quantitative

test of _our

previous

attempt to characterize the

dependence

^of

M~

on c and wo, which _was based _on the

application

of conformation space renormalization group

theory

for semi-dilute

polymer

^solutions to static

light scattering

data from

lecithin-cyclohexane

solutions

ill-

The basis for this

analysis

_was the connection between the scattered

intensity

and the osmotic

compressibility.

In the limit of

Q

_-

0, (dH/dc)~'

is related _to the static _structure factor

S(Q) by

the relation

[231.

N~

_{~ n i}

S(0)

=

k~

T

(4)

M dc

which

together

with

equation (2) yields

(~~~

₌

M~ S(0)

= M~~~

(5)

S(0)

for

polymers

can be written _{as a} function of _a reduced concentration x

= a

(c/c* )

_c.A2

M~

,

where _a is _a constant and

A~

^{is the}

(weight-average)

^{osmotic second} virial coefficient

[241.

An

explicit

functional form for

S(0

₌

f (X)

has for

example

been calculated

by

^Ohta et al.

using

the renormalization group method, and

good

agreement between

theory

and

light scattering

data _was found for classical

polymers

with different values of

M~

for _an extended range of concentrations

10~~

< X

<

10~,

where X is

given

for

polydisperse polymers by [251 A~ cM~

X

=

(6)

In

~~

9

Mn

I

₈

$-o~ '~~~~

~

A xc j

^' _~A

~ ~

-

'

~t

i io ioo iooo

c

[mglml]

iooo

'

'~ /l~~<f~~ ~

~

~ J~

~W

~

AA ~

- ,

~

A

- /

loo

f

~ SLP

i io ioo iooo

c

[mg/ml]

Fig. ^3. f,(A) and fhlB) _{a~ a} function of the concentration

c of the dispersed phase for lecithin/cyclohexane w/o microemulsions at different values of

M~o. The dashed lines connect points ^with equal values of _w-o and _are drawn _{as a} guide to the eye only. The solid line

corresponds

to a power law of

the form f c~'. where

x 0.7 for f, and

x = 0.65 for f~,

respectively

(li I_{i wo} 6.0 (cl

w>o =

8.0 (o) _w>o 12.0 _: (.) _{M-o =} ^14.0.

A

major problem

in a direct

application

of

polymer theory

to our systems comes from the

concentration and wo

dependence

^of the micellar size distribution. Since both

A~

^and

c* thus

depend

on w~o and _{c, we}

incorporated

a power ^law of the form

M~

=

Bi

c"

(7)

for the

(.-dependence

of the average

particle

size and _a

scaling

law of the form

R~

^{M~' in the relations for}

A~, given by

j~2 ^3/2 A~ 1 4 _w ^~'~ N

~

~ V'

(8)

Ml

and

c*,

which _can be

approximated by

~

4

wN~ R(

^~~~

This leads to

A~ c"l~

" ~~~, and the reduced concentration X is then

given by

X

=

2.10B)~"~"B~cl~~~~~~'+'1 (10)

Equations (7)

and

(lo)

allowed _us to

explicitly

take into account the _«

equilibrium polymer

_»

features in the renormalization group ^treatment

ii I.

We _were then able _to construct _a universal _curve

(M~/RT)(dH/dc)

i>ersiis a reduced

concentration X c/c* from the

experimentally

determined values of the osmotic compress-

ibility

if _a power law of the form of

equation (7)

_was used for the

c-dependence

of the average

particle

size

ill-

While the exponents v =

0.57 _± 0.03 and

a =

1.2 ₌ 0.3 and the parameter

82

_~

(0.8

_± 0.3 _x 10~ ^~

cm~ mol~~

~ ~J g~ ^~ were found _to be

independent

of wo, the values of

B,

which _are summarized

again

in table I reflect the

pronounced

water-induced micellar

growth ill-

From theoretical calculations

using

mean field models, _a varies between

a =

1/2 based either _on law of mass-action _or

Flory-Huggins

lattice model

calculations,

and

a w0.6 when

using

_a

scaling theory approach

for semi-dilute solutions

[21.

Therefore

a =

1.2 appears to be very

large

when

compared

to the theoretical

predictions

from _mean field models and

scaling

_arguments.

However, preliminary

results from Kerr effect measurements

[261

with

lecithin/cyclohexane

w/o microemulsions also indicate _a power law

dependence

of the

panicle

size _on concentration with _an exponent ^{which is of the} same order _as obtained in the

previous light scattering study.

Table I.

wo-dependence of

constant

Bi (wo)

in relation

M~

₌

Bi

c~ determined

from

SLS

e.<periments (from Ref. ill)-

wo B

i

~g~ ~/flloll~ ^~

4.0 4.5 _x 10~

6.0 6.0 x105

8.0 1.6 _x lob

12.0 7.6 _x lob

14.0 1-o x 107

We thus have _a

complete description

of the wo- ^and

c-dependence

of

M~

and X. In _a next step

we can now use this information and _test if _a self-consistent

interpretation

of the SLS and DLS results is

possible

within this theoretical framework. The theoretical basis for this is the

dependence

of

f~/R~

~

on the

overlap

_parameter X for

polymers

in

good

solvents, where

R~~_~ is the radius of

gyration

^which the

polymer

would have in the limit _c

-

0,

which _can be written _as

f~/R~

~ ⁼

F~(X). (ll)

An

explicit

functional for F

(X)

has been calculated with renormalization group methods

by

Nakanishi and Ohta

[271. Experimentally,

one does indeed find _a universal _curve for

f~/R~,~

versus X for

synthetic polymers,

which shows _no

systematic dependence

_on the

molecular

weight

_or the solvent. However, while the theoretical

F~~~~~(X)

based _on the

renormalization group

theory

calculations

by

Nakanishi and Ohta

reproduces

the correct

shape

of the

experimental

_{master curve,} the calculated

prefactor

is

clearly

too low

[211.

Instead of

using

the theoretical form of

F~ ~~~(X)

for _our purposes

only,

_we have also extracted the

experimentally

observed

F~ _~~~(X)

from reference

[211.

Both

F~, ~~~(X)

^and

F~,~~~(X)

are

reproduced

in

figure

4a.

We _{can now see} if _{we are} able to construct such _a master _curve from the data shown in

figure

3a. Therefore _we need _to calculate

R~~

~(wo, c)

for all values of wo and c

investigated.

This _can be done _as follows _{we can} calculate

M~(wo, c)

and

X(wo, c)

for all

samples

investigated

in the _current

study

without any

adjustable

parameter

using

the results from the

previous

renormalization group

analysis

summarized above

(Eqs. (7, lo)

and Tab.

I).

From

M~(wo,

cl, we can then calculate the

weight

average ^contour

length L~(wo,

c

).

This _can also be done without _any

adjustable

parameters ^since we know the local

packing

and the _mass per unit

length M~

of the micelles from

previous

SANS measurements

[12, 28, 29].

In _{a next} step

we can then determine the ideal

single

coil values of R~,

~(L~(w>o, c), i~)

from

equation (12) [17,

30,

311

~~ _~~ L ¹⁺ ~

(R~(L)~)

₌

~ ~ ~~

(12)

6 ₊ 5 _{~ + s}

where _e

=

0.176,

and

(13)

lw

^1/2

R~

~ ⁼

G(L, 0). (R~(L)~)

^dL

⁽¹³⁾

o

and

using i~= ^l101 [?81

and _a

Schulz-Flory

distribution for the size distribution

n(L).

The

intensity

distribution function for _a

given

^value of L _at

Q

-

0, G(L, 0),

_as used in

equation (13)

is described

by equation (14), [321

G(~'°~" w~~ ~).~~dL ^(~4~

o

The choice of the size distribution is both

supponed by previous light

and neutron

scattering experiments

in dilute solutions

[121

_as well _as based _on theoretical

grounds,

since it matches the

predictions

of the micellar

growth

model, I.e.,

n(M) exp(- M/M~)

and

M~/M~

~ 2,

where

M~

is the

number-average

^molecular

weight [21.

We _now have the theoretical values ofR~, _~

(wo,

c for all values of wo and c

investigated,

and

we can obtain the

corresponding

ratios of

f~/R~~

^for all the measured values of

f~(w,o, c)

^{shown in}

figure

3a. These ratios _are

plotted

as a function of

X(wo, c)

in

figure 4a,

and _{we see at once} that the different individual data _sets in

figure

3a which exhibit _{a very}

o

~

A

''. .

~n

~ ".

~

_{_i}

SLP '".

CD "

",

-1.0 0.0

'g(x)

A~A

B

~

~ ".

'_

A

~3i

-

~

Cfi '~,

a

-2

-1.0

'g(x)

Fig. 4. Al

f,/R~,

~

as a function of reduced concentration X for

lecithin/cyclohexane

w/o microemul- sion~ at different values of

M>u : (Al _wo 6.0 _: (c)

M>o 8.0 (o) _M-o 12.0 _: (.)

M>o 14.0. Also

shown is the theoretical _curve f~/R~

~ = F,_ _RGI (Xl i,s X from renormalization group theory as the dotted line, and the

experimentally

observed

f,/R~

~

F,

~,~

(X) _vs. X for

« classical

» polymers _as the solid line [211. B)

f~/R~

o

as a function of reduced concentration X for lecithin/cyclohexane M>/o microemulsions _at different values of_M.o (li _w-o

= 6.0 (cl _w<u 8.0 (o) _wo

= 12.0 j.) _w>o 14.0. Also shown _is

the theoretical _curve

fh/Rh

a =

Fh

RGT (X) _v-I X from renormalization group

theory

_as the dotted line, and the

experimentally

observed

fh/R~

~

F~ ~,~(X)

i-s. X for

« classical

» polymers as the solid line [21].

pronounced

_wo and _c

dependence collapse

onto a

single

« universal _» master _curve. Our values of

f~/R~

~

closely

follow the function F

~, ~~~

(X) previously

found for

synthetic polymers

except for the lowest values of X, where the

polymer

data also exhibits the

largest

_scatter.

Figure

4a

shows that _{we can} not

only interpret

the measured osmotic

compressibility

with _a combination of

S(0) given by

renormalization group

theory

and _{a c-} and

wo-dependent

^{micellar size}

distribution,

_as

successfully

demonstrated

previously [I],

but that this

approach

_can be extended _{to a}

quantitative description

of the full

scattering

_curve I

(Q ).

It demonstrates that the

Q-dependence

of

S(Q)

measured for

polymer-like

lecithin _reverse micelles is

fully

consistent

with the behavior of

synthetic polymers.

It is

important

to

point

out that, _once we have

calculated the _set of constants

Bi(wo)

and

B~,

the construction of the

plot

of

f~ (wo,

c

)/R~

~

(wo,

c

)

versus X

(wo,

c is

performed

with _no

adjustable

parameters, ^{and that} we

are thus able to

predict

the

Q-dependence

of

I(Q)

on absolute scale.

In _{a next} step we can now test if the _same

approach

_can ^be

applied

to the

dynamic

structure factor also. Renormalization group ^treatment for

dynamic properties

such _as

f~

^has not been

performed

as

extensively

_as for static

properties

such _as

f~

or

(dH/dc)~',

_{but there}

are

calculations available such _as the _one

by

Shiwa which

predicts

that

f~/R~,

~ ⁼

F

~(X (15)

where

R~

~

is the

hydrodynamic

radius of the

polymer

coil in the limit _c

- 0

[331.

While earlier measurements

by

Wiltzius _et al.

[341

^did suggest ^that a

plot

^of

f~/R~

~ versus X does not

yield

_a

universal curve and that

k~,c

^should be used instead, where

k~

is defined

by

D~~~=D~,~. (l +k~,c+. ),

_a careful

analysis

of the

large

data _set available for

polystyrene

as the most

widely

examined

(semi)-flexible polymer

showed that

plotting f~/R~

~

versus X does indeed

yield

_a universal _curve

[211.

However, the calculated

prefactor

is

again

too small, and _we have thus

plotted

in

figure

4b the theoretical _curve

F~, ~~~(X)

^from

renormalization group

theory

_as well _as the

experimentally

observed master curve

F~, ~~~(X)

^{which we extracted from the data}

given

in reference

[211.

A

major difficulty

in the construction of _an

experimental

master curve

using

the DLS data from

lecithin/cyclohexane

solutions in

figure

3b arises due to the lack of _an

appropriate

theoretical relation

R~ ~(L

=

f(L, i~)

^for semi-flexible chains with excluded volume effects

(analogous

to

equation (12)

for

R~

~) and the

experimental

observation that

Rh

reaches the

asymptotic regime (R~ M~,

_v

=

0.588)

much _more

slowly

than

R~.

^{We have therefore used}

an

empirical approach _R~,

~ ⁼

Cte

M°

~~

as found for

synthetic polymers

in

good

solvents

[21, 351,

in which _we write

j~ ^{0 05}

~h, o(~' _~p) ~h, _{wc(~' ~p)}

~

f (~~)

P

where

R~~~

is the

hydrodynamic

radius of _a worm-like chain with _contour

length

L and

persistence length i~

as

given by equations (49-52)

in reference

[361.

The additional _term

(L/2 i~)°

^°5 represents an « ad hoc _» correction for the influence of excluded volume effects _on

the molecular

weight dependence

^of

R~

M~ and

changes

the

scaling

_{exponent v =} 0.5 of

(Gaussian)

worm-like chains _{to v}

=

0.55 found

experimentally

^for

synthetic polymers

in

good

solvents.

Using equation (16),

_{we can now} calculate the ideal

single

coil values

R~ ~(wo, c)

from

equations (14)

and

(17)

w

l-

~~.

°

jo

R~(L, i~) ^~~~'

^{°~ ~ ~~~~}

and calculate the

corresponding

ratios of

f~/R~

~

for values of _w-o and _c

investigated analogous

to the

procedure

^{for the} static correlation

length

described above. The

plot

of

JOURNAL DE PHYSIQUE II T 4 N' 2 FEBRUARY 1994 j2

f~ (wo,

c

)/R~

~(w>o

c)

_i>er,ms X

(wo,

c is shown in

figure 48,

and _we see that all individual data sets from

figure

3b

collapse

onto a

single

master curve. The data

points

for leci-

thin/cyclohexane

solutions _are in

quite good

_agreement with the _curve

F~ _~~~(X)

for

synthetic polymers, although

^the

polymer

_curve does _not extended _to low

enough

reduced concentrations to

judge

the behavior _at low X.

Figure

4b thus shows that _{we can} not

only quantitatively

describe the static

light scattering

data

using

the

application

of

polymer

renormalization group

theory,

but that this

approach

also works for the

dynamic light scattering

data.

Conclusions.

We have

presented

a detailed

analysis

of the static and

dynamic

structure factor of

polymer-

like lecithin _reverse micelles. Giant micellar aggregates ^{exist in solutions with} wo m lo which results in _a very

large

static correlation

length _f~

at low concentrations _{c w c}

*,

and the

intensity I(Q) closely

follows the theoretical

prediction

for the

Q-dependence

of

polydisperse

semi-

flexible chains with excluded volume interactions. This

provides

a firm basis for the

application

of _a power law of the form

R~ M°~~~

_to these systems. ^The

dynamic

structure factor exhibits both contributions from _centre of _mass diffusion and internal chain

dynamics,

which result in _a

Q~-dependence

of the initial

decay (r(Q II

at

Q fh

* I. At _cm

c*,

the

static and

dynamic

correlation

lengths

decrease with concentration _as

expected

^for semi-dilute

polymer

solutions, and due to the

relatively

limited

Q-range

of the

light scattering experiments

the

Q-dependence

of the

intensity /(Q)

reduces to a

simple

Lorentzian

scattering

law, and

(r(Q))/Q~

is

practically Q-independent.

We

previously

demonstrated that _{one can}

directly apply

the results from conformation space renormalization group

theory

for semi-dilute

polymer

solutions _to

«

equilibrium polymers

_»

such as worm-like micelles or

microemulsions,

if _one takes into _account the fact that their size distribution is concentration

dependent ill-

Based _on the

assumption

of _a power law of the form

M~

c~ for the

c-dependence

of the micellar size, we were able _to construct a universal

curve

(M~/RT) (dH/dc)

versus a reduced concentration X~ c/c* from the

experimentally

determined values of the osmotic

compressibility.

We then concluded that _{one can} thus obtain information from static

light scattering experiments

on the concentration

dependence

^of both the micellar size distribution and the intermicellar interaction effects,

I.e.,

_on the

weight

average molecular

weight M~(c)

^and the static _structure factor

S(0, c).

The present

study

now

provides

an additional

independent

test and shows that this

approach

can be extended _to the

Q-

and

c-dependence

of the static

(S(Q))

and

dynamic (S(Q, t))

structure factor. The concentration

dependence

of the initial

Q-dependence

^{of the}

intensity

I

(Q ),

which _can be described

by

a correlation

length f~,

and of the initial

t-dependence

^of the

intensity

autocorrelation function

g~(t),

which _can be described

by

_a correlation

length f~,

_are in

quantitative

_agreement with the behavior found for

synthetic polymers

^such as

polystyrene

_over the entire range of concentrations studied. We _can construct universal _master

curves fot the dimensionless

quantities f~/R~

~

and

f~/R~,~

as a function of the reduced concentration X

only,

which

provides

_a further successful _test of the

previously

obtained information on the concentration and

composition dependence

of the micellar size distribution.

Acknowledgments.

This work was

supponed

in part

by

the

Ponuguese

National Fund for

Technological

and Scientific Research

through

the Grant

BD/1256/91-IC.

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