Small angle neutron and X-ray scattering study of the formation and structure of micelles of CTAB in formamide

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Small angle neutron and X-ray scattering study of the formation and structure of micelles of CTAB in

formamide

T. Perche, X. Auvray, C. Petipas, R. Anthore, I. Rico, A. Lattes, M. Bellissent

To cite this version:

T. Perche, X. Auvray, C. Petipas, R. Anthore, I. Rico, et al.. Small angle neutron and X-ray scattering

study of the formation and structure of micelles of CTAB in formamide. Journal de Physique I, EDP

Sciences, 1992, 2 (6), pp.923-942. �10.1051/jp1:1992189�. �jpa-00246612�

Classification

Physics

Abstracts

61,10 61,12 82.70

Small angle neutron and X-ray scattering study of the formation and structure of micelles of CTAB in formamide

T. Perche

('),

X.

Auvray ('),

C.

Petipas ('),

R. Anthore

('),

I. Rico

(2),

A. Lattes

(2)

and M. C. Bellissent (~)

(l) Groupe

de

M6tallurgie Physique

(*), Facult6 des Sciences et des

Techniques,

76134 Mont Saint

Aignan

Cedex, France

(2) Laboratoire des Interactions Moldculaires et Rdactivit6s

Chimiques

et

Photochimiques (**),

Universitd Paul Sabatier, l18 route de Narbonne, 31062 Toulouse Cedex, France

(3) Laboratoire Ldon Brillouin, CEA-CNRS,

CEN-Saclay,

91191 Gif-sur-Yvette Cedex, France (Received J3 November J99J, accepted J6 December J99J)

Abstract. Small

angle

neutron and

X-ray scattering

was used _to

study

micellization of

cetyltrimethylammonium

bromide (CTAB) in

partially

deuterated _or

hydrogenated

formamide from the absolute values of scattered intensities.

Highly charged

aggregates ^{of around 6}

monomers were observed at CTAB concentrations above the _cmc

(2.8

9b wt at 60 °C). These

aggregates

along

with

spherical particles (2

_nm radius)

containing

29 _{monomers were}

consistently

observed at concentrations above 8 9b. These

particles

_were considered to be micelles _as

they

had similar structure, albeit of smaller size to those observed in water (2.7 _nm,

aggregation

number 90). They also had _a higher

charge

in formamide than in water (degree of ionization 0.55 in

formamide and 0,14 in water). With increase in surfactant concentration, the micelles

elongated, although

the radius of the

cylinders

in the two dimensional

hexagonal phase

remained close to 2 nm. The

importance

of interactions of

polar

head with solvent molecules of

high dipole

moment

and dielectric constant is discussed. The less spontaneous self-association of surfactant molecules in formamide than in water poses the

problem

of the definition of the _cmc.

Introduction.

Ordered

lyotropic phases

of ionic surfactants have been observed in non-aqueous

polar

solvents in various

binary

_systems. The sequence of

phases depends

_on the solvent/surfactant

pair.

At

increasing

surfactant concentration, two sequences of

phases

_are observed with ionic

surfactants _i

micellar

phase (I~)

_w two dimensional

H~ hexagonal phase (p6m)

_w

Q~

cubic

phase

of space group Ia3d

wL~

Iamellar

phase

in

systems

^such as

CTAB/formamide, ethylene

glycol, glycerol [1, 3], cetylpyridinium

bromide

(CPBr)/N-methylsydnone [4],

sodium

dodecylsulfate (SDS)/formamide [5, 6]. Recently

_a cubic

Ii mesophase

between

I~

and

(*) URA 808 CNRS.

(**)

URA 470 CNRS.

H~

^is

reported

in _some

binary

_systems in formamide and

glycerol [7]

such _as

cetylpyridinium chloride/formamide

_or

glycerol

_;

isotropic phase

_w

L~

lamellar

phase

with

CTAB/N-methyl formamide, N-methylsyd-

none

[1, 3, 4], SDS/glycerol, ethylene glycol [6].

With _some surfactants such _as lecithin

[8, 9]

or

diodecylphosphatidyl

^choline

[I al, only

the

L~ phase

has _so far been observed. This is also

the _case in the temary

systems SDS/decanol/glycerol [11]

and aerosol OT

(AOT)/formamide/toluene [12].

In the

binary system AOT/formamide,

inverse

hexagonal

and cubic

phases

have been observed

[12].

The formation of microemulsions in

polar

non-aqueous solvents has also been

reported, although

doubt has been cast on such observations

[13-18].

The _most

commonly employed

solvents have been

formamide, glycerol

and

ethylene glycol

which _are all characterized

by

_a

high

cohesion energy as measured

by

the ratio

yv~~'~

where _y is the surface tension and _v the molar

volume,

_a

high dipole

moment MD, and

hydrogen bonding.

This latter factor has been considered _to be _a

prerequisite

for the

formation of aggregates ^of surfactants

[19],

which is

supported by

the observation that

only

the

L~ phase

is formed in solvents such _as

N-methylformamide

and

dimethylformamide

in which

hydrogen

bonds _are weak _or absent

[1, 3, 6],

This

stipulation

for

hydrogen bonding

has been

brought

into

question by

the observation of the normal sequence

I~

_w

H~

_w

Q~

_w

L~

in the

CPBr/N-methylsydnone system [4].

This solvent does _not form

hydrogen bonds, although

it has both

high

cohesion energy

(yv~

^"~

=

l.4 _x

l~f

J,m~ ~) and

high dipole

_moment

(p~

= 7.3

D) [19].

Studies of micellization of non-ionic

[20-22]

and ionic

[1, 15, 19, 23-31]

surfactants in _non- aqueous

polar

^{solvents have}

produced

rather

contradictory

results. The presence of micelles

immediately

before formation of the

H~ phase

is well

recognized, although

their presence in the

isotropic phase

when the

only

ordered

phase

is

L~

has not been demonstrated. The

process ^of

aggregation

of molecules of surfactants with

increasing

concentration is also not the

same as that observed in water.

We report ^here a

study

of micellization of CTAB in formamide

by

small

angle

neutron and

X-ray scattering.

The

plots

of surface tension _versus concentration at temperatures ^{above the} Krafft

point (43 °C) [32]

show _a

discontinuity

for _a surfactant concentration of 2.8 fb at 60 °C

[24].

This is defined _as the critical micellar concentration

(cmc).

Wamheim

[27] _reports

_{a cmc}

of 4 fb at 60 °C with a-deuterated

CTAB, although

the presence of deuterated chains could

account for this

discrepancy.

The cmc is around 70-fold that observed in _water. The

longitudinal

relaxation _rate in proton ^NMR at 60 °C alters little with increase in concentration between 0, I mole,l~ ^'

(3 fb)

and 0,4 mole. l~ ^'

(12 fb) [I ].

The coefficients of self-diffusion and the

longitudinal

relaxation _rates of formamide molecules

[26]

indicate the presence ^{of small} aggregates.

Aggregate

size appears ^to increase _up between 0. mole.l~ ^' to 0.4 mole,l~ ^' to an

equilibrium

micelle size that remains constant up ^to a concentration of I mole,l~

',

_{which is}

close _to that observed in _{water at} the _same temperature

(60 °C).

Recent determinations of

2H-spin

lattice and

spin-spin

nuclear

magnetic

relaxation rates in solutions of formamide _or mixtures of _water and formamide

containing

20 fb of CTAB at 60 °C have demonstrated the presence of

spherical micelles,

albeit smaller than those observed in water

[27].

Thirty

_{years ago,} the small

angle X-ray scattering

studies of Reiss-Husson and Luzzati

[33]

allowed to

distinguish

between the _two structural models of the ionic micelles _: unilamellar vesicles _or true

spherical

micelles. SAXS form factors for both

shapes

of

aggregates

are very similar except ^{for the scale and invariant. This}

study corresponded

also _to introduction of the Guinier

focusing

_camera in

physical chemistry

of surfactants

[34].

It is the _same for the accurate determination of the structural model of the ionic micelles in non-aqueous

polar

solvents.

In _a

previous

small

angle X-ray scattering study [15],

_we demonstrated the presence of aggregates at concentrations above the _cmc. In the present

study,

these results _were

confirmed

by

small

angle

neutron

scattering

at concentrations between the _cmc and 20 fib surfactant. The method _we used to calculate the absolute

intensity

is described in detail in

appendix

I and 2. Determination of the

scattering

intensities of neutrons and

X~rays

define the first stages ^{in the}

aggregation

of CTAB in

formamide,

_as well as the _structure and size of the micelles which form in solutions at above 10 fib surfactant. Some

apparently contradictory

results _can be accounted for

by

_{a process} of micellization akin to the micellization of surfactants in _water and other solvents with _a

high

_cmc. The

importance

of interactions of

polar

head with solvent molecules of

high dipole

moment and dielectric constant, and the definition of the _{cmc are} discussed.

Materials and methods.

REAGENTS.

Solvents.

2/3

deuterated formamide

(HCOND~)

referred _{to as} FD

(Merck,

99 fib min.

pur.),

and

hydrogenated

formamide

(HCONH~)

referred to _as FH

(Merck analytic,

99.7 fib min.

pur.)

were used without further

purification.

To avoid contact with _water all

samples

were

prepared

in _a

glove

box.

Surfactant. Cetyltrimethylammonium

bromide

(CTAB)

_was from Merck

(99

fib min.

pur.).

The Krafft

temperature

^{in formamide is 43} °C

[32].

The _cmc is defined _as the concentration above which surface tension remains constant is 9.0 _x 10~^~ mole.l~ ^'

(2.8

fib

wt)

at 60 °C

[24].

TECHNIQUES.

Small

angle

neutron

scattering (SANS).

Small

angle

neutron

scattering

measurements were

made in PACE spectrometer ⁱⁿ Laboratoire Ldon Brillouin

(CEN-Saclay).

The

sample

detector

separation

D _was 1.20 _m and the

wavelength

of incident beam _was A

=

0.468 _nm.

The

scattering

vector range was : 3.36 _x 10~ nm~ _< q < 3.49 nm~ ^' with q =

(q(

= 4 _w sin o/A where 2 o is the

angle

between the incident and the scattered beam.

The multidetector consisted of 30 concentric

rings; giving improved

_accuracy in the

determination of the

scattering

intensities _at the wider

angles.

Samples

_were contained in I _mm thick

quartz

^{cells maintained} at 48 ±1 °C. The overall accuracy in the measurement of incident and transmitted beams _was around _± I fib. The

scattering

_spectrum of _a

sample

of

H20

_was used to establish the absolute scale of

scattering

intensities. The way of

obtaining

the absolute

intensity I(q)

is described in the

appendix [35, 39].

The

plot

of

I(q) (A7) along

^{with their}

respective

uncertainties _are

compared

to the

plots

calculated from models.

Small

angle X-ray _scattering (SAXS).

An

X-ray

tube of fine focus and 1.5 kw power with _a

copper anticathode _was used. The incident beam

(CUK«I)

_was monochromatised and

focused

by

_{a quartz} bent

crystal.

^The

samples

contained in quartz

capillaries

_were maintained

at

(50

_±

0.I)

°C in _a thermostated bath. The detector _was

placed

346 _mm away from the

sample.

A

cylinder

with

beryllium

windows maintained under

primary

vacuum was

placed

between the

sample

and the linear localization detector

(Inel-France)

with _a resolution of 0.3 _mm.

The accessible domain of the

scattering

vector q was : 2.5.10 nm~ w q w 3.4 nm~ ~.

The diameters of the

capillaries

_were measured

optically,

and selected in _a range

(1.50

±

0.02)

_mm with similar attenuation when

empty.

^{The conversion of measured}

X-ray

intensity

into the absolute

intensity

is described in

appendix

2

[40, 41] (A8).

Characterization

of

the solutions. A _neutron

scattering study

was carried out with various mixtures of

hydrogenated

^and

partially

deuterated formarnide in the

following proportions

_:

75 fib FD-25 iii FH

(75 FD/25 FH)

50 fib FD-50 fib FH

(50/50)

25 iii FD-75 fib FH

(25 FDRS FH)

The concentrations

by weight

of the solutions of CTAB _{were :} 3.3 fib, 5.0 fib, 10.0 fib and 15.0 iii

(cmc

m 2.8 fib

wt).

Table I lists the various characteristic values of these solvents. For

X-ray scattering, only

solutions of

hydrogenated

formamide

(2.8 fb,

5.0 fib, 10.0 iii and 20.0 fib) were studied.

The molecular

scattering length

b

=

Z,

f

where Z is the number of electron per

molecule, f

is the

scattering length

of the electron, which for FH:

scattering length bm

= 6.77 _x 10~ ^'~ _cm and

scattering length density

_pm

=

10.23 _x

10~°cm~~

Table 1.

Solvent FD

75FD/25FH

50/50 25FD/75FH FH

Volumic _mass

(g/cm3)

1,19 1.17 1.16 1.14 1.13

Molar _mass

(g)

47.0 46.5 46.0 45.5 45.0

Nb of molecules

n per cm3

(x 102')

15.24 15.21 15.18 15.15 15.12

b

(x 10-'2 cm)

+ 3.136 ₊ 2.615 ₊ 2.095 ₊ 1.574 ₊ 1.053

p

(x 10'° cm-2)

₊ 4.78 ₊ 3.98 ₊ 3.18 + 2.39 ₊ 1.59

b is the _mean coherent scattering length _(b~~j~~~~ £b~~~~,~).

p is the scattering length density (p _n. b).

The coherent scattering length of atoms _were taken from Koester and Yelon [42].

Table II lists the

scattering lengths

^for CTAB

by

neutrons and

X-rays.

The contrasts between the solvent and _a

sphere

^of radius R

containing just

molecules of CTAB _are schematized in

figure

I. This

diagram

does not rule out _a more

complex

model of

micelles,

but shows that it is not

possible

to eliminate _contrast between the micelle and solvent

by altering

the level of deuteration of the solvent.

Results.

SCArrERiNG cuRvEs. For _{a monomer} concentration above _cmc, the _curves of

scattering

intensity I(q) display

_a diffuse band whose

peak q$ position

is

only slightly

affected

by

contrast, the incident radiation

(neutrons Figs. 2a, b,

_{c or}

X~rays Fig. 2d)

_or concentration

(Fig. 2).

However, the intensities of the diffuse band _are

markedly

affected

by

^all these

Table II.

Neutrons

X-rays

b

(CH~)

_x 10-'2 _cm 0.0834 2.256

b

(CH~)

_x 10-'2 _cm 0.4575 2.538

b

(N(CH3)3)

_x

^10-'2

cm 0.4425 9.588

b

(Br)

_x

10-'2

_{cm +} 0.6790 9.870

b

(CTAB)

x

101°

_{cm -1.472 55.836}

p

(CH2)

x 10'° cm-2 _{0.030 8.25}

p

(CH~)

x 10'° cm-2 0.80 4.45

p

(N(CH3)3)

x 10'° cm-2 0.43 9.37

p

(Br)

x 10'° cm-2 _{+ 1.73 25. ll}

p

(CTAB)

_x

^101°

cm-2 0.24 9.17

The volumes employed in the calculations _were [43, 45]

V°(CH~) 27.35 _x 10~^~nm~, V°(CH~) 57.06 _x 10~~nm~.

V°(N~ (CH~)3) ₌ 102.3 _x 10~^~nm~, V°(Br~

= 39.3 _x 10~^~nm~.

p(«io~°cm-2)

~----

FH Rayons X

---~

p-- ^-FD

~--- -75FD/25FH

~- ^{50/50 Neutrons}

A

~- 25FD/75FH

~---

-FH

Ap mininum

Fig.

I.

Scanering lengtll

densities of _a

dry

micelle in mixtures of FH and FD for neu~ons and

X-rays.

' lqi (cm-1)

25FH-75FD

. -50

25FD-75FH FH

o o,5 1-o 1.5 2.o 2.5 3.o q nm-I

a)

1(cm'l)

2,o

m.

m

m

m m

m m

~

m o.5

b)

Fig. 2. -

Scanering intensity ^lots

I

10

methods, b)

: various

oncentrations of CTAB in FD. Curves

calculated with S~n. c) SANS _:

3.3 9b and 5 9b CTAB in FD. d)

SAXS : 5 _9b

and

¹⁰ 9b CTAB

curves sing the model for igures b-c-d.

1(q)icm.1)

. ~ W

~ W 5%

. ~ W

W

W .

. .

~

. . . ~

o q

factors. There is _a difference of almost _an order of

magnitude

between the scattered intensities of neutron and

X~rays

for the 5 fib and lo fib solutions. The scattered intensities from solutions of low concentrations are distributed in the

reciprocal

_space, and _are much

more localized for solutions _at concentrations above lo fib

(Figs. 2b-d).

The set of _curves

(SANS)

obtained with different _{contrasts are} of similar

shape.

This shows

the

importance

of the structure factor

S(q)

in the appearance of a diffuse band in the _curves of scattered

intensity. Only

the

high

contrast neutron

scattering

_curves and the

X-ray scattering

curves have been

utilized, although

_we checked that the

proposed

^model could account for

the _{curves at} all values of contrast.

ANALYSIS _OF EXPERiMENTAL cuRvEs. The method described

by

Wa«

[46]

to show the

diffuse _cmc

region

in

dodecyltrimethylammonium

bromide/water

system

can be used for _a

preliminary analysis independent

of theoretical model. From the

integrated intensity Q*

=

lI(q)

_q

2dq

=

2

«ipc~~~

_p~~i~)2

_{~o,(i ~o,) (i)}

~

the volume of

homogeneous scattering particles

~P' _can be deduced. These

particles only

contain _monomers of

scattering length density

_pc~~~

(Tab. II).

Hence the number of

micellized _{monomers can} be calculated. If the _curves

I(q)

_q ^~

=

f(q)

tend to a constant value at

large

_q, the interface between

scattering particles

and solvent is

sharp [47-48]

and flat at the

scale q~ ^'

[49].

The

asymptotic

value is

expressed by

the Porod's law _: rim

v(q)

_q~)

= 2

«(p

c~~~ p~~iv)~ s/v

(2)

S/V is the total interracial _{area per} unit volume. One calls

ip

the Porod's

length [49]

ip

=

(4 V/S) @'(I @')

and _:

I

p ⁼

(4/w) Q

*/lim

Iq~

For

spherical homogeneous monodisperse particles

_:

VW'IS

=

Rp/3, Rp

is the Porod radius.

For diluted

solutions,

~b'«I Rp=3/4ip=3Q*/(wlimlq~. ₍₃₎

The

shape

of the _curves

I(q)

_q ^~

=

f(q) depends markedly

_on the surfactant concentration in the solution

(Fig. 3),

which shows that the _structure of the

scattering particles

in _concen-

tration-dependent.

Two types ^{of behaviours} can thus the

distinguished

_:

for _monomer concentration below lo fib where the _curves

I(q),q~= _f(q)

do _not

exhibit _a

bump,

and the

asymptotic

value is _not attained in the

experimental

_{range of q ;}

for _monomer concentrations

equal

to an above

loili,

the

scattering

_curves

I(q),

_q ^~

=

f(q)

have _a

bump

_q~, before

reaching

the horizontal Porod limit. The

algebraic

sum of the _areas between the _curve

I(q)

_q ^~ ₌

f(q)

and the

straight

line

corresponding

to the Porod limit is

approximately

zero

(the negative part compensating

the

positive part).

These two solutions therefore contain microstructured

scattering particles

with

sharp

and smooth

interfaces of curvature much

greather

than qj~j~

(qj~~

m 0.3

urn) [47, 49].

The Porod radii

[3]

_are 1.5 and 1.3 _{nm at} 15 fb and lo fb

respectively (SANS)

and 1.2 _{nm at} lo fib

(SAXS).

These structured

scattering particles

whose size is

relatively independent

of

concentration _are

regarded

as micelles.

At the lower

concentrations,

the

particle-solvent

interface is either _more diffuse than that observed at

high concentration,

or its _curvature is of order to

qjjj~

^{and the Porod's}

plateau

is not observed in

experimental

_range. These solutions contain small

scattering

clusters.

1.q4

lO~6(nm-5)

15%

.

lO%

.

m

"~.

^5%

~

.~

:

""

'

,,,~jjt~«j~ilii)

~

O

Fig.

3. Porod limit

q~l

_(q

versus q

plot

for different concentrations in FD _: (.) 15 9b, (m) 10 9b, (A)

5 9b, (+) 3.3 9b.

The number of _monomers c' involved in the micelles _are calculated from the

integrated intensity Q* Ill.

The contribution of

qmm

w

q~I(q) _dq

in the calculation of

Q*

_can

only

be

evaluated from _an

analytical expression

off

(q),

which is

possible

if

I(q)

falls _{as a} function of

q~~.

This

analytic

form _was that

employed

in the calculations for solutions of concentration

10,

15 f61

c = 0.32 mole.l~

(10

fib) c cmc = 0.23 mole.l~ c'

= 0.12 mole.l~ ^'

c =

0.49 mole.l~ ^'

(15

fib) c cmc =

0.40 mole.l~ ^' c'

= 0.16 mole.l~ ^'

The number of micellized _monomers is lower than

(c -cmc).

Excellent agreements are

obtained for

decreasing _part

of the _curves I/c' at lo fib and 15 fib _are indicated that the micelle sizes _are identical for these _two solutions

(Fig. 4).

At low

resolution,

intermicellar

interferences _are

significant

and the _curves I/c' _are not identical.

This

preliminary analysis

shows the

complexity

of the solutions

investigated

which the

proposed

theoretical model of the _set of _curves of

scattering intensity I(q)

must take into account.

MODEL _OF MICELLJZATION.

Principles of

the model. The

solvophobic

nature of the chains in formamide and the

solvophilic

character of the

polar

heads and counterions would indicate _a similar micellar

structure in FH _as in _{water : a} two shell model of

spherical

micelles is

used, (Fig. 5), although,

for

example,

the better

penetration

of molecules of formamide into the _core and the different conformations of the

chains,

different

polar head-polar

head and

polar

head-counterion interactions _may

produce

different sized micelles.

The

intensity

scattered

by monodisperse spherical particles

in interaction is

given by

:

I(q)

=

nP(q) S(q)

where _n is the number of

particles

_per unit

volume,

P

(q)

is the form factor of the

particle

and

S(q)

is the _structure factor

arising

from

interparticular

interferences. For _a two

density

1«lo~~cm+~mole"I)

C'

W

/~~

"

~..

°~~

. W

W~

+~

"~ ~"

~

+~~

W~~#~#

~

o.5 1.o

model _:

P(q)

"

(4/3 "R/arc(p

core

ppol) lP(R

^core) + 4/3

"Rip

core

psolv) *(R))

^~

(4)

R~~r~ is the radius of the

aliphatic

_core, R is the overall radius of the

micelles,

p~~~~, p~~i ^and p~~i~ are the

scattering length

densities of _core, outer

layer containing polar

heads and condensed counterions and molecules of bound FD

molecules,

and solvent

respectively, ~P(R)

is the

Rayleigh

function _:

~P(R)

=

3

[(sin (qR) qR

_cos

(qR) ]/(qR)

^~

S(q)

is calculated

using

the D-L-V-O- model

by

a

computer

_program ^described

by Hayter

and Penfold

(R.M.S.A.) [43, 50] assuming

that the

particles

_are

spherical

and identical

(or relatively

small

dispersed

in

size).

The differences in

intensity

_{as a} function of _c

(Fig. 2b)

must be accounted for

by

^either a

change

in

shape

of the

scattering particles (size

or

geometry),

_{or a}

change

in number

(modifying

their

interactions)

_{or a} combination of both.

Three

independent _parameters

which _are involved in

P(q)

and

S(q)

form the basis of the model _:

the

aggregation

number

N, highly dependent

on the

shape,

is the

major

factor in

determining

both

peack position q$

of the

I(q)

_curve and its

intensity

_;

the number h of solvent molecules in the

polar

outer

layer,

which is involved in the calculation of the excluded volume.

Only

the upper limit of h _can be determined with reasonable accuracy

(m lo) giving

_a maximum solvation

layer

about 0.55 _nm thick. The value of h is somewhat

arbitrary

in the range 5 _to lo

(in

the model

proposed

h is put at

7).

Canet

reported

values

ranging

from 8 _to 9

[26]

_;

the effective

charge

Z which

depends

_on the interaction between

polar

heads and

solvated counterions.

The value and the

position

of the

peak

in the

scattering

_curve in fact rule out a model of identical

homogeneous spheres

of radius

equal

to the

length

of the stretched out

aliphatic

chain

(i~

m

2.2

nm),

_or small

ellipsoids

_or rod like

undistinguished

from

spherical particles

of

larger

radii than

i~

for all concentrations of micellized _monomers.

Interpretation by

_a two

population

model. The

position

of the

peak

can be obtained from _a model of _a

sphere

of radius close to the Porod radius

Rp,

but the theoretical and

experimental scattering

intensities

only

_agree for _a concentration of micellized _monomers

c(

below

(c cmc)

in the lo fib, 15 fb _or 20 fib solutions. The lack of agreement ^{cannot be resolved}

by

assuming

_a continuous distribution of micelle size. This tends _to confirm the conclusions derived from the values of

integrated intensity Q*.

The

experimental

curves cannot be

accounted for

by

_a

single population

of

scattering particles

in the 10 fib, 15 fib and 20 fib solutions. The

experimental

_curves obtained

by X-ray

and neutron

scattering

which coincided if

g(0.468 nm)

=

1.2,

_or K

=

0.7 cm~ ^'

((A5)

and

(A7))

have been

interpreted

in terms of two

population

model _:

a

population

of small aggregates present ^{in solutions} at all

concentrations,

which in

some case is the _one

population

_at low concentration _;

a

population

of micelles present

only

at

higher concentration,

when the Porod's

plateau

is observed

Adequate

fits of

I(q)

_curves

(Figs. 2b-d)

_were obtained

by calculating

the scattered

intensity

from micelles

alone,

and then

adding

the

intensity

scattered

by

small

aggregates (the

scattered

intensity

from _a 5 fb solution

weighted by

the ratio of _monomers involved in these

small aggregates : c cmc

c().

Discussion

of

the micellar model. The small

aggregates

are not necessary structured.

However

they

are modeled

assuming compact particles

at two shells _as

spherical

micelles. The best agreement ^{between the calculated and}

experimental

absolute intensities

(SANS

and

SAXS)

_was obtained for micelles and aggregates ^{with the} characteristics listed in table III

(Fig, I).

Table III.

C C-cmc C( _R

~

Pwi P

care Z N n' _{« p}

mole-I-1 nr~l~ _{nm x} 101° _x 10t° _{6 1.6} 10t~ _1.6

x

cm-2 cm-2 _involved C.nm-2

0.490 0.400 3.9 3.0 ₊ 3.77 0.37 0.55 16 _*29 3.3 0.33

0.325 0.235 * 0.1 3.9 3.0 ₊ 3.77 0.37 0.55 16 _*29 2.3 0.33

5 0.159 0.069 2.3 1.8 ₊ 3.80 0.37 0.83 5 _** 6 4,1 0.30

3.3 0.108 0.018 2.3 1.8 ₊ 3.80 -0.37 0.83 5

** 6 1.8 0.30

The scattering length densities given in this table _are for neutrons (SANS) with deuterated forrrarride (FD).

C(is the _monomer concentration _* involved micelles in 10 fb and 15 9b solutions _; ** involved in aggregates ^{in 5 9l} and

3.3 fb solutions.

6 degree of ionization 6 is the fraction of counterions _« bound _».

n' is the number of aggregates of size N

* micelles _** small aggregates.

« is the surface _{area per} polar head at the aliphatic mediurn/solvent interface _{: «} 4

wR~~/N.

P is the surface charge density of the micelle _{: p} z/4 wR_~.

The

good

fit of the

falling

_part of the _curves

(q

_{~ q}

$)

where

S(q)

is close to

unity,

and which takes account of the fomJ factor led _{us to} include _an

equal

number of solvent molecules _to those of

aggregated

chains in the _core of the micelle. This may be

justified by

the less

solvophobic

character of chains with

respect

to formamide than water, which is in line with the

predictions

of Wamheim

[27].

The

probability

of

finding

solvent molecules in _a

layer

of

thickness dr and radius _r is

given by

the ratio of the volume of this

layer

to the _core volume :

dP(r)

=

(4 wr~dr)/(4/3 _wR/~~~).

These molecules of formamide _are

probably

close to the

polar

head. The notion of

probability

of localization of solvent molecules in the _core is less

arbitrary

than

assuming

that I _or 2

CH~

groups of the

aliphatic

chain

penetrate

^the

polar

outer

layer.

Given the small difference in contrast between

polar

outer

layer

and solvent for the

scattering length

densities of neutrons

(Fig. 5)

and the order of

magnitude

of the

experimental

resolution

(w/q~~~

ml-o

nm),

the form factor of _two

density

micelles _cannot be

readily

differentiated from that of

homogeneous spheres

whose diameter and

scattering length density

_are assumed _to be those of the

aliphatic

core

(cf.

Tab.

III).

The _contrast between

solvent and

particle

for

X-ray scattering

is

essentially

due _to the presence of Br~ ions

(Tab. II).

The fit between the

experimental

and calculated _curves is obtained

by condensing

most of the Br~ ions in the

polar

outer

layer.

The values of the

X-ray

scattered intensities

support

a two

density

model. Micelles in formamide _are structured _as

they

_are in water.

The small difference in contrast between the

polar

outer

layer

and the

FH/FD

mixtures

accounts for the

similarity

of

shape

of the

I(q)

_curves in

figure

2a. In the

expression

of the

form factor

(4),

the term (p~~~~

p~i) m(p~~r~

p~~i~) ^is

preponderant,

and the

intensity

becomes

proportional

to (p~~~~ p~~i~)~. ^{We verified that the model could} account for the set

of _curves shown in

figure

2a.

Since the D-L-V-O-

theory

does not

apply

to a

multicomponent

system, ^{and since the}

interacting particles

have different

charges

and _a distribution in sizes

(or

a discontinuous size

distribution),

the further from

reality

will be the model based _on

Hayter's

_program. The program ^described

by

Belloni

(HNC) [51],

which is in

principle

better

suited, requires knowledge

of the

charge

and size of the

scattering particles.

these parameters are

generally

not known in

complex

solutions. In addition to

that,

the _mean distances

I

_{between micelles} estimated from the volumic fraction

~b'

of the micellized _{monomers are}

equal

to 3.4 R and 2.7 R for lo and 15 fib solutions

respectively.

The distance between the surfaces of the micelles is now

only

1.4 _nm for the 15 fib

solution,

which is

only slightly higher

than twice the

Bjerrum length (0.5 nm).

Thus _one would

expect

^there to be

significant

differences between the calculated and

experimental

_{curves at} low values of q since the calculation of

S(q)

is

approximate.

These differences are the _same for neutrons and

X-rays

in the 10 fib solutions and increase with

increasing

concentration. It is

impossible

to model

accurately

the _curve of scattered

intensity

from _a 20 fib

solution,

which shows the

preponderance

of

S(q)

_{as q} tends to zero.

The presence ^of a dear~cut

peak

in the scattered

intensity,

_{or a} marked fall in

intensity

_{as q}

tends _{to zero,} is indicative of the existence of _a

repulsive

interaction

potential

between the

micelles,

which adds to the excluded volume term. This fall in

intensity

cannot be

entirely

attributed to _an excluded volume effect since the strong electrostatic

repulsions

have _a

stronger

^effect on the fall in

intensity

_{as q} tends _{to zero.} Thus the determination of the

charge parameter

^Z

depends

on the

modeling

of this part ^{of the} curve. For _a lo fib

solution,

the

intensity I(q)

scattered from the micelles alone _can be calculated

by

the

following expression (Fig. 2d) [52]

:

1(q)

=

nP

(q)

S ~~~q)

S~~~q)

is determined

by adequate fitting

the theoretical and

experimental

curves

(SANS

and

SAXS).

A value of 0.15 mole.l~ of micellized _monomers in the range

c]

=

0. I I mole. l~ and

(c-cmc)=0.23mole.l~'

is

found,

the

charge

Z is

unchanged.

This correction of

S(q)

takes account of the excluded volume underestimated because of the presence of

aggregates. ^{The values of}

charge

listed in table III _are thus

significant.

The surfaces _area per

polar

head in

spherical

micelles in formamide

(«

m I

nm~)

is above that found with almost

spherical

micelles in water

(«

m 0.7

nm~).

The micelles have _a

high degree

of ionization 3 ~ 0.5.

It _can be _seen from the results listed in table III that the 10 fib and 15 fb solutions contain identical micelles of small size

containing only

29 _monomers.

They

have _a

relatively

_narrow

size distributions since the calculated radius R and the Porod radius

R~

^do not

depend

on

concentration

[53]. R~

^{calculated from}

homogeneous spheres

is

nearly equal

to R~~r~

(Tab. III)

the most difference is 0.3 _nm. The concentration

c]

of micellized _monomers calculated from the theoretical and

experimental

absolute

scattering

intensities is almost

equal

to c' calculated from

Q* Ill.

The fraction of _monomers

cj(c cmc)

involved in the micelles increases with

increasing

concentration

(40

fib and 47 fib for the 10 fib and 15 fib solutions

respectively). However,

the number of micelles and the number of small aggregates also increase with

increasing

concentration.

The 5 fib and 3.3 fib solutions _are modeled

by

structured

compact

aggregates

containing

6

monomers and with _a

high degree

of ionization

(3

~

0.8). However,

the

density

of

charge

_on the surface of the

aggregates

^is little different from that on the

spherical

micelles. The size and the surface _area per

polar

head _are

only approximative

values

considering

that there is _a

large dispersion

in aggregate

size,

the

high

value of

0f2.75nm2)

indicates that the interface

between the aggregates ^{and the solvent is} not

particularly sharp

_; their curvature is the _same order _to

qj~[.

Their

scattering

is distributed

throughout

the

reciprocal

_space, ^the Porod's limit is not observed

(Fig. 3)

and their contribution to the

integrated intensity Q*

is

negligible.

The presence of _two

populations throughout

the domain of concentrations studied shows that aggregates ^did not grow

progressively

^with increase in concentration. Their presence up to the formation of the _two dimensional

H~ hexagonal phase

at 45 fib _{can account} for the discontinuities between the

position

of the

peak q$ just

before the transition and the

position

of the

H~ Bragg

line 10

[3].

This

discontinuity

is not observed in water. We assumed that in formamide _as in _water all the micelles which form _an ordered arrangement are of the _same size

[54]. Comparison

of the diameter of the micelles

(3.9nm)

and the parameter a~ of

H~,

which ranges from 4.6 _nm

(at

45 fib) to 4.4 _nm

(at

74 fib) shows that the diameter of

the

cylindrical

micelles is almost

equal

to the diameter of

spherical

micelles. The minimal thickness of the

layer

of free solvent ranges from 0.7 _nm to 0.5 _nm, while the diameter of the molecule of formamide is around 0.5 nm.

Discussion.

Our

findings

_on the onset of

aggregation

of CTAB in formamide showed that aggregates ^form

at a monomer concentration between 3 fib and 8-9 fib. The results of the calorimetric

study

of

CTAB in formamide

[23]

showed that the

aggregation

number is small

(N

= 6 _{± at} 45

°C)

over a

large

concentration range of 3 to 9 fib and that the association progresses over all this range. The differential

enthalpie

of solution of

crystalline

CTAB in formamide _at 45 °C

decreases at

higher

concentration than 9 fib.

Micelles of well defined size and

shape

_are also formed _at

higher concentration,

which exist in

equilibrium

with the

aggregates.

^{The diameter of}

spherical

micelles is around 4 _nm

(the

core radius is under

i~)

and is thus less than that of

spherical

micelles in water

(around

5.4

nm).

This value of 4 _{nm was} also observed for the diameter of the

cylindrical

micelles of the

H~ hexagonal phase

which form at

higher

CTAB concentrations

[3].

The

growth

of the micelles with

increasing

concentration _occurs above _a radius of 2 _nm via _a

change

in

shape

rather than _an increase in radius as

suggested by

Wamheim

[27].

The

aggregation

number _was found _to be around

30,

_a third of that observed in water. The size and the

degree

of ionization of micelles of CTAB

[27]

and CTAF

[28]

in formamide calculated from diffusion coefficients

are the _{same as} those found

by

small

angle scattering.

The

aggregation

number N is 30 and the counterion

binding

is 0.55~0.50 about this is 0.8 with N around 6. In water, the counterion

binding

is determined to 0.71

[55]

or 0.86

[56],

the estimated

degree depending

on the

experimental technique employed.

The relation between effective

charge

and radius has been

predicted by

Mitchell

[57]

and Evans

[58].

The radius of curvature of the interface

(2 nm)

_appears to be characteristic of the micelle interface in

formamide,

and is

independent

of the nature of the counterion. The size of the CTAB micelles in formamide does not

comply

the

simple

rule _: the radius of the

spherical

micelles _{at cmc} is determined

by

the

length

of the extended

alkyl

chain. The

polyethylene

oxide

alkyl

ethers C~

E~

in formamide _are

compared C~

~

E~

ⁱⁿ water

[23]

_{; so} the _core radius of CTAB micelles should be _to 1.7 _nm and the detemJined value is lower

(1.5

_nm, Tab.

III)

_; but the

repulsive

interaction of the ionic

polar

heads _are

higher

than between nonionic

polar

heads.

The

equilibrium

size of the micelles results from

competition

between the

solvophobic

action of the

aliphatic chains,

which favors

micellization,

and the

repulsive

interactions of the

polar heads,

screened

by

solvent molecules and

variously

solvated counterions. The _area per

polar

^head is

larger

in formamide than in water

(«

₌ 0.66

nm~),

and _so the distance between

neighboring polar

heads is greater ⁱⁿ formamide than in _water and _{so more}

aliphatic

_core is in contact with formamide.

However,

the surface tension ymjc ^{between the} core and formamide

(27.3 mN.m~')

is below the surface tension

y~~o/C

^{between core and water}

⁽⁵⁰

^mN.m~ ~)

[27].

The variation in surface free energy is

Ag~

₌

y(« «o),

_«o is the _area

occupied by

the

polar

head

+N(CH~)~. Assuming

that «o is the surface _area of _a cross~section of _a

sphere

of volume

v(+N(CH~)~), _Ag~

_can be calculated for micelles in _{water or} formamide

[59]

_:

mom 0.26

nm~

(Ag~)~~o

= 20 _x 10~ ^~~ mJ

m 0.13 eV

(Ag~)~~

₌ ¹⁹ _x 10~ ^~~ mJ

m 0.12 eV

Using

the _same

approximation [59],

_an

approximate

value for the

change

in free energy

Ag~

_{per monomer} due _to electrostatic

repulsions

_can be obtained from the

charge

_on the

micelles and the

potential

_:

ze

~ko =

Ze/(4

_{w e}

R)

_or

Ag

~ ⁼

l/N _~k

d(Ze)

=

Z

~e~/(N

8 _{w e}

R)

o

or in formamide _:

(Ag~)~~

₌ 0.03 eV and in _water

(Ag~)~~o

m 0.006 eV

taking

N

m

90,

R

m 2.7 _nm and _a

degree

of dissociation to 0.14

[56].

The difference between

(Ag~)

in water and formamide is

mainly

the result of the

high degree

^of dissociation in FH. This

approximate

calculation also shows that

(Ag~)

^is small

compared

to

(AgJ,

which supports ^the conclusions of Wamheim _on the

importance

of surface tension in the formation of micelles

[27].

The value of the _sum of

Ag~

^and

Ag~

^{is similar in}

water and formamide. The difference is _not

significant

in view of the

approximations

made. If the

only

_{parameters to} consider _{were y,} e~ and the

dipole

moment p~ of the

solvents,

micellization and formation of

lyotropic phases

would _{occur more}

readily

in

N-methylfor-

mamide

(MFA) (dielectric

constant e~ =

180,

_y

=

12.5

mN.m~'

_{and p~}

= 3.8

D)

than in

formamide _or in _water

(p~~~

=

3.7 D and

p~~~o

= 1.8

D) [2].

In fact

only

the lamellar

phase

is observed in MFA

[2, 3].

Thus _even if the electrostatic

repulsion

between

polar

heads is

highly

screened

by

interaction between the

dipole

moments of MFA and the

highly

localized

charge

of ⁺

N(CH~)~,

this

screening

cannot be

compensated by

the weak

solvophobic

action of

aliphatic

chains and the steric effect from the added

methyl

_group in the solvent. A similar situation arises in

N-methylsydnone (e~

₌

144,

~LD =

7.4 D and yv~ ^~/~ ₌ l.4 _x

llf J.m~~)

in

which CTAB

only

forms the

L~ phase.

On the other hand in

N-methylsydnone,

CPBr

micellizes and the whole sequence of

lyotropic phases

is observed

[4].

The

change

ⁱⁿ free energy due _to the

solvophobic

effect of

aliphatic

chains in this solvent is the _same for CTAB and CPBr in _a first

approximation,

but the

solvent-polar

head interaction is of the

dipole- dipole type.

^{The steric effect is} more

favorable,

and there is sufficient

screening

for self-

association, giving

rise to interfaces with _a noticeable curvature.

The cohesion energy of the solvent may ^{also be} an

important

factor in the

hydrophobic

effect. This effect in formamide with _a 16 carbon atom chain is

probably

close _to that observed

in _water with _an 8 _atom carbon chain

[I].

The formation of aggregates at low concentration

that _{are a} kind of

premicellar aggregation

is observed in all

binary surfactant/solvent

_systems in which association is not

spontaneous.

^{This is the} case for water,

despite

_a low value of _cmc, and

polar

non-aqueous solvents. For

example

the molar conductance of _an aqueous solution of

dodecyltributylammonium

^bromide

(CI~NBU~Br)

close _to the _cmc

(5

x

10~~

M.l~ _{~) as a}

function of

c'/~ displays

_a curvature rather than _a break in

slope, indicating

_a

gradual

_onset of micelle formation

[46].

In this _case, the

aggregation

number N varies with _c

(16

at I fib and 34

or 41 _at 8

fb).

The concentration of free _monomers determined from

Q

^* varies with _c. These

authors concluded that the «diffuse _cmc» and the increase in concentration of free

monomers are both manifestations of

weakly cooperative

micellization. The surface _area per

polar

head is

high (1.20 nm2) compared

to that of

C12N(CH~)3Br (0.63 nm2),

the difference

JOURNAL DE PHYSIQUEI -T 2, N'6, JUNE 1992 36