1261. Classification. Physic-s. Abstiar.ts. 64.70. 61.30E. 46.30J. Isotropic-to-nematic transition in wormlike micelles under shear. Jean-Frangois. Berret,. Denis.
J.
Phys.
France
II
(1994)
4
1261-1279
1994,
AuGusT
1261
PAGE
Classification
Physic-s
Abstiar.ts
64.70
61.30E
46.30J
Isotropic-to-nematic
transition
in
micelles
wormlike
under
shear Jean-Frangois Groupe
Berret,
(Rer.eived
20.lattuaiy
Abstract.
We
wormlike
micelles.
semidilute
Cl-j
s-') for
plateau #jj~, «
in
the
remains
However,
the
critical
«(#)-behavior
throughout phases
the
coexist)
#j~
picture
as
of
a
the
at
a
first-order
weakens
becomes a
Moreover, of
time
phase
been
be
These
L In
(in
transition
interpreted quantitatively in finally compared to are described who the nonlinear rheology shear-banding type. could
observed
results
terms recent
of
the
of
~b~,
the
concentrations
shear
above
transition
«(t)
micelles
in
terms
Cates of
some
the
micelles
the and at
In
agreement
the
transient
and
at occur
measured
nematic constant
with
the
behavior
growth
one-dimensional
and
that
of
rate
first-order.
to
ceases
isotropic
both
stress
investigated in detail. domain metastability), of
nucleation
clearly
orientation
predictions by Spenley,
transition
the
is
suggesting
dilution, the
(where
domain the
typically higher isotropic-to-nematic by a true characterized
transition
and
wormlike
Both
Rheological
j
rate
is
#
For
homogeneous
of
behavior
Nacl~bfine.
surfactant
shear
first~order
#).
al
Below
progressive and in the two-phase
transient has
upon
for
transition
regime, increasing
second-order.
of elongated cetylpyridinium
investigated.
were
a
The
stress
concentrated
the
of
0.5M
in
diluted
exhibit
6 %.
shear
solutions made
are
=
micelles
the
/994)
surfactant
of
(T 25 °C ) high enough (for
shear
m
May
II
micelles
temperature
~b
Montpellier II,
de
scrutiny
Sal-)
(Na+,
steady
of
character
characteristic
function
In
under
entangled
of
wormlike
of
indicates
sample.
placed
ambient a
rheology
nonlinear
and
to
«jj~.
4~ it
rather
ar.cepted
dependence
first-order
concentration
/994,
concentrations
rate
constant
Ap;il
sahcylate
at
suriactant
shear
Universitd
regimes
solutions
the
all
(*),
solutions
submitted
When
Porte
Condensdes
linear
sodium
performed
%.
I-lo
transition
and
Grdgoire
and
/3
surfactant
concentrated
were
%-30
~
Roux
reitised
the
on
The
and
experiments 4
/994,
report
(CP+,
chloride
#
C.
de Dynamique des Phases Montpellier Cedex 05, France
34095
than
Denis
process.
(Ref. [71) instability of
MacLeish
mechanical
Introduction.
spite
termed
(+)
of
an
U-R-A-
complexity, exhibit polymersj
apparent
equilibrium
2~3_
surfactant a
very
solutions
simple
of
linear
elongated wormlike viscoelasticity. In
micelles the
(also
semidilute
1262
JOURNAL
PHYSIQUE
DE
II
8
N°
regime of entangled network, the linear rheology is dominated by reptation (as for ordinary polymers) and by reversible breaking and recombination of the chains. The relaxation two times and r~ related the previous mechanisms the dynamical control to to response an r~~~~ external applied shear. As far as the breaking time r~ is much shorter than the reptation one, the linear function G(t) in measured relaxation experiments follows almost stress response an Maxwell behavior: G(t)=Goexp(-t/r~) where modulus Go is the plateau and pure terminal relaxation This Maxwell time. behavior observed has been r~ the on a large variety of materials forming equilibrium polymers (1-61. the
In
present
rheology
paper, wormlike
of
give
we
comprehensive which,
a
for
account
solutions,
micellar
in
an
to the
contrast
investigation of the nonlinear linear regime, is much less
understood.
first
Our
motivation
nonlinear
viscoelastic
semidilute
regime.
was
theoretical
recent
a
behavior
of
predictions
by Spenley,
work
wormlike
Cates
submitted
micelles
(referred
steady
a
[71 of the
McLeish
and
to
in
shear
the
theory in the following) are based on a so~called reptation~reaction model developed previously by Cates [81 and solved reference in constitutive [71 using the equations of the polymer dynamics [91 adapted to reversibly breakable chains. main of this The model results the following are as shear reaches threshold value ii 2.6/r~, a mechanical instability of shear-banding type a within the micellar solutions. This instability is characterized above #j by a plateau of occurs the shear the height c~* 2 Go/3. The plateau persists over a finite stress )-range, at c~ ii j < ii. Above j~, the shear (# linearly increase again with stress starts to respect to c~ I. Actually, the existence of a plateau in the c~(j) behavior level 0.6 Go has been at a The
of
these
authors
to
SCM
as
=
=
eisus response. all the G (G') data sets can be fitted accurately by a semicircle of radius Go/2 (as shown by the continuous lines), according to 2.3
LINEAR
A
RHEOLOGY.
detailed
of the
account
linear
viscoelastic
reference
=
=
"
(G' This
relaxation
viscoelastic time
r~.
behavior The
is
that
viscoelastic
Go/2)~ of
a
+
(I
=
Maxwell
parameters
G(/4
G"~
Go,
[19]
element T~
and
~o
characterized
Go =
r~
lim =
4oo
300
~
. .
/~
~~l
~~
~
00
21.9 $ ~ ~
~
0
200
400
G'
by
a
unique
)G*(w ))/w
as
1266
w
JOURNAL
0
-
easily
then
are
[61,
reference
derived
and
range (0.4 w ~b w 6 fb ) with this deviations from range, features of the concentrated around
r~
10 fb
~b
4-dependence
their
quantities
three
the
PHYSIQUE
DE
with
scale
in
exponents
II
N°
shown
is
with
agreement
However,
the
[Sal]/[CPy] T
by
Cn
~o
.'
noticeable
most
relaxation
time
O
~
q
,
.
j
~
o
~ ~
.
fl-
o
.
~
the
terminal
~
&
. _
cd
in
semidilute
°
.
~p
the
.
.
~'
~£
emphasized
in
.
_
~
As
25 fG.
25° C
=
the
0.5
=
4.
predictions [101. Beyond
theoretical
predicted power laws occur. regime are the maxima exhibited by the static viscosity around ~b
and
figure
in
concentration
surfactant
the
8
.
~
e :
oo
~o
~
0001
o
lo
i
i
§
Fig.
of
surfactant
the
regime,
semidilute
TR-data
2.4
(empty
with
circles)
4
was
pass
0.5 =
double
to
fG-21fb.
5
At
low
=
characteristic
the
c~,m)
at
c~~m
increases
However,
plateau ii,. For
a
sets
The
RHEOLOGY.
steady shear logarithmic scale, )
with
close
~b,
ji,
c~
rate
whereas
elastic
that
Note
plateau
solutions
(see straight line).
~
in
Inset
the
equilibrium
variation
:
of the
CPCI/Sal
circles)
polymers. terminal
regime~
concentrated
figure
in
Go (full
modulus
of
of the response several 5 for
nonlinear shown
both
surfactant
as
relaxation and
~o-
solutions Plotted
concentrations.
a
the
In
on
a
analogous behavior for increases linearity, reaches )-independent plateau (of height a j,m and then, at much higher rates increases again. Note that ji,m, the shear rate at which the plateau sets in, is lowered. shear
the
inspection abruptly and
up 8 fb
is
flow
and
CPCI/Sal
for
~b
against concentration. through a maximum.
NONLINEAR
(R
scales
~o
~jj
concentration
rR=~o/Go
time
~b
viscosity
zero-shear
4.
function
the
of a
true
stress
c~
c~(ji)
if )-data
exhibits
shows,
discontinuity
of
an
two
slope
overall
types
of
behavior. in
occurs
the
8 fG,
the
)-behavior
at
Above
c~(j
plateau regimes is much range ~b ~ plateau region has completely clearly 516-sample, the smoother rounded. For the and intermediate reminiscence inflexion point occurring in the disappeared the only of it is an have checked be emphasized of figure been should finally that the data points 5 It to j-rangethe
transition
between
the
Newtonian
and
the
N° 8
ISOTROPIC-TO-NEMATIC
TRANSITION
T
250 ~
=
loo
i$ '~
,°°.~.i.I,ii( (°(°]
o° °&~ ~~
o°.°b~ °
~
~o
.°a~
o°
lo
j
.
~
°
OOOOOOOOOOOOj~ OOO
~O*
o* O
~OOO o
.*o°°
.*o°°
~
~
~'
~A
~
~
°
(~
1267
MICELLES
~~
~O
~o
g
WORMLIKE
IN
°
4 4 4
° °
o* o°
~
=
5.t %
=
6_3 % lo %
"
]
~°~
()e
~
o-i
Fig.
5.
Shear
stationary
be
limit,
500
The
other
I.e.
after
s
(see
shear
the
of
the
stress
points
data
equilibration
an
values
of
All
%-21§b.
5
=
stationary
the
than
dependences
rate
4
concentrations
in
shear
measured
«
plateau
the
that
stress,
(I')
rate
larger
time
loo
lo
i
Shear
than
is,
few
a
and
data
one
set
is
obtained
linear
from
j
~
for
in
typically
times
the
longer
compared to each are respectively. Indeed,
j
al
=
samples,
21fG
=
as
obtained
were
seconds.
equilibration
after
dynamical and the steady shear viscosities ~ (w figures 6a and 6b for the ~b 14 fG and ~b is no general why dynamic and steady reason
system,
hundred
solutions
micellar
(see
section).
next
in
there
CPCI/Sai text)
from
regime
=
shear
should
data
whereas
measurements
the
identical
be
other
in
is
one
taken
any in
highly nonlinear conditions. However, for many solutions of polymers [191, it systems such as behave similarily as a function of their happens that both dynamic and steady shear viscosities This is known as the Cox and Merz empirical rule. In this it is respective context, arguments. worth mentioning that in the case of CPCI/Sal equilibrium polymers, dynamic and steady shear viscosities fin. strongly deviate from each other, namely for j ~
loco loo
o
o°~~~°°~°°00...~~ o
~ ~_
~ .
~
f
°..
~
°~,
~
f
°o.~
~
O
~
/
>
/
.
~
~
fi
IO
.
.~
~
~°
.
q(~
.
~
n(?)
fl(@)
.~ °
°
~
.
.~
.~
.
~~
loo
a
q(t0)
O
~
OOOOOOOOOOO,o
~
~
°
/
°
~°
°~
l
0.I Shear
100
10 rate
Id)
and
10
0.I
angular frequency (rad.K~)
Shear
rate
(d)
and
b)
a)
Fig. 6.Comparison between respectively plotted against wand ~(#) Y
"
with Yj/N.
respect
to
100
angular fTequency (rod.f')
~(w)
the
#, are
dynamical for
the
observed
4
and
the
14 fb (a)
steady and
shear
4
=
precisely
at
the
viscosities,
(b)
~(w)
and
~(#)
solutions.
Deviations
of
isotropic-to-nematic
transition,
at
21% =
1268
2.5
TRANSIENT
section,
limit,
it I.e.
solely exhibits
still
after
very
stationary
the
and
figures
in
STRESS
stress
stress
response c~ 7b for the
and
7a
stationary 8 % and in the plateau region
5
performed
is
obtained
were
for ~b ~ from constant
received
as
experiment cone-and-plate
shear
follows
as
Shear is then apparatus. recorded time. over
fG-sample
j
for
the
in
measurements
rate
firstly the sample is left abruptly switched on at Typical wit )-variations
:
(t j is 14
previous
the
In
REGIME.
PLATEAU
figure
of
N° 8
II
Actually,
times. c~
THE
IN
c~(j )-data
the
The
in the
rest
j
illustrated
are
at
value
constant
a
shear
behavior.
equilibrate
to
the
transient
a
SHEAR
that
long equilibration
c~,~),
=
DE
THE
mentioned
been
(c~
OF
BEHAVIOR
has
PHYSIQUE
JOURNAL
=1-5
'
s~
j,,,
Below
the
is rapidly, namely within r~(w I s), as shown by the I and reached very is clearly identified. As figure 7a. For shear rates exceeding j,~, an overshoot denoted shearing is switched «(t) first rapidly up to a value the increases soon as on, below relaxes slowly down to its c~(0) (the rising times remains s), and then the stress 10 s-200 s, depending on the stationary limit c~j~. Typical time scales for this kinetics are observe also be noted in figure 7b that at j 5 s~ ' not only do we shear rate applied. It must This feature is presently not understood. overshoot, but it is followed by a small undershoot. an carefully undershoot is not observed long as the it has checked that this However, been as 50 %). shear ji (corresponding to la (0j ji,~ is moderate c~j,~)/c~jm rate excess The stationary limit «1/~ 170 Pa This transient behavior further deserves comments. same
s-'-data
2
limit
in
=
