Transient behavior and relaxations of the L3 (sponge)
phase: T-jumb experiments
G. Waton, G. Porte
To cite this version:
G. Waton, G. Porte. Transient behavior and relaxations of the L3 (sponge) phase: Tjumb experiments.
Journal de Physique II, EDP Sciences, 1993, 3 (4), pp.515-530.
<10.1051/jp2:1993148>. <jpa-00247851>
HAL Id: jpa-00247851
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Submitted on 1 Jan 1993
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J.
Phys.
II
France
(1993)
3
515-530
1993,
APRIL
515
PAGE
Classification
Physics
Abstracts
42.20J
66.10C
82.70D
Transient
61.25
behavior
T-jump
relaxations
and
L~ (sponge)
the
phase
:
experiments
(I)
G.
Waton
ii)
Laboratoire
Pasteur,
4
and
G.
d'Ultrasons
Blaise
rue
(2)
Porte
de
Dynamique des
67070Strasbourg Cedex,
et
Pascal,
(2) Groupe de Dynamique des Phases
Eug~ne Bataillon, 34095 Montpellier
(Received
26
sponge
times
r~,
October
We
report
phase
after
AbstracL
the
of
and
r~
variable
intemal
variables.
We
accepted
1992,
here
on
a
abrupt
an
r~ having very
while r~ and r~
propose
is
Montpellier II,
time
jump.
investigation
resolved
obtain
We
of the
for
evidence
magnitudes. We show that
correspond to the relaxations of
interpretation of the
transient
behavior
an
quite weakly
Universit6
first
026,
Case
Place
29December1992)
temperature
different
r~
both
jump essentially puts the
interpretation, r~ corresponds to the
degree of symmetry of the
structure
actually accounts at least qualitatively
dependences of r~ and r~ close to the
Louis
France
systematic
temperature
transition
05,
Universit£
France
(*),
Condens6es
Cedex
Complexes (*),
Fluides
membrane
relaxation
and
r~
the
to
in L~
of
that
under
its
experimental
sponge-to-lamellar
for
based
related
observation.
transition
to
the
on
idea
that
the
to
this
of
the
Moreover,
that
to
r~
connectivity.
temperature
conserved
a
According
tension.
of
of
relaxation
thermodynamic
fluctuations,
density
behavior
distinct
unconserved
transient
concentration
of
is
transient
three
picture
This
the
temperature
that
indicate
this
order.
Introduction.
The
anomalous
experimental
isotropic phase L~
realization
of
an
in
an
infinite
amphiphilic
fluid
with no long-range
order
space
the static
factor and
structure
some
system
membrane
(sponge phase) provides
multiconnected
along
an
the
ideal
three
Equilibrium properties such as the phase
stability and
dynamical properties close to equilibrium have
been
extensively
studied
and
in the
of being well
In
comparison,
understood.
are
course
understanding
non-equilibrium properties is in a rather primitive stage [5, 6]. The present
article
aims at providing
further
experimental insights into this problem. More precisely, here
evidence
the T-jump technique in order to
and
the
characteristic
relaxation
we
measure
use
times
be
for
dynamical
properties
of
that
relevant
the
the
phase.
L~
can
directions
(*)
of
Associd
CNRS.
[1-4].
516
JOURNAL
N°
II
4
out of equilibrium by a
equilibrium
state at this
goes
new
the
In
change
in
is
achieved
by
sudden
temperature.
temperature
new
case,
discharging a capacitor through the conducting brine swollen L~ sample. And the relaxation of
is
using time resolved
the
observed
of the intensity
scattered
by the
structure
measurements
light very
sample. This procedure is here appropriate since L~ phases are
known
scatter
to
strongly specially at high dilutions.
fluid
exhibit
Structures
in complex
often
several
degrees of freedom at large scale, each of
time. In the
them having a
characteristic
relaxation
favorable
these times are sufficiently
cases,
different
in magnitude
and
each
degree of freedom
will
relax
after the
shortest
are
ones
completely
annealed
and while the longest ones
still quenched. The
different
contributions
are
thus be
resolved
separately.
can
backgrounds : we recall the intemal
variables
that
define
In section I we
the
sum
up
some
of
their
thermodynamic
phases and the scaling
behavior
of
characteristic
state
sponge
relaxation
times.
The experimental facts are reported in section 2. In section 3, we
propose
an
interpretation of the
transient
oscillation
of I(q)
in
of the
surface
tension
terrns
excess
transiently imposed to the
membrane
by the AT step. The
of this
interpretation is
relevance
The
of
idea
T-jump experiment
PHYSIQUE
DE
a
sudden
increase
of its
further
discussed
in
section
given
is to put the
to follow
abruptly
structure
how it
and
temperature
the
present
to
its
4.
Background.
1.
According
self
the
to
assembles
into
isotropically
and only
two
MiIner
et
al.
in
a
multiconnected
in
[7],
by
I) the
density
the
we
itself
to
distinct
two
characterized
area
commonly
admitted
for L~, it is
convenient
consider
the
to
figure I. The basic features are the following. Most of the amphiphile
bilayer defining an infinite surface free of rims and seams. The surface is
structure
drawing
schematic
of
that
the
of
membrane
amphiphiles #),
it) the connectivity density
Fig. I),
iii) the degree of asymmetry
vi
values
assume
average
n
the
in
subvolumes
a
per
three
directions
three
intemal
unit
volume
per
unit
Y'between
volume
subvolumes
~'~vi+v~
I.
Schematic
structure
of
the L~
phase.
2'
v,
divides
space
analysis
given time
in
the
of
a
is
:
(proportional
of
it
following
the phase at
of
variables
(density
And
space.
and
v~.
v,
Fig.
of
Accordingly
thermodynamical state
and
«
and
to
handles
the
»
volume
or
«
fraction
passages
of
»
in
v~.
~~
N°
SPONGE
4
equilibrium,
At
the
equilibrium
in
a
and
R
of
the
invariance
scale
general
structure).
inside/outside
structure
bilayer
the
0. (Here
locally
is
do
we
=
symmetrical
the
broken,
very
of
cases
the
surface,
the
so-called
scale.)
transiently
however
of
(concentration
global
be
times.
The
average
life
characteristic
indeed
has
where
related
forrn
the
mid
to its
variable
relevant
two
It is
by
deterrnined
at
can
conserved
a~ expressing
fi
the
to
:
~§
To exp
=
respect
dilute
symmetry
at
T~
with
equilibrium,
at
equilibrium, the
conservation
relation
and the
global scale. Note that while a is indeed a
amphiphiles), n and Y'are not. Accordingly, we expect
first one, r~, is the
time of the connectivity.
relaxation
given « passage ». According to [7] it
time r~ of one
Out
broken
517
EXPERIMENTS
consider
not
spontaneously
[8] is
symmetry
T-JUMP
:
by a non-linear
conservation
relation
[4, 5] (the geometrical prefactor is
related
are
Since
#
expect
we
PHASE
12)
B
rp~
where
is
activation
the
involved
energy
straightforward
frequency of
membrane
in one
elementary change
average
show
to
that
has
r~
the
r~
f
where
is
the
discussed
free
at
The
second
time,
of the
solvent
from
of
r~
bilayer
the
ar~
subvolume
So
Ap
Ap is the
0,
but
of the
obtain
Ap
=
between
Remembering
immediately see
which
#
that
that
relaxation
rq~
times
#
are
annealed
speculative
perrneability
factor
ar~).
and
scales
are
quenched (and
are
they
d~
rather
be
the
«
by
controlled
membrane.
assume
ar~
The
solvent,
brine
pores
»
the
in
the
core
expect
we
[7].
membrane
perrneability
a
transport
hydrophobic
parameter
(4)
a
subvolumes
I
and
2.
if
equilibrium
At
0)
=
~f~.
~~~
These
arguments
(5)
:
rq~
fi and
It is
symmetry.
can
we
Simple scaling
#~~(f~).
like
2
Ap=-fl'
We
it is
generally
more
=
Then,
:
difference
pressure
r~
time
dv,/dt
where
the
is
(3)
equilibrium.
at
scales
mechanism,
that
topology.
membrane
through the
imperrneable to
quite
equilibrium density of
average
transport
bilayer.
of
area
the
to
actual
E~
and
passage
T
phase
subvolume
I to
presumably
related
the
unit
per
q~,
that
relaxation
is the
being
rather
be
to
Whatever
r
the
the
R(a~flan~)
the L~
then
in
in
:
lk~
r~
=
density of
imply
energy
in [5]
length
form
collisions
here
=
arj
a~flaY'~~ #~
#~~ along
like
(from
a
specially important
therefore
(unconserved).
they are
since
must
be
(6)
d
aY'~
the
:
line
they define
considered
Attempting
presumably
usual
dilution
as
scaling
the
predict their
strongly system
to
time
conserved
argument
[5]),
we
arj).
(constant
scales
variables)
relative
below
and
magnitudes
dependent
(due
which
beyond
would
to
the
JOURNAL
518
Experimental
2.
PHYSIQUE
DE
N°
II
4
facts.
experimental set up has been
in detail in [9]. The amplitude AT of the T step
The
described
imposed to the sample (typically a few tenths of a °C) is adjusted by monitoring the charge
tension
of the capacitor. A
lamp is used to illuminate
the sample.
Incident
xenon-mercury
monochromatic
wavelengths are selected using interferential
filters : A
577 nm and 405 nm.
The
scattered
intensity is collected at two different angles off the incident beam. Combining
(q=0.50x10~ cm~', q=
wavelengths
four
of q's
and
angles,
values
available
are
2.05 x 10~ cm~ ~, q
0.72 x 10~ cm~', q
2.92 x 10~ cm~ ~) so that the q dependence of the
=
=
=
relaxation
times
estimated.
be
can
(0.2 M Nacl) for which we have
investigated is the system
CPCI/hexanoljbrine
The
system
previously collected the most complete set of structural data [I]. We studied three samples in
domain.
Their composition in CPCI and
the L~ monophasic
hexanol
(Tab. I) are chosen so that
dilution
line.
Comparing the data
they roughly correspond to the same
obtained
these
on
samples, we can estimate the dependence of the relaxation
function
of the
times as a
volume
0.072).
(#
0.026,
0.053,
Actually we
could
fraction
# of
membranes
#
#
not
investigate
concentrated
samples for which the response is too low and
therefore
the
more
signal-to-noise ratio does not allow an appropriate
resolution.
=
Table
L~
Composition
I.
L~
to
(g/lOOg)
samples.
T~~~L~ is
T~
,
2.28
2.72
O.053
8°C
3,ll
3.57
O.072
1°C
with
other
critical
has
system
On
systems
behavior
the
the
other
[lO],
the
related
to
its
the
to
the present
system does not show the
breaking at high
Y'symmetry
spontaneous
O-O I I, the L~ phase here rather stops swelling
brine.
So, at equilibrium, the L~ phase of the
the
beyond #
expelling
excess
of a symmetric
structure
hand,
dilution
=
the
L~
range
measurements
structure
of
the
sponge
three
of
final
of
at
relaxation
two
different
processes
T-jump experiment,
equilibrium values, thus
in
a
L~
L~ phase in
(Y'= O) all
temperatures,
and
so
we
over
its
samples investigated has
samples eventually phase
stability : more precisely, all three
(swollen
lamellar
phase) upon decreasing
with
the L~ phase
respectively
for each
sample are given
measured
temperatures
Usually,
_+
15°C
temperature
down
of
temperature
0.026
Upon increasing
simply phase separates
stability.
(g/lOOg)
n-hexanol
dilutions.
jump
the
.41
contrast
present
investigated
the
1.09
characteristic
and
=
transition.
GPO
In
of
=
could
The
temperature.
in
table
evidence
I.
We
domain
a
separate
transition
perforrned
interesting
of
limited
T-
slowing
approaching the L~ to L~ transition
temperature.
intensity varies monotonically from its initial
evolution
of the system
reflecting the monotonic
towards
when
the
scattered
N°
SPONGE
4
final
its
In
state.
typical
the
response
intensity
scattered
scattering
is quite
especially
present
insensitive
intensity
typically
very large
temperature
here, is the
to
the
behavior
is
by
scattered
few
a
:
the
519
note, that the initial and
This is actually
consistent
other.
each
to
that
EXPERIMENTS
samples,
figure
in
close
very
have
shown
are
which
data
T-JUMP
:
phase
sponge
2. First we
of
case
represented
is
PHASE
light
static
equilibrium
at
C.
per
so
a
:
of the
values
with
phases
sponge
variation
or
percent
different
very
final
But
what
is
(q ) observed
transiently in between
oscillation
identical
initial and final
values.
reaches in the
the two
almost
The amplitude of the
O.5 °C) about 30 fb of the equilibrium
favorable
(most dilute sample with AT
most
cases
amplitude of the effect appears to be roughly linear with the
of energy
values.
The
amount
injected by the electric discharge into the sample (I.e. linear with An as shown for an example
having a large enough amplitude, the
monotonic
variation
in figure 3. Owing to this
non
transient
behavior
of the
phase can be investigated with a
reasonable
sponge
accuracy.
impressive
oscillation
of I
~
We clearly
observe
respective
characteristic
experimental
conditions
variations
of
the
parameters,
the
magnitude
of the
other.
the
During
#
other
a
two
more
I(q)
ri
sum
I,
=
analyzed
high
below
2.05
be
resolution
changes
estimated.
x
=
time
first
lO~
shown
As
this
due
to
the
uneasy
in the
x
=
not
:
and
experimental
is
In
versus
that
to
set
well
lO~
The
up.
fitted
step is
situation
is
only
set
one
the
spite of this
difficulty,
example given in table II,
to
sample
better
with
the
exponential,
a single
following step 2 where
and
of
magnitude
of
be
proportional
to
the
it
the
fast to be
too
with
of
onset
~) this
cm~
the
the
experimental
of
sets
the
varying
behavior
all
For
shape
in
2.92
and q
(q ) increase
made
sign.
of
T.
times
increases
cm~
of the
the I
similar
relaxation
results
associate
we
of figure 2b
changes
from
rj, r~
r~
systematic investigation.
quickly. For the most
concentrated
remains
pattem
which
with
measurements
# and
q,
:
2b
many
of the
transient
variation
the
parameters
of the
the
in figure
performed
ranges
r~. We
deterrnine
to
intensity
(q
samples. Actually,
detailed
analysis is
variation
can
q
r~
r,,
order
time
and
experimental
up
the
the
within
in
general transient
amplitude and
step
at
distinct
times
three
We
0.072
three
order
appears
to
10~
i0'
the
0.
I
o.io
I
~
~
f
o
-
o.05
~
o.oo
-
io2
io~
time
1os
11o'
-o.05
itf
10~
mlcrosecondes
In
time
Fig.
AT
2.-a)
step.
0.5
AT
=
AT
step.
JOURNAL
The
°C. b)
Same
DE
b)
Evolution
of
horizontal
straight
sample
II
and
T
the
turbidity (intensity
line corresponds to
intensity
experimental
of the
Evolution
PHYSIQUE
i°~
microseconds
In
a)
1°~
3, N' 4,
APRIL
1993
scattered
conditions
over
at q
as
q's)
all
initial
the
2.92
=
in
as
x
figure
10~
2a
function
a
Sample #
value.
cm~
but
as
T
a
of
time
0.053,
=
function
the
after
35 °C,
T
=
of time
after
19 °C.
21
the
JOURNAL
520
PHYSIQUE
DE
N°
II
4
o
]
3
°
ol
f
~
E
?
°
2
o
o
o
0.2
0.4
AT
Fig.
Amplitude
3.
19 °C,
T
=
Table
q
II.
rj,
the
of
2.92
=
x
and
r~
oscillation
10~ cm~~
r~
happens
under
the
associate
L~
to L~
During
~~
of
linear
(mS)
the
AT
Sample
step.
O.053,
=
=
~2 (mS)
t3 (mS)
0.25
4.9
27
460
when
r,
close
conditions
with
step I
with
characteristic
(proportional
to
process
fluctuations
in
concentration
transition
to
estimated
the
diffusion
a
temperature
be
can
dependence
temperature.
step (step
with
no
enough
time
#~~
accuracy
measured
and
0.053,
=
17 °C.
T
490
quite
#
AT.
versus
24
same
spontaneous
function
2.5
to be
appreciable
a
O.51
Moreover,
~
it
as
roughly
is
different q~'s. Sample #
at
q2 (lO+lo cm"2)
q~
I(q)
of
effect
The
0.6
(°C)
q~~).
(dilute samples and low q's),
by quasi-elastic light scattering
Therefore
it
is
reasonable
having basically the
dynamics as
same
the
sample. On the other hand, ri shows
particular slowing down when T approaches
to
the
no
the
intensity
decreases
down
second
to
a
samples (#
O.026, #
O.053), the first two
times
sufficiently
different
characteristic
in magnitude making so the analysis
steps have
reliable.
At the
beginning of step 2, the
scattered
intensity starts decaying in a non-single
within
After
exponential
times
q-dependent.
while,
and
that
however,
the
are
way
a
single exponential and q-independent (see Tab. II for an example). This
becomes
I (q) decay
which
denote
characteristic
time,
hereafter
decreases
fast
increasing
we
upon
very
r~,
concentrations.
illustrate
have plotted in figure 4 the
In order to
this steep # dependence,
we
evolutions
of r~
evolutions
temperature for the two most dilute samples. Actually, these
versus
interrnediate
the
second
value.
For
the
two
2),
most
the
scattered
dilute
=
=
N°
SPONGE
4
PHASE
T-JUMP
:
W
~
o
Si
E
m
~
100
m
mm
m~
m
lo
0
EXPERIMENTS
521
522
JOURNAL
DE
PHYSIQUE
N°
II
o
~
,
«
~
#
~
$
4
~
o
o
2
o
~
o
lo
I
T
°C
a)
b)
.
Fig.
-a)
me
he
~
r2)~
close to
the
L~
to
L~
temperature
(see
Tab.1).
r~
4
N°
PHASE
SPONGE
4
2
T-JUMP
:
EXPERIMENTS
1
'~
@
~
m
~'
~'
~~
l
~
5
T
'
(°ci
523
524
JOURNAL
PHYSIQUE
DE
N°
II
4
1
8
~
°
6
~
o~~
d
$
4
°
lo
~
0
lo
5
20
15
A~
r~
Amplitude A~
8.
vanishes
lo
#
On
r~
as
T.
Same
of
function
a
35
sample
in
as
figure
(4
7
0.053).
T*.
close
to
to
scale
found
is
related
30
(°ci
T
Fig.
25
as
~
the
hand,
other
is
r~
found
to
in
vary
steeper
a
than
way
neither
to
or
even
we
r~ corresponds
r~(#~~) nor to r~(#~~). However, we must keep in mind that the scaling law for
assuming that the permeability of the
is the
in samples of
obtained
membrane
same
r~ (6) is
different
concentrations
condition
is in principle
satisfied
# (identical density of pores). This
composition of the membrane in surfactant and
dilution
line
where the
chemical
along an exact
cosurfactant
is exactly
In the
the
n-hexanol
has
appreciable
system,
constant.
present
an
of no
be
of
proportion
solubility in the brine
solvent
and we
know
what
is
to
means
sure
in building up the bilayer.
molecularly
dissolved
in the
solvent
and what is actually
involved
for our
dilution
there is very
little
chance
three
samples to belong to the same
Hence,
exact
line. Finally, the most
the
time
is
that
the
dependences
observed
#
present
steep
can
say at
we
qualitatively in favour of the idea that they correspond to r~ and
and
for
are
r~
r~
respectively.
r~,
#
~~
#
~
observation,
this
From
should
conclude
that
Interpretation.
3.
transient
The
monot6nic
identical
L~
initial
be
can
behavior
and
q~~
while
process
r~
r~
and
and
r~
therefore
typical
are
and
the
a
consistent
however
the
their
with
that
sample.
indirect
I
values
of the
other
two
independent
are
is
related
relaxation
the
r~
to
and
of
supported by
r,
average
while
the
of
to
of
r~ to the other two
observation
that r~
effect
on
the
three
variables
and
r~
for
any
variables
ri
quantity.
quantities. It
I,e.
show
Y'and
n
steep
r~(#~~)
and
the
q-component
time
of the
evolutions
in
osmotic
compressibility
of
the
n
found
to
the
to
tempting
therefore
(or conversely). This
dependences
#
versus
r~(#~~).
must
concentration
variables
sample.
We
are
a
here
idea
first is
like
vary
diffusion
hand,
other
r~
is
further
seems
in
fluctuations
probed
and
relate
to
which
keep
of
state
and Y'. The
corresponds
On
of the
intemal
is
vi
thus
is
a,
:
Interestingly,
vector,
wave
conserved
thermodynamical
that
intemal
not.
are
unconserved
scaling expectation
(q) essentially
measures
the
mentioned
we
the
a
the
Therefore,
I,
section
In
the
variable
conserved
a
values.
final
by
defined
unusual.
Instead
of the
classical
L~ after the T-jump is very
(q ), we here observe a large amplitude
oscillation
in
between
almost
of
of I
variations
mind
in
through
N°
4
The
of
purpose
of the
resistance
the
the
cell
are
of
one
interpret
mixture.
AT
+
much
before
final
values
the
on
relaxation
structural
can
any
susceptibilities related
of the
since I(q) is almost
identical
in the
very mild
variations,
transient
at least is subjected to strong
presumably
However,
525
discharge through the sample is indeed to generate a quick
Act~lally, the time
of the capacitor plus the
constant
extremely short (« I m ) so that the sample is homogeneously driven
the
temperature T
variation
temperature
is
EXPERIMENTS
T-JUMP
:
electric
in
new
variables
to
the
variation
temperature
to
PHASE
SPONGE
them
the
initial
final
and
of
intemal
three
this is
and
effect
The
occur.
to
states.
what
try
we
hereafter.
First, it will slightly enhance, the
effects.
We expect that the AT step has two
very quick
of wave
larger than
fluctuations
of the
membranes
amplitude of the therrnal
vectors
curvature
d~
of
the
structural
length
(diameter
characteristic
where d is the
see
average «
passage »
membrane
enhancement
of the
Fig. 2). Due to the high q range involved, the rise time for such
effect is
relaxation.
The
second
structural
roughness is certainly shorter than any large scale
hexanol)
both
(but
especially
of
the
which
chemical
membrane
related to the
components
more
solubilities
finite
have
AT
will
solvent.
time
to
thus
lead
This
effect
a
place
takes
The
solvent.
of
a
solubilities
portion
small
molecular
at the
level
the
total
that
so
with
increase
of
area
of the
consequence
the
the
characteristic
roughness
membrane
put
to
and
into
membrane
again its
expect
we
enhancement
temperature
the
of
and
under
membrane
tension.
transient
Besides
simple
AT
when
submitted
the
that,
known
accordingly.
This
electric
field
electric
field
is
However,
kV/cm
order
of
well
explode
of
similar
this
case,
which
is
In
a
transient
altemative
destruction
the
mixture
much
longer
so,
or
of
due
the
the
transient
the
structure
its
recovers
than
could
we
completely
multiconnected
local
pressure.
vigorous
by
achieved
birefringence
observed
time
the
to
transient
the
the
streaming
relaxation
oscillate
due
where
excess
be
can
characteristic
longest
membrane
the
onto
experiment,
imagine that
It is
occur.
electric
of the free ions
pressure
the bilayer. In the present
the
moving
the
minutes
experiment.
discharge.
that
of
discharge might plausibly
fields, lipid bilayers
electric
of the
moderate
to
L~ might
a
sonication.
effects
other
step,
reveals
capable
is
of
structure
few
to
brine
the
dissolution
Both the
unobservable.
very small and
basic
dissolution
have
common
as
a
be
partial
the
in
in
the
after
a
T-jump
the
of L~ is not totally destroyed after the
electric
structure
supported
by
is
proportional
the
fact
that
the
roughly
guess
response
electric
dissipated in the sample (Fig. 3). The possibility remains
that
however
to the
energy
infinite
small pieces of bilayers are tom off the
by the pressure
strike
induced by the
membrane
transient
electric
small pieces
immediately close up in the forrn of
field.
The
would
then
vesicles
decorating the remaining infinite
reminiscent
of the
surface.
Also, the holes
tom
pieces would
extremely short times (small holes). Then the retum
anneal
within
back
to
equilibrium requires that the small vesicles reintegrate into the infinite
membrane.
This implies
fusion of
therefore
local
and
would take quite a long time typically of the order of
membranes
So,
here
again,
this
mechanism
leads
transient
reduction
of the total
of bilayer
to a
area
r~.
available
So
in
We
is
the
end
infinite
under
surface
:
initial
we
is
Note
mean
here
just
that,
the
to
transient
be
very
the
therefore
and
this
that
parameter
of this
surface
tension.
membrane
hereafter
assume
that
further
infinite
the
therrnodynamic
At the
believe
we
This
tension
excess
along
relaxed
the
same
and
area
value
as
before
below,
that
the
a~~~
remains
is
tension.
and
unobservable),
AT
step
represents
after
(same
the
out
structural
the
fi~~~, fi
for
more
membrane
as
no
#)
and
effective
«
equilibrium
of
changes.
large scale
having integrated
wavelength therrnal ripples of the surface (see Refs. [5, II ]
frame,
although we expect a
reduction
of the
of
true
area
discharge, the effective area (b~~~) remains
unchanged as long
short
leading
the
steps
next
quick step (basically
here
under
it
put
»
or
structure
it is
but
area
density
smoothed
out
details). Within
(d) just after
large
scale
of
now
of
the
this
the
structural
JOURNAL
526
changes
have
effective
«
Any
place.
taken
and
»
the
time
scales
spite
of
true
«
value
much
shorter
enhancement
of the
a~~~ in
express
forrn
the
and
r~
d~ ')
a*
where
The
absence
of
of the
of
(7), it is
(around #
membrane.
fluctuations
Y'and
length
To
[8], the
in
this
make
Hamiltonian.
implies
simple
forrn
for
I(a,
Y')
n,
a
for
=Bo
represents
and
therefore
invariance
reduce
But
in
through
tension
possibility is quite clear when
the symmetric picture and
the I
of
towards
or
the
the 2
subvolume
Accordingly,
surface.
the
Y'are
with
structure
the
geometrical
a
ultimately
and
a~~~
tension
the
phases
of
the
free
relax
same
factor
of
ha
fluid
we
the
of
here
of
tension.
excess
of
discussed
up
discussed
simple
length
a
at
at
Landau
in
[5]
:
have
dramatic
no
on
the
consequences.
other
hand, B
the
forrn
:
(9)
)~+B'Y'~+DY'~
optimum value of n at
entirely specified by
Y'
light intensity.
build
phase, but,
expand B in
the
therefore
Y')
the
the
susceptibility
the
scattered
phase
the
of
symmetry
amplitude
(8)
which
~~~' ~'~
I(a,
of
connectivity
order unity.
concentration
the
to
of
Ba~
+
=
local
membranes
density
of
part
(enhances)
renorrnalizes
energy
~~
+B"~
and
there
to
the
is
of
terra
no
symmetry
in
Y',
I(a,
where
is
~
3
=
frozen.
as
release
On
fixed
a
and
Y'.
definition
the
Note
of
the
the couplings
optimum
value
that
Y').
And
a
tension.
excess
intrinsic
the
local
expresses
of the
increase
average
square
corrections
[5, 8]
neglect the logarithmic
we
independent of the actual large scale
structure
t
variables
and
We
depends on the intemal
n
I(a,
its
(7)
ao is
where
between
towards
from
area
a
where
A is
B
decrease
~'~)
a0
density
and
an
that
invariance
scale
the
considered
This
Starting
towards
(1
a*
f
where
be
Y'(r) and the
quadratic coupling between
tension
ultimately drives the variations of
picture
rigorous, it is
convenient
more
The
very
a
that
is
statement
our
the
between
Y'simply
O) will
=
generally,
owing to
More
misfit
the
partially
to
effective
to
of
some
must
Y'(r).
I.
either
the
~
O,
=
in
obvious
In
Y'
of
figure
surface
area
Y'(r)
linear
terra
a
fi and
a~~
area
release
possibility
of
reduction
effective
everywhere
with
4
:
the
represents
but
transient
the
to
effective
both
fluctuations
drawing
aeff
density
of
the
rq~,
remains
«
considering again the schematic
making a parallel displacement
actually yields a symmetrical
can
related
AT) (< d~~~(T)) will
there
q's (q
low
density
average
+
than
constraints,
these
is
N°
II
densities.
the
b~~~(T
tension
the
sense,
area
»
allowing
mechanism
equilibrium
new
this
In
PHYSIQUE
DE
potential
a
geometrical
Combining
ao.
(8),
9~)
Y')
and
(IO),
~'~
Y'~.
According
to
the
scale
Y')
takes
~a3(1
=
+
the
a
forrn
9~2)
:
(to)
can
equilibrium large scale
and
structure,
therrnodynamic
specify the appropriate
=f-»a
(11)
characterizing
factor
(9)
i(a,
(~,~~~'
l~
order
we
the
:
~P
SPONGE
4
N°
is
which
values
for
minimum
concentration
of
a
the
PHASE
equilibrium
the
determines
sample
p).
~~
Then
~P
can
of
taken
be
(1
ao
=
and
R
and
ao
is
al
the
appropriate
minimum,
is
=
(14),
In
independent
it is in
and
variables
that
o)
no,
~~
ha
+
~~
,
some
Hamiltonian
linear
are
of the
be
B'
Y'~
purpose
is
in
different
:
it is to
understand
the
such
p)ao+
(A
that:
effect
(D
+
expressed
and
variables
initial
of the Y'~ term
influence
the
to
:
diagonalized
is
combinations
(13)
,
due
pattem
of the
~'~
4
Bo
n
Y'~
temls
and
Gaussian.
the
to
~
in
a,
and H is
principle possible (although probably quite complicated
derive
and
the I(q)
compute (a(O) a(r))
to
so
our
(12)
v/
no
Specifying ao
manipulations
H.
after
neglect
can
we
susceptibilities
at
(14)
three
of
Y'. Far
In this
from
limit,
couplings between
equilibrium.
surface
transient
tension
a,
on
(14). Just after the AT step, the « true » area density is forced towards a
value
while the connectivity density is still quenched at its initial
transient
value, this
being at the origin of the tension.
Corresponding to these
transient
constraints
on
introduce
fi, we
Lagrange multipliers and define H,
two
lower
misfit
and
H,
where
y
(t)
connectivity
represents
which
transient
the
maintains
y(t)a+
=H+
and
tension
fi at its
initial
(15)
v(t)n
(t) is the
v
in
value
transient
spite of
the
potential of
Hi takes
chemical
tension.
Then
)
ao
the
the
:
+
a
4
reference
no
lfi~ to
But
form
actual
the
:
Hamiltonian
equilibrium
the
conditions,
critical
a
appropriate
be
+
+
the
(imposing
~a(
~~
no
ao
obtains
one
n
ao
~Po(ao,
=
n
Y'(= O)
no=
1
and
+
form
the
~P
A~P
Let
:
a
Bo
a,
of
values
527
EXPERIMENTS
T-JUMP
:
2
n
Bo
a
Bo
B(
B'ao~
~
ao
=
+
(y
(t
+
3
B(
l(
y
(t ) ao
2
Bo
v
al
(t ) no )
v
(t
+
B'
no
al
a
(17)
JOURNAL
528
transient
the
tension
Besides,
t
the
~~
(T
are
here
~~
AT) and
+
and
Aala
versus
following
mB'(1
forced
the
y(t)
eliminate
we
=
v(t).
and
equilibrium
transient
B(
Then
the
takes
+
a
~~
(T
=
+
a
AT
~~
AT)
(T
0)
+
(18)
a
a
AT
fl'~(o)
fl'~(t)
Afl'~(t)
:
AY'~(t)))
~~
A
a
B
~~
A
:
3
2~~
+
a
and
O)
+
that
form
B(
with
(T
imposing
and
An/n
n
a
simple
4
that, as expected from the former qualitative discussion,
we
see
expressed by y(t) (and v (t)) essentially modifies the susceptibility
susceptibilities of the other two independent variables involving also n and a
unaffected.
remain
Minimizing H,
values
N°
II
(16) and (17),
Considering
of
PHYSIQUE
DE
=
with
scattered
Just
expression
convenient
this
and
the
intensity.
the T-jump (t
after
represented
by
O),
of
good position
a
membrane
abruptly
is
nothing
Since
analyse
to
shifted
happens
else
the
time
by
down
the
at
of
evolution
amount
an
beginning
very
AT
O) is still at its initial
B( of Y'is thus
=
in
are
area
(<O).
a
/(t
the
=
~~
A
we
value
susceptibility
Y'~(t
that
so
shifted
O)
=
by
down
(18), the inverse
large provided that
Y'have
the tendency to
intensity. However, on
According
O.
=
that
amount
an
to
be
can
fluctuations
larger than I.
Therefore
therrnal
of
the
correlatively (owing to the coupling) so does the scattered
Thus
increasing
short
time
scales
(« rq~) the
membrane
is imperrneable
the
solvent.
to
( fl~q Y'_ ) implies transport of solvent over distances of the order of q~ ~. Correlatively, some
~
of
membrane
is also
transported from places where 1l'~ is higher towards places
amount
area
where it is lower.
Therefore
increase in I (q) essentially takes a time of the
the
corresponding
(~q~~) that can be
diffusion
order
of the
time
measured
by classical
quasi-elastic light
This is basically
scattering at the
what
observe
during the step
wavevector.
same
we
r, in figure 2.
Accordingly, we expect the higher q's
of Y'to
increase
faster than the lowers
fluctuations
fluctuations
of
Y'are
first.
q's. So just after the AT-jump only the highest q's
enhanced
Afterwards,
time
this
excitation
progressively along lower q's and
propagates
on,
as
goes
monotonically and therefore
and
the
correlatively Y'~(t) increases
relaxes
more
more
excess
tension.
increasing terra
This feed back effect actually
appears in (18) where the monotonically
B"/B'
is
much
increase
A
and
Y'~(t
compensates
and
more
initial
the
more
effect
~~
of A
time
goes
modes
of
t
~
I(q)
longer
(at
larger
a
diffusion
t
~
the
<
the
rq~,
tail
than
(q)
I
remain
of the I
r,,
A
that
of
at
a
dependent
perrneability of
q
Then
all
are
also
the
the
very
excited
the
the
observation
vector
wave
and
the
single
not
low q (such
together
(q) decay during the step
r~
to
B(
as
of
should
(Dq~)~
the
the
same
starts
initial
the
decreasing
of
steps
However,
efficient
~
r q~
of Y'
accordingly
increases
becomes
that
excitation
the
observation
and
with
time
due
exponential.
membrane
characteristic
and
of
variable
conserved
Y'susceptibility
AT
Y'~(t ) keeps increasing
scattered
Y'remains
Y'modes
therefore
becomes
range
rq~,
variable.
coefficient)
Accordingly,
t
r,)
than
time
smaller
as
conserved
which
the
intensity
the
long
As
decay
becomes
Tp
Then
ri
after
vectors
wave
(Eq. (18)).
at
Thus,
on.
the
on
~
Y'is
and
where
t
no
D is the
characteristic
partial tension
be q independent
the
when
time
relaxation.
and
single
N°
PHASE
SPONGE
4
exponential. Again,
what is reported in
So,
end
the
at
the
initial
the
drift
of
step
tension
of Y'~. As
time
So, during
progressively
the
in
value
the
misfit
a
experimental
4.
Discussion.
fact
that
is
r~
tation,
based
density
B"/B'
the
on
surface
tension
connectivity density,
Moreover, it
very large.
is
induced
is
q
actually
the
far
where
state
possible by
as
characteristic
and
vanishes
time
variables
all
Correlatively, I (q) should reach its
Again this picture is consistent with
independent and #~~ dependent.
large amplitude
of the
identical
almost
as
with
r~.
observation
the
was
two
and
values.
time
exponential,
single
scattered
the
with
relax
to
progressively
§ and fi
within
relaxed
and fi is
a
nicely
agree
equilibrium
interrnediate
an
allowed
now
equilibrium
way
challenging experimental fact
light intensity
between
The
fi is
(q) decay actually
the I
on
between
between
final
single exponential
the
misfit
further,
on
their
towards
the
to
529
EXPERIMENTS
reached
has
structure
related
goes
r~, the
step
shift
qualitative expectations
experimental section.
r~,
surface
r~.
final
these
the
T-JUMP
:
initial
and
final
oscillation
values.
of the
interpre-
Our
misfit
by the transient
between
the
surface
capable of explaining this fact provided
at
accounts
qualitatively
least
for
the
area
that
time
evolution
of
collective
diffusion
of
I(q).
So,
that
expect
we
I)
is
rj
diffusion
the
corresponding
its single
time
the
to
precisely
exponential tail) is rq~ the
more
relaxation
time of the
the
relaxation
time
of the connectivity. If our
is
r~
r~
interpretation is correct, the T-jump technique then
powerful technique
appears
as
a
very
providing detailed
inforrnation
dynamic
of
the
phase.
the
on
sponge
little bit
further.
However,
points must be
discussed
A basic
assumption of our
at
some
interpretation is that the respective susceptibilities of Y'and n have very different magnitudes
(B"/B'» I ). At the present time, we do not know
how
these
quantities
to
two
measure
independently.
However,
discussed
static
factor
of
length in [8], the
structure
at
as
L~ phases at least bears two
components having different q dependences : one arises from the
indirect
contribution
(via the quadratic coupling) of
Y'fluctuations
other
arising
and the
one
from the
direct
fluctuations
of # (membrane
Static light scattering
concentration).
measureperforrned on various systems [5, 8, lO] (among which the present one) have shown that
ments
the
Y'contribution
is always
appreciable. Since the coupling between
Y'and
# is quadratic
conditions
(and therefore
weak) this implies that, even far from critical
(moderate # range) the
susceptibility of Y'is very much larger than that of #. This
observation
is
indeed
in
favour
of
assumption but we
definitely in the absence of any
conclude
estimate
of the
cannot
our
susceptibility of n.
characteristic
features
of the I (q)
oscillation.
B" actually controls, in our picture, all
The
Since the amplitude AI/I of the
higher is B ", the stronger the transient
tension.
transient
effect
is
related
it)
r~ (or
symmetry and iii)
concentration
fluctuations
of
that
to
amplitudes.
On
the
the
transient
other
hand,
tension,
shown
as
r~
Since
a2flan~
~
L~
-
L~ phase
transition
density
temperature
(L~)
somewhere
of
and
a
with
r~
r~
structure
low
hesitates
connectivity
section
will
deterrnine
lower
:
(a~flan~)~~
again directly
controlled
actually first order, we
the
very
r~
a
B"
lower
a
preceding
more
density
by
the
actual
may
and
consider
(L~).
We
more
value
that
between
therefore
Although the
approaching the
high connectivity
of B".
when
a
expect
B"
to
vanish
So, the experimental
observations
(but close to) the
transition
temperature.
(Fig. 7) and vanishing A~ (Fig. 8) are again
consistent
with
picture.
The
our
(Fig. 5) is not as straightforward. In order to interpret rigorously its evolution
below
diverging
situation
B", r~ is
transition
is
=
that
expect
we
in
530
JOURNAL
T,
with
effect
has
one
on
procedure
r~ in figure
work
to
which
II
N°
4
completely the time evolution of (1l'~(t)) including the feedback
y(t) (see expression (15), (16) and (18)). This is a difficult
presently beyond our
capabilities. So, although the
evolution
of
out
tension
transient
the
PHYSIQUE
DE
is
comparison with that of r~, we can stress no definite
statement
description
time.
the
present
at
on
our
interpretation fails to provide a very clear explanation.
where
There is a point
however
our
This point is the level of the
intensity in the
interrnediate
transient
equilibrium state at
scattered
the end of step r~. In figure 2, this level is clearly below the initial and final levels, and this
its
5
natural
seems
consistency
observation
in
with
pertains
whatever
larger
interrnediate
state
is
enhancement
of
Y'fluctuations,
values.
Indeed,
monotonic
one
than
might object
connection
between
the
values
both
I(q)
that
the
of
#,
Y'(o~ and
should
this
q
Y'(~~
be
so
above
expectation
average
scattered
and
square
is
T.
we
rather
In
our
expect
than
implicitly
amplitude
interpretation, 1l'~ in the
that, consistently with this
below
based
of
the
on
the
initial
and
final
assumption
Y'fluctuation
and
of
a
the
(a~a_q). It is nevertheless
Actually, the
intensity
measures
changes
in
the
Y'fluctuations
through
the
in H (or H,) that couples
to
terrns
monotonic
Y'~, n and a. Since these couplings are quite complicated, the postulated
connexion
should
be questioned in
detail.
Nevertheless,
it
unlikely
that
complex
more
seems
more
a
connexion
explain the systematic puzzling
would
observation.
Another
possibility is that the
AT steps
sufficient
that
actually
induce
produce an appreciable AI(q)
to
are
response,
a
transient
tension large enough to trigger the
symmetriclasymmetric
transition.
In this
excess
picture, the initial increase (r, ) of I (q) would correspond to a spinodal decomposition related
transiently negative value of B(. The lower level of I(q) after r~ (or r~) could be
to the
explained by the well known fact [8] that the asymmetric L~ scatters
light less than the
symmetric
However,
this
scenario,
involving the symmetry
breaking at large scale,
one.
implies that some well defined
threshold
exists for the AT step beyond which the
transition
is
triggered. The experimental
observation
indicates
(Fig. 3) rather
linear
variation
the
a quite
order
investigate this point in more
detail,
perforrned
oscillation.
In
I(q)
to
we
some
that a
threshold
actually
under
measurements
very low AT steps (less than O.I °C). It
seems
exists
but the signal to noise
ratio of the
under
excitation
such a low
is not good
response
experimental work is to be done in order to
enough for any definite
Clearly,
statement.
more
clarify the puzzling point reported in this last paragraph. We are presently in the
of
course
improving the design of the experimental set up and we hope that we shall soon be able to
investigate the very low AT range.
measured
indirectly
intensity.
sensitive
References
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86
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II