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77
CROSS-OVER
IN
POLYMER
SOLUTIONS
B.
FARNOUX,
F.
BOUE,
J.
P.
COTTON,
M.
DAOUD,
G.
JANNINK,
M.
NIERLICH
and
P.
G.
DE
GENNES
(*)
DPh-G/PSRM,
CEN
Saclay,
Boite
Postale n°
2,
91190
Gif sur Yvette,
France
(Reçu
le
18
avril
1977,
révisé
le
21
juillet
1977,
accepté
le
8
septembre
1977)
Résumé.
2014
La
fonction
de
corrélation
de
paire
P(r)
pour
les
polymères
en
solution
a
été
mesurée
par
diffusion
de
neutrons
aux
petits
angles
dans
l’intervalle
3
RG ~ r ~
I
où
RG
est
le
rayon
de
giration
et
l la
longueur
du
monomère.
A
la
température
thêta
cette
fonction
est
décrite
par
la
loi
de
Debye
1/r.
En
bon
solvant
(limite
haute
température)
et
à
la
limite
de
la
concentration
nulle,
S.
F.
Edwards
prédit
que
cette
fonction
est
uniformément
proportionnelle
à
r-4/3.
Cependant
le
résultat
expérimental
montre
que
pour
des
concentrations
assez
élevées
ou
pour
des
températures
intermédiaires
la
fonction
P(r)
présente
les
deux
comportements.
On
trouve
qu’ils
sont
séparés
par
des
longueurs
de
cross-over
r*
qui
dépendent
de
la
température
et
de
la
concentration.
Le
scaling
de
r*
est
relié
au
scaling
de
la
longueur
de
corrélation 03BE
et
du
rayon
RG
dans
le
diagramme
température-concen tration.
Abstract.
2014
Using
a
small-angle
neutron
scattering
experiment,
we
measured
the
pair
correlation
function
P(r)
in
polymer
solutions
in
the
interval
3
RG ~ r ~ l,
where
RG
is
the
radius
of
gyration
and
I
the
step
length.
At
the
theta
temperature,
this
function
is
known
to
follow
the
characteristic
Debye
law
P(r) ~
r-1.
In
good
solvents
(high
temperature
limit)
and
in
the
limit
of
zero
polymer
concentration
this
function
is
uniformly
proportional
to
r-4/3,
as
predicted
by
S. F.
Edwards.
We
observe
however,
that
at
higher
concentrations
or
intermediate
temperatures,
P(r)
exhibits
both
characteristic
behaviours,
depending
on
the
range
of
r.
The
cross-over
distances
r*
which
separate
the
patterns
are
found
to
depend
upon
concentration
and
temperature.
The
scaling
of
r*
is
related
to
the
scaling
of
the
screening
length 03BE
and
the
radius
RG
in
the
temperature-concentration
diagram.
LE
JOURNAL
DE
PHYSIQUE
TOME
39,
JANVIER
1978,
Classification
Physics
Abstracts
36.20
-
64.00
-
61.40
1.
Introduction.
- We
examine
the
monomer-
monomer
pair
correlation
function
of flexible
polymer
coils
dispersed
in
a
solvent.
The
statistics
of
such
coils
is
often
given
in
terms
of
the
step
length
1
and
the
average
squared
radius
of
gyration (
RG
).
However,
in
the
reciprocal
space
defined
by
the
scattering
vector
where
A
is
the
radiation
wavelength
and
0
the
scatter-
ing
angle,
there
is
a
domain
in
between
the
Guinier
range
qRG
1
and
the
submonomer
range
ql
>
1
which
is
very
appropriate
for
the
investigation
of
polymer
statistics.
In
this
intermediate
range
scattering
experiments
reveal
the
asymptotic
behaviour
of
the
pair
correlation
function
in
the
limit
of
infinite
molecular
weight.
In
this
range
universal laws
can
be
considered,
in
contrast
to
the
more
detailed
[1]
infor-
mation
obtained
from
diffraction
patterns,
which
is
specific
to
each
polymer
species.
Earlier
results
were
reported
concerning
these
laws
[2,
3].
Recent
neutron
scattering
experiments
have,
however,
brought
new
evidence.
We
therefore
find
it
appropriate
to
give
here
a
general
survey
of
these
results.
The
observation
which
we
wish
to
discuss
is
the
cross-over
or
change
of
behaviour.
A
well-known
example
of
cross-over
in
polymer
solutions
is
found
in
the
scaling
of
RJ)
with
molecular
weight
M.
For
the
isolated
coil
in
a
good
solvent,
theory
[4,
5]
predicts
that
the
coil.is
swollen
with
respect
to
the
random
walk
configuration
when
M
is
sufficiently
great.
Below
a
given
value
M*,
the
coil
returns
to
a
configuration
which
has
the
essential
(*)
Collège
de
France,
Paris.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197800390107700
78
observable
characteristics
of
a
random
walk.
The
vicinity
of
M *
is
characterized
by
a
change
of
beha-
viour
of the
coil
configuration.
The
space
M,
T,
C
(Fig.
1),
where
T
is
the
tempe-
rature
and
C
the
monomer
concentration
(g
cm- 3)
is
in
fact
partitioned
into
regions
in
which
RG
has
characteristic
scaling
laws
in
these
three
variables
(Table
I)
(Refs.
[6,
30]).
As
one
goes
from
one
region
to
another,
the
scaling
law
changes
smoothly
from
one
pattern
to
another.
There
are
thus
several
cross-
overs
in
the
polymer
solution
diagram
[7].
Although
the
change
of
behaviour
in
the
cross-over
region
was
acknowledged
in
the
literature
[29],
it
was
never
studied
as
such.
There
are
two
reasons
why
a
special
interest in
this
problem
is
being
developed
at
the
present
time.
FIG.
1.
-
Temperature
concentration
diagram
for
flexible
polymer
solutions.
i
= T201303B8
is
the
reduced
temperature
and
C
the
monomer
concentration.
This
diagram
is
partitioned
into
different
regions ;
region
I’
is
the
Flory’s
theta
or
tricritical
range.
By
increasing
the
temperature
there
is
a
cross-over
to
the
region
I,
the
dilute
or
critical
range.
The
cross-over
line
is
given
by
equation
(2.6).
Region
II
is
the
semi-dilute
(critical)
range
and
C*
is
the
cross-
over
line
(eq.
(3.2)).
Region
III
is
a
tricritical
domain
(theta
semi-
dilute
regime)
limited
by
lines
C** -
1 !
symmetric
with
respect
to
C
axis.
Regions
IV
and
V
correspond
to
the
domain
where
the
chains
are
totally
collapsed
and
demixing
occurs
(after
Daoud
and
Jannink
ref.
[6]).
1)
Recent
progress
[8]
in
the
theory
of
critical
phenomena
has
shown
that
there
are
characteristic
exponents
associated
with
the
cross-over
between
critical
and
tricritical
behaviour
[9].
2)
Recent
progress
in
neutron
scattering
tech-
niques
[10,11,12]
has
allowed
a
precise
observation
of
the
coil
pair
correlation
function,
in
dilute
as
well
as
in
semi-dilute
solutions,
over
the
entire
intermediate
TABLE
1
Scaling
of
the
squared
end
to
end
distance
R 2
M is
the
molecular
weight,
C
the
concentration
(g
cm-3)
and
the
reduced
temperature
1:
=
(r -
0)/0.
(After
Daoud
and
Jannink
[6].)
range
of
reciprocal
space
q.
The
variable q
is
not
just
one
more
parameter
like
M,
T
or
C.
It
in
fact
bears
a
precise
relation
to
the
cross-over
phenomena
which
helps
us
to
understand
a
particular
aspect
of
polymer
statistics.
The
first
is
currently
being
studied
and
will
be
published
elsewhere
[13].
We
will
be
concerned
here
with
two
experimental
observations
of
the
pair
correlation
function
in
the
temperature-concentration
diagram.
2.
Remarks
on
the
pair
corrélation
function.
-
The
pair
correlation
function
for
a
set
of
N
points,
repre-
senting
the
monomers
of
a
single
polymer,
each
of
which
is
at
a
position
ri
(i
=
1,
...,
N)
is
defined
to
be
were ( )
denotes
the
average
over
all
configurations.
It
is
customary
to
consider
the
second
moment
of
this
function,
which
is
related
to
the
average
squared
radius
of
gyration
R G.
However
there
is
more
information
in
(2.1)
than
in
RG
and
we
shall
be
particularly
interested
in
the
r
dependence
of
this
function.
The
Fourier
transform
of P(r)
is
directly
measurable
in
a
scattering
experiment.
The
usual
problem
associated
with
(2.2)
is
the
accurate
calculation
of
the
inverse
Fourier
transform
from
a
set
of
data,
which
are
limited
in
reciprocal
space
by
the
experimental
conditions.
In
the
case
where
the
scatterer
is
a
single
polymer
chain
of
N
segments
(we
shall
denote
the
scattering
law
by
S1(q)),
there
is
a
more
subtle
relation
between
(2.2)
and
the
terms
of
the
sum
(2.1),
arising
from
the
linear
arrangements
of
the
monomers
and
which
can
easily
be
seen
from
the
alternative
way
of
writing
(2.2)
(valid
when
edge
effects
are
negligible).
79
Here
the
running
index n
measures
the
distance
between
two
monomers
(i, j )
along
the
chain.
It
is
sometimes
called
the
chemical
distance,
in
contrast
to
the
actual
distance
rj.
As
will
be
shown,
the
para-
meter n
plays
an
important
role
in
the
understanding
of
cross-over
in
polymer
solutions.
The q
dependence
of
the
scattering
law
S1(q)
is
determined
directly
by
the
scaling
of
rn >
with
n.
Suppose
that
there
exists
a
particular n,
called
nc,
such
that
the
scaling
below
ne
is
different
from
the
scaling
above
nc’
We
may
split
(2. 3)
into
two
significant
terms
where
Obviously si
reflects
the
scaling
below
nc,
i.e.
the
correlations
of
smaller
chemical
distances.
If
we
consider
a
value q
of reciprocal
space
in
the
range
we
see
from
inspection
of
(2.4),
that
the
contribution
of
S1
to
Sl
is
greater
than
Sli.
Conversely,
in
the
range
The
contribution
of
Sli
is
dominant.
Thus
from
the
analysis
of
S1 (q)
we
are
able
to
derive
the
behaviour
of rn )
with n,
as n
increases
through
nc’
There
is
a
large
body
of
evidence
supporting
the
hypothesis
that
the
scaling
of r’ >
changes
around
a
characteristic
value
nc’
1)
In
a good solvent, in the
limit
of zero
concentration,
it
is
well-known
[4]
that
the
excluded
volume
interac-
tion
v
swells
the
coil
only
if
This
condition
expresses
in
fact
the
Ginzbourg
crite-
rion
of
the
critical
phenomena
[14].
It
can
be
written
in
the
neighbourhood
of
the
theta
temperature
0
[4]
where
i
is
the
reduced
temperature
1:
=
(T -
0)/0.
The
important
step
to
the
understanding
of
the
cross-over
phenomena
is
the
extension
of
condi-
tion
(2. 6)
to
any
chemical
distance n
N.
At
a
given
temperature
i
for
which
(2. 6)
holds,
there
is
a
charac-
teristic
ne
such
that
all
distances
rn >
(n >
ne)
are
swollen
and
all
distances rn >
(n
ne)
are
unperturbed.
Defining
then
the
mean
end-to-end
distance
is
where v
is
the
excluded
volume
exponent
(v
=
3/5).
Using
(2. 7)
and
(2. 8)
formula
(2. 9)
yields :
which
is
a
known
result
[6,
7].
2)
In a
good
solvent,
but
in
the
semidilute
regime,
the
excluded
volume
interaction
v
is
screened
beyond
a
characteristic
value
ncc
by
the
finite
density
of
coil
segments.
Here,
the
distances
rn )
for n
ncc
are
swollen,
and
the
distances
r’ >
for n
>
ncc
are
unperturbed,
in
contrast
to
the
change
of
behaviour
around
n,,.
Previous
calculations
[15]
have
shown
that
ncc
varies
with
segment
concentration
C
as
Defining
we
have
or
by
(2.11)
and
(2.12)
which
was
observed
in
a
small-angle
neutron
scattering
experiment
[15].
3)
Combining
1)
and
2)
in
a
semidilute
solution
we
have
until
now,
the
existence
of nc
and
ncc
was
derived
as
a
theoretical
hypothesis,
which
agrees
with
the
obser-
vation
of
the
law
(2 .10)
and
(2.14).
In
the
next
section
we
present
experimental
evidence
for
the
existence
of
these
different
behaviours.
3.
Experimental
study
of
température
and
concen-
tration
cross-overs.
-
We
have
investigated
by
Small-
Angle
Neutron
Scattering
cross-overs
predicted
by
the
theory
[6]
in region
1
(dilute
solutions)
and
region
II
(semidilute
solutions)
of
the
temperature-concentra-
tion
diagram
(Fig.
1).
The
special
features
of
the
scattering
technique
are
amply
described
in
recent
papers
[10,11,12].
All
measurements
are
performed
on
a
small-angle
scattering
spectrometer
of
the
Laboratoire
Léon-Brillouin
set
on
a
cold
neutron
guide
of
the
EL3
reactor
at
Saclay
[10].
The
incident
6
LE
JOURNAL
DE
PHYSIQUE.
-
T.
39,
N°
1,
JANVIER
1978
80
TABLE
II
Samples
used
in
cross-over
studies
(p)
Samples
E
to
J
contain
0.005
g
cm-3
of PSD
chains
of same
molecular
weight
as
PSH.
(b)
The
theta
temperature
of this
system
is
38 °C
[27].
(C)
See table III.
wavelength,
defined
by
a
pyrolitic
graphite
mono-
chromator,
is Â
=
4,70
±
0,04 A.
Precise
definition
of
angular
divergency
is
achieved
by
using
linear
grids
and
is
fixed
at
a
value
of
12
min.
The
scattering
vector
range
set
by
inequalities
(1.1)
lies
between
Samples
are
solutions
of
polystyrene
in
different
solvents
and
are
listed
in
table
II.
The
useful
signal
is
obtained
by
subtracting
the
scattered
intensity
of
the
proper
solvent
from
the
scattered
intensity
of
the
solution.
In
the
following
studies
we
are
concerned
only
with
the q
dependence
of
the
scattering
functions
and
not
with
the
precise
determination
of
the
cons-
tants.
3. 1
CONCENTRATION
CROSS-OVER.
-
This
cross-
over
is
observed
in