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Submitted on 1 Jan 1985
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Model calculations for wetting transitions in polymer
mixtures
I. Schmidt, K. Binder
To cite this version:
I. Schmidt, K. Binder. Model calculations for wetting transitions in polymer mixtures. Journal de
Physique, 1985, 46 (10), pp.1631-1644. �10.1051/jphys:0198500460100163100�. �jpa-00210111�
1631
Model
calculations
for
wetting
transitions
in
polymer
mixtures
I.
Schmidt
and
K.
Binder
Institut
für
Physik,
Johannes-Gutenberg
Universität,
Postfach
3980,
D-6500
Mainz,
F.R.G.
(Reçu
le
13
mai
1985,
accepté
le
6 juin
1985)
Résumé.
2014
Nous
étudions
des
mélanges
binaires
partiellement
compatibles
de
polymères
flexibles
en
présence
d’une
paroi
qui
adsorbe
préférentiellement
un
des
composants.
Utilisant
une
approximation
de
champ
moyen
du
genre
Flory-Huggins,
nous
montrons
que,
dans
le
cas
générique,
à
la
coexistence
des
deux
phases,
la
paroi
est
toujours
«
mouillée
»,
c’est-à-dire
recouverte
d’une
couche
d’épaisseur
macroscopique
de
la
phase
préférée.
Nous
montrons
aussi
que,
sur
la
courbe
de
coexistence,
la
transition
vers
l’état
non
mouillé
se
passe
à
des
fractions
de
volume
d’ordre
1/~N
(N
est
la
longueur
de
la
chaîne).
Nous
trouvons
à
la
fois
des
transitions
de
mouillage
du
premier
et
du
second
ordres
et
nous
étudions
la
variation,
à
travers
la
transition,
de
l’épaisseur
de
la
couche
de
surface,
de
l’énergie
de
surface
et
d’autres
quantités
reliées.
Nous
discutons
à
la
fois
la
validité
de
l’approximation
de
grande
longueur
d’onde
que
nous
utilisons
et
les
effets
possibles
de
fluctuations
sur
le
«
mouillage
critique
».
Nous
comparons
nos
résultats
à
des
simulations
numériques
de
Monte
Carlo
du
mouillage
dans
des
modèles
de
type
Ising.
Nous
mentionnons
brièvement
le
rapport
de
nos
résultats
avec
des
travaux
antérieurs
et
avec
de
possibles
conséquences
expérimentales.
Abstract. 2014
Partially
compatible
binary
mixtures
of
linear
flexible
polymers
are
considered
in
the
presence
of
a
wall
which
preferentially
adsorbs
one
component.
Using
a
Flory-Huggins
type
mean
field
approach,
it
is
shown
that
in
typical
cases
at
two-phase
coexistence
the
wall
is
always
«
wet
»,
i.e.
coated
with
a
macroscopically
thick
layer
of
the
preferred
phase,
and
the
transition
to
the
non
wet
state
occurs
at
volume
fractions
of
the
order
of
1/~N
(where
N
is
the
chain
length)
at
the
coexistence
curve.
Both
first
and
second
order
wetting
transitions
are
found,
and
the
variation
of
the
surface
layer
thickness,
surface
excess
energy
and
related
quantities
through
the
transition
is
studied.
We
discuss
both
the
validity
of
the
long
wavelength
approximation
involved
in
our
treatment,
and
pos-
sible
fluctuation
effects
for
«
critical
wetting
»,
comparing
our
results
to
Monte
Carlo
simulations
of
wetting
in
Ising
models.
The
relation
of
our
results
to
previous
work
and
possible
experimental
consequences
are
also
briefly
mentioned.
J.
Physique
46
(1985)
1631-1644
OCTOBRE
1985,
Classification
Physics
Abstracts
05.20
-
61.40K -
64.75
-
81.20S
1.
Introduction.
A
surface
of
a
container
enclosing
a
fluid
mixture,
which
has
a
miscibility
gap
and
is
held
at
a
compo-
sition
corresponding
to
one
of
the
coexisting
phases,
may
be
completely
wetted
by
the
other
phase.
This
phenomenon
of
«
complete
wetting
»
is
predicted
to
occur
in
all
binary
mixtures
close
enough
to
their
critical
point
of
unmixing
[1].
Changing
the
state
of
the
system
such
that
one
moves
along
the
coexistence
curve
to
a
region
of
low
mutual
solubility,
one
under-
goes
a
o wetting
transition »
from
the
wet
state,
where
the
surface
is
coated
with
a
macroscopically
thick
layer
of
the
phase
preferred
by
the
wall,
to
a
non-wet
state,
where
this
thickness
is
microscopically
small.
This
transition
typically
is
first
order
but
may
also
be
second
order
(o
critical
wetting
»
[2-24]),
the
order
of
the
transition
changes
at
a
wetting
tricritical
point
[2-4];
if
the
wetting
transition
is
first
order
there
should
also
exist
a
precursor
pheno-
menon
in
the
one
phase
region
where
the
thickness
of
the
adsorbed
surface
layer
jumps
from
a
small
to
a
large
but
finite
value
(«
prewetting »
[1]).
While
there
thus
is
an
enormous
theoretical
acti-
vity
on
this
problem,
experimental
work
so
far
reports
observations
of
first-order
wetting
transitions
only
[25-33].
It
does
not
seem
easy
to
find
systems
suitable
for
the
observation
of
critical
wetting.
In
order
to
understand
this
problem
further
it
clearly
is
necessary
to
look
into
the
details
of
specific
systems
more
closely.
This
is
one
motivation
why
it
should
be
interesting
to
study
mixtures
of
linear
flexible
macromolecules
which
is
the
subject
of
the
present
work :
in
polymer
mixtures,
the
degree
of
polyme-
rization
(in
our
description,
the
« chain
lengths >>
NA,
NB
of
polymer
species
A,
B)
can
be
varied
in
a
controlled
fashion
over
a
wide
range
(for
a
practical
example,
see
e.g.
[34]),
while
the
basic
interactions
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198500460100163100
1632
between
monomers
(as
well
as
monomer-wall
inter-
actions)
should
be
essentially
independent
of
chain
length
(apart
from
end
effects).
In
addition,
since
the
chemical
structure
of
these
macromolecules
is
so
complicated,
one
may
well
have
somewhat
different
effective
interactions
than
for
the
case
of
mixtures
of
small
molecules.
Since
in
(partially)
compatible
polymer
mixtures
the
effective
energy
responsible
for
unmixing,
measured
as
usual
by
the
Flory-Huggins
parameter
[35]
x,
is
much
smaller
than
unity
[energy
being
measured
in
units
of
KB
T,
T
being
the
tempe-
rature
and
KB
Boltzmann’s
constant],
there
is
no
reason
to
assume
that
interactions
between
the
monomers
and
the
wall
will
be
similarly
small.
Consequently,
the
region
where
the wall
is
wet
is
not
restricted
to
the
vicinity
of
criticality.
One
thus
expects
to
observe
the
wetting
transition
in
a
region
where
the
mutual
solubility
of
the
mixture
is
small.
Thus
one
expects
drastic
surface
effects
in
thin
films
of
mixed
polymers.
In
addition,
the
growth
of
the
wetting
layer
into
the
bulk
may
be
an
important
mechanism
of
phase
separation,
when
one
studies
mixtures
of
polymers
quenched
into
the
two-phase
regions,
which
is
a
problem
of
current
theoretical
[36,
37]
and
experimental
[38]
interest
Additional
kinetic
effects
are
expected
to
be
associated
with
the
(first-
order)
wetting
transition
itself :
in
the
wet
phase,
the
non-wet
state
may
be
metastable
until
one
reaches
a «
surface
spinodal » [23],
and
vice
versa.
For
mixtures
of
small
molecules
the
limit
of
metastability
in
the
bulk
(standard
«
spinodal
»)
is
of
little
physical
significance
[39],
because
of
fluctuation
effects,
and
thus
it
is
questionable
whether
surface
spinodals
are
significant
then.
For
polymer
mixtures,
on
the
other
hand,
the
mean
field
character
of
the
unmixing
transition
[40]
also
has
the
consequence
that
the
bulk
spinodal
gets
a
well-defined
meaning,
for
NA,
NB -+
oo
[37];
we
also
expect
that
surface
spinodals
are
significant
then.
In
addition,
since
polymers
are
rather
slow
objects
one
might
also
more
easily
follow
the
decay
of
metastable
surface
states
[23].
In
section
2,
we
formulate
our
model
and
derive
some
general
results,
while
section
3
gives
typical
numerical
results
for
the
quantities
of
interest
Our
treatment
is
closely
related
to
that
of
Nakanishi
and
Pincus
[23],
but
unlike
the
latter
authors
we
do
not
restrict
the
treatment
to
the
vicinity
of
the
bulk
critical
unmixing,
as
it
is
very
unlikely
that
the
wetting
transition
occurs
there;
thus
we
also
disagree
with
their
conclusion
that
critical
wetting
is
nearly
impossible
for
polymer
blends.
In
section
4,
we
then
discuss
fluctuation
effects
on
critical
wetting
for
polymer
mixtures,
and
for
the
sake
of
comparison
discuss
some
recent
Monte
Carlo
results
on
critical
wetting
in
the
Ising
model
[41].
Section
5
summarizes
our
conclusions.
2.
Phenomenological
theory.
We
start
by
writing
down
an
expression
for
the
bulk
free
energy
of
the
polymer
mixture
in
the
long
wave-
length
approximation
[23,
36,
37],
for
a
semi-infinite
system
(z
>
0)
{ surface
area
with
surface
element
dA
located
at
z
=
0 }
Equation
(1)
is
based
on
considering
a
simple
cubic
lattice
of
spacing
a,
each
cell
of
which
is
either
taken
by
one
effective
unit
of
A
(there
are
NA
such
units
along
an
A
chain)
or
B, 4Y
is
the
volume
fraction
of
A,
and
Ap
the
chemical
potential
difference
(for
convenience
we
work
in
the
grandcanonical
ensemble
although
in
practice 0
rather
than
Ap
is
the
experi-
mentally
controlled
parameter).
The
interaction
para-
meter X
(leading
to
unmixing
for X
>
0)
is
found
to
depend
on
temperature
T
and
volume
fraction 45
(and
even
on
chain
length
if
end
effects
are
included)
[42],
but
we
shall
ignore
the
dependence
of
x
on 0
and
NA,
NB
in
the
following,
and
also
assume
the
polymers
to
be
monodisperse.
More
accurate
theories
of
polymer
mixtures
avoiding
this
Flory-Huggins
approach
[35]
are
possible
[43],
but
are
rather
com-
plicated
and
involve
many
parameters,
and
hence
will
not
be
considered
here.
The
lattice
spacing a
of
the
Flory-Huggins
lattice
can
be
related
to
the
effective
Kuhn
step
lengths
(J A’
(JB
of
the
chains
(defined
from
gyration
radii
Rg r
=
aA,.,/NA/6,
Rg r
=
aB..,INB16)
as
[36, 37, 44]
however,
in
the
following
we
shall
simplify
the
problem
by
implying
complete
symmetry
NA
=
NB,
UA
=
7p
( =
a,
independent
of
4»,
deferring
a
discussion
of
asymmetry
effects
to
the
end
of
this
section.
It
is
also
important
to
recall
that
even
under
the
chosen
assumptions
equation
(1)
is
quantitatively
reliable
only
in
the
long
wavelength
limit
however,
as
will
be
discussed
below
(see
also
[45]),
equation
(1)
provides
a
qualitatively
reasonable
description
for
the
present
problem
also
when
equa-
1633
tion
(3)
is
not
fulfilled
In
order
to
show
this
we
shall
treat
the
limit
NA,
NB -+
oo
by
the
method
of Helfand
and
coworkers
[46,
47]
for
the
study
of
interfacial
properties
[equation
(1)
corresponds
to
the
treatment
of
polymer
interfaces
in
reference
[48]].
A
more
com-
plete
treatment
applying
the
numerical
techniques
of
Noolandi
and
Hong
[49]
is
left
to
future
work.
The
perturbing
effect
of
the
surface
is
described
by
an
additional
contribution,
the
«
bare »
surface
free
energy
F:b),
which
is
assumed
to
depend
on
the
local
volume
fraction 0
* 4Y(z
=
0)
at
the
surface
only
A
simple
explicit
expression
for
fs(b)
is
obtained
using
linear
and
quadratic
terms
In
ø,
analoguous
to
equation
(1),
if
ø1
is
small,
and
again
omitting
unimportant
constant
terms
where p,
plays
the
role
of
a
chemical
potential
diffe-
rence
favouring
species
A
in
the
surface
layer,
and g
represents
the
change
of
interactions
near
the
surface
(including
the
effects
due
to
«
missing
neighbours
»,
etc.).
Hence g
can
be
positive
as
well
as
negative.
We
shall
discuss
the
consequences
of
choosing
a
more
general
form
for .
occasionally,
and
will
also
briefly
comment
on
long-range
surface
pertur-
bations
where .
z)
decays
towards
zero
for
z -+
oo
in
power-law
form,
as
it
happens
for
van
der
Waals
forces
between
the
surface
and
the
molecules
in
the
mixture
[16-18].
If
instead
1
- 01
1,
we
ma
ex
and
b
_ - ’
1
- -
1
1
-
2,
may
expand
f
= -
P?(1 -
4»1) -
2 9(
4»1)2,
which
apart
from
an
unimportant
constant
again
reduces
to
equation
(5).
As
usual
we
disregard
any
inhomogeneities
in
directions
parallel
to
the
surface,
and
find
the
equili-
brium
solution
from
minimizing
the
total
free
energy
{ A
is
the
total
surface
area }
where
The
bulk
solution
0 . =-
4Y(z
-+
oo)
is
found
from
equations
(6),
(7)
disregarding
all
surface
and
gradient
terms,
and
hence
described
by
the
equation
At
the
coexistence
curve
Ap
=
0
in
this
symmetric
case.
By
the
notation
X(O.,
AjM)
we
have
already
indicated
that
it
is
more
convenient
to
use
ø 00
and
AM
as
independent
variables
rather
than
x
and
AM.
The
concentration
profile
near
the
surface
is
now
obtained
as
the
solution
of
the
following
equation
[«
phase
portrait »
[2,
3]]
Following
standard
treatments
[1-3]
the
surface
excess
free
energy
F.
is
then
obtained
as
Minimizing
this
result
with
respect
to
«P1
yields
a
boundary
condition
at
the
surface,
1634
Using
this
equation
together
with
equation
(9)
for
z
=
0,
one
finds
an
explicit
solution
for
«P1,
with
and
where
G
is
the
free
enthalpy
per
site
and
the
solution
yielding
the
minimum
value
of F8
has
to
be
taken.
Equation
(10)
can
be
re-written
as
Next
we
define
a
local
«
surface
layer
susceptibility >> Xl, I
as
[50]
and
similarly
Keeping
ø 00
and
Ap
fixed
in
equation
(15)
means
that
the
Flory-Huggins-Parameter
x
is
kept
constant
[Eq.
(8)],
and
hence
this
corresponds
to
a
derivative
at
constant
temperature
as
usual
[50],
since x
is
controlled
by
the
temperature.
Using
equation
(12)
for fi
and
equation
(5)
for .
it
is straightforward
to
derive
explicit
expres-
sions
for
X,
and
X, ’.
These
quantities
are
of
particular
interest,
since
the
prewetting
critical
point
can
be
found
from
the
condition
xi,l
=
0;
if
this
occurs
at
the
coexistence
curve
(Ap
=
0)
we
have
a
wetting
tricritical
point
One
can
also
ask
how 45,
responds
to
a
change
of
temperature
(and
hence
change
of
0.
with
other
parameters
staying
constant),
A
further
quantity
of
interest
is
the
excess
internal
energy
due
to
the
surface,
which
becomes,
Finally
we
consider
the
solution
for
the
concentration
profile
itself,
which
results
from
equation
(9)
as
This
expression
shows
that
the
profile
is
exclusively
determined
by
the
bulk
properties
of
the
mixture
-
the
boundary
condition
at
the
surface
enters
only
through
the
lower
integration
limit :
one
thus
finds
just
the
same
type
of
profile
which
would
exist
also
between
the
bulk
co-existing
phases
0(’)
=
0 ,
ø x
= 1
-
0. ;
the
distinction
only
is
that
the
surface
can
cut
off
this
interfacial
profile
at
any
value
0,
intermediate
between
ø 00
and
1
-
ø 00.
Of
course,
if
01 -+
1
- ø 00
in
equation
(19)
z -+
oo.
Thus,
if
01
>
1
- 0.
(we
assume 0.
Ov ,,,i,
=
l /2,
so
that
we
assume
an
A-rich
phase
in
the
container),
the
profile
decreases
first
from
0,
to
1
- 0.
as
z -+
oo,
while
the
interface
from
this
B-rich
layer
at
the
sur-