Beverly Sackler. Faculty of. Exact. Sciences,. Tel-Aviv. University,. Ramat-Aviv. 69978, Tel-Aviv, Israel. (~) Institut. Charles. Sadron, 6 rue. Boussingault,. 67083.
J.
Phys.
ii
France
(1993)
3
JANUARY1993,
121-138
121
PAGE
Classification
Physics
Abstracts
68.45-61.25-82.65
adsorption
Polymer solid
monolayers
surfactant
on
and
heterogeneous
surfaces David
(~) School Tel-Aviv
(~)
of Physics University,
Institut
(Received
Abstract. where
the
adsorbing
29
monolayer
Astronomy,
and
Sadron,
July 1992,
6
Raymond
Boussingault,
rue
accepted
October
6
soluble for
interactions
Exact
of
Sciences,
Cedex,
France
flat
We
a
fixed
1.
is
increase
on
average
of the
neutral
surfactant
for
surface
adsorption
chemical
polymer
between
potential) and
or
surfactant.
insoluble Even
(at
if the
isolated polymer chain adsorbs via a adsorption of a polymer solution can induce phase transit-ions iii an insoluble monolayer that phase separates into dense regions where the polymer adsorbs and dilute regions from which the polymer is depleted. Phase transitions induced by polj,trier adsorption can also for soluble monolayer. For a surface with periodic occur quenclied heterogeneities we calculate the adsorbed of polymer as a function of the amount wavelength of the heterogeneity.
local
polymer
Faculty
but heterogeneous surface, in both cases always find that the heterogeneity of the annealed heterogeneity example of surface with a
surfactant attractive
Strasbourg
67083
adsorption. As an monolayers (at
enhances
either
Sackler
Israel
1992)
lve study polymer adsorption on heterogeneity is quenched or annealed.
surface
Beverly
and
69978, Tel-Aviv,
Ramat-Aviv
concentration)
fixed
a
Charles
consider
we
(~) and Jeaii-Fr.ail&ois Joaiiny (~)
AiideInian
concentration.
an
The
Introduction.
adsorption fi.oin solution plays a major role in a number of industrial problems such lubrication, stabilization, adhesion and and has been the subject of as numerous experinieiital and theoretical Over the last few years a rather studies. detailed description of the and properties of adsorbed polymer layers on "ideal" surfaces (in the of structure sense Polymer
colloidal
an(I cheiuically homogeneous) has been developed [1-5]. adsorption of polymers is a cooperative efsect, the polymers which are attracted adsorbed by the surface strongly and form thick adsorption If the layer. adsorption arc a takes place from semi-dilute polyiuer solution, the thickness of this layer is of the order of a length ( of the bulk polymer solution. the correlation On the other hand, if the adsorption proceeds fi.oiii a dilute solution, the thicl;tress is of the order of the radius of gyration of an isolated cliaiii and can, thus, be as large a~~ a few hundred angstr6ms. In the adsorbed layer, the polymer chains form loops with a broad distribution of sizes and the of the structure
perfectly Since
flat
the
JOURNAL
122
layer
self-siniilar;
is
distance
from
the
the
monomer
surface.
This
PIIYSIQUE
DE
profile decays
concentration
self-similar
N°1
II
structure
[3]
as
inverse
an
well for
accounts
law of the
power
of the
many
observed
polymer layers [2, 5]. neither homogeneous in composition nor Most real surfaces smooth. Although are, however, influence the adsorption of polymer solutions, these the heterogeneity may strongly non-ideal theoretical situations have received relatively little attention [6-12]. The aim of this paper is to study the effect of chemical heterogeneities on polymer adsorption while keeping the adsorbing properties
surface
adsorbed
of
flat.
types of adsorbing
Two
surfaces
[12]. A heterogeneous
faces
quenched
considered:
are
surface
quenched
is
if the
and
disorder
annealed heterogeneous surcharacterizing the adsorbing A typical example is a sense.
degree of freedoiu in the thermodynamic chemical irreversibly attached impurities which laterally diffuse on cannot disordered surface for the the surface. A quenched environment represents a difserent local adsorbing iuodel it by interaction which has local fluctuations about ive surface iuonoiuers, a value. randoiu quenched disorder first calculate the of For properties [13], one must mean a adsorbed polymer layer for a given realization of the surface interaction and then the average these properties ~vith respect to the disorder. On the other hand, if the heterogeneity is annealed, the disorder of the adsorbing surface equilibriuiu. This is, for instance, the of chemical heterogeneities is at thermodynamic case that fi.eely diffuse laterally on a solid surface. Another example which we consider in can details is a surfactant monolayer at an air-water interface. The annealed heterogeneity is then surface
solid
is
"fi.ozen"
a
with
surface
the
surfactant
concentration
surfactaiit-polymer properties
such
coupling its
as
phase
per
unit
affects
transitions
important issue to be adsorption as well as
The
area.
the
polyiuer and
surface
addressed the
is how
surfactant
the
surface
tension.
organized as follows. In the next section, we present general considerations paper polyiuer-surfactant therinodynaiuics of mixed Then, in section the 3, we study systenis. on the adsorption of a single polyiuer chain on a surfactant monolayer. Although this is a rather academic problem, it gives solve physical insight into the problem of surfactant-polymer inSection 4 is devoted adsorption of polymers from semi-dilute solutions teractions. to the on monolayers. The descripannealed heterogeneous surfaces, and iuore specifically on surfactant solution properties field level along the lines of the tion of the polymer reiuains at a mean is
Our
density functional apl)roach of polymer adsorption [14, 15]. Section 5 considers pequenched heterogeneous surfaces both in the case where the surface is neutral on and in the where it is strongly adsorbing. The last for the polj a~, concentration, 6c 0, If t
0), the equivalent homogeneous For negative repel the polyiuer. When i is decreased, there is a first order transition between depletion and adsorption at a,,alue to- We distinguish between three liiuits: For
this
effect
to
-ci~/1(/~ -3a2/81(/~ > 7 > -a2/21(/~
314(-17ci14)~/~li
ci~
"
0 > 7
to"-271(/~-3ci~/4 to
-ci~/21(/~
-7~14/ci~
=
»
»
(3.8)
7.
monolayer undergoes a phase separation and negative, the pure surfactant iuonolayer state used to derive equation (3.8) is that of the denser coexisting phase. As was excluded volume interactions mentioned above, it is possible to include for the polymer chain. Ilowever, since the qualitative picture does not change by including these interactions, ,ve do not treat this effect in the present paper. The adsorption of a single polymer chain on a quenched disordered surface has been studied and the chain is large but finite, there by several authors [6-10]. If the surface is infinite the ,vith a sufficiently large is always a region on infinite surface attractive fluctuation for the chain to adsorb. Thus, the polyiuer chain always adsorbs on a surface with quenched heterogeneities, but this is very specific to adsorption of a single finite chain and will not be that
Note
the
discussed.
further
adsorption
Polyi»ei
4.
In
when i is
reference
this
section
(Gibbs)
discuss
we
adsorption
the
mouolayers:
suifactaut
on
of
polymers
semi-dilute
from
a
semi-dilute
polymer bulk
solutions.
solution
on
soluble
(Langiuuir) iuonolayers. The equilibrium and interfacial properties of the combined from the total free energy of the polymer-surfactant systeiu can be obtained mixed system: Ft discussed in sections 2 and Whereas F~ and Fp~ have been F~ + Fp + (,s. 3, the interfacial fi.ee energy (, for a seiui-dilute polymer solution has to be specified. Within field approach [3, 15] that ignores excluded volume correlations, it can be written (in a mean and
insoluble
"
units to
the
of kBT
and
local
per
unit
surface
(4.I)
concentration
describes cb
"
1b/
a
"
a
functional
polymer
#~(z
oc). -
profile in The
polymer
of the
cp(z) by 1~2(z)
=
contact
profile
order
parameter
1~(z)
related
cp(z)
~(l~~
/~~ (~lb)~ ~
~P Equation
as
concentration
iuononier
~l~)~) ~Z. with
obtained
a
from
(~.i)
polymer reservoir at infinity of equation (4.I) decays towards
JOURNAL
128
the imposed bulk polymer chains. excluded
value
The
volume
c~
co + bc and
=
the
It is
convenient
~a~("bC
z/(
e
u
/
fl The
#~ = with
e
+
2
free
energy
#(v
0),
=
67(
(V#)~
=
fl
is
and
t, 14 and
ci,
2
We
then
polymer
mininiize
one
(#~
l)~) of
surfactant
the
#s and
value
surface
fi.ee
fl
thus
can
du
the
#(
& 6c 2
with
fi.ee
fl is is
where
At this
free
separately 4.I
iiuplicit and
energy insoluble for
ADSORPTION
insoluble, the
an
theriuodynaiuic
the
surfactant
ON
function
and
AN
concentrat.ion
+
as
separates
and
d~fl W When
6c
=
0, the
monolayer
u
polymer
the free
profile 0, #(u)
=
(4A)
2
order
#s and
energy, for the
coth(u+[),
=
by #~
parameter
obtain
surface
relation
a
value
equation (4A) adsorption of =
where coth
I
the
between
6c
of the
)16c~
+
+
surfactant
polymer
concentration
~14 6c~ 6c
surfactant
(4.6)
given by equation (4.5).
concentration
adsorption.
This
becomes
fi.ee
When
MONOLAYER.
is is
obtained now
from
discussed
the
monolayer
is
and monolayer remains constant The stability of the monolayer is fl; the monolayer becomes per unit area negative. This defines a spinodal becomes
in
molecules
conserved of the
iuetastal)le
=
)il.
~14 6c~
(4.5)
LANG&iuiR
the
parameter.
order
by studying the convexity of the free
which
z
of the
concentration
the
if the
line
at
function
a
is + 4)
unstable
derivative
0)
=
3tfa~2~fip~,
%
=
obtained
second
with
inonolayers.
surfactant a
=
~[4s
~ &
determines
INSOLUBLE
of
6af, I
e
4
the
parameter
to
value of the
the
soluble
is
&
+
2
concentration
surfactant
of the
equilibrium, completely
number
total
I)(#j
ibc~
+
Miniiuizing
6c.
respect
fl
energy
(is
=
~fi)
~fi2(z
lengths:
and
polymer profile (#(u)),
concentration
energy surfactant the
total
the
express
surfactant
(4.2)
Ft/(a2(~fi(/3kBT),
=
6c +
We
the
concentration
(4.3)
the
to
between
tb/tbb.
+
profile (#>(u)) yields the usual order polymer solution on an ideal solid surface seiui-dilute a I is an integration constant related to the surface value respect
a
obtains:
functional
a
the
of the
positive
z/(a~/3u1~()~/2
=
7:
with
7)#~.
+
concentrations
4
By properly rescaling Ft, 14 + 314(a~~~fip~, and f
elasticity
the
interaction
with
surface
the
dimensionless
introduce
to
for
accounts
volume
interaction
direct at
"
term
conditions.
nionoiuers
FPS
N°1
II
excluded
the
in
u
Fps
term
gradient
The
describes
term
good solvent includes only the
parameter
coupling
The
concentration
infinity.
at
~fib
second
PIIYSIQUE
DE
energy
energy
regions
unstable
I ~ ~ j ~~2 ~ unstable
WI
/~2
at
a
@
~
positive
value
~~ ~~
I =
is
jj ~~
~~ ~
i +
jj
~~'~~
N°1
IiDSORI~'rION
POLYAfER
polyiuer
the
where
order
paraiueter
the
on
Is
I
ON
surface
j(I +
=
129
AIONOLA"ERS
SURFACTANT
#~ is
@i).
(4.9)
phase by the construction of the free value of bc, the standard tangent energy comiuon a (4.7). If I is io value higher than the spinodal value given by equation transition at a occurs larger than io, the monolayer retrains homogeneous and the polymer does not adsorb. If I is phase onto smaller surfactant than io> the monolayer phase separates into two phases: a dense and a dilute surfactant which the polymer adsorbs phase from which the polymer is depleted. with a of a interaction In the strong adsorption liniit, &~ » 14> the energy monomer molecule is of order kBT, and a first order transition takes place at surfact.ant When
by changing the precise location of
decreased
is
transition.
The
transit-ion, to is positive
the
The
surfactant
independent In
the
polymer
of the
dense
the
obtained
(4.10) concentration bc
in
&3 /314
=
the
bulk.
which
is also
surface
order
concentration.
polymer
the
adsorption
is
enhanced
and
the
is
parameter
#~ The
order
first
&~/1814.
=
dense
bulk
phase,
surfactant
monolayer undergoes a function of 6c) is as a #. For fixed
independent of the polymer phase at the transition is
and
in
concentration
(io
transition
io At
the
temperature,
this
experimental
relevant
cb((4s -1).
quantity
is
=
the
+ &~ /314
(4. II)
polymer surface independent
r
excess
f/~(cp(z)
=
cb)dz
=
polymer bulk concentration ot~~ corresponding to a very strong adsorption. and it is of the order of r Although we have discussed here possible phase transitions in Langmuir monolayers inadsorption, ,,,e have not phase diagram of duced by polymer constructed the global surface polyiuer-surfactant iuixture. Sortie general be found in reference [17]. the considerations can calculated using the full free energy of the surfactant In principle, the phase diagrams can be monolayer valid in the whole range of and not only a Landau expansion. The concentrations phase diagram of the monolayer contains two lines, the "special transition" line transition between adsorption and depletion for the polyiuer and a binodal line representing the phase In
the
surfactaiit
phase
§
dense
it is
of the
=
of the
surfactaiit.
concentration
co
transitions factant
The
"special
monolayer
of the
transition" and
has
only depends on the approximated here by
nominal
line been
a
vertical
sur-
line
temperature-surfactaiit concentration binodal line is shifted by plane. The surfactant coupling t-o t-he adsorption of the polymer. The spinodal line of this transition is given by equation (4.7) ,vith 6c 0 ,vhere t is the inverse osiuotic compressibility of the monolayer calculated derivat.I,>e of the surfactant froiu the second free energy F~. Using the Bragg-Williams form of the fi.ee energy, find that the spinodal line is always shifted to higher temperature by ,ve polymer and that the critical corresponding maximum the adsorption of concentration to the than for the pure the of the spinodal line has a higher value surfactant. general, In two lines of the phase diagram surfactant concentration which be either smaller larger meet at a or can concentration of the surfactant than the critical phase transition.
in
the
the
=
4.2
ADSORPTION
surfactant
imizing attractive
ON
the
surface for
the
soLuBi~E
A
concentration fi.ee
is
a
energy,
polymer, I-e-,
GIBBS
non-conserved 7 is
parameter
order
equation (4A). ,vhen
In
NIONOLAYER.
large,
In
the
the
case
surface
a
and
soluble its
where free
monolayer, the by minmonolayer is strongly has a single minimum
value the
energy
Gibbs is
obtained
130
JOURNAL
adsorption
the
and
creased free
belo,v
these
small
is
first-order
When is 6c
and
This
the
result
of r that
Note
surfactants in
the
on
polymer
be
can
values
As 7 is dethe
concentration),
bulk
exchange of stability change in surfactant
an
the
oft
surfactant.
and
of the
between concen-
at
polymer
the
[12].
lo,,,er
oft, the transition
adsorption
concentration
is r
excess
order
oft, the
values
to-
values
surface
first
The
e
life do not take influence of the
the
is
but
solution
aqueous
an
is
virtually
of
surfactant
no
from
polymer
of the
interaction
interface.
adsorption
Polyuier
5.
and
and
solution
surfactant
there
the preferred state for t adsorption vanishes, at large
reference
condition
a
and
in
in
=
solution
such a
in
(independent
high
At
N°1
adsorption is weak. For a critical value to> there is a discontinuous and a much the iuonolayer towards a higher concentration stronger
cia2cb/t
=
obtained order
oft
function
a
decreased.
II
polymer
between
is also
equation (4.10) the
coupling
the
&~ /(4.33/214),
=
As
i is
the
average
&/(2i)
=
§c
value
when
transition
adsorption.
by
enhanced minima.
t,vo
niiniiua
t,vo
tration
a
has
energy
is
critical
PIIYSIQUE
DE
heterogeneous
queuclied
on
surfaces.
A typical example of quenched heterogeneities of the adsorbing surface. to the case contaminations which heterogeneities due to chemical terminally attached to a are solid surface. If the amplitude of the heterogeneity is small, it can be expressed as a sum of semi-dilute Fourier modes. discuss the adsorption of a polymer solution onto a solid We first surface ,vith one single q-iuode periodic heterogeneity. Two considered: (I) a strongly cases are adsorbing surface ,vith a small periodic iuodulation; and (it) a neutral surface whose average transition line ("special bet,veen adsorption and desorption. In the is just on the transition" latter extend results randoiu heterogeneity which is a sum of all q-modes. true to our case ,ve a We
turn
now
would
be
STRONG
5.I
ADSORPTION
polymer solution approximation by the
of
ON
the
~/~BI~ first
The be
teriu,
noted,
and
the
Fp,
ho,vever,
gradient
"
in
/ ((
is the that
terra
,vith
contact
functional
F
Fp
=
(~~/~)~ ~
bulk
since
(V~fi)~
HETEROGENEOUS
PERIODIC
A
a
Jl~)~j ~~
l~(~~~
both
given
is
~~
in
(perpendicular to the surface). The second term, Fp~, is the interaction between the polymer and the (assui»ed attractive) 7 coupled to the polymer the surface, concentration ~fi). on As our example of a periodic heterogeneous surface, we choose a coupling varies periodically along the x-direction with a wavevector q.
In
the
weak
strong adsorption heterogeneities for
liniit
1/71
which
67
is of
ii
+
+
671f> and
heterogeneity liniit the polymer concentration profile can be found perturbatively concentration profile order in the heterogeneity strength, the in 6f. At zeroth I), coth(u where the #o(u) surface, ideal homogeneous for + an same as 21ii b/( is such that sinh 2i extrapolation length I perturbation to this profile is periodic with a wavevector q, the order first order =
=
=
the
parameter
be
can
written
as
# Substitution
lo + 6f #1(u)
=
equation (5.3) gives
in
~
d#i
where Far
«(
a,vay
correction
fi.oin
the
the
order
to
#1(u) where sionless
A is
an
~
j~(o)
~
(5 5)
surface, the order parameter is # parameter profile is then given by
solid
=
A
e~~~("+~~ (coth~(u
integration
decay length K[
decay length is of the (q( » I) the decay length this
integration
constant
distances
order
of the
is of the
A
can
e
r/cbf.
a
'~~
surface.
At
(5.6)
small
with
the
in the
«ii
limits:
two
i~
«ii
«
£e~~~
Kii
»
1.
r.
We
I.
In
«
I),
wavevectors
«ii
» I and
dimen-
(q(
wavevectors
length f, whereas at large period of the heterogeneity q~~
of the
The
~
correlation
estimated
«
I
I
(5.7)
bKi
relevance
For
+
vanish.
must
exponentially
decreases
solid
the
bulk
order
be
=
ofexperimental f
from
I)
+
#i
and
I
=
coth(u
+ Ki
=
A
quantity
far
I)
+
correction
This
constant. at
A
excess
j)
k~.
4 +
e
+
#il
for
lllil~~~+
ill (0)
+
$
(5A)
kw.
cos
following equation
the
~~
? ~~~~
The
obtained
(5.3)
expansion as an is exactly the
surface
equations
smooth,
667f.
In the
The
dimensionless
the
0
fi
=
qf, and
+
0
=
f
is
on
=
+I(w)#s
parameter
and
as
same
k
and
131
#s.
flat
otherwise
are
)) 6f
z/(,
%
u
# + ~fil~bb with a surface surface is only heterogeneous but is
MONOLAYERS
SURFACTANT
is the
surface
homogeneous
excess
to
surface
"
define the
here presence
a
dimensionless of
a
periodic
heterogeneity +o~
r
In
the
limit
of
small
,vavevector
=
ro
26f
+
(«ii
qb < I,
to
t =
#o41 du.
coskw
« +
(5.8)
I)
&I
cos
kw.
(5.9)
JOURNAL
132
DE
PIIYSIQUE
N°1
II
(b7/71) cos qx. As soon as the wavechange in the surface is br/ro excess heterogeneity is larger than the extrapolation length b, qb > I, the fluctuation of the surface follows the heterogeneity of the solid surface. In the strong adsorption limit excess considered here, the extrapolation length b is of the order of the size a. In practice, monomer all heterogeneities of the soli(I surface will thus fluctuations of the surface create excess. of an adsorbed polymer layer to a heterogeneity is different from the Note that the response of a thin wetting film fluctuations for which it has been shown [21, 22] that with a response wavelength smaller than a finite healing length (in general larger than the film thickness) are damped. For the liquid fiIni, the daiuping of the surface corrugations is due to surface tension between the film and the there is no Iii the polymer adsorption problem, interfacial vapor. tension bet,veen the adsorbed layer and the bulk solution. This result can easily be understood looks at the local adsorbed layer. Locally, the adsorbed layer has the if one of an structure seiui-dilute polymer solution, it thus responds of perturbations with a wavelength to structure a length. The heterogeneity produces a perturbation on the correlation larger than the local ,vhere the local correlation length is the extrapolation length b. Hence, it perturbs the surface profile as soon as the wavelength is larger than b. concentration relative
The
length
"
of the
within the Although this result has been obtained mean-field approximation, the same physpicture prevails ,vhen excluded voluiue interactions taken into This has been account. are de Gennes free energy checked quantitatively using the Cahn appropriate for polymers in a
ical
good solvent. the
In
liniit
is
excess
small
,,.a,,elengtli heterogeneities (qb
of very small and in the
approximation
field
mean
»
I),
the
fluctuation
decay length is the
its
in
bulk
the
surface
correlation
length f
~ ~~'~~~
4+~~(2 5.2
NEUTR,iL
PERioDic
AND
A
suRFAcEs.
DISORDERED
neutral
surface
with
chemical
het-
preference to the polymer adsorption. a absence of heterogeneity, the concentration profile is flat, the reduced order parameter In the vanishes ro between the polymer is # I and the surface 0. The coupling constant excess here only the limit and the surface is given by equation (5.I) ,vith 71 ii consider 0. We where the aiuplitude 67 of the heterogeneity is small. The or(ler parameter profile is also obtained froiu equation (5. I ). In order to find this profile, expand # in Fourier series we
erogeneities
is
surface
wltose
averaged
I)ehavior
=
has
no
=
"
"
#
~j
+
=
#n(u)
(5.ll)
nkw.
cos
n
We
then
expan(I
in
If only'
order
the
polymer profile
t-he first
in
ii
harnionic
po,vers
does
not
where The
K(
%
first
4 + k~
order
as
in
the
correction
previous to the
~
ii
+
cos
reduced
vanish.
$)41
("1
)
of the
kW
"
=
It
heterogeneity 6§. Up the equation
to
first
satisfies
0
0
(5.12)
section.
polymer profile
41(u)
=
((
is
thus
exp(-«iu).
(5.13)
POLYMER
N°1
To
first
order in
order
6§
6f,
in
in order
to
the
ADSORPTION
surface
obtain
lo and #2 have
components
requires only the
surface
lo
of
to
3&12
d~4i~~
4j(2)
of the
surface
excess
given by
It is
eXp(-2Kiu)
~j
d2u
°
calculation
#i~
6f,
in
expand up to second only the first two Fourier
reasons
The
133
thus
must
one
(6f)2.
order
second
MONOLAYERS
symmetry
order
to
and
For
excess.
contributions
contribution
vanishes
still
excess
the
SURFACTANT
ON
1
This
leads
to
~j? 4i~~
Therefore,
w>e
can
deterniine
bi~
~
2Ki
(5.14)
3bi~
j2 ~~
$
"
d#[~~
~
~
3 + k2
surface
i-educed
the
4(3
~-2~iu
k2)
+
+OJ
f
excess
(5.15)
K(
+
du[-#(
=
~j2 r
Thus, and If
although the average and ro surface is neutral heterogeneity enhances adsorption, as in the the of heterogeneity is small, q( < I, ,vavevector
The
surface
correlation
roughly
adsorption length (. If the
becomes
excess
length of the independent
the
thus
oft
heterogeneities
=
/ dp (b7(0)b7(p ii of the
surface
P
G'(q correlation
correlat.ion
The r is
surface of the
length is of length (, is excess
various
can
the
be
w,avevectors
«
r > 0
section
3.
case
decreases
the
with
in bulk
concentration
(5.18) the
larger
is
wavevector
is
neutral,
length on the wavelength larger
the than
a
random,
but
,vhich
for
siiuplicity
equal
is
bulk
the
it
characterize
we
is
than
taken
to be
to
profile (, and
correlation
by
its
power
Lorentzian
A2 =
bulk
the
concentration
surface
(5.19)
+11
heterogeneity
surface
surface
discussed
correlation
q~
The
heterogeneous
,vith
~
bulk
r
surface
periodic
is not
e~q
the
on
annealed
when
average
local
the
0,
"
~~~~~~~~
If the
and
(5.16)
=
siuall
very
solution.
layer follo,,>s
heterogeneity
spect.ruin G(q
tions
k2).
+
4«1
the
r
is
»(2
=
2#(~~]
+
2
o
~.
is
supposed
to
be
much
smaller
than
the
(. obtained
in
For
q.
a
each
first
approximation vector
wave
the
by summing contribution
to
over
the
the
surface
contribuexcess
given by equation (5.lfi).
/ )( ~i' )/ ~
r
=
G(q ).
(5.20)
134
JOURNAL
In the
limit
is
where
(
energy with a non vanishing cubic teriu in bc. The coefficient 13 of this cubic term is positive if co > ck and negative if co < ck. lIere, ,ve discuss single chain adsorption on such a monolayer in the neutral where 7 thickness adsorbed polymer chain D is a 0. The of the case To + ciao function of the local concentration in the iuonolayer. coefficient 13 It is independent of the and is given by the values found at the critical point. The total free energy function of a as the surfactant is obtained along the same lines as equation (3A), concentration
When
the
reference
separation
concentration
Landau
the
=
adsorbs is
,vhen
continuous:
6c
=
surfactant
t < t~
=
the
not
the
of
critical
concentration
free
surfactant
~~)
j6c~(t
+
)13 6c~
larger
concentrations
a~ and the change in the
1 ic)
=
is
for
is
=
Ft If13 is positive, (for
co
expansion
/13.
thickness
The
as
)i
D* ~-
the
than
tc
critical
value),
the
due
to
concentration
monomer
of the
(Al
(14 6c~.
+
adsorbed
layer diverges
at
polymer chain adsorption
the
the
transition
/13]~~
(A2)
ci
with
a
different
po,,>er
If13 is negative, ([or
corresponding
to
the
la,v
divergence
adsorption
than
when
concentrations
surfactant
of
the
polyiuer
13
"
smaller chain
0, equation (3.5). than is
the
critical
discontinuous
value), and
the occurs
transition at
to
"
JOURNAL
136
a~+21(/(914). is 6c
=
The
juiup
in the
The
thickness
-213/(314).
PHYSIQUE II
DE
concentration
surfactant of the
finite
Appendix
expansion
surface zero
long
as
13
as
quenched
for
I'>lx)) adsorption,
polyiuer
the
polymer r
where
A is the
~vhereas
for
surface
is
area,
annealed
the
quenched
(tJ)h
e
=
f (~b2jr)
A
rq
case
P(h),
Hamiltonian
degree
simple
where
case
distribution
about
of
quenched
for
and
disorders
annealed
llre~~~ioih iie~~~ioih
[13].
~~~~
~
~~~ =
is
the
~~~
taken
average
theriual
~~ ~~
)~
~~~0
average
with
the
on
order
~fi
bet,veen
the
and
the
7ii
"'IIainiltonian"
interaction
coupling
paraiueter
As in
appearing surface
monomer
the
polymer
surface
Gaussian
the
over
taken
fll~fiexp(-7io)tJ/ JD~fiexp(-7io).
+
polymer-surface bilinear
jB2)
dr
r~ is
(CJ)o is the
and
7io° (CJ)o 7io depends
the
as
is
#~(r). represents
~bi)
averaged differently
cure
fllh P(h)L?/ fllh P(h)
distribution
"Hamiltonian"
The
a
x')
defined
excess
~~
where
in
Gaussian
(Bl)
surface
~~
riLy
possible
0.
~~
the
is
excess.
uncorrelated
an
A~61x
=
=
~
for
disorder
surface
have
surface
mean
l'ilx)'i(x'))
For
0.
To and rq > To- We quenched disorder enhance and below we outline proof that r~ > rq > To within this is a general result a of the cuniulant expansion. the the impurity degrees of freedom (h) in equations (85, 86), out averages over
Carrying for
one
ro, ra
suggesting disorder,
are
Annealed surface
than
excess
the
surface
quenched
the
This
case.
surface
in
excess
itself, yields rearrange enhancement is based
can
an on
even
the
presence
higher fact
that
polymer reaches its bulk condistance ( ft.oni the surface. correlation centration within a Due to the polymer elasticity, the length f is independent of the surface disorder. In addition, for semi-dilute polymer solutions, the polymer concentration epic) is not the order parameter ~fi(r) but the square of it [14, 15]. The inequality ,ve found is strongly based this fact since the disorder like a surface acts on parameter). random interaction and not as a randoiu field [13] (coupled linearly to the order One should, ho,vever, keep in mind that the results obtained here heavily rely on the mean field approxiniation which ignores excluded volume correlations and thus does not take fully the
disorder
into
account
assumed
is
the
only
polymer
of the
enhanceiuent
an
,vhere
lie
to
the
on
surface
and
the
that
fluctuations.
concent.ration
References
ill Napper
D.
II., Polymeric hI. A.,
Cohen-Stuart
[2] [3] de
Geiiies
P.
Auvray L.,
[G]
Ilawaguclii AI.,
[6]
Cotton
Baumgirtner
A.,
M. E.,
[7]
Cates
[8]
Ed,vards
[9]
llone
S.
Cosgrove T.,
G., Adu.
[4]
Ball
F.,
D., Ji II.,
J.
Stabilization
Gall.
P.,
fitolecules
&iuiliukitmar
J.
Prigs.
Aliitltukiiuar Pincus
P. A.,
B., Adu.
(1987)
27
20
J.
Fiance
hi.,
J.
Ghem. 49
Phys.
fifaciomolecules
Phys. 20
Int.
Sci.
202.
(1983)
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Chem.
Gall.
189.
(1988)
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AI.,
Dispersions (Academic
Colloidal
Vincent
Sci.
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A.,
Takahmhi
R.,
of
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1465.
(1987)
3082.
2009. 89
(1988)
2435.
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(1986)
143.
London, 1983).
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Ji H.,
[10]
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Hone
M.,
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[11] Odijk T., Andelman
[13]
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[14] [16]
[19] [20] [21]
[22] [23] [24]
S.
Gennes
Cabane
F., P. G.,
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B., Duplessix R.,
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24
in
Phys.
(1989)
(1991)
Houches
(1965) 613;
85
Scaling Concepts
22
1458.
1875.
Maciomolecvles
Ill-Condensed
Pioc.
N°1
2600.
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(1990)
II
6040.
1978, R. Balian, R. Maynard
ibid 88
(1966)
Fiance
48
(1987)
Gennes
P.
G., J. Phys.
Ghem.
and
G.
Toulouse
265.
Polymer Physics (Cornell University
Press,
Ithaca, 1979).
651.
(1990) 8407. Chari Il., Hossain T., J. Phys. Ghem. 95 (1991) 3302. Lionti-Addad S., Dimeglio J. M., Ltingmuir 8 (1992) 324. Chemistry of Surfaces (Wiley Interscience, Adamson A. W., Physical D., Joanny J.-F., Robbiiis M. O., Europhys. Left. 3 (1988) Andelman Andelman D., Joamiy J.-F., Phys. Rev. A 43 (1991) Robbins AI. O., Mcconnell H., J. Phys. Ghem. 91 (1987) 1715. Subramaiiiam S., Sammon J., Phys. Rev. Lett. 64 (1990) 1903. Seul M.,
[17] de [18]
C.,
T.
J.
(1988)
21
R.,
Ball
Macromolecules
(North-Holland,
Edwards
[15] de
W.,
D., Joanny
[12]
Eds.
Macromolecules
Barford
PHYSIQUE
DE
94
New 729. 4344.
York, 1990).