Eur. Phys. J. E 5, 337–351 (2001)

THE EUROPEAN
PHYSICAL JOURNAL E
EDP Sciencesc Società Italiana di FisicaSpringer-Verlag 2001

Scaling theory of mixed amphiphilic monolayers
Shigeyuki Komura1,2,a and S.A. Safran2,b1
2

Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan
Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel
Received 29 December 2000 and Received in ﬁnal form 19 March 2001
Abstract. We predict the elastic properties of mixed amphiphilic monolayers in the swollen state within theblob model using scaling arguments. First the elastic moduli and the spontaneous curvature of a bimodal
brush are determined as a function of the composition and the relative chain length. We obtain simple
and useful scaling functions which interpolate between the elastic moduli of a pure short-chain brush anda pure long-chain brush. By using the analogy between block copolymer interfaces and polymeric brushes,
the eﬀect of mixing on self-assembled diblock copolymer monolayers is investigated in the swollen state.
We calculate various interfacial properties, such as the equilibrium surface coverage, interface curvature,and the mixing free energy as a function of the composition. In general, we ﬁnd a nonlinear dependence
on the composition, which deviates from the simple linear averaging of the properties of pure components.
Our results are used to discuss a recent experiment on the eﬀect of amphiphilic block copolymers on theeﬃciency of microemulsions.
PACS. 36.20.-r Macromolecules and polymer molecules – 68.05.-n Liquid-liquid interfaces – 82.70.-y Disperse systems; complex ﬂuids

1 Introduction
There are many technological applications of selfassembled amphiphiles such as their use as stabilizers of
two immiscible solvents, or equilibrium vesicles for encapsulation. Accurate control of the interfacial properties isthus necessary to utilize these materials. The most basic and important physical property of an adsorbed amphiphilic layer at a liquid-liquid interface is its preferredinterfacial curvature. A formal expansion of the free energy per unit area of a bent surface up to quadratic orderin the curvatures is written as [1,2]Fb (c1 , c2 ) =κ(c1 + c2 − 2c0 )2 + κG c1 c2 .
2

(1.1)

Here c1 and c2 are two principle curvatures, κ and κG are
the bending and Gaussian moduli, respectively, and c0
is the spontaneous curvature. The quantities (c1 + c2 )/2and c1 c2 are the mean curvature and Gaussian curvature,
respectively. The free energy expresses the fact that themean curvature that minimizes Fb has a value c0 whenκG = 0. Much progress has been made in understanding
the thermodynamics, structural, and dynamic propertiesof microemulsions and surfactant membranes using theconcept of curvature energy [3–5].
abe-mail: komura@comp.metro-u.ac.jp
e-mail: sam.safran@weizmann.ac.il

In addition to short-chain surfactants, long-chain polymers can sometimes also show surfactant-like behavior [6].
Examples include block copolymers, where two or moreincompatible polymers are chemically joined together [7].
When mixed with two incompatible homopolymers or
small-molecule solvents which are also selective for thetwo blocks, these copolymers can function as compatibilizers and thus make stable microemulsions [8–11]. Due totheir temperature stability and high eﬃciency in reducing
the interfacial tension, it is sometimes preferable to use
these long-chain polymeric surfactants.

1.1 Monodisperse interfaces
The curvature elasticity of monodisperse diblock copolymer monolayers or bilayers has been widely investigatedtheoretically. There are several approaches to this problem. One of them is to utilize the analogy between block
copolymer interfaces and polymeric brushes, namely,
chains that are irreversibly anchored by one end to an
impenetrable surface [12–14]. (A short review of theoretical works on brushes will be provided in the next section.) Once the stretching energy of a brush is known,
the stretching energy for a copolymer layer can be simply obtained by adding the stretching energy of the two
diblock sections. Wang and Safran [15] calculated the elastic moduli of the diblock copolymer monolayers under melt

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The European Physical Journal E(no solvent) condition by using the results from the selfconsistent ﬁeld theory (SCFT) of a brush [16–20]. They
also studied the onset and morphology of various equilibrium emulsion phases for an A and B homopolymer
mixture with AB diblock copolymers [15].
Similar calculations have been done for swollen monolayers of diblock copolymers which can be obtained if thetwo solvents are good for one of the blocks and poor forthe other, and if the molecular weight of the solvents are
small compared to the blocks. This case was ﬁrst considered by Cantor [21]. Leibler later treated the problem
of a random dispersion of A and B homopolymers by asmall amount of AB block copolymer [22,23]. In a different paper by Wang and Safran [24], within the SCFT
employing a scaling-augmented free-energy functional [16,17], the elastic moduli and the spontaneous curvature formonolayers of diblock copolymers under swollen condition
was calculated.
In the case of self-assembled monolayers, the surface
density adjusts itself to minimize the total free energy
which includes both the chain stretching contribution andthe interfacial energy [15,24]. This is diﬀerent from graftedpolymer brushes whose surface density is ﬁxed a priori. References [15,24] showed that, for both the melt
and swollen states, the surface density decreases with themolecular weight of the chain, whereas the interface curvature increases with the molecular weight ratio of the two
blocks. Hence the control of the overall molecular weightor asymmetry leads to the manipulation of the interfacial
properties. However, synthesis of many diﬀerent types ofcopolymer chains, each with diﬀerent molecular weights orsymmetries, is a formidable task and may be impractical.

1.2 Mixed interfaces
Another way to control the interfacial properties is to mix
diﬀerent types of amphiphilic molecules through composition variation. For low molecular surfactants, it was shownthat a mixture of surfactants can spontaneously form large
bilayer vesicles, whereas the single components do not [25].
The appearance of equilibrium vesicles in mixtures of surfactants is explained by a phenomenological model [26–28]
where the spontaneous curvature c0 is dependent on themonolayer composition (so-called “curvature instability”).
This idea was further extended by Dan and Safran [29,30]
to the amphiphilic interfaces consisting of mixture of diblock copolymers in the melt state.
Recently, a very interesting experiment by Jakobs
et al. has shown that amphiphilic block copolymers(PEP-PEO) dramatically enhance the solubilization capacity of medium-chain surfactants (Ci Ej ) in ternary(oil/water/surfactant) microemulsions [31]. By adding
small fractions of amphiphilic block copolymers, the minimum amount of surfactant (Ci Ej ) needed to form a
one-phase microemulsion is greatly reduced. More recent
small-angle neutron scattering experiments demonstrated
that the copolymers sometimes form mushroom conformations (where the polymer chains do not overlap each other)on both sides of the surfactant monolayer [32]. In that paper, the observed solubility enhancement is attributed to
the variation of the elastic moduli κ and κG , which has
been calculated by Lipowsky and coworkers for mushroom
conformations of anchored (end-grafted) polymers [33,34].
The assumption of a uniform curvature distribution iscorrect only when the mushrooms just begin to overlap.
At relatively higher copolymer concentrations in the experiment, however, the copolymers start to form brushes
and the mushroom picture breaks down [32]. In this case,the monolayers in the microemulsions can be regarded
as brushes comprising a mixture of short (surfactant)and long (polymer) amphiphilic diblock copolymers in theswollen state. The curvature elasticity of mixed brushes inthe swollen case has not yet been studied, although there
is a calculation for the melt case [19]. It is therefore desirable to consider the theory of the elastic moduli of mixed
diblock copolymer monolayers in the swollen state.
1.3 The present work
In this paper, we discuss the elastic properties of mixed
amphiphilic monolayers within the blob picture. By using scaling arguments, we ﬁrst extend the results in references [33,34] to a bimodal brush (a mixture of shorter andlonger chains) in order to calculate its elastic moduli andspontaneous curvature in the swollen state. In contrast to
the fairly complicated results for the elastic moduli of a bimodal brush in the melt state [19], we obtain a simple anduseful scaling functions which interpolate between those ofa pure short-chain brush and a pure long-chain brush [35].
We then make use of the analogy between block copolymer
interfaces and polymeric brushes and add the stretchingenergy of the two diblock sections to calculate the stretching energy of mixed amphiphilic monolayers composed of
two types of diblock copolymers. Using arguments similar to those of reference [30] (that treats the melt case),
we discuss the eﬀect of mixing on self-assembled diblock
copolymer monolayers in the swollen state. By taking into
account the fact that the surface density adjusts itself tominimize the total free energy, we calculate the equilibrium surface coverage, interface curvature, and the freeenergy as a function of the composition. In general, we ﬁnd
a nonlinear dependence of these quantities on the composition, which deviates from the simple linear averaging of
the pure component monolayers.
Our work can be regarded in part as an extension of
the work by Dan and Tirrell who considered the phasebehavior of monodisperse diblock copolymer microemulsions [36]. Within a scaling model, they showed that symmetric copolymers form lamellar interfaces, whereas asymmetric copolymers aggregate in spherical microdomains.
In the present work, we concentrate on the eﬀect of mixing on bimodal diblock copolymer microemulsions.
This paper is organized as follows. We ﬁrst give a brief
review of the theoretical work on end-grafted brushes. In
section 3, we discuss the elastic properties of a bimodal
brush in the swollen state within the scaling theory. We
calculate the free energy of the bimodal brush for the ﬂat,

Shigeyuki Komura and S.A. Safran: Scaling theory of mixed amphiphilic monolayers

spherical, and cylindrical cases. Using this result, we extract the elastic moduli and the spontaneous curvaturefrom a curvature expansion. The eﬀect of mixing on selfassembled amphiphilic monolayers is considered in section 4. Various surface properties are calculated there by
the minimization of the chain stretching energy. The paper
ends with a summary and a discussion of the experiments
by Jakobs et al. [31,32] in light of our result.

2 Previous work on monodisperse brushes
Before explaining our calculations, we brieﬂy summarize
some previous theoretical work on monodisperse polymeric brushes [6]. This is useful since the analogy between
polymeric brushes and block copolymer interfaces is utilized later in this paper. A brush on a ﬂat surface was ﬁrst
considered by de Gennes [12,13] within a mean-ﬁeld theory. When all the chains are of the same molecular weight
N (monodisperse brush), he showed that, as a function ofmolecular weight N and surface coverage Γ (see Eq. (3.2)
below), the height of the brush scales as N Γ 1/3 and thefree energy per unit area scales as N Γ 5/3 . Subsequently,
Alexander introduced the blob picture for a polymer brushand developed a scaling theory [14]. He predicted the samescaling for the height N Γ 1/3 , whereas the free energy perunit area is shown to scale as N Γ 11/6 . The diﬀerence inthe free energy reﬂects the fact that the mean-ﬁeld theory
overestimates both repulsive and attractive terms in thesame manner.
An important development in this ﬁeld was advancedby Milner and his coworkers [16,17] by applying selfconsistent ﬁeld theory (SCFT) to polymeric brushes basedon the work of Semenov [37]. The SCFT exploits thefact that in the limit of strong stretching, the partition function of the brush is dominated by the “classical paths” of the chains, and random-walk ﬂuctuations
about these paths can be neglected. Their work goesbeyond the Alexander-de Gennes theory in several aspects; for example, it was shown that the density proﬁle is parabolic [16,17] rather than a step-function [12–14]. However, for chains at moderate concentration in a
not-too-good solvent (referred to as the “moderate density
case”), the scaling behavior of the brush height and thefree energy is the same with the mean-ﬁeld theory by deGennes mentioned above. (Hence this case is sometimes
called as the mean-ﬁeld version of SCFT.) The SCFT
of swollen brushes has been also considered by using ascaling-augmented free-energy functional [16,17].
More recently, Lipowsky and his coworker investigatedthe eﬀects of the addition of anchored polymers on bilayer
membranes and calculated the changes in the bending
moduli and the spontaneous curvature due to the polymers [33,34]. By considering anchored polymers in themushroom regime (low polymer coverage) and the brushregime (high polymer coverage) separately, they foundthat, in both cases, the bending modulus κ increases,
whereas the Gaussian modulus κG decreases. Similar theoretical results have been also reported in other works [38,33939]. Note that these results are in contrast to the previous theoretical works on the eﬀect of adsorbed polymers
on membranes [40–43], where the bending modulus decreases and the Gaussian modulus increases due to thepolymers. The eﬀect of adding polymers to both sides ofthe membrane have been considered in references [44,45].
The increase in the bending modulus of the polymer decorated membranes has been conﬁrmed by several experiments [46–52].

3 Mixed swollen brushes: a scaling theory
In this section, we consider a bimodal brush in the swollen
state using a scaling argument in which each chain istreated as a string of blobs [53]. Brushes are formed whenthe mean distance between the polymer chains is smaller
than the radius of the polymer coil so that they overlap
each other. Our goal in this section is to calculate the elastic moduli and the spontaneous curvature of a swollen bimodal brush by extending the scaling arguments in references [33,34] to a bimodal brush. We obtain simple scaling
functions which interpolate between the bending moduli
of a pure short-chain brush and a pure long-chain brush.
Our results will be compared with those derived using the
SCFT for mixed brushes [18,19].
We consider a bimodal brush containing a fraction φof longer chains of molecular weight NL and a fraction(1 − φ) of shorter chains of molecular weight NS (NS <NL ). If the total number of polymer chain (including bothlonger and shorter chains) is denoted by X, there are Xφlonger chains and X(1−φ) shorter chains. Both chains are
assumed to have the same segment length a. We deﬁne ameasure of the molecular weight diﬀerence between thetwo chains asα≡

N L − NS
.
NS

(3.1)

Hence NL and NS are related by NL = (1+α)NS . The sizeof a polymer chain is characterized by the mean end-toend distance R ≈ aN ν of the polymer in solution, whereν is the correlation exponent. In this paper, we are mainlyinterested in polymer brushes under good solvent conditions and consider the swollen brush case. Hence, within
the Flory approximation, the corresponding exponent ν
has the numerical value ν  3/5.
The distance between the grafting points is denoted
ξ0 . Then it is convenient to introduce the reduced surface
coverage Γ given by
 2
a
Γ ≡
,
(3.2)
ξ0
which gives the dimensionless number of polymer chains
per unit area. The reduced overlap coverage scales asΓov ∼ NS−2ν . The mushroom regime, applicable to nonoverlapping chains, extends from Γ = 0 up to Γ = Γov [33,34]. For Γ > Γov , the chains overlap to form a brush andthe polymers experience an additional loss of entropy arising from the steric conﬁnement by the neighboring chains.

340

The European Physical Journal E

ξ 0 / φ 1/2

By solving this equation for h1 and using (3.2), we obtain
1−1/νh1 = NS a1/ν ξ0h2

N L - NS

NS

h1

(3.4)

This expression for the height is the same as equation (46)in reference [34] and scales as NS Γ 1/3 for ν = 3/5, which isin agreement with results for monodisperse brushes mentioned in section 3.
In the second, upper layer, (NL − NS ) monomers are“dawdling” beyond h1 . These extra (NL − NS ) monomers
are analogous to a chain of (NL − NS ) segments attached
to a wall at h1 . However, the eﬀective grafting distanceis larger than that of the ﬁrst layer, and is given by the
distance between long chains, ξ0 /φ1/2 , due to the twodimensional nature of the surface. Similar to (3.3), theheight of the second layer h2 (see Fig. 1) satisﬁes the following relation:

ξ0
Fig. 1. Scaling picture of a bimodal polymer brush on a ﬂat
surface. The height of the ﬁrst layer (closer to the grafting
surface) is h1 and that of the second layer is h2 . In the second
layer, (NL − NS ) monomers are “dawdling” beyond h1 . The
distance between the anchor points is ξ0 on the grafting surface.
In the second layer, the blob size increases discontinuously toξ0 /φ1/2 where φ is the fraction of longer chains.

= aNS Γ (1−ν)/2ν .


NL − N S = h 2

ξ0 /φ1/2
a

1/ν

1
.
ξ0 /φ1/2(3.5)

Hence we obtain
1−1/ν (1−ν)/2ν

h2 = (NL − NS )a1/ν ξ0φa(NL − NS )Γ

φ(1−ν)/2ν (1−ν)/2ν

=
.

(3.6)

From (3.4) and (3.6), h1 and h2 are related by
The free ends of the chains must be segregated verticallyby molecular weight and the brush consists of two layersof blobs as shown in Figure 1. This is the case consideredin this section.
However, it is important to note that we cannot use
the brush model in the limit of only a few long chains.
In our theory, we will assume that the long chains on topof the short ones form a brush, which is only true if thelong chains are close enough to overlap. For the second
upper layer one may deﬁne an eﬀective coverage Γ  = Γ φwhere Γ is given by (3.2). Hence Γ  vanishes for φ → 0.
∼The overlap coverage for the second layer scales as Γov−2ν
(NL − NS ). If the long chains are too dilute, they areeﬀectively isolated and do not form a brush. For this case,the mushroom (on top of a brush) picture is relevant andwill be discussed in section 5 separately.
3.1 Flat surface
Consider a bimodal brush on a ﬂat surface for which thefree ends of the shorter chains are found at distances closer
to the grafting surface than those of the longer chains as
in Figure 1 [18,19]. The ﬁrst layer, which is closer to thegrafting surface, consists of blobs containing both shorterand longer chains, whereas the second layer consists ofblobs containing only longer chains. In the ﬁrst layer, eachblob is of radius ξ0 and contains (ξ0 /a)1/ν segments. The
number of blobs is equal to h1 /ξ0 where h1 is the heightof the ﬁrst layer (see Fig. 1) and satisﬁes 1/ν
ξ0
1
NS = h 1
.
(3.3)
a
ξ0

h2 = αφ(1−ν)/2ν h1 ,(3.7)

which will be used later to eliminate h2 .
Next, we consider the free energy which is proportional
to the number of blobs. Let us denote the total area of thegrafting surface by A. There are A/ξ02 and A/(ξ02 /φ) chainsin the ﬁrst and the second layer, respectively. Notice that
A/ξ02 is equal to the total number of chains X. Since thenumber of blobs is proportional to h1 /ξ0 and h2 /(ξ0 /φ1/2 )in the ﬁrst and the second layer, respectively, the total
free energy of the system (aside from a numerical factorof order unity) is given by   


A
h1
A
h2
F̃0 = T+=
ξ02
ξ0
ξ02 /φ
ξ0 /φ1/2
  
h1 
A1 + αφ(1+2ν)/2ν ,
(3.8)
T 2
ξ0
ξ0
where we have used (3.7) to eliminate h2 , and T is thetemperature measured in energy units. (The Boltzmann
constant kB is contained in T .) The above energy accounts
for chain stretching and excluded-volume interactions of
the bimodal brush. By using (3.2) and (3.4), the free energy per unit area F0 = F̃0 /A is then given by
F0 =

T h1
T
f0 (α, φ) = 2 NS Γ (1+2ν)/2ν f0 (α, φ),
ξ03
a

(3.9)

where we deﬁne the function f0 (α, φ) by
f0 (α, φ) = 1 + αφ(1+2ν)/2ν .

(3.10)

Shigeyuki Komura and S.A. Safran: Scaling theory of mixed amphiphilic monolayers

341

1ξ2s ( r )
0.8

0.6

g

0

h 2s
0.4

h1s

NL- NS

r
ξ0

NS

0.2

R
0

0

0.2

0.4

φ

0.6

0.8

1

Fig. 2. Plot of g0 ≡ [f0 (α, φ) − 1]/[f0 (α, φ = 1) − 1] = φ11/6
as a function of φ (fraction of longer chains). The function
f0 (α, φ) is deﬁned in (3.12).

For ν = 3/5, (3.9) and (3.10) becomes
F0 =TNS Γ 11/6 f0 (α, φ),a2(3.11)

Fig. 3. Scaling picture of a bimodal polymer brush on a spherical surface of radius R. The height of the ﬁrst layer (closer
to the grafting surface) is h1s and that of the second layer is
h2s . In the second layer, (NL − NS ) monomers are “dawdling”
beyond h1s as in Figure 1. The distance between the anchor
points is ξ0 on the grafting surface. The blob size scales asξ1s (r) = ξ0 r/R in the ﬁrst layer and ξ2s (r) = ξ0 r/(Rφ1/2 ) in
the second layer, where r is the distance from the center of thesphere (see (3.14)).

obtain the free energy of mixing as

andf0 (α, φ) = 1 + αφ11/6 ,∆F0 =(3.12)

respectively. Notice that f0 = 1 for either α = 0 (monodisperse brush) or φ = 0 (pure short-chain brush). In thislimit the above scaling behavior coincides with that given
by Alexander [14] or by equation (47) in reference [34](note also [54]). It is easy to see that for φ = 1 (purelong-chain brush), we have F0 = (T /a2 )NL Γ (1+2ν)/2ν as
it should be.
The behavior of the function g0 ≡ [f0 (α, φ) − 1]/[f0 (α, φ = 1) − 1] = φ11/6 is shown in Figure 2. Notice that g0 does not depend on α, and the slope of g0
vanishes for small φ. For φ near unity, the curve drops
more steeply than a linear interpolation between the purebrushes (φ = 0 and 1). These facts indicate that, as far
as the free energy of the ﬂat bimodal brush is concerned,
adding a small fraction of shorter chains is much more
eﬀective than the addition of a small fraction of longer
chains as has been pointed out before [18,19,29,30].
One can compare (3.11) with the free energy of a bimodal brush obtained within the SCFT [18]. For the moderate density case, their result of the free energy per unit
area scales as NS Γ 5/3 [1+αφ5/3 ]. The small diﬀerences between our results and those of SCFT are of order Γ 1/6 andφ1/6 and are due to correlation eﬀects that are neglectedin the mean-ﬁeld version of the SCFT used for the case ofmoderate density [18].
By subtracting the free energy of two separate assemble of brushes made of the pure NL and NS chains, weT(NL − NS )Γ 11/6 φ11/6 − φ .
2a(3.13)

As long as Γ is ﬁxed, this free energy is minimized for
φ = (6/11)6/5 ≈ 0.483.

3.2 Spherical surface
In order to obtain the elastic moduli and the spontaneous
curvature, one has to extend the scaling argument for abimodal brush grafted on spherical and cylindrical surfaces. A polymer brush on a sphere or a cylinder consistsof concentric layers of blobs. As in the ﬂat surface case,a bimodal brush on curved surfaces also consists of twolayers. This is illustrated in Figure 3 for a spherical surface. Let us denote the size of blobs in the ﬁrst and thesecond layer by ξ1i (r) and ξ2i (r), respectively, where r is
the distance from the center of the sphere, and i = s andi = c for spherical and cylindrical geometries, respectively.
In each of the layers, the size of the blobs increases withthe distance from the grafting surface. In the ﬁrst layer
the surface area is given by S(r) ≈ Xξ1i (r)2 , where X isthe total number of polymer chains, whereas in the second
layer it is S(r) ≈ Xφξ2i (r)2 since the second layer consistsof Xφ longer polymer chains. These relations hold bothfor the spherical and the cylindrical cases.
For a sphere of radius R, S(r) = 4πr2 and X =4πR2 /ξ02 where ξ0 is the distance between grafting points.
Using these relations, we obtain the blob size as a function

342

The European Physical Journal Eof concentration fromξ1s (r) =

ξ0 r
,
R

ξ2s (r) =

ξ0 r
,
Rφ1/2

(3.14)

in each of the layer. Each blob contains [ξ1s (r)/a]1/ν and[ξ2s (r)/a]1/ν segments in the ﬁrst and the second layer,
respectively.
As described in appendix A, we follow a similar procedure as in the ﬂat surface and obtain the free energy perunit area h1TFs (R) = 2 NS Γ (1+2ν)/2ν fs,(3.15)
a
R
with



x
ν
fs (x) = log 1 +
x
ν
+φ



1 + (x/ν) 1 + αφ(1−ν)/2ν
log. (3.16)x
1 + (x/ν)

3/2 νFor either α = 0 (monodisperse brush) or φ = 0 (pureshort-chain brush), the above result reduces to equation (50) in reference [34]. (We did not subtract the freeenergy of the ﬂat brush (3.9).) Our results generalize thisto the case of the mixed polymer brush. We note here thatthe equations such as (A.3), (A.6), (3.15) will not changeeven if we include additional dimensionless coeﬃcients in
NS (NL ) or Fs [34].
3.3 Cylindrical surface
Here we repeat a similar argument for a bimodal brush ona cylindrical surface of radius R and length D. The surface
area of a cylinder is now S(r) = 2πDr and X = 2πDR/ξ02 .
Hence the blob size varies as
1/2

ξ1c (r) =

ξ0 r
,
R1/2

1/2

ξ2c (r) =ξ0 r,
R1/2 φ1/2(3.17)

in each of the layer. Each blob contains [ξ1c (r)/a]1/ν and[ξ2c (r)/a]1/ν segments in the ﬁrst and the second layer,
respectively.
As described also in appendix A, we follow a similar
procedure as in the spherical surface and obtain the freeenergy per unit areah1
TFc (R) = 2 NS Γ (1+2ν)/2ν fc,(3.18)a2R
with
fc (y) =ν/(1+ν)


1+ν
y
−1
1+
ν


1
1+ν 
y 1+αφ(1−ν)/2ν
+φ3/2
1+
y
ν
ν/(1+ν) 

1+ν
y
− 1+
.
ν
1
y

For either α = 0 (monodisperse brush) or φ = 0 (pureshort-chain brush) the above result reduces to equation (53) in reference [34]. Our results generalize this tothe case of a mixed polymer brush.
We comment here that the same scaling picture can bealso applied to a bimodal brush grafted to the inside of asphere or a cylinder as long as the innermost blob is larger
than a. (See the more detailed discussion in reference [34].)

3.4 Small curvature expansion
Here we determine the elastic moduli and the spontaneous
curvature of a bimodal brush by expanding the free energies in (3.15) and (3.18) up to second order in h1 /R. For
the spherical geometry, we haveT
Fs (R) ≈ 2 NS Γ (1+2ν)/2ν
a


1
1
f1 (α, φ)x+ 2 f2 (α, φ)x2 +· · · (3.20)
× f0 (α, φ) −2ν3νwith x = h1 /R and Γ is the surface coverage. For cylindrical geometry, we ﬁndT
Fc (R) ≈ 2 NS Γ (1+2ν)/2ν
a


1
ν +2
2
× f0 (α, φ)− f1 (α, φ)y+ 2 f2 (α, φ)y +· · · (3.21)2ν
6ν
with y = h1 /2R. In the above, f0 (α, φ) is given by (3.10),
and f1 (α, φ) and f2 (α, φ) are deﬁned as
f1 (α, φ) = 1 + 2αφ(1+2ν)/2ν + α2 φ(2+ν)/2ν ,(3.22)f2 (α, φ) = 1 + 3αφ(1+2ν)/2ν + 3α2 φ(2+ν)/2ν
+α3 φ3/2ν .

(3.23)

Since f1 = f2 = 1 for either α = 0 (monodisperse brush)or φ = 0 (pure short-chain brush), (3.22) and (3.23) coincide in this limit with equations (59) and (60) in reference [34], respectively. Notice also that for x → 0 andy → 0 (small curvature limit), both (3.20) and (3.21) reduce to the free energy of a bimodal brush on a ﬂat surface(3.9).
Both (3.20) and (3.21) show a minimum as a function of 1/R, whereas the original expressions (3.15) and(3.18) decrease monotonically. For the cylindrical case, thesecond-order expansion (3.21) exhibits a minimum atymin =

3ν f1 (α, φ).
2ν + 4 f2 (α, φ)(3.24)

ν/(1+ν)(3.19)

Hence the above expansion breaks down for h1 /2R  ymin .
Once we have obtained the free energy of a brush bent
into the inside or outside of a sphere or cylinder, we cancompute the elastic moduli and the spontaneous curvature

Shigeyuki Komura and S.A. Safran: Scaling theory of mixed amphiphilic monolayers343

1

in (1.1) using the following relations:κ = R2 Fc (R) + Fc (−R) − 2F0 ,(a)(3.25)

1 2
R Fs (R) + Fs (−R) − 2F0 ,2
1 
κc0 ≡ k = R Fc (−R) − Fc (R) ,4

0.8

2κ + κG =(3.26)(3.27)

where F0 is the free energy of the ﬂat brush given by (3.9).
We then ﬁnally extract κ, κG , and c0 as
ν+2
T NS3 Γ 3/2ν f2 (α, φ),12ν 2
1
κG = − T NS3 Γ 3/2ν f2 (α, φ),6ν
1 T 2 (2+ν)/2νN Γ
κc0 ≡ k =f1 (α, φ),8ν a S
κ=

g

1

0.6

0.4

(3.28)
0.2(3.29)
0(3.30)

where f1 (α, φ) and f2 (α, φ) are given by (3.22) and (3.23),
respectively. These are the main new results of this paper.
The calculated bending modulus is positive whereas theGaussian modulus is negative. This means that bending
into a saddle-shaped surface costs energy. For ν = 3/5,
the above equations become65
T NS3 Γ 5/2 f2 (α, φ),108
5κG = − T NS3 Γ 5/2 f2 (α, φ),18
5 T 2 13/6k=N Γ
f1 (α, φ),24 a S
κ=

0

0.2

0.4

0.2

0.4

φ

0.6

0.8

1

0.6

0.8

1

1

(b)
0.8

0.6

g

2

(3.31)(3.32)

0.4(3.33)
0.2

where
f1 (α, φ) = 1 + 2αφ11/6 + α2 φ13/6 ,f2 (α, φ) = 1 + 3αφ11/6

2 13/6+ 3α φ(3.34)
3 15/6

+α φ

.

0

0

(3.35)For either α = 0 (monodisperse brush) or φ = 0 (pureshort-chain brush) which implies f1 = f2 = 1, the above
results reduce to the elastic moduli and the spontaneous
curvature obtained in reference [34]. We also note that for
φ = 1 corresponding to a pure long-chain brush, (3.22) and(3.23) become f1 = (1 + α)2 and f2 = (1 + α)3 , respectively. Hence (3.28) to (3.30) become those for monodisperse brush with NS being replaced by NL . (Notice that
NL = (1 + α)NS .)The general behavior of the functions deﬁned g1 ≡
[f1 (α, φ) − 1]/[f1 (α, φ = 1) − 1] and g2 ≡ [f2 (α, φ) − 1]/[f2 (α, φ = 1) − 1] as a function of φ are shown in Figure 4(a) and (b), respectively, for ν = 3/5 and three different values of α. Similar to Figure 2, the dependenceof the elastic moduli on the volume fraction φ is not a
simple linear interpolation between the modulus of a pure
short-chain brush and a pure long-chain brush. We ﬁndthat the eﬀect of mixing a small number of long chains ina brush of mostly short chains is much less eﬀective thanthe mixing of a small number of short chains in a brushof mostly long chains. This is because the long chains areφFig. 4. Plots of (a) g1 ≡ [f1 (α, φ) − 1]/[f1 (α, φ = 1) − 1]and (b) g2 ≡ [f2 (α, φ) − 1]/[f2 (α, φ = 1) − 1] as a functionof φ (fraction of longer chains) for three diﬀerent values of
α = 0.1 (solid line), 1 (dashed line), and 10 (dotted line). The
functions f1 (α, φ) and f2 (α, φ) are given by (3.34) and (3.35),
respectively.

strongly aﬀected by the free volume introduced even asmall number of short chains. Moreover, in the present
scaling theory, the ratio κ/κG is a constant and given by
κ
ν+2
.
=−
κG
2ν

(3.36)For ν = 3/5, we have κ/κG = −13/6 ≈ −2.167.
In the case of a monodisperse brush (f1 = f2 = 1),
we can compare the above expressions with those obtained by Milner and Witten for monodisperse brushes
in the moderate density case using SCFT [19]. Theirresults are κ = (9/64)(12/π 2 )1/3 T N 3 Γ 7/3 , κG =−(3/35)(12/π 2 )1/3 T N 3 Γ 7/3 , and k ∼ (T /a)N 2 Γ 2 which

344

The European Physical Journal E

are of the same sign and have the same molecular weight
dependence (∼ N 3 ) as those obtained within the present
scaling theory. However, the dependence on Γ is diﬀerent
by a factor of Γ 1/6 . As mentioned before, this diﬀerenceis due to the correlation eﬀects which are neglected in themean-ﬁeld version of SCFT.
For a bimodal brush, the elastic moduli of a mixed
brush has previously been calculated only for the meltcase [19,30,35], and we can not directly compare this with
our results. It is important to stress that α and φ dependence of κ and κG are exactly the same and are represented by a single function f2 (α, φ) in the swollen case.
This is not the case for the melt state, although it is reported that the ratio κ/κG is constant to within a few percent [19,30,35]. In general, the replacement of a small fraction of long-chain molecules by short-chain molecules dramatically changes the elastic moduli both for the swollen
and the melt cases. Equations (3.34) and (3.35) can be
expanded for φ ≈ 1 asα(3.37)
f1 (α, φ → 1) ≈ (1 + α)2 − (22 + 13α)(1 − φ),6α
f2 (α, φ → 1) ≈ (1+α)3 − (11+13α+5α2 )(1−φ).(3.38)
2
Hence the bending modulus and other quantities decreases
linearly with (1 − φ) for φ ≈ 1 in the present swollen caseas well as in the melt case [19]. For a small fraction of
long-chain molecules, the bending modulus increases as
φ3 in the melt case [19], whereas it increases as φ11/6 inthe swollen case (see (3.35)) which has an even stronger
eﬀect than the melt case. (See, however, the discussion insection 5.2.)

4 Mixed amphiphilic monolayers
Up to now we have discussed the elastic properties of abimodal brush within the scaling theory. Using the obtained results for polymer brushes, we investigate here
the eﬀect of mixing on the interfacial properties of selfassembled amphiphilic monolayers in the swollen state.
Typical examples of such systems are mixtures of two AB
amphiphilic diblock copolymers or surfactant moleculeswith diﬀerent overall molecular weights and/or asymmetries, in a blend of highly incompatible solvents. For simplicity, we discuss mixed AB diblock copolymer monolayers between two immiscible solvents.
We consider the case where the solvents are inversely
selective, i.e., the two solvents are, respectively, good forone of the blocks and poor for the other. Then the unfavorable interaction between incompatible block-solvent pairs
drives the copolymer chain to the liquid-liquid interface,
and the system thereby reduces the interfacial tension.
When the incompatibility between the solvents is highenough, the A-B junction points are constrained to theliquid-liquid interfaces. Since for large incompatibilities,
the width of this interface is narrow compared to the chainsize, one can assume that the two blocks form brush-likeregions on both sides of the interface as shown in Figure 5

φ

A

A
N SA

1/Γ
N SB
B

φB
Fig. 5. Schematic diagram showing a mixed diblock copolymers at a liquid-liquid interface. When the copolymers aggregate at liquid-liquid interfaces, the A block forms a brush in
one of the solvents and the B block in the other solvent. The
two solvents are, respectively, good for one of the blocks andpoor for the other. Hence each block is assumed to be in theswollen state. 1/Γ is the dimensionless area per chain

as long as the copolymer concentration at the interface
is high enough. Hence one can calculate the elastic properties of mixed diblock interfaces by using the results forpolymer brushes discussed in the previous section.
Mixed diblock monolayers in the melt state has been
considered by Dan and Safran [30]. Here we apply theirargument to the mixed monolayers in the swollen state.
According to references [15,24,29,30], the stretching energy of a copolymer monolayer can be simply obtained byadding the stretching energies of the two block sections.
For each of the blocks, the stretching energy per unit area
is given by (1.1) which can be rewritten asκFb (c1 , c2 ) = F0 −2k(c1 +c2 )+ (c1 +c2 )2 +κG c1 c2 , (4.1)
2where F0 = Fb (c1 = c2 = 0) is the free energy of the brushon a ﬂat surface given by (3.9). For our later calculations,
we consider the free energy per chain for each of the block
given by F̄b = Fb /Γ and write it in the form
1F̄b (c1 , c2 ) ≡ Γ 5/6 F̄0 + Γ 7/6 k̄(c1 + c2 )2+Γ 9/6 κ̄(c1 + c2 )2 + κ̄G c1 c2 ,(4.2)

where Γ is the surface coverage given in (3.2). By using theresults (3.9) and (3.28) to (3.30) in the previous section,F̄0 , k̄, κ̄, and κ̄G can be deﬁned by the unbarred quantities

Shigeyuki Komura and S.A. Safran: Scaling theory of mixed amphiphilic monolayersby omitting the Γ dependence asT
NS f0 (α, φ),a2
5T 2k̄ ≡ −
N f1 (α, φ),
6a S
65
T NS3 f2 (α, φ),
κ̄ ≡
216
5
κ̄G ≡ − T NS3 f2 (α, φ).
18
F̄0 ≡

(a)

A

(b)
φ

(4.3)

A

(4.4)
A

(4.5)

NS

(4.6)
B

Here we have used the value ν = 3/5, and f0 (α, φ),f1 (α, φ), and f2 (α, φ) are given by (3.12), (3.34), and(3.35), respectively.
It is important to notice that, in the case of selfassembled monolayers, the surface coverage, Γ , is not ﬁxed
a priori as in the grafted brush, but adjusts itself to minimize the total free energy [2,15,24]. In the present case,the total free energy consists of the chain stretching contribution (4.2) and the interfacial energy. The interfacial
energy is linearly proportional to the area of the liquidliquid interfaces on which the A-B junction points areconstrained. Let us take the liquid-liquid interface as thecenter of curvature so that the curvatures of the B blockbrush have signs opposite to those of the A block brush.
Then the mixed free energy F̄m including both the stretching energy and the interfacial energy can be written asγF̄m (Γ, c1 , c2 ) ≡ + F̄bA (c1 , c2 ) + F̄bB (−c1 , −c2 ) =Γγ1+ Γ 5/6 F + Γ 7/6 δ(c1 + c2 )
Γ
2




B
+Γ 9/6 (κ̄A + κ̄B )(c1 + c2 )2 + κ̄A
G + κ̄G c1 c2 , (4.7)
where γ is the interfacial tension,
F ≡ F̄0A + F̄0B ,

φ

345

(4.8)

and
δ ≡ k̄ A − k̄ B = 11/6  A 2  A 13/6
5 T  A 2
NS
1 + 2αA φA
φ
+ α
−
6a




13/6 
 B 2

11/6
2
1+2αB φB
. (4.9)− NS+ α B φB
In the above, the quantities with the superscript A or B
are the corresponding quantities for the A and B blocks,
separately. Namely, F̄0J , k̄ J , κ̄J , and κ̄JG (J = A, B) are deﬁned in (4.3) to (4.6) for NS = NSJ , α = αJ , and φ = φJ ,respectively. The spontaneous curvature contribution of
the interface is proportional to δ. Since φJ denotes thefraction of longer J blocks in the J brush, there are, in
general, two possible cases [30]. The ﬁrst case is a situation where the same chain contains both the longer A
block and the longer B block as typically shown in Figure 6(a). In this case, φA = φB holds. The second caseis that one chain contains the longer A block, while theother chain contains the longer B block as shown in Figure 6(b). In the latter case, φA = 1−φB holds. Comparing

NS

A

+

+

NS

NSB

φB
φB
Fig. 6. Two representative mixtures of block copolymers considered in the text. (a) The two chains have the same asymmetry but diﬀerent overall molecular weights, so that αA =αB = 0 and φA = φB . (b) The two chains have the same
overall molecular weights, but diﬀerent asymmetries, so that
NSA = (αB /αA )NSB and φA = 1 − φB .

the mixed free energy for the melt case (Eq. (6) in reference [30]) with that for the swollen case (4.7), we see that
(4.7) has a lower order Γ dependence. This leads to thediﬀerent scaling behaviors of various interfacial properties
as discussed below.
We note that φJ is controlled by the experimentally
imposed composition of the mixture as long as all the
copolymer chains are localized at the liquid-liquid interface. However, both the surface coverage Γ and the interface curvature c1 and c2 are free to vary, and these quantities are determined by the minimization of the mixed
free energy F̄m in (4.7). Consider a spherically deformedinterface with c1 = c2 = c. In this case (4.7) becomesF̄m (Γ, c) =γ
+ Γ 5/6 F + Γ 7/6 δc + Γ 9/6 Kc2 ,Γ(4.10)

where

  B

B
K ≡ 4κ̄A + κ̄A
G + 4κ̄ + κ̄G ,(4.11)

which provides the overall bending modulus. As canbe seen from (4.5) and (4.6), κ̄A and κ̄A
G depend onBB
f2 (αA , φA ), while κ̄B and κ̄B
on
f
(α
,
φ
). In the case2
Gof αA = αB and φA = φB ≡ φ, for example, the φ dependence of the overall bending modulus K is given by
Figure 4 as before, and the softening eﬀect can be seenfrom (3.38).
Minimization of (4.10) with respect to both Γ and c(without any constraint) gives the equilibrium condition∂ F̄m (Γ, c)= 0,∂Γ

∂ F̄m (Γ, c)= 0.
∂c(4.12)

346

The European Physical Journal E

The equilibrium surface coverage is determined, to thelowest order, by the balance between the interfacial energy
and the ﬂat monolayer energy. Hence we have
Γ (0) =

6γ
5F

6/11
,

(4.13)

which scales as γ̃ 6/11 (NSB )−6/11 where γ̃ = γa2 /T is thedimensionless interfacial tension. The equilibrium interface curvature c is determined by the balance between
the spontaneous curvature that depends linearly on δ =k̄ A − k̄ B (see (4.9)) and the bending modulus K (see(4.11)). To the lowest order, we havec(0) = −δ  (0) −1/3
δ
Γ
=−
2K
2K



5F
6γ

2/11
,

(4.14)

which scales as (1/a)γ̃ −2/11 (NSB )−9/11 . Note that the curvature vanishes for zero spontaneous curvature. Hence a
ﬂat lamellar structure is stable only when δ = 0. Using theabove results, the lowest-order mixing free energy turns
out to be5/11
11  (0) 5/6
11 6γ
(0)
F =
Γ
F =
F,
(4.15)
6
6 5F
which scales as (T /a2 )γ̃ 5/11 (NSB )6/11 . Although thepresent results are for mixed monolayers, the scaling behaviors in (4.13), (4.14), and (4.15) are similar to thosefor monodisperse monolayers [21,36] as far as the surface tension and the chain length dependences are concerned. For mixed monolayers in the melt conditions,on the other hand, the corresponding quantities scaleas Γ (0) ∼ γ̃ 1/3 (NSB )−1/3 , c(0) ∼ γ̃ −1/3 (NSB )−2/3 andF (0) ∼ γ̃ 2/3 (NSB )1/3 [30]. Calculating higher-order terms
in a dimensionless small parameter δ 2 /(KF ), the equilibrium surface coverage, curvature, and mixing free energy
per chain are obtained as


3δ 2
∗
(0)
Γ =Γ
1+
,
(4.16)
22KF


δ2
∗
(0)
1−
c =c
,
(4.17)
22KF


15δ 4
3δ 2
∗
(0)
−
1−
.
(4.18)
F̄m = F
22KF
1936K 2 F 2
To see the eﬀect of mixing on the interfacial properties of swollen monolayers, we examine here two typical mixtures of block copolymer monolayers as considered in reference [30]. Notice again that αJ and φJ(J = A, B) are the molecular weight diﬀerence between
the two chains (see (3.1)) and fraction of longer chainsfor each of the blocks, respectively. i) The two chains
have the same asymmetry but diﬀerent overall molecular weights, so that αA = αB = 0 and φA = φB as shownin Figure 6(a). ii) The two chains have the same overall molecular weights, but diﬀerent asymmetries, so that

NSA = (αB /αA )NSB and φA = 1 − φB as shown in Figure 6(b). In the latter case one chain contains the longer
A block, while the other chain contains the longer B block.
∗acIn Figures 7 and 8, we have plotted Γ ∗ , c∗ , and F̄mcording to (4.16), (4.17), and (4.18), respectively, for thesetwo cases. In Figure 7 (case i)), we ﬁxed NSA = 2NSB andconsidered three diﬀerent values of αA = αB = 0.1, 1, 10;
all the plots are shown as a function of φ ≡ φA = φB . In
Figure 8 (case ii)), we ﬁxed αA = 1 and considered three
diﬀerent values of αB = 0.1, 1, 10; all the plots are shown
as a function of φ ≡ φA = 1 − φB .
Since the surface coverage is related to the overall interfacial area in the system (see section 5.3), Γ ∗ can be regarded as a measure of the emulsiﬁcation eﬃciency of thecopolymer. When αA = αB is not so large ( 1) in case i),
we ﬁnd that adding a small fraction of shorter chains toa copolymer monolayer is more eﬀective in changing thesurface coverage than addition of a small fraction of longer
chains (see Fig. 7(a)). This is because the stretching energy of the longer chains is considerably reduced by thepresence of shorter chains which act as spacers. However,this dramatic eﬀect of adding a small fraction of shorter
chains is suppressed for larger αA = αB = 10. The dependence of the equilibrium surface coverage Γ ∗ on φ isnon-monotonic in case ii), and each curve obtains a maximum at a ﬁnite composition (see Fig. 8(a)). The maximum
point shifts to higher φ as αB is increased.
The equilibrium curvature c∗ is inversely proportional
to the microemulsion droplet size. When αA = αB = 0.1(i.e., when the polymers have almost equal molecularweights) in case i), the optimal curvature c∗ is almost constant for small φ, whereas it decreases almost linearly for
φ close to unity (see Fig. 7(b)). However, c∗ is not verymuch aﬀected if a small fraction of either short or long
chains are added when αA = αB = 1. The behavior of c∗for αA = αB = 10 (i.e., the two polymers diﬀer greatly intheir molecular weights) is completely diﬀerent. It drops
sharply for small φ and stays almost constant for φ  0.5.
Next we discuss the φ dependence of c∗ in case ii) in which
the two chains have the same overall molecular weights but
diﬀerent asymmetries. From Figure 8(b), we see that for
αB = 0.1 and 10, c∗ depends only weakly on φ, whereasc∗ changes its sign at φ = 0.5 for αB (= αA ) = 1. For thelatter case, c∗ just changes its sign for φ = 0 and φ = 1,
as it should.
∗
All the curves of the reduced free energy F̄min case i)increase monotonically (see Fig. 7(c)). On the other hand,
all the curves in case ii) obtain a single minimum at a ﬁnite
composition (see Fig. 8(c)). In other words, there are noinﬂection points in both cases. Therefore the system does
not tend to phase separate and the mixed monolayer ispreferred. This is due to the fact that the interactions between chains of diﬀerent asymmetry or molecular weights
are always attractive [30]. The attractive interaction originates from the relaxation in the stretching energy by mixing shorter chains. A similar discussion has been presented
by Cantor [21]. Although the scaling behaviors of various
interfacial properties are diﬀerent in the present swollencase and the melt case [30], the eﬀect of mixing and theShigeyuki Komura and S.A. Safran: Scaling theory of mixed amphiphilic monolayers

347

1.2

0.7

(a)

(a)
1
0.5

Γ

Γ

∗

∗

0.8

0.6
0.3
0.4

0.1

0

0.2

0.4

φ

0.6

0.8

0.2

1

0

0.2

0.4

φ

0.6

0.8

1

0.2

0.4

φ

0.6

0.8

1

0.2

0.4

φ

0.6

0.8

0.2

(b)

(b)
0.1
0.16

0.12

c*

c*

-0.1

0.08
-0.3
0.04

0

0

0.2

0.4

φ

0.6

0.8

1

-0.5

0

(c)

11

13

(c)
9

9

7

Fm*

Fm*

11

7

5

5

3

3

0

0.2

0.4

φ

0.6

0.8

1
∗

Fig. 7. Plots of equilibrium (a) surface coverage Γ (in unitsof γ 6/11 a12/11 T −6/11 (NSB )−6/11 ), (b) interface curvature c∗ (inunits of γ −2/11 a−15/11 T 2/11 (NSB )−9/11 ), and (c) monolayer free∗energy F̄m(in units of γ 5/11 a−12/11 T 6/11 (NSB )6/11 ) when thetwo chains have the same asymmetry but diﬀerent overallmolecular weights (case i) in the text or Fig. 6(a)). All thegraphs are shown as a function of φ ≡ φA = φB . The molecular weight is ﬁxed NSA = 2NSB . Diﬀerent curves correspond
to αA = αB = 0.1 (solid line), 1 (dashed line), and 10 (dottedline), respectively.

1

0

1
∗

Fig. 8. Plots of equilibrium (a) surface coverage Γ (in unitsof γ 6/11 a12/11 T −6/11 (NSB )−6/11 ), (b) interface curvature c∗ (inunits of γ −2/11 a−15/11 T 2/11 (NSB )−9/11 ), and (c) monolayer free∗energy F̄m(in units of γ 5/11 a−12/11 T 6/11 (NSB )6/11 ) when thetwo chains have the same overall molecular weights, but different asymmetries (case ii) in the text or Fig. 6(b)). All thegraphs are shown as a function of φ ≡ φA = 1 − φB . The
molecular weight is chosen as NSA = (αB /αA )NSB with αA = 1.
Diﬀerent curves correspond to αB = 0.1 (solid line), 1 (dashedline), and 10 (dotted line), respectively.

348

The European Physical Journal E

underlying physics as discussed above, are rather similar
in both cases.
Finally, we remark that the case of a cylindrical monolayer can also be calculated by putting c1 = c and c2 = 0
in (4.7) and minimizing with respect to Γ and c as in
the spherical case. One can obtain the equilibrium surface coverage, interface curvature, and the free energy by(4.16), (4.17), and (4.18) with δ and K being replaced by(k̄ A − k̄ B )/2 and κ̄A + κ̄B , respectively. However, cylindrical monolayers are unlikely to form because the freeenergy is always higher than that of spherical monolayers as long as there is no constraint on the volume of the
microemulsion phase [30,36]. But if there are conservation
constraints on the solvent, then we get the transition fromspheres to cylinders or to lamellae [2].

5 Summary and discussion
5.1 Main ﬁndings
In this paper, we applied the blob model and scaling arguments of a bimodal brush, to predict the elastic and interfacial properties of mixed amphiphilic monolayers. We
ﬁrst extended the results in reference [34] for a ﬂat, bimodal brush consisting of shorter and longer chains inthe swollen state. Next, we calculated the elastic moduli
and the spontaneous curvature as a function of the composition and the relative chain length. We obtained simple scaling functions which interpolate between the elastic
moduli of a pure short-chain brush and a pure long-chain
brush. We then used the analogy between block copolymer interfaces and polymeric brushes in order to calculate the stretching energy of mixed amphiphilic monolayers. Following the argument in reference [30], we discussed
the eﬀect of mixing on self-assembled diblock copolymer
monolayers in the swollen state. Since the area per chain
adjusts itself to minimize the total free energy in such asystem, the free energy per chain is minimized with respect to both the surface coverage and the curvature. We
then calculated the equilibrium surface coverage, interface curvature, and the mixing free energy as a functionof the composition, and discussed their behavior for twocases of mixed, block copolymer monolayers. For a certain
choice of parameters, we ﬁnd that adding a small fractionof shorter chain to a copolymer monolayer is more eﬀective in changing the surface properties than addition of asmall fraction of longer chains. In general, we ﬁnd a large
deviation of the various surface properties from the simple
linear averaging of the pure component monolayers. Thisnonlinear dependence on the composition is due to theeﬀectively, attractive interactions between diﬀerent typesof chains, because the addition of shorter chains leads toa relaxation in the stretching energy of the longer blocks.
These properties are qualitatively the same as those of
mixed brushes in the melt state [30], although the scaling behavior is diﬀerent between the melt and the swollencases.

5.2 Applicability of the present theory
Although we have mainly discussed the elastic propertiesof mixed monolayers composed of AB diblock copolymers,
one can apply our theory to mixtures of longer-chain surfactant molecules and shorter-chain surfactant molecules(cosurfactants) as considered in references [55,56]. Our result is consistent with their ﬁnding that the bending modulus is signiﬁcantly reduced by the replacement of longchain molecules by short ones. In fact, this phenomenon
is experimentally observed in ﬂuctuating ﬂexible bilayers composed of surfactant-cosurfactant mixtures [57,58].
Strictly speaking, however, the blob picture used in thispaper is valid only for long chains and the model applies to
ﬂexible diblock copolymer monolayers. Hence it should be
kept in mind that the results concerning small surfactants
are merely suggestive.
At the beginning of section 3, we mentioned that thebrush model is inappropriate in the limit of only a few
long chains. Notice that f1 (α, φ) and f2 (α, φ) in (3.34)
and (3.35), respectively, are nonanalytic for φ → 0. This
reﬂects the invalidity of the model in this limit. If thelong chains are too dilute, the mushroom (on top of abrush) picture is relevant for the upper second layer. As
described in section 2, the variation of the elastic moduli
κ and κG for mushroom conformations of anchored (endgrafted) polymers has been calculated in references [33,34,
38,39]. Using their results, the elastic moduli of a bimodal
brush for small φ can be written asκ(α, φ → 0) ≈65
T N 3 Γ 5/2
108  S

1
π
αNS
Γ φ,
+
1+
T
12
2
6

κG (α, φ → 0) ≈ −5
1 αNS
T NS3 Γ 5/2 − T
Γ φ.
18
6
6

(5.1)
(5.2)

The ﬁrst term in each of the equations corresponds tothe elastic modulus for a pure short-chain (see (3.31) and(3.32)). The mushroom part of the long chains is assumed
to be ideal and its gyration radius is given by (αNS /6)1/2 .
Note also that Γ φ is the eﬀective coverage for the second
upper layer, and both of the above expressions change
linearly with φ, which is in contrast to the brush case.

5.3 Relation to experiment
Our theory may be used to interpret recent experiments on
the eﬀect of amphiphilic block copolymers on ternary microemulsions as described in the Introduction. Experimentally, the amount of surfactant molecules (Ci Ej ) which is
suﬃcient to form a one-phase microemulsion can be dramatically reduced by the addition of a small amount ofamphiphilic block copolymers (PEP-PEO) [31,32]. The
neutron scattering experiments showed that the aboveﬁnding is connected to the increase in the structural lengthscale d (typical size of an oil or water domain) of the microemulsion [31]. For small concentrations of copolymers,

Shigeyuki Komura and S.A. Safran: Scaling theory of mixed amphiphilic monolayers

mushroom picture predicts that the observed solubility
enhancement is attributed to the variation of the elastic
moduli [32]. For relatively high copolymer concentrations,however, the brush regime is entered and the mushroom
picture ceases to be valid. In this case our theory can beapplied. The surfactant molecules and amphiphilic block
copolymers in the experiment correspond to shorter andlonger chains in our theory, respectively.
We now brieﬂy explain why the change in the bending modulus aﬀects the structural length scale of the
microemulsion [2]. We denote the number of surfactant
molecules per unit volume by n. Then the domain size dis roughly related to n by [59]
d∼

Γ
.
na2

(5.3)

This domain size should be compared with the persistence
length 'p of the monolayer deﬁned by [59,60]
 κ
'p ∼ a exp
.
(5.4)
T
The interface remains ﬂat over distances smaller than 'p
but is crumpled at larger length scales. For 'p > d, themonolayers tend to be parallel to each other, whereas for'p < d, the monolayers are wrinkled and can form microemulsions [59,61–64]. Therefore, as a rough estimateof the minimum surfactant concentration nc needed tosolubilize oil and water, we set 'p ∼ d and obtainΓκ
nc ∼ 3 exp −
.
(5.5)
a
T
Since nc decreases exponentially with κ, the eﬃciency
boost can reﬂect an increase in κ [31,32]. It may be thenconvenient to deﬁne an emulsiﬁcation enhancement factor
η given by η ≡ exp(∆κ/T ), where ∆κ is the change of thebending modulus due to the presence of the added longer
chains.
When the volume fraction of the block copolymer is
suﬃciently small, each copolymer may be considered to
be in the mushroom regime. The change of the bending
modulus in the mushroom regime has been calculated byseveral authors in references [33,34,38,39]. In that case,the enhancement factor can be given (ignoring some numerical factors)ηmush ∼ exp NL Γ ,(5.6)
for NLNS . However, it is important to notice thatthe polymer mushroom inﬂuences the properties of theinterface only on a scale of the order of polymer size. The
uniform distribution of the curvature is correct only whenmushrooms just begin to overlap [32]. When the volume
fraction of the copolymer is high enough to form brushes,
one should use the expression of the bending modulus calculated in this paper or in reference [34]. Using (3.28), theenhancement factor is given by 18/11 
,(5.7)
ηbrush ∼ exp NL3 Γ 5/2 ∼ exp NL

349

NS and we have used (4.13). Notice that ηbrushfor NLhas a much stronger dependence on NL and Γ than inηmush . Hence the solubility enhancement should be much
more dramatic in the brush regime. When NL = 100, forexample, the overlap coverage is Γov ≈ 0.004. At this surface coverage, the enhancement factor in the mushroom
and brush regimes are ηmush ≈ 1.49 and ηbrush ≈ 2.75, respectively. A combination of these two regimes would account for the eﬀect of the added block copolymers on the
microemulsions in both the mushroom and brush regimes.
We have greatly beneﬁted from the discussions and correspondence with N. Dan and R. Strey. SK thanks the Ministry of
Education, Science and Culture, Japan for providing ﬁnancial
support during his visit to Weizmann Institute of Science, Israel. Support from the exchange program between the Japan
Society for the Promotion of Science (JSPS) and the Israel
Ministry of Science and Technology is also gratefully acknowledged. SAS is grateful to the Israel Science Foundation for its
sponsorship of the Center on Self Assembly and to the donorsof the Petroleum Research Fund administered by the American
Chemical Society.

Appendix A. Bimodal brush on a sphere anda cylinder
In this appendix we derive the free energy of a bimodal
brush on a sphere and a cylinder. First we consider the
spherical surface as illustrated in Figure 3. The brushheight of the ﬁrst layer h1s (R) is implicitly given by thefollowing equation:


R+h1s

NS =

dr
R

ξ1s (r)a1/ν 1.
ξ1s (r)(A.1)

From (3.14), this leads to
h1 = νR
1/νh1s (R)
1+
−1 ,
R

(A.2)

where h1 is the height of the ﬁrst layer in the ﬂat caseand is given by (3.4). Solving this equation for h1s (R), we
obtain


ν
h1
−1 .
(A.3)
h1s (R) = R 1 +νRThis equation is the same as equation (48) in reference [34]which treats the monodisperse case. For the second layer,
the height of the brush h2s (R) satisﬁes the equation
NL − NS =

R+h1s +h2s

dr
R+h1s

ξ2s (r)a1/ν 1. (A.4)ξ2s (r)From (3.14), this leads toh2 = νRν
h2s (R)
h1
+ 1+
R
νR

1/ν 

h1
− 1+
νR


, (A.5)

350

The European Physical Journal E

where h2 is the height of the second layer in the ﬂat caseand is given by (3.6). Solving this equation for h2s (R), we
obtainν ν 
h1 + h2h1− 1+
h2s (R) = R 1 +
.
(A.6)
νR
νR
Similar to the ﬂat surface, the total free energy F̃s ofa bimodal brush on a spherical surface is proportional to
the number of blobs. Thus it can be calculated (to withina constant of order unity) asF̃s = T

 R+h1sA
1dr
2
ξ0ξ1s (r)R R+h1s +h2s

A
1
dr
+ 2
,
ξ0 /φ
ξ2s (r)
R+h1s

(A.7)

where ξ1s (r) and ξ2s (r) are given by (3.14), h1s and h2s are
given by (A.3) and (A.6), respectively. After some calculations, the free energy per unit area deﬁned by Fs = F̃s /A
is given by (3.15).
Next we consider the cylindrical surface case. The
brush height of the ﬁrst layer h1c (R) implicitly satisﬁes
an equation analogous to (A.1) with h1s and ξ1s (r) being
replaced by h1c and ξ1c (r), respectively. Using (3.17) andsolving for h1c (R), we have
2ν/(1+ν)(1 + ν)h1
h1c (R) = R 1 +
−1 .
2νR

(A.8)

This equation is the same as equation (51) in reference [34]which treated the monodisperse case. Similarly, the heightof the second layer h2c (R) satisﬁes an equation analogous
to (A.4). We then solve for h2c (R) to obtain2ν/(1+ν)(1 + ν)(h1 + h2 )h2c (R) = R 1 +2νR2ν/(1+ν) (1 + ν)h1
− 1+
.
2νR

(A.9)

We calculate the the total free energy F̃c analogous to(A.7), and convert to the free energy per unit area Fc =F̃c /A. Then we obtain (3.18).

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