EUROPHYSICS LETTERS

15 December 2003

Europhys. Lett., 64 (6), pp. 844–850 (2003)

The unbinding transition of mixed ﬂuid membranes
S. Komura 1 (∗ ) and D. Andelman 2 (∗∗ )1
Department of Chemistry, Faculty of Science, Tokyo Metropolitan University
Tokyo 192-0397, Japan
2
School of Physics and Astronomy, Raymond and Beverly Sackler Facultyof Exact Sciences, Tel Aviv University - Ramat Aviv 69978, Tel Aviv, Israel(received 5 August 2003; accepted in ﬁnal form 8 October 2003)
PACS. 87.16.Dg – Membranes, bilayers, and vesicles.
PACS. 68.05.-n – Liquid-liquid interfaces.
PACS. 64.60.-i – General studies of phase transitions.

Abstract. – A phenomenological model for the unbinding transition of multicomponent ﬂuid
membranes is proposed, where the unbinding transition is described using a theory analogous
to the Flory-Huggins theory for polymers. The coupling between the lateral phase separationof inclusion molecules and the membrane-substrate potential provides a rich phase behavior.
Our model describes the ﬁrst-order nature of the unbinding transition in multicomponent membranes as was observed in a recent experiment. In particular, we predict diﬀerent scenarios ofphase coexistence between bound and unbound membrane states.

Introduction. – Adhesion of membranes and vesicles is responsible for cell-cell adhesion
which plays an important role in all multicellular organisms. In general, bio-adhesion isgoverned by the interplay of various generic and speciﬁc interactions [1]. Speciﬁc interactions
act between complementary pairs of proteins such as ligand and receptor, or antibody andantigen. Well-studied examples of such coupled systems are biotin-avidin complexes [2], or
selectins and their sugar ligands [3].
The problem of adhesion of multicomponent membranes is intimately related to that of domain formation. Experimentally, adhesion-induced lateral phase separation has been observedfor various systems [4–6]. It was reported that adhesion molecules aggregate spontaneously
and form tight-adhering domains. From the theoretical point of view, this problem has been
considered in [7] using a phenomenological model, where the inter-membrane distance is coupled to the concentration of sticker molecules on the two adhering membranes. In a diﬀerent
approach, a lattice model for a multicomponent membrane in contact with another substrate
was proposed [8], and studied via detailed Monte Carlo simulations [9, 10]. More recently, awork combining these two approaches has been published [11].
In a recent experiment by Marx et al. [12], the role of long-range repulsions due to thermal
ﬂuctuations (Helfrich repulsion) of adhering membranes has been addressed. By using Reﬂection Interference Contrast Microscopy [4], a multicomponent bilayer membrane with added(∗ ) E-mail: komura@comp.metro-u.ac.jp(∗∗ ) E-mail: andelman@post.tau.ac.ilc EDP SciencesS. Komura et al.: The unbinding transition of mixed fluid membranes

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Fig. 1 – A mixed ﬂuid membrane adhering to a substrate. Black ﬁlled ovals indicate inclusions such asproteins or lipopolymers. The height of the upper membrane leaﬂet from the substrate is , whereas
the thickness of the bilayer membrane is δ.

lipopolymers (modiﬁed DOPE lipid with polyethylenoxide) and cholesterol is examined in thevicinity of an attractive substrate. Analyzing the probability distribution of the membranesubstrate spacing for various multicomponent membranes, a phase separation between two
distinct lipopolymer-poor and lipopolymer-rich states having two diﬀerent spacings from thesubstrate was suggested, where both states are unbound from the substrate. Although this phenomenon is dynamic in nature, it was interpreted as a lateral phase segregation induced by theHelfrich repulsion. However, a clear physical description for the appearance of such a phase coexistence has not been given. Moreover, a ﬁrst-order unbinding transition scenario is required
to account for the multiple time scales in the time series of the membrane ﬂuctuations [12].
In this letter, we propose a simple phenomenological model for a multicomponent (mixed)
ﬂuid membrane which can undergo simultaneously a lateral phase separation and an unbindingtransition. The model is motivated by the experiment [12] and relies on the coupling between
the inclusion concentration and the membrane-substrate spacing. The lateral phase separationof the inclusion aﬀects the second virial coeﬃcient of the unbinding transition which is taken
into account in analogy to the Flory-Huggins theory for polymers [13]. Our model exhibits
various types of phase coexistence, including a phase separation between bound and unboundstates as well as between two unbound ones. The former phase coexistence indicates theﬁrst-order nature of the unbinding transition, as was anticipated in the experiment [12].
Unbinding transition. – Fluid membranes in a lamellar stack or close to a substrate experience steric repulsion arising from their reduced undulation entropy due to the conﬁnementeﬀect. The corresponding interaction energy per unit area has been given by Helfrich [14] asvs () =b(kB T )2,κ( − δ)2

(1)

where kB is the Boltzmann constant, T the temperature, κ the bending rigidity of the membrane having thickness δ, and  the average height of the upper membrane lipid leaﬂet fromthe substrate (see ﬁg. 1). The numerical prefactor b was calculated as b = 3π 2 /128 in [14],but its value is still debatable in the literature [15]. The combination of the above steric
repulsion and other direct microscopic (van der Waals, electrostatic, hydration) interactions
determines whether the membrane binds to the substrate or unbinds. Using functional renormalization group techniques, Lipowsky and Leibler have shown that this unbinding transition
is of second order [16, 17]. The average spacing  diverges as the strength of the attractive
van der Waals interaction W (Hamaker constant) approaches a critical value from above, i.e., ∼ (W − Wc )−ψ with ψ ≈ 1.0. Here Wc is the critical strength of the Hamaker constant.
While a simple superposition of the Helfrich repulsion, eq. (1), and other direct interactions
gives an incorrect (ﬁrst-order) description of the unbinding transition, a simple theory for the

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unbinding transition in a bulk of lamellar phase was considered by Milner and Roux [18], and isbrieﬂy reviewed below. Following the spirit of the Flory-Huggins theory for polymers [13], theHelfrich estimate of the entropy is taken into account accurately, whereas the other interactions
are approximately incorporated via a second virial term. This approach is justiﬁed since theperturbative eﬀect of the direct interactions is included through a virial expansion rather than
the simple superposition of the interactions. The resulting free energy per unit volume of alamellar stack isb(kB T )2 3
w − µw w,(2)
gMR (w) = −kB T χ̄w2 +κδ 3
where the membrane volume fraction w = δ/ ≥ 0 cannot be negative. Note that  represents
the lamellar repeat distance, χ̄ is the second virial coeﬃcient, and µw the chemical potential
which accounts for the conservation of the membrane volume fraction w. The second virial
coeﬃcient is predicted to behave as χ̄ ∼ (W − Wc ). For χ̄ > 0 the free energy (2) describes aﬁrst-order unbinding transition as µw is lowered.
To consider the unbinding behavior of a single membrane (see ﬁg. 1) as in the experiment [12], eq. (2) is multiplied by the lamellar repeat distance . We also note that themembrane average distance  from the substrate is not a conserved quantity. Then the freeenergy per unit area of a single membrane can be expressed asb(kB T )2 2
w .
(3)
κδ 2
Minimization of f (w) with respect to w shows that this free energy has a continuous secondorder transition at χ̄ = 0 between a bound state (w > 0) for χ̄ > 0 and an unbound state(w = 0) for χ̄ < 0. A similar free energy for a single membrane undergoing the unbindingtransition was proposed by Helfrich [19]. Equation (3) is the form of the free energy which
will be used in our model.
f (w) = −kB T δ χ̄w +Unbinding transitions of mixed membranes. – So far, the unbinding transition of a singlecomponent membrane has been discussed. In the case of multicomponent membranes, thelateral phase separation aﬀects the direct interactions between the membrane and the substrate, and hence their unbinding behavior. For simplicity, we consider a two-component
membrane adhering to a substrate as in ﬁg. 1. The overall membrane state is characterized
by its average distance  from the substrate. The internal degree of freedom, on the other
hand, corresponds to the membrane composition. In addition to the lipid component, that is
the main building block of the membrane, we introduce a second component or an “inclusion”
representing additional proteins, cholesterol, or lipopolymers residing on the membrane. Let
the concentration of these inclusions be denoted by Φ (0 ≤ Φ ≤ 1). Here we discuss the casein which the interaction between two inclusions is attractive leading to a condensation transition. Then, below a certain temperature Tc , the mixed membrane undergoes a ﬁrst-ordertransition, where an inclusion-poor phase coexists with an inclusion-rich phase as observed
experimentally [20]. The ﬁrst-order transition terminates at a critical point, having a criticalconcentration Φc . The concentration diﬀerence is then deﬁned as φ = Φ − Φc [21].
We now combine the unbinding transition and the lateral phase separation of the inclusion
close to the critical point. The proposed free energy per unit area of a single mixed membrane
undergoing the unbinding transition is1
1
1
f (φ, w) = −µφ + tφ2 + φ4 − χw + w2 + γφw,
2
4
2

(4)

with the constraint w ≥ 0. Here all energy terms have been scaled by 2b(kB T )2 /κδ 2 , and are
now dimensionless. The ﬁrst three terms in eq. (4) depend only on φ, and correspond to theS. Komura et al.: The unbinding transition of mixed fluid membranes

847

Landau free energy of a two-component membrane undergoing a lateral inclusion-lipid phaseseparation [22], µ is the chemical potential conjugate to φ, and t ∼ (T − Tc ) the reduced
temperature. The quartic coeﬃcient has been chosen to be positive and is arbitrary set to 14
without loss of generality. The next two terms depend only on w, and represent the unbinding transition of a single membrane as have been described by eq. (3). In the above, χ is thescaled virial coeﬃcient.
The last term is the lowest-order coupling term between φ and w with a dimensionlesscoupling coeﬃcient γ > 0. The physical meaning of this bilinear term is as follows. Whenthe mixed membrane is quenched into the two-phase region, an inclusion-poor phase (φ < 0)coexists with an inclusion-rich phase (φ > 0). This can lead to diﬀerent direct interactions andhence diﬀerent second virial coeﬃcients χ for each of the domains. We model this situationby considering an eﬀective virial term in eq. (4) as −χeﬀ w = −(χ − γφ)w, which leads to thecoupling term φw.
In the above model, we have assumed that the phase separation of the inclusion molecules
does not aﬀect the homogeneous bending rigidity of the membrane. In other words, only
the direct microscopic (van der Waals, electrostatic, hydration) interactions are modiﬁed by
the phase separation, whereas the steric interaction remains unchanged. This is valid as longas the size of the inclusion molecule is close to the membrane thickness δ. We note thata diﬀerent scenario was suggested in [21, 23], where ﬂuid membranes with spatially varying
bending rigidity have been considered. In these models, the coupling between membrane
curvature and inclusion concentration leads to a lateral phase separation.
Minimizing the free energy (4) with respect to w gives w = 0 (unbound) for φ > χ/γ andw = χ − γφ > 0 (bound) for φ < χ/γ. By substituting back the value of w into eq. (4), thefree energy as a function of φ only becomes
1
1
f (φ) = −µφ + tφ2 + φ4 ,2
4for φ > χ/γ,

1
1
1
f (φ) = − χ2 + (γχ − µ)φ + (t − γ 2 )φ2 + φ4 ,2
2
4for φ < χ/γ,

(5)

(6)

depending on the value of χ/γ. Next, this free energy is minimized with respect to φ to ﬁnd
the equation of state. The two-phase region is obtained by the Maxwell construction, and thewhole phase diagram is calculated numerically.
Phase behavior. – Hereafter we set γ = 1 without loss of generality. Then the phasebehavior of the above model depends only on the value of χ. For χ > 0, a second-ordertransition line (critical line where φ = χ) separates the unbound phase from the bound one,as shown in ﬁg. 2 for χ = 1. The critical line ends at a critical end-point (CEP —denoted by
a ﬁlled square in ﬁg. 2) on the ﬁrst-order transition line. This ﬁrst-order line itself ends atan ordinary critical point (CP —denoted by a ﬁlled circle in the ﬁgures) corresponding to aliquid/vapor-type CP between the two bound phases (B1+B2). Below the CEP temperature,
the ﬁrst-order line separates the bound phase from the unbound phase (B1+U).
The CP between the two bound phases is located at (φc , tc ) = (0, 1). Hence, the criticaltemperature is increased from tc = 0 to 1 due to the coupling between the inclusion concentration φ and the membrane-substrate distance  = δ/w. In other words, the phase separationis enhanced by the adhesion for χ > 0. This result is in accordance with previous theoreticalmodels [7, 9, 10]. Between the CEP and the CP of the two-phase region (B1+B2), the two coexisting values of φ lead to diﬀerent membrane-substrate distances given by  = δ(χ − γφ)−1 .
Since the  value for B1 is smaller than that for B2, the phases B1 and B2 correspond to the

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(b)

(a)
1

1

B2
B1

B2

B1+B2

t

B1

U

t

0

0

U
B1+U

-1

-1

0

1

φ

1

-1

2

µ

Fig. 2 – The phase diagrams for χ = 1 and γ = 1 as a function of (a) inclusion concentration φand reduced temperature t, and (b) inclusion chemical potential µ and reduced temperature t. The
continuous line is a ﬁrst-order line, whereas the dashed line is a second-order one. The critical point
and the critical end-point are indicated by a ﬁlled circle and a square, respectively. The bound (w > 0)
and the unbound (w = 0) phases are denoted by B and U, respectively. Below the critical point, there
is a coexistence region either of two bound phases (B1+B2), or bound and unbound phases (B1+U).

“tight” and the “loose” bound phases, respectively, as discussed in [7]. At χ = 0 the CEP
meets with the CP and becomes the double critical end-point.
When χ becomes negative, the second- and ﬁrst-order lines join up at a tricritical point(TCP —denoted by a ﬁlled triangle in ﬁg. 3). In addition to the coexistence between thebound and the unbound phases (B+U1),
a new two-phaseregion between the two unbound√√phases (U1+U2) appears for −1/ 2 < χ < −2/3 3. As an example, ﬁg. 3 gives the phase
diagrams for χ = −0.5. In this case, the TCP is located at (φtcp , ttcp ) = (−0.5, 0.25), and the

0.5

0.5

(b)

(a)

U1

0

U1

U1+U2

t
B

Tr

0

U2

Tr

t

U2

B+U1

B

-0.5

-0.5

B+U2
-1

-1

0

φ

1

-0.5

µ

0

-1

Fig. 3 – The phase diagram for χ = −0.5 and γ = 1 as a function of (a) inclusion concentration φand reduced temperature t, and (b) inclusion chemical potential µ and reduced temperature t. The
same notation of diﬀerent lines and symbols is used as in ﬁg. 2. The critical point is indicated by aﬁlled circle, the tricritical point by a ﬁlled triangle, and the triple point by Tr.

S. Komura et al.: The unbinding transition of mixed fluid membranes

849

CP at (φc , tc ) = (0, 0). Notice that once the membrane unbinds from the substrate due tothe Helfrich repulsion, the lateral phase separation between the two unbound phases occurs
simply because of the temperature quench (t < 0). There is also a triple point (Tr) at which
the bound phase and the two unbound phases coexist (B+U1+U2). Below this triple pointtemperature, there is a region of two-phase coexistence between the bound and the unboundphases (B+U2).
√
For χ < −1/ 2, a similar (φ, t)-phase diagram as in ﬁg. 2 can be obtained (not shownhere) with the critical line being located at φ = χ < 0. Below the CP at (φc , tc ) = (0, 0), thetwo unbound phases coexist until t reaches the CEP temperature. Then a region of two-phasecoexistence between the bound and the unbound phase appears below the CEP. For χ < 0
the interaction is repulsive on average, but can become attractive when the phase separation
becomes suﬃciently strong. In this case, the eﬀective second virial coeﬃcient χeﬀ = χ − γφturns out to be positive for negative enough values of φ.
In the model, the unbinding transition becomes ﬁrst order below CEP or TCP for any
value of χ. This result is in accordance with the experimental prediction [12]. Our theory also
describes the coexistence between the two unbound states where the membrane has an inﬁnite
separation from the substrate. In the experiment, the ﬁnite separation of the membrane wasattributed to the gravitational eﬀect [12]. Within our model, the gravitational energy per unit
area should enter as a g/w term in eq. (4) with g > 0. In the presence of such a term, we obtain a coexistence between the two bound states with ﬁnite separations, which is in agreementwith the experiment. However, the bound membrane goes continuously away to inﬁnity andunbinds for g → 0 when φ is large enough. This is the limit we have examined in this paper.
The present results can be compared with those obtained in other theoretical works. In
a previous phenomenological model for the adhesion-induced lateral phase separation [7], wehave considered only the phase coexistence between two bound phases. Hence a possible unbound state of the membrane was not taken into account. A phase coexistence between boundand unbound states has been predicted independently in [9,10] by a mean-ﬁeld treatment of alattice model and Monte Carlo simulations. In those works, the topology of the phase diagram
depends on several quantities such as the sticker binding energy, the potential range, or thestrength of the cis-interaction between the stickers. However, the coexistence between the two
unbound phases has not been predicted within the lattice model. A ﬁnal remark is that thesequence of the phase behavior obtained from our model is similar to that for sponge phases
in surfactant solutions when the symmetry-breaking parameter is varied [24].
In summary, we have proposed a phenomenological model for the unbinding transitionof a two-component ﬂuid membrane. The coupling between lateral phase separation andthe membrane-substrate potential explains some of the recent experimental results. Further
extensions of the present study, such as the eﬀect of ﬁnite tension on the membrane, oradhesion between two ﬂuctuating mixed membranes are currently under investigation.
∗∗∗
We thank H. Diamant, S. Marx, D. Roux, E. Sackmann, H. Shirotori, and
T. Weikl for useful discussions and comments. Support from the Ministry of Education,
Culture, Sports, Science and Technology, Japan (Grant-in-Aid for Scientiﬁc Research No.
15540395) for SK, and from the Israel Science Foundation under grant No. 210/02 and theAlexander von Humboldt Foundation for DA is gratefully acknowledged. DA also would liketo thank the Japan Society for the Promotion of Science (JSPS) for a travel grant.

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