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The interplay of adhesion and phase separation is studied theoretically for two-component membranes that can phase separate into two fluid phases such as liquid-ordered and liquid-disordered phases. Many adhesion geometries provide two different environments for these membranes and then partition the membranes into two segments that differ in their composition. Examples are provided by adhering vesicles, by hole- or pore-spanning membranes, and by membranes supported by chemically patterned surfaces. Generalizing a lattice model for binary mixtures to these adhesion geometries, we show that the phase behavior of the adhering membranes depends, apart from composition and temperature, on two additional parameters, the area fraction of one membrane segment and the affinity contrast between the two segments. For the generic case of non-vanishing affinity contrast, the adhering membranes undergo two distinct phase transitions and the phase diagrams in the composition/temperature plane have a generic topology that consists of two two-phase coexistence regions separated by an intermediate one-phase region. As a consequence, phase separation and domain formation is predicted to occur separately in each of the two membrane segments but not in both segments simultaneously. Furthermore, adhesion is also predicted to suppress the phase separation process for certain regions of the phase diagrams. These generic features of the adhesion-induced phase behavior are accessible to experiment.

Multi-component membranes consisting of a small number of lipids provide simple model systems for biological membranes, which contain a huge number of different lipid and protein components. Since membranes are essentially 2-dimensional systems, they can attain different thermodynamic phases and undergo phase transitions between these phases. From a biological perspective, the most interesting phase transitions are provided by transitions between two distinct

The simplest examples for fluid-fluid coexistence in membranes are presumably found in binary mixtures of cholesterol and a single phospholipid. Indeed, a variety of spectroscopic methods such as deuterium nuclear magnetic resonance [

For ternary mixtures consisting of an unsaturated phospholipid, sphingomyelin, and cholesterol, as originally studied in the context of sphingolipid-cholesterol rafts [

In this paper, we consider the effect of adhesion onto the phase behavior of multi-component membranes. We first emphasize that many adhesion geometries lead to a segmentation of the membranes. Examples are provided by the adhesion of vesicles, by hole- or pore-spanning membranes, and by membranes supported by chemically patterned surfaces. In all of these cases, the adhesion leads to two distinct membrane segments that experience different environments. These environments attract the different molecular components of the membrane with different affinities,

We will focus on the simplest example for fluid-fluid coexistence as provided by two-component membranes and study the adhesion effects by generalizing a generic lattice model for binary mixtures. This lattice model corresponds to a semi-grand canonical description and depends on the relative chemical potential for the two molecular species. Using this model, it is relatively easy to see that the adhesion-induced segmentation of the membranes leads, in general, to two distinct phase transitions in the two membrane segments. In order to obtain theoretical predictions that are accessible to experiments, we then consider the mole fractions in the two membrane segments and show how one can obtain the phase behavior in terms of these mole fractions. One important parameter turns out to be the affinity contrast, which describes the different molecular interactions between the two membrane segments and their environments.

We show that the fluid-fluid coexistence region as found for the two-component membrane in a uniform environment is replaced, for any nonvanishing affinity contrast, by two distinct coexistence regions, which are separated by an intermediate one-phase region. The relative sizes and positions of these different regions are shown to depend only on a relatively small number of parameters, namely temperature, mole fraction of one molecular species, area fraction of one of the membrane segments, and affinity contrast. For the generic case of a nonzero affinity contrast, our theory predicts that phase separation can only occur separately in each of the two membrane segments but not in both segments simultaneously. Furthermore, adhesion is also found to suppress the phase separation process within certain regions of the phase diagrams.

Our paper is organized as follows. First, Section 2 contains a brief review of fluid-fluid coexistence in two-component membranes and Section 3 describes the lattice binary mixture as a generic model for phase separation in two dimensions. In Section 4, we discuss several adhesion geometries such as vesicle adhesion and pore-spanning membranes, and describe how these geometries lead to a segmentation of the membranes into two different membrane segments. These segments experience distinct environments, which can be characterized by relative affinities. In Section 5, we introduce the lattice model for the adhering membranes and show that the relative affinities of the two membrane segments lead to shifts of the relative chemical potential. It is then relatively easy to conclude that the adhering membranes undergo two phase transitions. In order to obtain theoretical predictions that are accessible to experiments, we then replace in Section 6 the relative chemical potential by the mole fraction _{a}_{a}

At constant pressure, the phase diagrams of multi-component membranes depend on the ambient temperature _{a}_{b}

one of which can be varied independently.

Thus, phase diagrams of two-component membranes are conveniently described in the (_{a}

The membrane undergoes phase separation into the _{t}_{c}_{t}_{c}

with the two binodal lines

A relatively simple but instructive model for two-component membranes is provided by lattice binary mixtures, in which the configurations of the two molecular components are described by occupation numbers [

To proceed, let us first consider a large membrane segment with total area

Since the discretization lattice is characterized by a single lattice constant

Within the framework of the lattice binary mixture, the molecular configurations of the two-component membrane are now described in terms of occupation numbers

The mole fraction _{a}_{i}

The total number _{a}_{a}_{i}_{i}_{b}

Thus, for the lattice binary mixture, the total number of _{a}_{b}

If two neighboring sites _{aa}_{bb}_{ab}

and the total configurational energy ℰ{

where we have introduced two chemical potentials _{a}_{b}_{a}_{b}

with the relative chemical potential

The term _{b}_{B}

The form (

The lattice binary mixture just described is equivalent to the two-dimensional Ising model if one expresses the molecular configurations {_{i}_{i}

and the dimensionless ordering field

The Ising system undergoes phase separation for _{c}_{c}

for the Ising model on a square lattice. As we approach the line _{i}_{i}_{i}

The corresponding phase transition in the lattice binary mixture occurs at the relative chemical potential

As we approach the transition value Δ_{αβ}_{a}_{i}

respectively, with the function ϒ (_{a}_{a}_{,}_{β}_{a}_{a}_{,}_{α}_{a}

The semi-grand canonical ensemble of the lattice binary mixture is not particularly convenient from an experimental point of view since the relative chemical potential Δ_{a}_{a}

The change from Δ_{a}_{a}

First, it follows from thermodynamic stability that the chemical potential _{a}_{a}_{b}_{b}_{a}_{a}_{a}_{b}_{a}

More precisely, the function Δ_{a}

and stays constant with

It then follows that _{a}_{,}_{β}_{a}_{,}_{β}_{a}_{,}_{β}_{a}_{,}_{β}

The adhesion of membranes and vesicles often leads to different membrane segments, in which the molecules experience distinct environments. Such a segmentation is found both for adhering vesicles and for solid-supported membranes. As explained in the following subsections, these adhering membranes often consist of two segments that are exposed to two different environments.

The adhesion of vesicles has been studied for a long time, see, e.g., [

In general, a vesicle that sticks to a planar substrate surface exhibits two membrane segments: an unbound segment

The two membrane segments meet at the contact line of the vesicle. Along this line, the unbound membrane segment has the contact curvature radius [

which depends on the membrane’s bending rigidity

The strong adhesion regime considered here corresponds to the situation, in which

_{co} is much smaller than the vesicle size _{ve}. This inequality is equivalent to

where the quantity

Let us now focus on giant vesicles that are conveniently studied by optical microscopy. The size _{ve} of such vesicles usually exceeds 20 _{co}_{*}≃ 0.5

For a lipid bilayer, the bending rigidity ^{−19} J or 24 _{B}_{o}_{o}_{*} ≃ 2 × 10^{−4} mJ/m^{2} or |_{*} ≃ 0.5 _{B}^{2}. Thus, the strong adhesion regime should typically apply as long as the adhesion is mediated by a large number of molecular interactions. In this regime, the vesicle spreads onto the surface as much as possible. For fixed vesicle volume
_{eff} as shown in

The total surface area
_{eff}, see

which can be controlled by osmotic deflation and inflation. Because we focus here on the adhesion-induced segmentation of the membrane, it will be convenient to choose the area fraction

of the unbound membrane segment
^{[2]} of the bound segment
^{[2]} = 1 − ^{[1]}. The relations between ^{[1]} and the other geometric parameters such as the reduced volume

The area fraction ^{[1]} turns out to be an important parameter in order to describe the phase diagram of multi-component membranes. This parameter can vary within the range

The limiting cases correspond to a flat pancake with
^{[1]} = 1. For a hemi-sphere, the area fraction ^{[1]} = 2^{[1]} is uniquely determined by the reduced volume

A variety of methods has been developed in order to immobilize membranes on solid or rigid substrate surfaces. Here, we will discuss several support geometries that lead to two membrane segments and, thus, can again be characterized by the area fraction ^{[1]} of the segments
^{[1]} = 0 and ^{[1]} = 1.

Two examples for partially supported membranes are provided by hole-spanning membranes, also known as black lipid membranes, see, e.g., [

For these adhesion geometries, the membrane is again divided up into an unbound segment
^{[1]} of the hole- or pore-spanning segment can now vary in the range

where the limiting case ^{[1]} = 0 corresponds to a membrane supported by an adhesive surface without a hole or pore. For the present geometry, the area fraction ^{[1]} must always be smaller than one in order to firmly attach the membrane to the substrate surface. If the membrane spans several holes or pores, it contains several unbound segments, and the area

Another adhesion geometry of interest are membranes adhering to a chemically patterned surface as depicted in

The area fraction ^{[1]} of the segments

The two limiting cases ^{[1]} = 0 and ^{[1]} = 1 correspond to uniform substrate surfaces with strong and weak adhesion, respectively.

The different adhesion geometries depicted in _{a}^{[}^{m}^{]} and _{b}^{[}^{m}^{]}. We use the sign convention that

acting on the

We will now characterize each membrane segment by its relative affinity

If membrane segment
_{a}^{[}^{m}^{]} = _{b}^{[}^{m}^{]} ≡ 0 which implies Δ^{[}^{m}^{]} = 0 as well.

Because of the sign convention (^{[}^{m}^{]} depends on the relative size of the interactions between

We will now extend the lattice binary mixture as described in Section 3 to adhering membranes with two membrane segments as in

Consider an adhering membrane partitioned into two segments,
^{[1]} and Ω^{[2]}. Sublattice Ω^{[1]} for segment

lattice sites whereas sublattice Ω^{[2]} for segment

such sites. As before, the symbol ^{[1]} of segment

The configurations of _{i}_{i}_{a}^{[}^{m}^{]} and _{b}^{[}^{m}^{]} for the

with the relative affinity Δ^{[}^{m}^{]} as in (_{U}^{[}^{m}^{]} in (

where the interaction energy ℰ_{int}^{[}^{m}^{]} describes the molecular interactions between the

which is identical with the expression (^{[}^{m}^{]} corresponding to segment

The configurational energy of an adhering membrane consisting of two membrane segments is then given by

where the additional energy term ℰ^{db}{^{[1]}{^{[2]}{^{db}{

Therefore, in this large membrane limit, we are left with two membrane segments,

The configurational energy (_{αβ}_{aa}_{bb}_{c}

As long as the relative affinities Δ^{[}^{m}^{]} of the two membrane segments are different, ^{[2]} ≠= Δ^{[1]}, the two critical values _{αβ}^{[2]} and _{αβ}^{[1]} are different as well and the two segments undergo two distinct phase transitions. Therefore, the adhesion-induced partitioning into two membrane segments leads to two distinct phase transitions for Δ^{[2]} ≠= Δ^{[1]}.

In order to make theoretical predictions that are accessible to experiment, we will now describe the phase behavior of the two membrane segments in terms of their mole fractions. Since the two membrane segments experience different environments and, thus, different relative affinities, they will, in general, differ in their compositions. Therefore, we have to distinguish the mole fraction _{a}^{[1]} in segment
_{a}^{[2]} in segment
_{a}^{[1]} and _{a}^{[2]}, we need two independent relations between these two variables. As shown in the following section, one relation is obtained from the partitioning of the total number of

One relation between the two mole fractions _{a}^{[1]} and _{a}^{[2]} is provided by the partitioning of the molecules between the two membrane segments. Because the total number |Ω| = |Ω^{[1]}| + |Ω^{[2]}| of molecules is fixed within the adhering membrane, the mole fractions _{a}^{[1]} and _{a}^{[2]} satisfy the relation

where _{a}^{[1]} and ^{[2]} = 1 − ^{[1]}, the relation (

Note that this partitioning relation depends only on two parameters, the area fraction ^{[1]} of segment
_{a}

Comparison of the configurational energy (^{[}^{m}^{]} for segment
^{[}^{m}^{]} = _{a}^{[}^{m}^{]}),

As explained in Section 3.4, thermodynamic stability implies that the function _{a}^{[}^{m}^{]}) is monotonically increasing for 0 _{a}^{[}^{m}^{]}_{a}_{,}_{β}

and continues to increase monotonically for _{a}_{,}_{α}_{a}^{[}^{m}^{]}_{a}_{,}_{β}_{a}_{,}_{α}

Since the two membrane segments can exchange

or

where we introduced the

between the two membrane segments with Δ^{1,2} = −Δ^{2,1}. Thus, chemical equilibrium as described by (_{a}^{[1]} and _{a}^{[2]}. This relation becomes particularly useful if one of the two segments undergoes phase separation because we can then replace one of the _{αβ}

which attains the value

and increases monotonically with

It is not difficult to see that any two mole fractions _{a}^{[1]} and _{a}^{[2]} that satify this equation and, thus, represent a solution of it do not depend on the shift _{αβ}^{2,1}. Indeed, we can add any constant to the functions Δ

If segment
_{a}^{[1]} must have a value within the range

and Δ_{a}^{[1]}) = 0 as in (

We now have to distinguish two cases corresponding to zero and nonzero affinity contrast Δ^{2,1}. If Δ^{2,1} = 0, the molecules experience the same relative affinities in both membrane segments and thus have no preference for either segment. We then conclude that the whole membrane undergoes phase separation and that _{a}^{[2]} = _{a}^{[1]} = _{a}

On the other hand, if Δ^{2,1} ≠ 0, the chemical equilibrium relation (_{a}^{[2]}) ≠ 0, _{a}^{[2]} = _{a}_{,*}^{[2]} satisfies the implicit equation

and stays constant as long as the mole fraction _{a}^{[1]} lies within the coexistence region (

Because Δ_{β}_{α}_{a}_{,*}^{[2]} of the uniform spectator phase in segment
^{2,1} and satisfies

as well as

Furthermore, this mole fraction exhibits the limiting behavior

as well as

Therefore, as one increases the affinity contrast Δ^{2,1} from large negative to small negative values, the mole fraction _{a}_{,*}^{[2]} of the spectator phase in segment
^{2,1}, from _{a}_{,*}^{[2]} = 1 to _{a}_{,}_{α}^{2,1} = 0, this mole fraction jumps from the value _{a}_{,}_{α}_{a}_{,}_{β}^{2,1} from small positive to large positive values, the mole fraction _{a}_{,*}^{[2]} decreases monotonically from the value _{a}_{,}_{β}_{a}_{,*}^{[2]} = 0.

Now, consider the situation, in which segment
_{a}_{,*}^{[2]} of segment

and that Δ_{a}^{[2]}) = 0 as in (_{a}^{[1]} in segment

which implies

where _{a}_{,*}^{[1]} denotes the mole fraction of the uniform spectator phase in segment

Using again the general properties of the function Δ_{a}^{[1]} = _{a}_{,*}^{[1]}^{2,1}. More precisely, as one increases Δ^{2,1} from large to small negative values, the mole fraction _{a}_{,*}^{[1]} increases from _{a}_{,*}^{[1]} = 0 to _{a}_{,}_{β}^{2,1} = 0 from _{a}_{,}_{β}_{a}_{,}_{α}_{a}_{,}_{α}_{a}_{,*}^{[1]} = 1 as the affinity contrast Δ^{2,1} is increased from small to large positive values. Thus, the mole fraction _{a}_{,*}^{[1]} satisfies

and

Furthermore, this mole fraction exhibits the limiting behavior

as well as

From an experimental point of view, it is most useful to describe the phase behavior in terms of overall composition and temperature. The corresponding phase diagrams can be derived by combining the partitioning relation (_{a}^{[1]} and (ii) the equalities (_{a}^{[2]} = _{a}_{,*}^{[2]} of the uniform spectator phase into the partitioning relation (_{a}_{a}_{a}^{[2]} and the equalities (_{a}^{[1]} = _{a}^{[1],*} into the partitioning relation (

The two coexistence regions for the two membrane segments are separated by an intermediate one-phase region as long as the affinity contrast Δ^{2,1} does not vanish, ^{[}^{m}^{]} = _{a}^{[}^{m}^{]} − _{b}^{[}^{m}^{]}. Thus, for Δ^{2,1} ≠ 0 or relative affinities Δ^{[2]} ≠ Δ^{[1]}, the adhesion-induced partitioning into two segments leads to two distinct two-phase coexistence regions within the (_{a}

The phase diagrams of the adhering membranes in the (_{a}^{[1]} of segment
_{a}^{2,1}. The function Δ_{aa}_{ab}_{bb}_{B}_{ab}_{aa}_{bb}

Therefore, for the lattice model considered here, the phase diagrams of the adhering membranes depend only on four parameters: (i) overall mole fraction _{a}^{[1]}, a purely geometric parameter, and (iv) affinity contrat Δ^{2,1}.

The four-dimensional parameter space is most easily explored via two-dimensional slices. In the following, we will display two-dimensional phase diagrams that depend on the overall mole fraction _{a}_{c}_{c}^{[1]} and of the affinity contrast Δ^{2,1}, see _{a}^{[1]} = 0 and ^{[1]} = 1 of the area fraction as well as for vanishing affinity contrast Δ^{2,1} = 0. In the following, we will use the term “uni-env membrane” as an abbreviation for “membrane in a uniform environment”.

The phase diagrams in _{a}_{,*}^{[1]} and _{a}_{,*}^{[2]} as given by (

In all phase diagrams displayed in

In _{a}_{c}^{2,1} = Δ^{[2]} − Δ^{[1]} keeping the area fraction ^{[1]} of segment
^{[}^{m}^{]} are related to the molecular interaction potentials _{a}^{[}^{m}^{]} and _{b}^{[}^{m}^{]} via Δ^{[}^{m}^{]} = _{a}^{[}^{m}^{]} − _{b}^{[}^{m}^{]} which implies that

Because we use the sign convention that attractive interaction potentials are negative, see (_{a}_{a}

For ^{2,1} as shown in ^{2,1} as shown in

The phase diagrams in ^{[1]} ≤ 1. In ^{[1]} for large, positive values of the affinity contrast Δ^{2,1}. For ^{[1]} = 0, the adhering membrane consists of segment
^{[1]}, a narrow blue region appears at small values of _{a}^{[1]} and, thus, the relative size of segment
^{[1]} = 1, the adhering membrane consists only of segment
^{[1]} for fixed affinity contrast Δ^{2,1}, the phase diagram in the (_{a}_{c}^{[1]} including the limiting values ^{[1]} = 0 and ^{[1]} = 1. In the latter cases, we recover the coexistence region for the uni-env membrane as depicted by the broken line in

All phase diagrams shown in ^{2,1}, by two coexistence regions, a blue one for segment

In _{a}_{c}^{2,1} = Δ^{[2]} − Δ^{[1]} keeping the area fraction ^{[1]} of segment
^{[}^{m}^{]} are expressed in terms of the molecular interaction energies _{a}^{[}^{m}^{]} and _{b}^{[}^{m}^{]}, we now find that

Because we use the sign convention that attractive interaction energies are negative, see (_{a}_{a}

The phase diagrams for negative values of the affinity contrast are intimately related to those for positive values. This relation can be understood as follows. Instead of changing the sign of the affinity contrast Δ^{2,1}, we could also interchange the names of the _{a}_{b}_{a}_{b}

The variation of the area fraction ^{[1]} for fixed affinity contrast ^{2,1} leads to a smooth evolution of the coexistence regions in the (_{a}_{c}^{2,1} from small positive to small negative values, or ^{[1]}.

As an example, consider the phase diagram for ^{2,1} and ^{[1]} = 0.7 as shown in _{a}_{a}_{a}^{2,1} = 0, this one-phase region has disappeared and the red and blue regions have merged into the single coexistence region for the uni-env membrane, see broken lines in ^{2,1}, the blue and the red coexistence regions swap their relative positions: the relatively broad blue coexistence region is now located at 0 ≲ _{a}_{a}

The abrupt changes of the (_{a}_{c}^{2,1} = 0. Indeed, as long as we do not cross Δ^{2,1} = 0, a continuous change of the affinity contrastΔ^{2,1} leads to a smooth variation of the two coexistence regions within the (_{a}_{c}^{2,1} as obtained from the mean-field approximation to the lattice model defined in Section 5.2.

In this paper, we first emphasized that the adhesion of membranes often leads to two membrane segments, denoted by

Our theoretical analysis was based on the configurational energies ℰ^{[1]} and ℰ^{[2]} for the two membrane segments as described by the expression (^{[1]} and ℰ^{[2]}, the interactions of the ^{[1]} and Δ^{[2]} as defined in (_{a}_{b}^{[1]} ≠ Δ^{[2]} as follows from the relations in (

In order to obtain theoretical predictions that are accessible to experiments, we then considered the mole fractions _{a}^{[1]} and _{a}^{[2]} in the two membrane segments and showed how these mole fractions determine the phase diagrams as a function of the overall mole fraction _{a}_{a}^{[1]}, and affinity contrast Δ^{2,1} = Δ^{[2]} − Δ^{[1]} as defined in (

For the generic case of nonzero affinity contrast, the phase diagrams for the adhering membranes contain two distinct coexistence regions in the (_{a}^{2,1} across the hyperplane defined by Δ^{2,1} = 0. The latter behavior is illustrated by the phase diagrams in

All phase diagrams shown in _{a}

The theory described here can be extended and generalized in several ways. First, it is possible to study the dynamics of the phase separation processes, which proceed via the formation and coarsening of intramembrane domains within the two membrane segments, by simulations of the lattice model. We have already performed preliminary Monte Carlo simulations that support the phase diagrams described in this paper. Second, one can apply the theoretical approach used here for the lattice binary mixture to any two-component membrane. One exampe is provided by binary cholesterol/DPPC mixtures, for which the phase diagram in

For two-component membranes exposed to two different environments as considered here, the phase diagrams depend on four parameters, three of which are easy to determine experimentally. Indeed, the mole fraction _{a}^{[1]} can be controlled by the design of the adhesion system. The remaining parameter provided by the affinity contrast Δ^{2,1} could also be determined experimentally. For adhering vesicles, for instance, one can measure the adhesion energy of the bound membrane segment for different membrane compositions, from which the relative affinities of the bound membrane segment and, thus, the affinity contrast can be deduced. Alternatively, one may also obtain these relative affinities from molecular dynamics simulations of atomistically resolved membranes.

In this appendix, we consider the spherical cap shape of an adhering vesicle in the strong adhesion regime as shown in

First, the area fractions
_{eff} via

and

Likewise, the reduced volume

Inverting this latter relation numerically, one obtains

as shown in ^{[1]} assumes the form

for 0 ≤ _{eff} = 0 and area fraction
_{eff} = ^{[1]} = 1. The relation (_{eff}.

For constant area
^{[1]}.

This study was supported by the Deutsche Forschungsgemeinschaft (DFG) via IGRTG 1524 on “Self-Assembled Soft Matter Nano-Structures at Interfaces”.

Generic phase diagram for mixed membranes consisting of cholesterol and a single phospholipid. The variable _{a}_{a}_{a}_{,}_{β}_{a}_{a}_{,}_{α}_{c}_{t}

(_{eff}; and (

(

Exact phase diagrams of the lattice model for adhering membranes as a function of mole fraction _{a}_{c}^{2,1} = Δ^{[2]} − Δ^{[1]} and fixed area fraction ^{[1]} = 0.7 of membrane segment
^{2,1} imply that the _{a}_{a}^{2,1} = 0. The broken line in (

Exact phase diagrams of the lattice model for adhering membranes as a function of mole fraction _{a}_{c}^{[1]} = 0.1^{2,1} is kept fixed at a large positive value as in ^{[1]} from ^{[1]} = 0 up to ^{[1]} = 1, the coexistence regions smoothly evolve from a single red region for segment

Exact phase diagrams of the lattice model for adhering membranes as a function of area fraction _{a}_{c}^{2,1} and fixed area fraction ^{[1]} = 0.7. Negative values of Δ^{2,1} imply that the

Mean-field phase diagrams of the lattice model for adhering membranes as a function of mole fraction _{a}_{c}^{2,1} ≡ ^{2,1}_{B}_{c}^{2,1} = −5, (^{2,1} = −0.5, (^{2,1} = −0.05, and (d) ^{2,1} = 0. Inspection of these subfigures shows that the blue and red coexistence regions for the two membrane segments change smoothly as the affinity contrast is increased from large negative to small negative values. Furthermore, in the limit of vanishing affinity contrast, the two coexistence regions merge into the single region for the uni-env membrane as described by the broken black lines in (

Spherical cap geometry: (_{eff} and (