To proceed, let us first consider a large membrane segment with total area A, which is disretized into a square lattice Ω with lattice sites i and lattice constant A. The total number of lattice sites will be denoted by

Since the discretization lattice is characterized by a single lattice constant A, both the a- and the b-molecules are described here by the same molecular area A.

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 X_a of the a-molecules is now given by the expectation value of the occupation number n_i, i.e.,

The total number N_a of a-molecules is given by N_a = ∑_in_i and the total number N_b of b-molecules by

Thus, for the lattice binary mixture, the total number of a- and b-molecules is fixed and given by N_a + N_b = |Ω|. This constraint on the two molecular numbers implies that the lattice binary mixture describes a semi-grand canonical ensemble, see further below.

If two neighboring sites i and j of the lattice are both occupied by a molecules, these molecules have the interaction energy U_aa. Likewise, two neighboring b molecules interact via the interaction energy U_bb and two neighboring sites occupied by one a- and one b-molecule contribute the interaction energy U_ab. For the unbound membrane segment, the configuration-dependent interaction energy is then given by

and the total configurational energy ℰ{n} has the form

where we have introduced two chemical potentials μ_a and μ_b for the a- and b-molecules. Since the molecular numbers N_a and N_b satisfy the constraint (7), this configurational energy is equivalent to

with the relative chemical potential

The term μ_b|Ω| on the right hand side of (10) can be omitted because it does not depend on the occupation numbers {n} and, thus, cancels out from all expectation values calculated with the statistical weight exp(−ℰ{n}/k_BT).

The form (10) of the configurational energy shows explicitly that the lattice binary mixture represents a semi-grand canonical ensemble [32,33], which depends on the total number |Ω|A of membrane molecules and on the relative chemical potential Δμ. This ensemble is appropriate here because membranes have a fixed surface area and we use the simplifying assumption that both molecular species have the same molecular area. Therefore, when the system exchanges a- and b-molecules with the corresponding chemical reservoirs, we have to remove b-molecules when we want to insert a-molecules and vice versa.

The lattice binary mixture just described is equivalent to the two-dimensional Ising model if one expresses the molecular configurations {n} in terms of the spin variables σ_i ≡ 2n_i − 1. This equivalent Ising model depends only on two parameters, the dimensionless temperature

and the dimensionless ordering field

The Ising system undergoes phase separation for H̄ = 0 and 0 ≤ T̄ < T̄_c, where T̄_c denotes the critical temperature. The exact value of the critical temperature is given by [34]

for the Ising model on a square lattice. As we approach the line H̄ = 0 from positive and negative values of H̄, the order paramter 〈σ_i〉 attains the values 〈σ_i〉 = + ϒ (T̄) and 〈σ_i〉 = −ϒ (T̄), respectively, with the spontaneous order parameter [34]

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

As we approach the transition value Δμ = μ_αβ of the relative chemical potential from below or from above, the mole fraction X_a = 〈n_i〉 approaches the values

respectively, with the function ϒ (T̄) as given by (15). These two mole fractions define the binodal lines X_a = X_a,β(T̄) and X_a = X_a,α in the (X_a, T̄) phase diagram of the lattice binary mixture. This phase diagram is symmetric with respect to the transformation X a → X a ′ = 1 2 - X a, which reflects the particle-hole symmetry of the lattice binary mixture.

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 Δμ does not represent an experimental control parameter. Instead of Δμ, one typically controls the mole fraction X_a. If we consider X_a as an independent thermodynamic variable, the relative chemical potential becomes a dependent variable that will be described by the functional relationship

The change from Δμ to X_a corresponds to a Legendre transformation from the semi-grand canonical ensemble to the canonical ensemble. Even though the form of the function G is not known explicitly (because we have no exact solution for the 2-dimensional Ising model in a finite ordering field, H̄ ≠= 0), thermodynamics implies some useful properties of G(X_a).

First, it follows from thermodynamic stability that the chemical potential μ_a of the a-molecules must be a non-decreasing function of X_a, see, e.g., [35]. Likewise, the chemical potential μ_b of the b-molecules must be a non-decreasing function of X_b = 1 − X_a, which implies that it must be a non-increasing function of X_a. It then follows that the relative chemical potential Δμ = μ_a − μ_b as a function of X_a must satisfy

More precisely, the function Δμ = G(X_a) has a strictly positive derivative,

and stays constant with

It then follows that G(X) is a monotonically increasing function of X for 0 < X < X_a,β( T̄), stays constant for X_a,β(T̄) ≤ X ≤ X_a,β(T̄), and continues to increase monotonically for X_a,β(T̄) < X < 1, where the two binodals are explicitly given by (17).

The adhesion of vesicles has been studied for a long time, see, e.g., [36,37] for one-component membranes and [1,38] for multi-component membranes. Here, we will focus on the strong adhesion regime, in which the vesicle attains the shape of a spherical cap. The same shape has also been observed for the strong adhesion of red blood cells [39,40].

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 q^[1] of the segments S [ 1 ]. Membranes supported by uniform substrate surfaces then correspond to the limiting cases q^[1] = 0 and q^[1] = 1.

The different adhesion geometries depicted in Figures 2 and 3 all lead to a partitioning of the membranes into two segments S [ m ] with m = 1, 2. Because these segments are exposed to different environments, the two molecular components a and b experience different interactions potentials, U_a^[m] and U_b^[m]. We use the sign convention that

acting on the a- and b-molecules, respectively, within the segment S [ m ].

We will now characterize each membrane segment by its relative affinity

If membrane segment S [ m ] represents an unbound segment not in contact with any adhesive surface, see segment S [ 1 ] in Figure 2 and Figure 3a, we put U_a^[m] = U_b^[m] ≡ 0 which implies ΔU^[m] = 0 as well.

Because of the sign convention (32), the sign of the relative affinity ΔU^[m] depends on the relative size of the interactions between a- and b-molecules in segment S [ m ] and the environment of this segment. Thus, we have

i.e., if the a- or the b-molecules within segment S [ m ] are more strongly attracted by its environment.

Consider an adhering membrane partitioned into two segments, S [ 1 ] and S [ 2 ]. The discretization of such a membrane leads to two sublattices, Ω^[1] and Ω^[2]. Sublattice Ω^[1] for segment S [ 1 ] consists of

lattice sites whereas sublattice Ω^[2] for segment S [ 2 ] has

such sites. As before, the symbol A denotes the area per lattice site and corresponds to the molecular area of the a- and b-molecules. The area fraction q^[1] of segment S [ 1 ] is now equal to

The configurations of a- and b-molecules within an adhering membrane are again described in terms of the occupation numbers {n_i}, where n_i = 1 represents an a-molecule as before. The configurational energy of this membrane now contains additional terms arising from the interaction potentials U_a^[m] and U_b^[m] for the a- and b-molecules with the corresponding environments. More precisely, membrane segment S [ m ] contributes the additional energy term

with the relative affinity ΔU^[m] as in (33) where the last n-independent term on the right hand side can again be omitted because it does not affect the statistical properties of the system. Adding the U-dependent energy term ℰ_U^[m] in (38) to the standard form (10) for the configurational energy of the lattice binary mixture, the configurational energy of segment S [ m ] becomes

where the interaction energy ℰ_int^[m] describes the molecular interactions between the a- and b-molecules within segment S [ m ]. This interaction energy now has the form

which is identical with the expression (8) apart from the summation that now includes only nearest neighbors 〈ij〉 within the sublattice Ω^[m] corresponding to segment S [ m ].

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

where the additional energy term ℰ^db{n} arises from the domain boundaries between the two membrane segments. The latter term can be ignored compared to the first two terms in (41) when we consider the limit of large membrane segments. Indeed, in this limit, the free energies obtained from the first two terms ℰ^[1]{n} and ℰ^[2]{n} increase as the segment areas A [ 1 ] ~ ∣ Ω [ 1 ] ∣ and A [ 2 ] ~ ∣ Ω [ 2 ] ∣ whereas the free energy arising from the domain boundary term ℰ^db{n} increases only as the length of the domain boundary, which is of the order of min ( ∣ Ω [ 1 ] ∣ , ∣ Ω [ 2 ] ∣ ).

Therefore, in this large membrane limit, we are left with two membrane segments, S [ 1 ] and S [ 2 ], each of which is governed by a configurational energy of the form (39). In the semi-grand canonical ensemble of the lattice binary mixture, these large segments are only coupled via the relative chemical potential Δμ, i.e., via the particle reservoirs for a- and b-molecules.

The configurational energy (10) for the standard lattice binary mixture leads to a phase transition at the relative chemical potential Δμ = μ_αβ = 2(U_aa − U_bb) for 0 ≤ T̄ < T̄_c as described by (16). Comparison of the configurational energy (39) for the membrane segment S [ m ] with the configurational energy (10) for the standard model then shows that this segment undergoes a phase transition at the relative chemical potential

As long as the relative affinities ΔU^[m] of the two membrane segments are different, i.e., as long as ΔU^[2] ≠= ΔU^[1], the two critical values μ_αβ^[2] and μ_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 ΔU^[2] ≠= ΔU^[1].

One relation between the two mole fractions X_a^[1] and X_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 X_a^[1] and X_a^[2] satisfy the relation

where X_a is the overall mole fraction of the a-molecules as before. Using the expression (37) for the area fraction q^[1] and q^[2] = 1 − q^[1], the relation (43) becomes

Note that this partitioning relation depends only on two parameters, the area fraction q^[1] of segment S [ 1 ] and the overall mole fraction X_a.

Comparison of the configurational energy (39) of membrane segment S [ m ] with the configurational energy (10) of the standard lattice binary mixture shows that the relative chemical potential Δμ of the standard model is replaced byΔμ−ΔU^[m] for segment S [ m ]. It then follows from (19) thatΔμ−ΔU^[m] = G(X_a^[m]), i.e., membrane segment S [ m ] is governed by the relative chemical potential

As explained in Section 3.4, thermodynamic stability implies that the function G(X_a^[m]) is monotonically increasing for 0 < X_a^[m]< X_a,β(T̄), attains the constant value

and continues to increase monotonically for X_a,α(T̄) < X_a^[m]< 1, where the two binodals X_a,β( T̄) and X_a,α( T̄) are explicitly given by (17).

Since the two membrane segments can exchange a- and b-molecules by lateral diffusion, they will reach a state of chemical equilibrium with

or

where we introduced the affinity contrast

between the two membrane segments with ΔU^1,2 = −ΔU^2,1. Thus, chemical equilibrium as described by (48) provides a second relation between the two mole fractions X_a^[1] and X_a^[2]. This relation becomes particularly useful if one of the two segments undergoes phase separation because we can then replace one of the G-terms by the constant value μ_αβ. Furthermore, it is convenient to define the shifted function

which attains the value

and increases monotonically with X outside of this interval. In terms of the shifted function ΔG, the chemical equilibrium relation (48) becomes

It is not difficult to see that any two mole fractions X_a^[1] and X_a^[2] that satify this equation and, thus, represent a solution of it do not depend on the shift μ_αβ but only on the affinity contrast ΔU^2,1. Indeed, we can add any constant to the functions ΔG(X) or G(X) without changing the solution to (52) or (48).

If segment S [ 1 ] undergoes phase separation, the mole fraction X_a^[1] must have a value within the range

and ΔG(X_a^[1]) = 0 as in (51). The chemical equilibrium relation (52) then simplifies and becomes

We now have to distinguish two cases corresponding to zero and nonzero affinity contrast ΔU^2,1. If ΔU^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 X_a^[2] = X_a^[1] = X_a, where the second equality follows from the partitioning relation (44).

On the other hand, if ΔU^2,1 ≠ 0, the chemical equilibrium relation (54) implies that ΔG(X_a^[2]) ≠ 0, i.e., that segment S [ 2 ] does not undergo phase separation but represents a spectator phase with uniform composition. The corresponding mole fraction X_a^[2] = X_a,*^[2] satisfies the implicit equation

and stays constant as long as the mole fraction X_a^[1] lies within the coexistence region (53) of segment S [ 1 ]. Note the subscript * that is used, here and below, to indicate the mole fractions of a spectator phase.

Because ΔG(x) increases monotonically with x both for 0 < x < X_β and for X_α< x < 1, the implicit equation (55) implies that the mole fraction X_a,*^[2] of the uniform spectator phase in segment S [ 2 ] decreases monotonically with increasing affinity contrast ΔU^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 ΔU^2,1 from large negative to small negative values, the mole fraction X_a,*^[2] of the spectator phase in segment S [ 2 ] decreases monotonically as a function of ΔU^2,1, from X_a,*^[2] = 1 to X_a,α(T̄) corresponding to the upper binodal. At ΔU^2,1 = 0, this mole fraction jumps from the value X_a,α(T̄) for the upper binodal to the value X_a,β(T̄) for the lower binodal. Finally, as one increases the affinity contrast ΔU^2,1 from small positive to large positive values, the mole fraction X_a,*^[2] decreases monotonically from the value X_a,β(T̄) to X_a,*^[2] = 0.

Now, consider the situation, in which segment S [ 2 ] of the adhering membrane undergoes phase separation. Repeating the arguments of the previous subsection, it then follows that the mole fraction X_a,*^[2] of segment S [ 2 ] satisfies

and that ΔG(X_a^[2]) = 0 as in (51). The mole fraction X_a^[1] in segment S [ 1 ] then satisfies the implicit equation

which implies

where X_a,*^[1] denotes the mole fraction of the uniform spectator phase in segment S [ 1 ].

Using again the general properties of the function ΔG(x), we find from (61) that the mole fraction X_a^[1] = X_a,*^[1]increases monotonically with increasing affinity contrast U^2,1. More precisely, as one increases ΔU^2,1 from large to small negative values, the mole fraction X_a,*^[1] increases from X_a,*^[1] = 0 to X_a,β(T̄), jumps at ΔU^2,1 = 0 from X_a,β(T̄) to X_a,α(T̄), and finally increases monotonically from X_a,α(T̄) to X_a,*^[1] = 1 as the affinity contrast ΔU^2,1 is increased from small to large positive values. Thus, the mole fraction X_a,*^[1] satisfies

and

Furthermore, this mole fraction exhibits the limiting behavior

as well as

The phase diagrams of the adhering membranes in the (X_a, T̄)-plane depend on the parameters that enter the partitioning relation (44) and the chemical equilibrium relations. The partitioning relation (44) contains only one parameter, the area fraction q^[1] of segment S [ 1 ], in addition to the overall mole fraction X_a of the a-molecules. The chemical equilibrium relations, on the other hand, involves the function ΔG(X) and the affinity contrast ΔU^2,1. The function ΔG(X) depends on the interaction parameters U_aa, U_ab, and U_bb in the configurational energy (40) via the dimensionless temperature T̄ ≡ 4k_BT=(2U_ab − U_aa − U_bb) as introduced in (12).

Therefore, for the lattice model considered here, the phase diagrams of the adhering membranes depend only on four parameters: (i) overall mole fraction X_a, (ii) dimensionless temperature T̄, (iii) area fraction q^[1], a purely geometric parameter, and (iv) affinity contrat ΔU^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 X_a and on the rescaled temperature T̄ = T̄_c = T/T_c for fixed values of the area fraction q^[1] and of the affinity contrast ΔU^2,1, see Figures 4–7 below. This choice is convenient because it allows a direct comparison with the phase diagram of the two-component membrane in a uniform environment. The latter phase diagram, which depends only on mole fraction X_a and on temperature, is recovered for the limiting values q^[1] = 0 and q^[1] = 1 of the area fraction as well as for vanishing affinity contrast ΔU^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 Figures 4–6 below are based on the exact binodals of the lattice binary model as described by (17) and on the exact values of the spectator phase mole fractions X_a,*^[1] and X_a,*^[2] as given by (63)–(66) and (56)–(59) for small and large values of the affinity contrast. Therefore, the phase diagrams in Figures 4–6 are exact as well. In contrast, the phase diagrams in Figure 7 are based on the mean-field approximation.

In all phase diagrams displayed in Figures 4–7 below, the two coexistence regions for the membrane segments are distinguished by their color: the coexistence regions of segment S [ 1 ] are blue whereas the coexistence regions of segment S [ 2 ] are red. Furthermore, the remaining white regions in the phase diagrams represent one-phase regions, in which the membranes attain uniform compositions without domains.

In Figure 4, we display the phase diagrams of adhering membranes in the (X_a, T/T_c)-plane as obtained from the lattice binary mixture for small and large positive values of the affinity contrast ΔU^2,1 = ΔU^[2] − ΔU^[1] keeping the area fraction q^[1] of segment S [ 1 ] constant. The relative affinities ΔU^[m] are related to the molecular interaction potentials U_a^[m] and U_b^[m] via ΔU^[m] = U_a^[m] − U_b^[m] which implies that

Because we use the sign convention that attractive interaction potentials are negative, see (32), the second inequality in (67) applies to systems, in which the b-molecules prefer to stay in segment S [ 2 ] while the a-molecules prefer to stay in segment S [ 1 ]. Therefore, as one increases the mole fraction X_a starting from X_a = 0, the a-molecules are first enriched in segment S [ 1 ], which then undergoes phase separation leading to the blue coexistence regions in Figure 4.

For large positive values of the affinity contrast ΔU^2,1 as shown in Figure 4a, the two coexistence regions differ only in their width but have the same shape. For small positive values of U^2,1 as shown in Figure 4b, the two coexistence regions provide a decomposition of the coexistence region of the uni-env membrane, the latter being depicted by the broken line in Figure 4a.

The phase diagrams in Figure 4 correspond to fixed area fraction q [ 1 ] = A [ 1 ] / A = 0.7. In general, this area fraction can vary in the range 0 ≤ q^[1] ≤ 1. In Figure 5, we show how the phase diagram evolves with increasing values of q^[1] for large, positive values of the affinity contrast ΔU^2,1. For q^[1] = 0, the adhering membrane consists of segment S [ 2 ] only and we are left with the red coexistence region only. For small nonzero values of q^[1], a narrow blue region appears at small values of X_a, see Figure 5a. As we further increase the area fraction q^[1] and, thus, the relative size of segment S [ 1 ], the blue region grows at the expense of the red region, see Figure 5b–d. Finally, as we reach the limiting value q^[1] = 1, the adhering membrane consists only of segment S [ 1 ] and we are left with the blue coexistence region only. Thus, as we change the area fraction q^[1] for fixed affinity contrast ΔU^2,1, the phase diagram in the (X_a, T/T_c)-plane evolves smoothly for all values of q^[1] including the limiting values q^[1] = 0 and q^[1] = 1. In the latter cases, we recover the coexistence region for the uni-env membrane as depicted by the broken line in Figure 5.

All phase diagrams shown in Figures 4 and 5 have the same topology: the single coexistence region for the uni-env membrane, i.e., the membrane in a uniform environment, is replaced, for nonzero affinity contrast ΔU^2,1, by two coexistence regions, a blue one for segment S [ 1 ] and a red one for segment S [ 2 ]. For T > 0, these two coexistence regions are separated by an intermediate one-phase region that lies within the coexistence region of the uni-env membrane. This generic topology of the phase diagrams has interesting consequences for experimental observations. Thus, consider a two-component membrane that is phase separated when exposed to a uniform environment. After this membrane is segmented by adhesion, the phase separation is restricted to one of the two membrane segments, i.e., phase separation can be observed either in segment S [ 1 ] or in segment S [ 2 ] but not in both segments simultaneously. Furthermore, for mole fractions and temperatures that correspond to the intermediate one-phase regions between the two coexistence regions of the membrane segments, the phase separation in the uni-env membrane is suppressed by the adhesion-induced segmentation.

In Figure 6, we display the phase diagrams of adhering membranes in the (X_a, T/T_c)-plane as obtained from the lattice model for small and large negative values of the affinity contrast ΔU^2,1 = ΔU^[2] − ΔU^[1] keeping the area fraction q^[1] of segment S [ 1 ] constant. When the relative affinities ΔU^[m] are expressed in terms of the molecular interaction energies U_a^[m] and U_b^[m], we now find that

Because we use the sign convention that attractive interaction energies are negative, see (32), the second inequality in (68) applies to systems, in which the a-molecules prefer to stay in segment S [ 2 ] while the b-molecules prefer to stay in segment S [ 1 ]. Therefore, as one increases the mole fraction X_a starting from X_a = 0, the a-molecules are first enriched in segment S [ 2 ], which then undergoes phase separation leading to the red coexistence regions in Figure 6.

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 ΔU^2,1, we could also interchange the names of the a- and the b-molecules. We would then obtain the same phase diagrams as in Figures 4 and 5 but with X_a replaced by X_b. If we now redraw these diagrams in terms of X_a = 1 − X_b, we swap the relative positions of the blue and red coexistence regions and recover the phase diagrams for negative values of the affinity contrast as shown in Figure 6.

The variation of the area fraction q^[1] for fixed affinity contrast U^2,1 leads to a smooth evolution of the coexistence regions in the (X_a, T/T_c)-plane as shown in Figure 5. In contrast, the coexistence regions undergo abrupt changes as we vary the affinity contrast ΔU^2,1 from small positive to small negative values, or vice versa, for fixed values of the area fraction q^[1].

As an example, consider the phase diagram for small negative affinity contrasts ΔU^2,1 and q^[1] = 0.7 as shown in Figure 6b. This phase diagram exhibits a relatively narrow red coexistence region for mole fractions X_a in the range 0 ≲ X_a ≲ 0.3 and a relatively broad blue coexistence region for mole fractions within 0.3≲ X_a ≲ 1, separated by a very narrow one-phase region. For vanishing affinity contrast ΔU^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 Figures 4 and 6. Finally, for small positive affinity contrasts ΔU^2,1, the blue and the red coexistence regions swap their relative positions: the relatively broad blue coexistence region is now located at 0 ≲ X_a ≲ 0.7 and the relatively narrow red coexistence region at 0.7 ≲ X_a ≲ 1. Thus, as we change the sign of the affinity contrast, the phase diagrams change in an abrupt and discontinuous manner.

The abrupt changes of the (X_a, T/T_c) phase diagrams arising from variations of the affinity contrast are limited to the vicinity of ΔU^2,1 = 0. Indeed, as long as we do not cross ΔU^2,1 = 0, a continuous change of the affinity contrastΔU^2,1 leads to a smooth variation of the two coexistence regions within the (X_a, T/T_c)-plane. This latter property is illustrated in Figure 7, which displays such a smooth variation for negative values of U^2,1 as obtained from the mean-field approximation to the lattice model defined in Section 5.2.