Simple Thermodynamic Description of the Micellar-Bilayer State Transition of Assemblies Composed of n-Octyl-β-D-glucopyranoside and 1,2-Dioleolyl-sn-glycero-3-phosphocholine Dispersed in Aqueous Media or Supported on Solid Substrates.

In the preceding paper, we investigated a mixed assembly composed of a nonionic surfactant, n-octyl-β-D-glucopyranoside (OG), and an amphoteric lipid, 1,2-dioleolyl-sn-glycero-3-phosphocholine (DOPC), formed on hydrophilized solid substrates immersed in aqueous solutions containing OG and DOPC. The experimental data could be interpreted in terms of the phase equilibrium; thus, the partition equilibrium profile of OG between the bulk solution phase and the supported assembly phase was obtained, as well as that between the bulk solution and the dispersed assembly. The partition equilibrium profiles suggested that micellar-bilayer state transitions occur both in the supported assembly and in the dispersed one in a roughly synchronized manner, even though there are significant discrepancies between them. In this paper, we propose a simple thermodynamic model for the micellar-bilayer transition of the dispersed and supported assembly of OG and DOPC, assuming that the micellar and bilayer states are also pseudo-phases distinct from each other. Using this model, we analyzed these partition equilibrium profiles and concluded that the transition in the supported assembly should mainly be attributed to the transition in the dispersed assembly, which is partly modified by the interaction energy between the supported assembly and the substrate.

Various studies on the partition equilibrium of surfactants between the bulk solution phase and the micellar or vesicular state assembly phase have been reported 12, 18 22 . Because the bulk solution phase is in equilibrium with the dispersed assembly phase, the chemical potential or activity of the surfactant in the dispersed assembly could be monitored during the micellar-vesicular state transition by quantifying the monomerically dissolved surfactant concentration in the bulk solution. These researchers pointed out that the transition occurs at specific compositions of the assemblies and that the boundary compositions of the coexisting micellar and vesicular state assemblies are different from each other.
In a previous study, we studied the formation of a molecular assembly composed of a nonionic surfactant, n-octylβ-D-glucopyranoside OG , and an amphoteric phospholipid, 1,2-dioleoyl-sn-3-glycerophosphocholine DOPC , supported on hydrophilic solid substrates immersed in aqueous solutions containing OG and DOPC 23 . We obtained the partition equilibrium profiles of OG between the bulk solution phase and the supported assembly phase, as well as that between the bulk solution and the dispersed assembly. A comparison of these two partition equilibrium profiles revealed that both the supported and dispersed assemblies exhibit micellar-bilayer transitions, and that there are significant discrepancies between them.
In this study, we assumed the phase-separation model as illustrated schematically in Fig. 1, where the micellar state and the bilayer state in the supported or dispersed assembly phases are respectively regarded as pseudo phases distinct from each other that are in equilibrium with the bulk solution phase comprising monomerically dissolved surfactant and lipid. Using this model, we analyzed the partition equilibrium profiles to describe the micellar-bilayer transitions in supported or dispersed assemblies in terms of chemical thermodynamics.
2 Formulation for the partition equilibrium profile of OG for a dispersed or supported assembly Figure 2 shows the partition equilibrium profile of OG between the bulk solution phase and the dispersed assembly phase, and that between the bulk solution and the supported assembly on a hydrophilized Ge substrate, which were obtained in a previous study 23 . In the partition equilibrium profiles, the concentration of monomerically dissolved OG in the bulk solution, C 1 OG, is plotted against the averaged mole fraction of OG in the dispersed assembly, X dis OG, or that in the supported assembly, X sup OG . The profile of C 1 OG vs. X dis OG open squares was determined using equilibrium dialysis while that of C 1 OG vs. X sup OG closed circles was determined by ATR-FTIR spectroscopy. X sup OG was directly estimated from the ATR spectrum on the Ge substrate, and Fig. 1 Phase separation model for the supported assembly phase and the dispersed assembly phase composed of a surfactant and a lipid. Each of the assembly phases can adopt either micellar or bilayer vesicular states, which are also separated into pseudo phases distinct from each other. The bulk solution phase comprises monomerically dissolved surfactant and lipid. Since these phases are in equilibrium with one another, the chemical potentials of the components are common in the entire system. C 1 OG was calculated from the ambient solution composition using the C 1 OG vs. X dis OG profile. In these experiments, we used aqueous solutions containing various concentrations of OG, whereas the DOPC concentration was maintained as 1.15 mM.
The C 1 OG vs. X dis OG profile is similar to that obtained using OG and L-α-phosphatidylcholine from egg yolk, as reported by Ueno 19 and Paternostre et al. 21 . According to them, the profile can be divided into three regions. The range of X dis OG 0.8-1 is the micellar state region, and that of X dis OG 0-0.6 is the vesicular state region. In the range of X dis OG 0.6-0.8, the micellar-vesicular state transition occurs, and the turbidity of the solution sharply increases when X dis OG is lower than 0.8, indicating the formation of vesicular particles. These interpretations are consistent with the phase-separation model, where the micellar and vesicular states are pseudo phases distinct from each other. In the transition region, three phases, namely the micellar state assembly, vesicular state assembly, and the bulk solution, coexist to restrict the C 1 OG constant 17.7 mM owing to the Gibbs phase rule. Furthermore, the micellar state assembly and the vesicular assembly should have their specific boundary compositions, as demonstrated in various studies 12, 18 22 , and X dis OG is only their average value. Therefore, X dis OG should be related to the mass ratio between the micellar and vesicular states, according to the lever rule. The partition equilibrium profile, as shown in Fig. 2, would be useful in treating experimental systems containing water, surfactants, and lipids Supporting Information I . We interpreted the partition equilibrium profile in the micellar or vesicular state region in a manner analogous to that of Holland & Rubingh 24 . At equilibrium, the chemical potential of the OG in the micellar or vesicular state assembly is equal to that of the monomerically dissolved OG, as follows:  Fig. 2 Partition equilibrium profiles of OG between the bulk solution phase and the supported or dispersed assembly phases 23 . The monomerically dissolved OG concentration, C 1 OG is plotted against the OG mole fraction in the supported assembly phase on a Ge substrate, X sup OG closed circles or that in the dispersed assembly phase, X dis OG open squares . These were obtained from ATR-FTIR spectroscopy and equilibrium dialysis, respectively. The partition equilibrium profile for the dispersed assembly phase could be divided into three regions; the micellar state region as fitted by a solid line; the vesicular state region as fitted by a dashed line; and the transition region represented as a dotted horizontal line C 1 OG 17.7 mM . These lines are described based on the phase separation model and the regular solution approximation, where phase boundaries of micellar and vesicular states are represented by open diamonds, the intersection points of the theoretical lines: X dis OG 0.803 and 0.616. As for the supported assembly phase, the profile for the bilayer state region is clearly observed and is fitted by a chained line, although it is modified from that for the vesicular state region of the dispersed assembly phase. C 1 OG at the transition region for the supported assembly phase is higher than that for the dispersed assembly phase, and is more inclined, asymptotically approaching the micellar state region.
The μ OG s with the superscripts o and a OG s represent the standard chemical potentials and activities of OG in the corresponding states, respectively. Here, we assume that the activity of the monomerically dissolved OG is represented by the molar concentration, C 1 OG, and that the activity of OG in the micellar or vesicular state assembly is the product of the activity coefficient, f OG , and the mole fraction of OG in the assembly, X dis OG. Consequently, we obtain Using the regular solution approximation 25 , the activity coefficients of OG in the micellar and vesicular state assemblies, f M OG and f V OG, are given by where β M and β V are the interaction parameters between OG and DOPC in the micellar and vesicular state assemblies, respectively. By substituting Equation 4 into Equation 3 and transforming the resultant equation, we obtain OG.

5
C M o OG and C V o OG correspond to the CMC and CVC for the micellar and vesicular state assemblies of pure OG, respectively. We calculated the regression lines for the micellar and vesicular state regions of the partition equilibrium profile for the dispersed assembly; these are represented in Fig. 2 as solid and dashed lines, respectively. Using these regressions, we could estimate the parameters for Equation 5 that are tabulated in Table 1. C M o OG was determined as 25.9 mM, which corresponds well with the CMC of pure OG 23 . C V o OG was revealed to be higher than C M o OG , ensuring that the pure-OG vesicular state assembly remained imaginary. As for the interaction parameters, we could compare the obtained β M and β V with those for various sets of binary surfactant mixtures, which Holland tabulated in his review 26 . Our β values are comparable to those for the set composed of nonionic and ionic surfactants; the absolutes are smaller than those for anionic-cationic sets 10-20 and larger than those for nonionic-nonionic sets 0-1 . The partition equilibrium profile for the supported assembly phase, C 1 OG vs. X sup OG , is roughly similar to that for the dispersed assembly phase and is also divided into the micellar state region, the bilayer state region, and the transition region, although significant modifications are observed. The transition region for the supported assembly has a higher C 1 OG than that for the dispersed assembly and has a steeper slope, gradually approaching the micellar state region; thus, the boundary of the micellar state region is more ambiguous. The profile for the bilayer state region is also characterized by fitting the theoretical curve similar to Equation 5 : The estimated parameters are presented in Table 1.
suggesting that the imaginary pure-OG supported bilayer is more unstable than the pure-OG vesicle. On the other hand, β S is more negative than β V , suggesting that the OG/DOPC interaction in the supported bilayer is more favorable than that in the vesicles.

Micellar-vesicular state transition described by thermodynamics
Using the partition equilibrium profile, we could derive the molar Gibbs free energies of the micellar and vesicular state phases as a function of their composition to interpret the micellar-vesicular state transition, as shown below. Such a diagram would enable us to conduct further systematic studies on their dependence on temperature, chemical structure of surfactant or lipid, and addition of the third component membrane protein or minor lipid , etc., to provide a more comprehensive understanding of the micellar-vesicular state transition.
We define the averaged molar Gibbs energies of the micellar and vesicular state phases, G M and G V as follows:  * Concentration of the monomerically dissolved OG, C 1 OG being in equilibrium with the corresponding assembly of pure OG. Interaction parameter between OG and DOPC in the corresponding assembly, based on the regular solution approximation.
Using Equations 3 and 4 , and similar equations, μ OG and μ DOPC in the micellar state are given as For those in the vesicular phase, similar equations are given as In  Figure 3 shows G M and G V as functions of X dis OG to account for the micellar-vesicular state transition. In the region of X dis OG 0.803-1, where the micellar state is observed, G M is lower than G V . In the vesicular state region of X dis OG 0-0.616, on the other hand, G V is lower than G M . These properties are mainly due to the energy difference between μ M o OG and μ V o OG, and that between μ M o DOPC and μ V o DOPC . In other words, the pure-OG micellar state is more stable than the pure-OG vesicular state, and the pure-DOPC vesicular state is more stable than the pure-DOPC micellar state. μ V o OG and μ M o DOPC cannot be estimated directly because the pure-OG vesicular state and the pure-DOPC micellar state are imaginary; however, they are related to molecular geometry of OG and DOPC, such as the hydrophilic-lipophilic balance.

Influence of the solid substrate on the micellarbilayer state transition in terms of thermodynamic quantities
As shown in Fig. 2, the partition equilibrium profile of OG for the supported assembly has characteristics similar to those of the dispersed assembly, although there are significant discrepancies. Because the supported assembly phase, the dispersed assembly phase, and the bulk solution phase are in equilibrium with one another, the chemical μ and n are the chemical potential and the number of moles, respectively. The subscripts refer to the components, and the superscripts refer to the phases blk, the bulk solution phase; dis, the dispersed assembly phase; and sup, the supported assembly phase . Under these restrictions, the partition equilibrium profiles between the bulk solution and the assemblies should be the same, regardless of whether the assembly is dispersed in the ambient solution or supported on the substrate. Therefore, the difference between the partition equilibrium profiles shown in Fig. 2 would result from the interaction between the assembly and the substrate, which is not considered in the Gibbs-Duhem equation. Now, we discuss how the interaction energy modifies the partition equilibrium profile for the supported assembly.
In a manner analogous to the Gibbs adsorption isotherm, we define the internal energy of the supported assembly phase, E sup , as where T is the temperature, and p is the pressure. S sup and V sup are the entropy and volume of the supported assembly, respectively. A is the area of the assembly/substrate interface, and ε sub is the interaction energy between the assembly and the substrate per unit area, corresponding to the surface tension as excess energy in the Gibbs adsorption isotherm. Assuming that the supported assembly is two-dimensionally uniform along the substrate surface, we integrate the right side of Equation 13 with A to obtain Therefore, the modification of the partition equilibrium profile due to the supporting substrate, as shown in Fig. 2, indicates that ε sub exhibits some dependence on the composition of the assembly. We tried to estimate ε sub between the bilayer state of the supported assembly and the Ge substrate as a function of the mole fraction of OG in the supported assembly phase, X sup OG , using Equation 15 . μ OG as a function of X sup OG is obtained from the theoretical curve for the bilayer state of the supported assembly in Fig. 2 and are expressed by Equations 8 and 9 , respectively, which restricts the relation between μ OG and μ DOPC , as shown in Fig. 4. Consequently, we obtained μ OG and μ DOPC as functions of X sup OG as shown in Fig. 5 a . At the same time, we estimated Γ OG and Γ DOPC as functions of X sup OG as shown in Fig. 5 b , using the total weight of the supported assembly per unit area, w sup , determined in a previous Fig. 4 Relation between chemical potentials of OG and DOPC, μ OG and μ DOPC in our entire system. A solid line corresponds to μ OG vs. μ DOPC calculated from the theoretical curves of the partition equilibrium profile for the dispersed assembly phase as shown in Fig. 2, using Equations 8 and 9 . At the point where μ OG 0.942 kJ mol 1 and μ DOPC 4.35 kJ mol 1 , the curve sharply bends, reflecting the micellar-vesicular state transition the left hand region of the point corresponds to the vesicular state region, and the right to the micellar state region . A dotted line corresponds to the imaginary curve assuming that the dispersed assembly were kept to be in the vesicular state regardless of the composition, and did not exhibit any transitions. study 23 , by complementing the missing data in the range of X sup OG 0-0.45 by the quadratic regression curve Supporting Information II . Using the data in Figs. 5 a and 5 b , we numerically integrated Equation 15 to calculate ε sub as a function of X sup OG as shown in Fig. 5 c . We defined ε sub as 0 at the point where X sup OG 0. In the range of X sup OG 0-0.632, ε sub decreases monotonically as X sup OG increases, indicating that the larger the amount of OG included was, the more the bilayer state of the supported assembly was stabilized by the Ge substrate. At the point of X sup OG 0.633, where the micellar-vesicular state transition occurs in the dispersed assembly in the ambient solution and μ DOPC vs. μ OG profile in Fig. 4 and μ DOPC vs. X sup OG in Fig. 5 a bend, ε sub sharply increases with X sup OG , indicating that the bilayer state is no longer stable. Such characteristics of ε sub X sup OG would result in a micellar-bilayer state transition in the supported as-sembly.
As shown in Fig. 5 c , the larger X sup OG was, the higher the affinity of the supported bilayer toward the Ge substrate was, as reflected by the negative slope of ε sub X sup OG in the bilayer state region. The variation in ε sub in the bilayer region is on the order of 10 9 kJ cm 2 , which corresponds to a difference of only a few kJ mol 1 between OG and DOPC regarding their contributions to ε sub . ε sub is considered to involve the bonding energy between the substrate and the OG or DOPC. Many researchers have demonstrated that supported lipid bilayers are formed on the substrate surface via a water layer with a thickness of several nanometers 27 31 . Therefore, the bonding OG or DOPC to the substrate surface was ascribed to a hydrogen bonding network, each of which was considered to be several tens of kJ/mol. Because the bonding energy also depends on the Fig. 5 a Chemical potentials of OG and DOPC, μ OG and μ DOPC as functions of the OG mole fraction in the supported assembly, X sup OG . μ OG solid line was calculated from the theoretical curve for the bilayer state of the supported assembly chained line in Fig. 2 , and μ DOPC dashed line was calculated from μ OG using Fig. 4. b Amounts of OG and DOPC in the supported assembly per unit area, Γ OG solid line and Γ DOPC dashed line as functions of X sup OG . These were calculated from the total weight of the supported assembly per unit area, w sup determined in a previous study 23 , with complementing the missing data in the range of X sup OG 0-0.45 by the quadratic regression curve Supporting Information II . c The interaction energy between the bilayer state of the supported assembly and a Ge substrate, ε sub solid line as a function of X sup OG . This was estimated by numerical integration of Equation 15 using data in a and b . Dotted lines in a and c represent the imaginary μ DOPC and ε sub , assuming that dispersed assembly were kept to be in the vesicular state regardless of the composition, as represented by a dotted line in Fig. 4. chemical properties of the substrate, the partition equilibrium profile in the bilayer region should vary according to the type of substrate, even though it might have a minor effect. In addition, ε sub would involve the energy required to deform the assembly structure to be fit for the flat surface. Sun & Ueno 4 and Ueno et al. 32 reported that the small unilamellar vesicles of phospholipids containing some amount of nonionic or amphoteric surfactants tended to grow by fusing with each other, suggesting that these surfactants promote the flexibility of the lipid bilayer structure. This effect would also contribute to the negative slope of ε sub X sup OG in the bilayer state region. Because the ambient solution system is much larger than the supported assembly phase, the chemical potentials of the entire system are determined from the partition equilibrium between the dispersed assembly phase and the bulk solution phase. When the dispersed assembly is in the vesicular state, the corresponding region of the μ DOPC vs. μ OG profile in Fig. 4 is roughly consistent with the partition equilibrium profile for the bilayer state of the supported assembly, resulting in a relatively small change in ε sub with the X sup OG . However, once the composition of the dispersed assembly passes through the micellar-vesicular state transition point, the μ DOPC vs. μ OG profile for the micellar state region is no longer consistent with the partition equilibrium profile for the bilayer state of the supported assembly, resulting in a large change in ε sub with the X sup OG . We also estimated the imaginary μ DOPC vs. μ OG profile, and the imaginary μ DOPC and ε sub as functions of the X sup OG , assuming that the dispersed assembly was always in a vesicular state regardless of the composition, as shown by the dotted lines in Figs. 4, 5 a , and 5 c . In this case, a marked change in ε sub does not occur until at least X sup OG 0.8, suggesting that the transition from the bilayer state to the micellar state in the supported assembly would be observed only at much higher X sup OG . Therefore, the micellar-bilayer state transition in the supported assembly is mainly attributed to the transition in the dispersed assembly.

Limitation of the Phase-separation Model
Our approach, based on the phase-separation model, involves several oversimplifications. According to the Gibbs phase rule, for example, when micellar and vesicular state assemblies coexist, C 1 OG should be constant. In practice, Fig.  2 shows that C 1 OG in the micellar-vesicular state transition region changes slightly with X dis OG; for example, when X dis OG 0.59, C 1 OG 16.4 mM. Furthermore, C 1 OG slightly increases as X dis OG increases, and is 18.8 mM when X dis OG 0.81. This phenomenon demonstrates the limitations of the proposed model.
We assumed that the mixed micellar or vesicular state assembly was a pseudo phase and that the number con-centration of the assembly particle did not influence the phase equilibrium. However, when we regarded the micellar and vesicular particles as chemical species, and gave them free energies derived from the configuration entropy, the chemical potentials of OG and DOPC would depend on the number concentration of the micellar or vesicular particles. In fact, the partition equilibrium profile shifts to a higher C 1 OG as the preparation concentration of DOPC increases, as noted in Supporting Information III. Therefore, for a stringent experiment and analysis, we should prepare a partition equilibrium profile using the desired concentration ranges of surfactants and lipids. In a previous study 23 , we prepared the partition equilibrium profile for the dispersed assembly and that for the supported one using a series of OG/DOPC solutions with C DOPC 1.15 mM. Furthermore, as shown in Fig. 3, the troughs of the molar Gibbs free energy landscape of the micellar and vesicular assemblies are shallow, and even at the midpoint of phase separation, namely X dis OG 0.72, the gain of free energy due to phase separation is only 0.1 kJ/mol. Therefore, any micellar and vesicular particles with X dis OG 0.6-0.8 could exist probabilistically, even though the probabilities would depend on their association numbers. The distribution of the compositions of the micellar and vesicular assemblies should also be considered.
Therefore, to confirm the validity of our micellar-vesicular phase-separation model, we also constructed a thermodynamic description of the assembly composed of OG and DOPC without explicitly assuming the micellar-vesicular transition.
As for the equilibrium between the dispersed assembly phase and the bulk solution phase, the chemical potential of the OG is described by and that of DOPC is described by In these equations, μ dis o OG and μ dis o DOPC correspond to the chemical potential of OG in the pure-OG micelles and DOPC in the pure-DOPC vesicles, respectively. In this treatment, the micellar state was not explicitly distinguished from the vesicular state. For this reason, the complicated behavior of the partition equilibrium profile in Fig.  2 can be ascribed to the activity coefficients in the dispersed assembly, f dis OG and f dis DOPC, as functions of X dis OG. From the C 1 OG vs. X dis OG profile in Fig. 2, we estimated μ OG and f dis OG as functions of X dis OG using Equation 16 , as shown in Fig. 6. We also estimated μ DOPC by numerically integrating In this equation, ΔG ideal and ΔG nonideal represent the contributions of ideal mixing and non-ideality due to intermo-lecular interactions, respectively. Figure 7 shows ΔG mix , ΔG ideal , and ΔG nonideal as functions of X dis OG. Although ΔG nonideal exhibits two distinct minima, the upward convex region between them completely cancels out the downward convex shape of ΔG ideal . Consequently, ΔG mix changes linearly in the range of X dis OG 0.6-0.8. We conclude that Fig. 7 is essentially identical to Fig. 3, and that the linear region of ΔG mix corresponds to the common tangent line in Fig. 3, that is, the ΔG mix curve is actually separated into two different types of curve at X dis OG 0.6-0.8, where the phase transition occurs.
Using the correlation between μ OG and μ DOPC as shown in Fig. 6 a , we investigated the influence of the solid substrate on the assembly in a manner similar to that described in Section 4. Figure 8 a shows μ OG and μ DOPC as functions of the mole fraction of OG in the supported assembly, X sup OG , and Fig. 8 b shows ε sub as a function of X sup OG . It should be noted that ε sub in Fig. 8 b was evaluated by defining ε sub 0 at X sup OG 1, whereas ε sub in Fig. 5 c was defined as ε sub 0 at X sup OG 0. For comparison, we also plotted ε sub in Fig. 5 c in Fig. 8 b , which was displaced vertically to be superimposed. Figure 8 b indicates that ε sub of the supported bilayer calculated based on the micellar-vesicular phase separation model and that without any explicit assumption agree well in terms of the slope in the  Despite various limitations, the Gibbs free energy diagram provides important insights into the micellar-bilayer state transition. For example, the transition point was determined mainly by the energy differences between the micellar and vesicular states composed of pure surfactant and pure lipid, which are closely related to their molecular geometries. Furthermore, our approach has versatility and extensibility, because the Gibbs free energy diagram could integrate additional experimental data, for example, the effects of the types of surfactant and lipid, the additional lipophiles including the third lipid and membrane protein, and the environment, including temperature, pressure, solvent composition, and solid substrates, as discussed in this paper. Therefore, it is feasible to modify our model to involve the law of mass action using statistical mechanics.

Conclusion
In a previous paper 23 , we investigated the phase equilibrium between the supported OG/DOPC assembly phase on hydrophilized substrates and its ambient aqueous solution system consisting of the dispersed assembly phase in the micellar or vesicular state and the bulk solution phase comprising the monomerically dissolved OG. The obtained partition equilibrium profiles of OG between the bulk solution phase and the dispersed or supported assembly phases suggest that a micellar-bilayer transition occurs both in the dispersed and supported assemblies, even though there are significant modifications.
In order to interpret the difference in the partition equilibrium profile depending on whether the assembly is dispersed or supported on substrates, we constructed a simple thermodynamic model to describe the micellar-bilayer transition, assuming a phase-separation model in which the micellar state and the bilayer state are pseudo phases distinct from each other. The micellar state region and the bilayer state region in the partition equilibrium profile were interpreted using the regular solution theory, which is often used in studies on mixed surfactant systems 24,26 , to obtain the Gibbs free energies of the micellar and bilayer states as a function of the composition. Based on this model, we successfully interpreted the partition equilibrium profile for the supported bilayer, considering the restriction of the chemical potentials of OG and DOPC due to the ambient solution system, and additional energy, ε sub , the interaction energy between the assembly and the substrate. As shown in Fig. 5 c , if the supported assembly retained the bilayer state over the entire range of X sup OG , ε sub would increase sharply at the transition point of the dispersed assembly. This suggests that the supported assembly cannot adopt the bilayer state under the thermodynamic restraint of the ambient micellar solution system and transforms to the adsorbed micellar state. Fig. 8 Thermodynamic analysis of the supported assembly without assuming the micellar-vesicular phase separation. a Chemical potentials of OG and DOPC, μ OG and μ DOPC as functions of the OG mole fraction in the supported assembly, X sup OG . μ OG open squares was calculated from the C 1 OG vs. X sup OG profile closed circles in Fig. 2, and μ DOPC closed triangles was calculated from μ OG using the correlation between μ OG and μ DOPC as shown in Fig.  6 a . b The interaction energy between the supported assembly and a Ge substrate, ε sub open circles as a function of X sup OG . This was estimated by numerical integration of Equation 15 using data in a and Fig. 5 b . A solid line represents ε sub of the supported bilayer, which was shown in Fig. 5 c , based on the micellar-vesicular phase-separation model, and displaced vertically to be superimposed on the open circles for comparison.