Coexistences of lamellar phases in ternary surfactant solutions

TL;DR

This paper theoretically explores the phase coexistences in binary and ternary lamellar surfactant solutions, revealing complex phase diagrams with multiple coexistence regions influenced by surface tension effects.

Contribution

It extends the free energy model to include membrane segment entropy and analyzes the resulting complex phase behaviors in surfactant solutions.

Findings

Binary solutions show phase separation and a critical point.

Ternary solutions exhibit various three-phase and two-phase coexistence regions.

Finite surface tension broadens coexistence regions.

Abstract

We theoretically investigate the coexistences of lamellar phases both in binary and ternary surfactant solutions. The previous free energy of a lamellar stack is extended to take into account the translational entropy of membrane segments. The obtained phase diagram for binary surfactant solutions (surfactant/water mixtures) shows a phase separation between two lamellar phases and also exhibits a critical point. For lamellar phases in ternary surfactant solutions (surfactant/surfactant/water mixtures), we explore possible phase behaviors and show that the phase diagrams exhibit various three-phase regions as well as two-phase regions in which different lamellar phases coexist. We also find that finite surface tension suppresses undulation fluctuations of membranes and leads to a wider three-phase and two-phase coexistence regions.

Equations

$$v_{\mathrm{s}}(\ell )=\frac{b(k_{\mathrm{B}}T{)}^2}{\kappa (\ell -\delta {)}^2}.$$

$$\bar{f}(\varphi )$$

$$+\frac{b(k_{\mathrm{B}}T{)}^2}{\kappa {\delta }^3}\frac{{\varphi }^3}{(1-\varphi {)}^2}-\bar{\mu }\varphi .$$

$$\bar{\chi }=-\frac{1}{2{\nu }^2}\int \mathrm{d}\mathbf{r}(1-\exp [-U_{\nu }(\mathbf{r})/k_{\mathrm{B}}T]),$$

$$f(\varphi )=a\varphi (\log \varphi -1)-\chi {\varphi }^2+\frac{{\varphi }^3}{2(1-\varphi {)}^2}-\mu \varphi ,$$

$$\mu =\log \varphi -2\chi \varphi +\frac{3{\varphi }^2}{2(1-\varphi {)}^2}+\frac{{\varphi }^3}{(1-\varphi {)}^3}.$$

$$\frac{{\partial }^{2}f(\varphi )}{\partial {\varphi }^2}=0.$$

$$\chi =\frac{1}{2}\left[\frac{1}{\varphi }+\frac{3\varphi }{(1-\varphi {)}^2}+\frac{6{\varphi }^2}{(1-\varphi {)}^3}+\frac{3{\varphi }^3}{(1-\varphi {)}^4}\right].$$

$$\frac{{\partial }^{2}f(\varphi )}{\partial {\varphi }^2}=0,\frac{{\partial }^{3}f(\varphi )}{\partial {\varphi }^3}=0.$$

$$\frac{\partial f(\phi)}{\partial\phi}\biggr{|}_{1}=\frac{\partial f(\phi)}{\partial\phi}\biggr{|}_{2}=0,~{}~{}~{}~{}~{}f(\phi_{1})=f(\phi_{2}).$$

$$g(\varphi ,\psi )$$

$$-{\chi }_{\varphi \varphi }{\varphi }^2-{\chi }_{\psi \psi }{\psi }^2-{\chi }_{\varphi \psi }\varphi \psi$$

$$+\frac{(\varphi +\psi {)}^3}{2(1-\varphi -\psi {)}^2}-{\mu }_{\varphi }\varphi -{\mu }_{\psi }\psi .$$

$$\displaystyle\mathbf{H}=\left(\begin{array}[]{cc}\dfrac{\partial^{2}g}{\partial\phi^{2}}&\dfrac{\partial^{2}g}{\partial\phi\partial\psi}\\[8.61108pt] \dfrac{\partial^{2}g}{\partial\psi\partial\phi}&\dfrac{\partial^{2}g}{\partial\psi^{2}}\end{array}\right).$$

$${\mathbf{H}}^{\prime }$$

$$H=0,H^{\prime }=0.$$

$$\frac{\partial g(\phi,\psi)}{\partial\phi}\biggr{|}_{1}=\frac{\partial g(\phi,\psi)}{\partial\phi}\biggr{|}_{2}=0,$$

$$\frac{\partial g(\phi,\psi)}{\partial\psi}\biggr{|}_{1}=\frac{\partial g(\phi,\psi)}{\partial\psi}\biggr{|}_{2}=0,$$

$$g({\varphi }_1,{\psi }_1)=g({\varphi }_2,{\psi }_2).$$

$$\frac{\partial g(\phi,\psi)}{\partial\phi}\biggr{|}_{1}=\frac{\partial g(\phi,\psi)}{\partial\phi}\biggr{|}_{2}=\frac{\partial g(\phi,\psi)}{\partial\phi}\biggr{|}_{3}=0,$$

$$\frac{\partial g(\phi,\psi)}{\partial\psi}\biggr{|}_{1}=\frac{\partial g(\phi,\psi)}{\partial\psi}\biggr{|}_{2}=\frac{\partial g(\phi,\psi)}{\partial\psi}\biggr{|}_{3}=0,$$

$$g({\varphi }_1,{\psi }_1)=g({\varphi }_2,{\psi }_2)=g({\varphi }_3,{\psi }_3).$$

$$v_{\mathrm{s}}(\ell ;\xi )=\frac{b(k_{\mathrm{B}}T{)}^2}{\kappa (\ell -\delta {)}^2}{\left[\frac{(\ell -\delta )/\xi }{\sinh [(\ell -\delta )/\xi ]}\right]}^2,$$

$$g(\varphi ,\psi ;x)$$

$$-{\chi }_{\varphi \varphi }{\varphi }^2-{\chi }_{\psi \psi }{\psi }^2-{\chi }_{\varphi \psi }\varphi \psi$$

$$-x^{2}G(x)\frac{(\varphi +\psi {)}^3}{(1-\varphi -\psi {)}^2}-{\mu }_{\varphi }\varphi -{\mu }_{\psi }\psi .$$

$$x=\left(\frac{\delta }{\xi }\right)\frac{1-\varphi -\psi }{\varphi +\psi },$$

$$G(x)=-\frac{1}{2{\sinh }^{2}x}.$$

Full text

I Introduction

One of the simplest mesoscale structures found in mixtures of water and surfactant molecules is the lamellar phase in which bilayers of amphiphilic molecules form roughly parallel sheets separated by water SO . In some surfactant/water binary systems, it is known that the lamellar phase can be swollen almost without limit. For example, in the mixture of C12E5 ($n$-akylpolyglycolether) and water, the repeat distance of the lamellar phase can exceed 3,000 Å SSRNO . The transition from the bound lamellar phase to the unbound phase is generally called the “unbinding transition” LS . Although it is very rare, the coexistence of two lamellar phases in thermodynamic equilibrium has been also reported for binary surfactant/water solutions typically containing DDAB (didodecyldimethylammonium bromide) Dubois92 ; Zemb93 ; Dubois98 ; Brotons05 . In this case, a higher-density condensed lamellar phase is in equilibrium with a lower-density swollen lamellar phase. However, the reason why such a lamellar-lamellar coexistence is so rare in binary surfactant solutions is still a matter of debate and even puzzling Silva07 ; Silva10 .

In contrast to binary mixtures, lamellar-lamellar coexistence is fairly common in ternary systems such as surfactant/surfactant/water mixtures Lis81a ; Lis81b ; Marques93 ; McGrath97 ; Ricoul98 ; Montalvo02 or polymer/surfactant/water mixtures Rong96 ; Deme97 ; Bryskhe01 . Here the surfactant molecules assemble into stacked bilayers, while they are organized as coexisting lamellar phases. The fact that ternary solutions typically exhibit a lamellar-lamellar coexistence can be explained if the bilayers with different components have different interactions across the water layer Noro99 . In such a lamellar-lamellar phase separation, it is known that a long-ranged repulsive interaction such as electrostatic interaction and/or steric repulsive interaction play an important role. The latter interaction is known as the Helfrich steric interaction which arises from the reduced undulation entropy of fluctuating membranes Helfrich78 ; SafranBook . In other words, the excluded volume of the neighboring membranes limits the configuration of a membrane and hence reduce its entropy.

For example, Harries et al. investigated phase separations of charged surfactants by taking into account both the electrostatic and non-electrostatic interactions within a mean-field theory Harries06 . They found that the lamellar-lamellar phase separation is controlled by non-electrostatic interactions between the counterions, and also by the interactions between the neutral and charged surfactants. On the other hand, lamellar-lamellar coexistences in charged membranes were described only by electrostatic interactions in the other work Jho10 .

For electrically neutral bilayer membranes, the combination of the steric repulsive interaction and other direct microscopic interactions, such as long-ranged van der Waals attraction and short-ranged hydration repulsion, determines whether membranes bind each other or unbind to have an infinite separation between them. Lipowsky and Leibler pointed out that a simple superposition of the Helfrich steric repulsion and other direct interactions within a mean-field level gives incorrect (first-order) description of the unbinding transition LL1 . An appropriate treatment of this problem using a functional renormalization-group method showed that the unbinding transition should be a continuous second-order transition (known as the critical unbinding transition).

Later, Milner and Roux proposed a theory for the unbinding transition in a bulk of lamellar phase following the spirit of a mean-field theory for polymers MR . In their argument, the Helfrich estimate of the entropy is taken into account accurately, whereas the other direct microscopic interactions are approximately incorporated as a correction to the hard-wall result for the second virial coefficient. Their theory correctly accounts for the second-order nature of the critical unbinding transition. Furthermore, it has been used to predict both the unbinding and preunbinding behaviors of a lamellar stack in binary surfactant solutions Komura06 .

In this paper, we investigate the coexistences of lamellar phases both in binary surfactant solutions (surfactant/water mixtures) and ternary surfactant solutions (surfactant/surfactant/water mixtures) within a mean-field theory. We consider a situation when the surfactant molecules are electrically neutral, or the electrostatic interaction is sufficiently screened in the presence of electrolyte. By taking into account the translational entropy of membrane segments, we extend the mean-field theory by Milner and Roux in order to properly account for the phase behaviors in binary and ternary surfactant solutions.

Based on the proposed phenomenological free energy, we first discuss the phase diagrams of binary systems in which we find a lamellar-lamellar coexistence that ends in a critical point, as found in one of the experiments Zemb93 . By considering three interaction parameters (virial coefficients) between different components, we further discuss ternary mixtures and explore possible types of ternary phase diagrams (Gibbs triangles). The phase behavior is very rich, and three-phase coexistences as well as two-phase coexistences between different lamellar phases are predicted for a certain range of the interaction parameter values. We investigate both symmetric and asymmetric cases in terms of the two surfactant/water interactions. In the symmetric case, the interactions (or the virial coefficients) between the same surfactant species are identical, whereas they are different in the asymmetric case. We also discuss the effects of finite membrane surface tension on the phase behavior of ternary surfactant solutions.

In the next section, we explain the extension of the mean-field theory by Milner and Roux. Using the extended free energy, we first calculate the phase diagrams of binary surfactant solutions. In Sec. III, we consider the phase behavior of ternary surfactant solutions. Various types of ternary phase diagrams are obtained both for the symmetric and asymmetric cases. In Sec. IV, we also calculate the ternary phase diagrams in the presence of finite surface tension acting on the membranes. The summary of our work and some discussions are given in the last Sec. V.

II Lamellar phases in binary mixtures

Bilayer fluid membranes experience steric repulsion arising from their reduced undulation entropy SO . The corresponding interaction energy per unit area of membrane was considered by Helfrich and is given by Helfrich78 ; SafranBook

[TABLE]

Here $k_{\rm B}$ is the Boltzmann constant, $T$ is the temperature, $\kappa$ is the bending rigidity, $\ell$ is the average repeat distance between bilayers, as shown in Fig. 1, and a constant $\delta$ is the membrane thickness that is used as the smallest cutoff length. Note that $\ell-\delta$ in the denominator corresponds to the inter-membrane distance in which membranes can undergo out-of-plane fluctuations. The numerical prefactor $b$ was calculated to be $b=3\pi^{2}/128\approx 0.23$ in the original work by Helfrich Helfrich78 , but its value is debatable in the literatures LZ ; NL93 ; NL95 . For example, Monte Carlo simulations in Ref. NL95 yielded a lower value of $b\approx 0.12$, almost a half of the above value. In the present study, the exact value of $b$ does not affect the results because we rescale all the energy densities by including the factor $b$ [see later Eq. (4)].

In order to describe the free energy of a lamellar stack in a binary surfactant/water solution, we first introduce the membrane volume fraction $\phi=\delta/\ell\geq 0$. Here we have assumed that all the surfactant molecules constitute bilayers. Extending the argument by Milner and Roux MR , we consider the following grand potential per unit volume of a lamellar stack:

[TABLE]

Here the first term represents the translational entropy of membrane segments, which was not considered before MR . This term, however, plays an essential role when we calculate the phase diagrams within equilibrium thermodynamics. In general, the translational entropy term should be given by $\phi\log\phi+(1-\phi)\log(1-\phi)$ SafranBook ; DoiBook , and the first term in Eq. (2) corresponds to its lowest order expansion in terms of small $\phi\ll 1$. However, since such a generalization does not result in any essential modification, we shall study the above grand potential in this paper. Our approximation is justified because the term $(1-\phi)\log(1-\phi)$ vanishes when $\phi\rightarrow 1$, while the third term in Eq. (2) diverges, as explained below.

The second term in Eq. (2) is the correction to the entropic hard-wall result, and $\bar{\chi}$ is the second virial coefficient obtained from

[TABLE]

where $\nu\approx\delta^{3}$ is the volume of the membrane segment, and $U_{\nu}(\mathbf{r})$ is the interaction between bits of membrane of volume $\nu$, and $\mathbf{r}$ is a three-dimensional vector. All the direct microscopic (van der Waals, hydration, electrostatic) interactions are taken into account through $U_{\nu}(\mathbf{r})$. The third term in Eq. (2) is due to the Helfrich steric repulsion, and the factor of $(1-\phi)^{-2}$ comes from the finite membrane thickness [see also Eq. (1)] DC . Finally, the chemical potential, $\bar{\mu}$, is needed for the conservation of the surfactant volume fraction $\phi$. A similar free energy to Eq. (2) was also proposed in other works to describe the unbinding transition Helfrich89 ; Komura03 ; Bougis15 but without the translational entropy term that we have introduced. We note again that the first translational entropy term in Eq. (2) is not accounted for by the Helfrich steric repulsion term which also has an entropic origin. Without the translational entropy term, the behavior of the free energy is thermodynamically inappropriate around $\phi\approx 0$ and one cannot describe correct phase behaviors.

It is convenient to rescale all the energy densities by $2b(k_{\rm B}T)^{2}/(\kappa\delta^{3})$. Then Eq. (2) can be presented in a dimensionless form as

[TABLE]

where $a=\kappa/(2bk_{\rm B}T)$ is a numerical factor of order unity, and hence can be set as $a=1$ in the following discussion for simplicity, while $\chi=\bar{\chi}\kappa\delta^{3}/(2bk_{\rm B}T)$ is a dimensionless interaction parameter. The equation of state is then given by minimizing the grand potential, $\partial f/\partial\phi=0$, and becomes

[TABLE]

Using the above grand potential density, we can obtain the spinodal from the condition Koningsveld :

[TABLE]

With the use of Eq. (4), it can be written as

[TABLE]

The conditions for the critical point is given by

[TABLE]

These conditions and Eq. (5) can be numerically solved to obtain the critical point as $(\chi_{\rm c},\phi_{\rm c},\mu_{\rm c})=(3.16,0.23,-2.76)$.

On the other hand, the thermodynamic equilibrium between the two coexisting phases denoted as “1” and “2” and characterized by $\phi_{1}$ and $\phi_{2}$, satisfies the following conditions Koningsveld :

[TABLE]

We have numerically solved the above set of conditions to obtain the phase diagrams.

The calculated phase diagrams in the $(\chi,\phi)$ and $(\chi,\mu)$ planes are shown in Fig. 2. The red solid lines are binodal lines, and the black dashed lines are spinodal lines. When $\chi>3.16$, the binary mixture separates into two lamellar phases characterized by different $\phi$ values indicated by the horizontal tielines (see also Fig. 1). Notice that there is a critical point at $(\chi_{\rm c},\phi_{\rm c},\mu_{\rm c})=(3.16,0.23,-2.76)$ where the two lamellar phases become identical. For $\chi<3.16$, on the other hand, the complete unbinding of the lamellae occurs upon swelling with excess water LL2-1 ; LL2-2 . In the $(\chi,\phi)$ phase diagram, such a transition occurs when we take the limit of $\phi\rightarrow 0$.

The calculated phase diagram in Fig. 2(a) resembles that obtained for DDAB/water binary mixtures Dubois92 ; Zemb93 ; Dubois98 ; Brotons05 if we assume that the interaction parameter $\chi$ is inversely proportional to the temperature. In these experiments, the existence of a critical point and associated critical phenomena were experimentally evidenced by various scattering methods Zemb93 .

III Lamellar phases in ternary mixtures

Next we consider lamellar phases in ternary surfactant/surfactant/water solutions in which bilayer membranes are composed of two different types of surfactant, say surfactant A and surfactant B, as shown in Fig. 3. Let us define the volume fractions of surfactant A and B by $\phi$ and $\psi$, respectively. Then the volume fraction of water (solvent) is automatically fixed by $1-\phi-\psi$ due to the incompressibility condition. As a generalization of Eq. (4), we consider the following dimensionless grand potential per unit volume for ternary mixtures:

[TABLE]

In the above, the first two terms represent the translational entropy of each surfactant component, the next three terms describe the different interactions characterized by the three dimensionless virial coefficients $\chi_{\phi\phi}$, $\chi_{\psi\psi}$, and $\chi_{\phi\psi}$ which are assumed as independent parameters. The first term in the third line corresponds to the Helfrich steric repulsion acting between mixed membranes. Notice here that the total surfactant volume fraction $\phi+\psi$ corresponds to the volume fraction of membranes. Several other assumptions that lead to this expression are separately discussed in Sec. V. Furthermore, $\mu_{\phi}$ and $\mu_{\psi}$ are the chemical potentials for the two surfactants. When, for example, surfactant B is absent and hence $\psi=0$, Eq. (10) reduces to Eq. (4), as it should.

In order to discuss the stability of the above grand potential with two independent variables, we consider the $2\times 2$ Hessian matrix of $g(\phi,\psi)$ given by Koningsveld

[TABLE]

At the spinodal, the Hessian defined as the determinant of the matrix, $H=\det\mathbf{H}$, vanishes, i.e., $H=0$. The critical point can be obtained by considering another $2\times 2$ matrix

[TABLE]

and its determinant $H^{\prime}=\det\mathbf{H}^{\prime}$. Then the conditions for the critical point are given by Koningsveld

[TABLE]

For ternary mixtures, the thermodynamic equilibrium between the two coexisting phases denoted as “1” and “2” and characterized by $(\phi_{1},\psi_{1})$ and $(\phi_{2},\psi_{2})$, satisfies the conditions Koningsveld :

[TABLE]

[TABLE]

[TABLE]

Similarly, for a three-phase coexistence between phases “1”, “2” and “3”, the following set of conditions should be satisfied Koningsveld :

[TABLE]

[TABLE]

[TABLE]

For lamellar phases under consideration, an example of three-phase coexistence is schematically presented in Fig. 3(b). In the following, we present the numerically calculated ternary phase diagrams (Gibbs triangles) for different interaction parameters.

We first consider the symmetric case between the two surfactants A and B, i.e., $\chi_{\phi\phi}=\chi_{\psi\psi}$. Figure 4 shows a ternary phase diagram when $\chi_{\phi\phi}=\chi_{\psi\psi}=3$ (symmetric) and $\chi_{\phi\psi}=-3$. This is the case when each surfactant/water binary solution does not exhibit lamellar-lamellar phase separation because $\chi_{\phi\phi}=\chi_{\psi\psi}<3.16$ (see Fig. 2). The obtained phase diagram is always symmetric with respect the line $\phi=\psi$. Due to the strong repulsion between the A and B components ($\chi_{\phi\psi}=-3$), there is a wide region of two-phase coexistence (red solid lines) with horizontal tielines (black solid lines). In the upper part of the triangle, there is a region of three-phase coexistence (blue triangle) associated with two wings of two-phase coexistence. These two-phase coexistence regions end in two corresponding critical points (black circles). The black dashes lines indicate the spinodal lines that appear inside the coexistence regions. When we make $\chi_{\phi\psi}$ larger such as $\chi_{\phi\psi}=3$ (not shown), only the two-phase coexistence region remains in the lower part of the triangle with horizontal tielines, and the three-phase region disappears.

In Fig. 5, we present the ternary phase diagrams when $\chi_{\phi\phi}=\chi_{\psi\psi}=4$ (symmetric), while the A/B interaction is changed as (a) $\chi_{\phi\psi}=5$, (b) $\chi_{\phi\psi}=3$, and (c) $\chi_{\phi\psi}=0$. These are the cases when each surfactant/water binary solution exhibits lamellar-lamellar phase separation because $\chi_{\phi\phi}=\chi_{\psi\psi}>3.16$ (see Fig. 2), as represent on the two S-A and S-B sides of the triangles. In the case of Fig. 5(a), these two-phase regions merge to form a single two-phase region with tilted tielines in the upper part of the triangle. In the lower part of the triangle, on the other hand, there is a region of two-phase coexistence with horizontal tielines. This two-phase region ends in a critical point. When $\chi_{\phi\psi}$ is made smaller as in Fig. 5(b), the upper two-phase coexistence region separates into distinct two-phase regions that also end in two corresponding critical points. For even smaller $\chi_{\phi\psi}$ as in Fig. 5(c), the three two-phase regions meet each other forming a three-phase coexistence region similar to Fig. 4. As a result, all the three critical points disappear.

In Fig. 6, we show the ternary phase diagrams for an asymmetric case of $\chi_{\phi\phi}=4>3.16$ and $\chi_{\psi\psi}=3<3.16$, while the A/B interaction is chosen as (a) $\chi_{\phi\psi}=4$ and (b) $\chi_{\phi\psi}=-3$. In this case, only the binary A/S mixture exhibits the phase separation while the binary B/S does not. Here Fig. 6(a) should be compared with Fig. 5(b). The two two-phase regions end in the receptive critical points. When $\chi_{\phi\psi}$ is made smaller as in the case of Fig. 6(b), the two two-phase coexistence regions are connected to each other with the appearance of a three-phase coexistence region. This three-phase region accompanies another small two-phase region and a critical point on the S-B side of the triangle. In this asymmetric case, the tielines are not horizontal and tilted especially in the upper part of the phase diagram.

IV Effects of surface tension

In this section, we consider lamellar phases in ternary surfactant solutions in which surface tension, $\sigma$, is acting on membranes. It is known that finite surface tension significantly suppresses membrane undulations, and the range of fluctuation-induced interaction between tense membranes becomes shorter. Although the calculation of this interaction is complicated in general, Seifert provided a simple self-consistent calculation Seifert95 . He showed that the energy per unit area of membrane in the presence of surface tension is given by

[TABLE]

where $\xi=(2k_{\rm B}T/\pi\sigma)^{1/2}$ is the characteristic length arising from the competition between the thermal energy and the surface energy. When the surface tension $\sigma$ is small ($\xi\rightarrow\infty$), the above expression reduces to Eq. (1) for a tensionless membrane, and recovers the long-range algebraic decay. When the surface tension $\sigma$ is large ($\xi\rightarrow 0$), on the other hand, Eq. (24) decays exponentially with distance $\ell$, consistent with the renormalization-group result LS and Monte Carlo simulations NL95 . The reduction of membrane undulations in the presence of surface tension was experimentally observed in Ref. Mutz89 .

Using Eq. (24) in the presence of surface tension, we consider a modified grand potential per unit volume for ternary surfactant solutions as DC

[TABLE]

Here the dimensionless quantity $x$ is defined by

[TABLE]

whereas the scaling function $G(x)$ is given by

[TABLE]

We note that $x$ depends also on $\phi$ and $\psi$, while $\delta/\xi$ is a dimensionless parameter that can be given externally.

Figure 7 shows the calculated ternary phase diagrams when $\chi_{\phi\phi}=\chi_{\psi\psi}=4$ (symmetric) and $\chi_{\phi\psi}=5$, while the parameter controlling the surface tension is (a) $\delta/\xi=10$ and (b) $\delta/\xi=20$. These phase diagrams should be compared with that in Fig. 5(a) for which $\delta/\xi=0$. As shown here with finite surface tension, the phase separation is dramatically enhanced and the upper two two-phase coexistence regions extend down to the middle part of the triangle. At the expense of the critical point in Fig. 5(a), there appears a large three-phase coexistence region in the middle part. This three-phase coexistence region is connected to the lower two-phase region with horizontal tielines. As we see in Fig. 7(b), the three-phase coexisting region further expands when $\delta/\xi$ is made larger. Therefore, surface tension promotes the phase separation between different lamellar phases. The fact that dense lamellar phases coexist with an excess water on the two S-A and S-B sides of the triangles is in accordance with the experiment Mutz89 . In the presence of finite surface tension, they observed membranes merging one by one into bundles at mutually adhering membranes. Also the water between the membranes was driven into a small number of compact water pockets Mutz89 .

V Summary and discussion

In this paper, we have investigated the coexistences of lamellar phases both in binary and ternary surfactant solutions. To calculate the phase diagrams, we have extended the previous free energy of a lamellar stack MR by taking into account the translational entropy of membrane segments. The obtained phase diagrams for a binary surfactant solution show a phase separation between two lamellar phases and also exhibit a critical point. For lamellar phases in ternary surfactant solutions, we have further extended the free energy to take into account the different interactions between three species and explored possible phase behaviors. The calculated phase diagrams include various coexistences between three different lamellar phases (three-phase regions) or between two lamellar phases (two-phase regions). A systematic change of the phase behavior has been observed by changing the interaction parameter between the two surfactant species. Finally, we have looked at the effects of finite surface tension which suppresses membrane fluctuations and leads to a wider three-phase coexistence region.

We stress again that the addition of translational entropy terms in the free energies [see Eqs. (2) and (10)] is essential in calculating the correct phase diagrams. Without these terms, one cannot obtain the coexistence between two lamellar phases having different repeat distances. The original free energy by Milner and Roux was considered in order to explain the unbinding transition in surfactant solutions MR . Although their phase diagram exhibits a coexistence between a lamellar phase and excess water (i.e., unbound lamellar phase), a coexistence between two distinct lamellar phases does not occur. We consider that these translational entropy terms should be included in addition to the Helfrich steric interaction which also has an entropic origin.

In the present work, we have assumed that the lamellar phase is the only lyotropic liquid crystaline phase that is formed for any temperature (interaction) and composition. In real surfactant solutions, however, this is certainly not the case, because typical phase diagrams contain other phases such as the micellar phase, the hexagonal phase, and the cubic phase SSRNO . Rather than reproducing realistic phase diagrams by considering all the possible phases in surfactant solutions, our purpose is to investigate in detail the competition between the Helfrich steric repulsion and other direct microscopic (van der Waals, hydration) interactions especially in ternary mixtures. This is why we have extended the free energy of a lamellar stack by Milner and Roux, and calculated various coexistences only between the lamellar phases. A similar theoretical approach was made by Noro and Gelbart who also discussed lamellar-lamellar phase separations in surfactant solutions Noro99 . We also note that the steric repulsive interaction acting between neighboring cylinders in the hexagonal phase is discussed in Ref. SO , which can be used to extend our treatment.

Another simplification in our work is that we have not taken into account the composition dependence of the bending rigidity $\kappa$ or the surface tension $\sigma$ when a membrane is composed of two surfactants A and B. Usually, a membrane made of an A/B surfactant mixture will show an intermediate bending rigidity between those of pure membranes. In principle, such a change in the bending rigidity affects the Helfrich steric interaction. One of the possible ways to describe the intermediate behavior is to linearly interpolate between the two pure limits. This approximation can be justified when the bending rigidities of the pure components are not so different. When they are very different, such as in membranes composed of surfactant and amphiphilic polymer, a nonlinear effect on the bending rigidity becomes important KS01 . Furthermore, a detailed discussion on the surface tension in a mixed membrane was recently given by some of the present authors Okamoto16 .

Although the phase diagrams calculated in this paper may not be simply compared with experimentally obtained ones because of the previously mentioned reason, it is still useful to discuss the ternary phase diagrams of glycolipid/cationic surfactant/water mixtures at room temperature Ricoul98 . In this ternary mixture, they found the two-phase regions between the coexisting lamellar phases on both sides of the Gibbs triangle. More interestingly, they further identified two corresponding critical points and also the region of the three-phase coexistence according to the phase rule. Such a phase behavior is very reminiscent to the phase diagrams in Fig. 5(b) and (c). Although not yet done, we expect that one can reproduce the experimentally obtained phase diagram by further tuning the three interaction parameters in our model.

In this paper, we have considered a situation in which the surfactants are electrically neutral or the electrostatic interaction is sufficiently screened. As is clear from Eq. (1), the Helfrich steric repulsion is important only when the membrane is flexible, $\kappa\simeq k_{\rm B}T$. For strongly charged and unscreened membranes (no electrolyte), on the other hand, the dominant repulsion originates from the electrostatic interactions between flat membranes. The interplay between the electrostatics and fluctuations of a stack of membranes (without van der Waals and hydration forces) was studied before by Pincus et al. Pincus90 . When electrostatic interactions are strong enough (Gouy–Chapman regime) compared with the Helfrich steric repulsion, they showed that out-of-plane membrane fluctuations become smaller than the inter-membrane separation $\ell-\delta$. In the other weaker electrostatic regimes, on the other hand, the suppression of membrane fluctuations is less important, and in some cases, the screened electrostatic interactions can be completely neglected Pincus90 . Our assumption for charged membranes is justified for such situations.

Our results indicate that the lateral phase separation in mixed membranes causes different inter-membrane distances. Recently much efforts have been made to study the statics and dynamics of multi-component lipid membranes Komura14 , mainly using giant unilamellar vesicles (GUVs) in the experiments Smith09 . In the future, it is interesting to study the phase behaviors of multi-lamellar vesicles composed of more than two types of lipid by taking into account the interactions between neighboring membranes. The present work would provide us with a useful theoretical guide for such a research direction.

Acknowledgements.

We thank T. Kato and Y. Kawabata for useful discussions. S.K. and R.O. acknowledge support from the Grant-in-Aid for Scientific Research on Innovative Areas “Fluctuation and Structure” (Grant No. 25103010) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan, and the Grant-in-Aid for Scientific Research (C) (Grant No. 15K05250) from the Japan Society for the Promotion of Science (JSPS). J.W. acknowledges support from the C.N.R.S. (Centre National de la Recherche Scientifique), the French ORT association, and ORT school of Strasbourg.