Spontaneous Formation of Vesicular Liposomes: Thermodynamics and Bending Energetics

Abstract

By means of combining bending elasticity theory with solution thermodynamics of small systems, we demonstrate that unilamellar vesicular liposomes can be thermodynamically stable with a wide range of average sizes depending on the various bending elasticity constants. The average vesicle size increases with increasing bending rigidity (k_c) and saddle-splay constant (${\overline{k}}_c$), and with decreasing spontaneous curvature (H₀). Bilayer aggregates predominate over micelles at lower values of the spontaneous curvature, in the regime of which H₀ favours large vesicles. However, small unilamellar vesicles may be favoured by low values of k_c, rather than high H₀. Mixing two amphiphilic components with different spontaneous curvatures gives rise to an explicit contribution that always brings down k_c. As a result, the amphiphilic component with high positive spontaneous curvature prefers to be located in the outer, positively curved, monolayer, whereas the other compound prefers to be located in the inner, negatively curved, layer. In contrast to unilamellar vesicles, geometrically open disks can only be thermodynamically stable close to the micelle-to-bilayer transition in a dilute solution of non-interacting bilayer aggregates. However, in more concentrated solutions, above the overpacking limit of vesicles, disks may be more favourable than vesicles due to more favourable packing conditions.

1. Introduction

Colloidal solutions are thermodynamically stable systems in which amphiphilic molecules (surfactants or amphiphilic lipids) self-assemble spontaneously to micelles, microemulsions, or bilayer aggregates. This is in contrast to colloidal dispersions, which are non-equilibrium systems consisting of one phase dispersed in another (i.e. two phases), like emulsions or dispersed vesicular liposomes. The stability of aggregates in colloidal solutions depends on the local curvature of the aggregates together with the entropy of self-assembling amphiphilic molecules. The latter effect tends to reduce the size of the self-assembled aggregates. As a result, the size and shape of a thermodynamically stable aggregate are determined by the curvature properties of the aggregate interface together with the entropy of self-assembly [1]. A positive spontaneous curvature is required for oil-in-water droplets or ordinary surfactant micelles to form in an aqueous solvent. Likewise, water-in-oil droplets or reversed micelles dissolved in a continuous oil phase may form by amphiphilic molecules with negative spontaneous curvature [2].

The simplest form of aggregate structure is a geometrically homogeneous spherical micelle or microemulsion droplet, with a homogeneous curvature all over its interface. Other aggregate shapes include non-spherical micelles, unilamellar vesicles, and bilayer disks. Such aggregates may be considered as composed of geometrical parts with different curvatures and may form under certain conditions related to the detailed geometrical shape. The formation of non-spherical elongated micelles has recently been theoretically treated in the so-called general micelle model by mean of combining thermodynamics of self-assembly with bending elasticity theory [3]. Bilayer aggregates are expected to form in an aqueous solvent by amphiphilic molecules with low spontaneous curvature (or in an oil phase by amphiphilic molecules with high spontaneous curvature). When mixing a micelle-forming surfactant with a bilayer-forming amphiphilic lipid, a reversible transition from micelles to bilayers is usually observed at a certain surfactant/lipid composition in the aggregates [4].

Bilayer aggregates in isotropic solutions are mainly shaped as either open circular disks or geometrically closed unilamellar vesicles. Both aggregate types are geometrically heterogeneous and considered as composed of different geometrical parts. For instance, a vesicle is defined as a (nearly spherical) geometrically closed bilayer consisting of two geometrical parts: a positively curved outer monolayer and a negatively curved inner monolayer (cf.Figure 1). A vesicular aggregate is sometimes denoted as a liposome when at least one of the components constituting the vesicles is an amphiphilic lipid. Liposomes made up of amphiphilic lipids are usually non-equilibrium structures in a colloidal dispersion that need input of energy to be able to form, for example by means of sonication and extrusion. However, by simply mixing two amphiphilic components, unilamellar vesicles sometimes form spontaneously in a thermodynamically stable colloidal solution, depending on the choice of components. Notably, vesicles have been observed to form spontaneously in several systems where two oppositely charged surfactants have been dissolved in water [5]. The size of unilamellar vesicles ranges from about 20 nm in diameter up to several hundreds of nanometer. These mixed cationic/anionic vesicles are usually found in the range 20–100 nm and referred to as small unilamellar vesicles (SUV). The size of cationic/anionic vesicles has been found to be sensitive to the chemical structure of components as well as surfactant concentration and ionic strength of an aqueous solvent. These vesicles tend to increase in size with decreasing total surfactant concentration and increasing ionic strength, and they become destabilized and transform into bilayer disks as the aggregates become sufficiently large [6, 7, 8].

Figure 1.

Schematic picture of a spherical unilamellar vesicles with bilayer midplane radius R_v and monolayer thickness ξ.

Spontaneously formed vesicles have also been observed in mixtures of phospholipid and bile salt surfactants [9, 10] and, more recently, in mixtures of phospholipid and certain amphiphilic drug surfactants [11]. In the latter case, ultrasmall vesicles with a diameter as small as less than 20 nm were discovered to form spontaneously. In all these systems, the vesicles were formed spontaneously from mixed micelles by means of simply diluting more concentrated micellar solutions. In contrast to the mixed cationic/anionic surfactant systems, the mixed drug surfactant/phospholipid vesicles are stable at high ionic strengths (physiological saline solution) and decrease, rather than increase, in size upon diluting the samples.

In this chapter, we theoretically investigate the spontaneous formation of unilamellar vesicles by means of combining thermodynamics of self-assembly and bending elasticity theory in a similar manner as has previously been done for micelles in the general micelle model [3, 12] and for microemulsions [2]. In particular, we compare the stability of unilamellar vesicles and bilayer disks from a thermodynamic point of view. We are able to conclude that in sufficiently dilute solutions, where inter-aggregate interactions are neglected, geometrically closed vesicles are the thermodynamically stable aggregate type of bilayers.

2. Thermodynamics of self-assembly

Thermodynamically stable equilibrium structures where N-free amphiphilic molecules (monomers) dissolved in an aqueous phase self-assemble to form a small bilayer system $B_N$ (vesicle or disk) can be treated in terms of a set of multiple equilibrium reactions

$$N\text{monomers}\rightleftharpoons B_N$$E1

The change in Gibbs energy in the self-assembly process may be written as a sum of two contributions:

$$∆G=N∆\mu -T∆S_{\mathit{agg}}$$E2

where the change in entropy for the process in Eq. (1), when self-assembling N free monomers at a volume fraction ϕ_free into a bilayer aggregate B_N with volume fraction ϕ_N, is given by

$${\mathrm{\Delta }S}_{\mathit{agg}}=-k\left(ln{\varphi }_N-Nln{\varphi }_{\mathit{\text{free}}}\right)$$E3

and k is Boltzmann’s constant. ΔS_agg must always be a negative quantity since the process of self-assembly is unfavourable in the absence of a specific driving force for the process in Eq. (1). The quantity Δμ comprises several residual contributions to the free energy of forming a single bilayer aggregate. The most important contribution to Δμ comes from the main driving force for the self-assembly process, the hydrophobic effect; that is, the principle that oil and water do not mix and the hydrocarbon-water interfacial area tends to become reduced.

Combining Eqs. (2) and (3) gives the following set of equilibrium conditions, one for each aggregation number N

$$∆G_N={\mathrm{E}}_N+\mathit{kT}ln{\varphi }_N=0$$E4

where T is the absolute temperature. For the sake of simplicity, we have introduced the free energy parameter E_N, defined as [13]

$${\mathrm{E}}_N\equiv N∆\mu -\mathit{NkT}ln{\varphi }_{\mathit{\text{free}}}$$E5

At equilibrium, ΔG_N = 0 and amphiphilic molecules are reversibly exchanged between bilayer aggregates and as free monomers. In accordance with Eq. (4), the volume fraction of aggregates with aggregation number N equals

$${\varphi }_N=e^{-{\mathrm{E}}_N/\mathit{kT}}$$E6

Summing up the different volume fractions in Eq. (6) gives the total volume fraction ϕ_bil of amphiphilic molecules self-assembled in bilayer aggregates, that is

$${\varphi }_{\mathit{bil}}=\sum _{N=1}^{\infty }{\varphi }_N\approx \underset{1}{\overset{\infty }{\int }}{e}^{-{\mathrm{E}}_N/\mathit{kT}}\mathit{dN}$$E7

Inserting the proper mathematical expression for the function E_N in Eq. (7) gives the size distribution of aggregates. Below we will derive expressions of E_N for bilayer vesicles and disks, respectively, using bending elasticity theory, to arrive at the proper size distribution for vesicular liposomes and bilayer disks.

3. Bending elasticity theory

The free energy per unit area γ at a single point of a self-assembled monolayer depends on the mean and Gaussian curvatures, H = (c₁ + c₂)/2 and K = c₁c₂, respectively, for a given small system of self-assembled amphiphilic molecules in a particular solvent at a given set of environmental conditions. A quantitative description has been proposed by Helfrich [14] in the so-called Helfrich expression, that is

$$\gamma \left(H,K\right)={\gamma }_0+2k_c{\left(H-H_0\right)}^2+{\overline{k}}_{c}K$$E8

where γ₀ is a constant with respect to curvature. Eq. (8) defines the three bending elasticity constants k_c (bending rigidity), H₀ is the spontaneous curvature, and ${\overline{k}}_c$ is the saddle-splay constant. The total free energy of a small bilayer system with aggregation number N can be obtained by integrating Eq. (1) over the entire interfacial area A, giving

$$E_N={\gamma }_{0}A+2k_c\int {\left(H-H_0\right)}^2\mathit{dA}+{\overline{k}}_c\int \mathit{KdA}$$E9

The Helfrich expression introduces three quantities related to different aspects of bending a surfactant-lipid monolayer, that is k_c, H₀, and ${\overline{k}}_c$. The three quantities are, in principle, possible to determine from experiments or from detailed model calculations [15, 16, 17, 18]. In these models, it is of essential importance that the hydrophobic tails in a surfactant/amphiphilic lipid monolayer are subjected to geometrical packing constraints that relate the area per amphiphilic molecule at the hydrophobic-hydrophilic interface (a) with the thickness of the hydrophobic part of the monolayer (ξ) and the molecular volume of the hydrophobic part (v). Geometrical packing constraints are taken into account by the following relation [19]

$$\frac{1}{a}=\frac{\xi }{v}\left(1-\mathit{\xi H}+\frac{{\xi }^2}{3}K\right)$$E10

The surfactant/amphiphilic lipid monolayers are usually treated as an incompressible medium with a constant molecular volume. Notably, the geometrical relation in Eq. (10) is exact within a second-order expansion in curvature and, as a result, the Helfrich approach is expected to be accurate for aggregates with comparatively high interfacial curvature.

It has previously been demonstrated that a rather abrupt transition from micelles to bilayer aggregates is predicted to occur according to the following rather simple expression [12, 15]

$$H_0=\frac{k_c{H}_0}{k_c}=\frac{1}{4\xi }$$E11

In accordance with Eq. (11), the location of the micelle-to-bilayer transition entirely depends on the spontaneous curvature H₀, as defined in the Helfrich expression in Eq. (8) and, as a consequence, micelles in a surfactant/amphiphilic lipid mixture are expected to predominate as H₀ > 1/4ξ, whereas bilayer aggregates predominate as H₀ < 1/4ξ.

Previous model calculations have demonstrated that the product k_cH₀ is more readily interpreted from a physical point of view than H₀ itself [15, 16, 17, 20]. Below, k_cH₀ denotes the effective spontaneous curvature, the quantity of which depends on molecular properties such as head group charge number, tail structure, and the hydrophilic-lipophilic balance (HLB) in a straightforward way. Hence, we are able to conclude, in accordance with Eq. (11), that micelles are favoured by high effective spontaneous curvatures k_cH₀ and low-bending rigidities k_c, whereas bilayer aggregates are favoured by high values of k_c and low values of k_cH_0.

4. Geometrically closed unilamellar bilayer vesicles

A spherical unilamellar vesicle consists of two distinct geometrical parts, a positively curved outer monolayer and a negatively curved inner layer, both with approximately same thickness ξ (cf.Figure 1). Denoting the radial distance that separates the two monolayers R_v, the hydrocarbon-water contact interface is located at a radial distance R_e = R_v + ξ and the corresponding interface of the inner monolayer is located at R_i = R_v – ξ. The free energy of forming a unilamellar vesicle out of free monomers in solution is obtained as a sum of contributions from the outer and inner monolayers, respectively [21, 22, 23]. Since the spherical interfaces of the two monolayers each have a homogeneous curvature, the free energy of a unilamellar vesicle can be simply evaluated as

$${\mathrm{E}}_{\mathit{ves}}=A_e{\gamma }_e+A_e{\gamma }_e=4\pi \left(R_e^2{\gamma }_e+R_i^2{\gamma }_i\right)$$E12

γ_e and γ_i are obtained from Eq. (8) by inserting the proper expressions, $H=1/R_e$ and $K=1/R_e^2$ for the outer layer and $H=-1/R_i$ and $K=1/R_i^2$ for the inner layer, that is

$${\gamma }_e={\gamma }_0+2k_c{\left(\frac{1}{R_e}-H_0\right)}^2+\frac{{\overline{k}}_c}{R_e^2}$$E13

$${\gamma }_i={\gamma }_0+2k_c{\left(-\frac{1}{R_i}-H_0\right)}^2+\frac{{\overline{k}}_c}{R_i^2}$$E14

where ${\gamma }_0={\gamma }_p-2k_c{H}_0^2$ and γ_p is the interfacial tension of a planar bilayer. Introducing the dimensionless vesicle radius defined as r_v = R_v/ξ, we may deduce the following expression from Eq. (12)

$$\frac{{\mathrm{E}}_{\mathit{ves}}}{\mathit{kT}}={\alpha }_v+8\mathit{\pi \lambda }{r}_v^2$$E15

where the dimensionless parameter

$${\alpha }_v=\frac{4\pi k_{\mathit{bi}}}{\mathit{kT}}$$E16

corresponds to the work of bending a planar bilayer into a geometrically closed bilayer vesicle (bending energy) with identical interfacial area. The bilayer bending constant is defined as

$$k_{\mathit{bi}}=2\left(2k_c+{\overline{k}}_c-4k_c\xi H_0\right)$$E17

Although vesicles are geometrically heterogeneous aggregates with two distinct geometrical parts, that is the positively curved outer layer and the negatively curved inner layer, the bending energy in Eq. (15) consists of one single term, equal to α_v, and does not depend on vesicle size. The reason for this is that one term in each expression for the outer and inner layer, respectively, cancels out in the derivation of Eq. (15).

The second term in Eq. (15) corresponds to the work of forming a planar bilayer with an identical interfacial area as the vesicle (stretching energy), where we have introduced the dimensionless interfacial tension λ defined as

$$\lambda =\frac{{\xi }^2{\gamma }_p}{\mathit{kT}}$$E18

In accordance with Eq. (7), the volume fraction of molecules aggregated in bilayer vesicles equals

$${\varphi }_{\mathit{ves}}=\frac{16\pi {\xi }^3}{v}{e}^{-{\alpha }_v}\underset{1}{\overset{\infty }{\int }}{r}_v{e}^{-8\mathit{\pi \lambda }{r}_v^2}d{r}_v=\frac{{\xi }^3}{\mathit{v\lambda }}{e}^{-{\alpha }_v-8\mathit{\pi \lambda }}$$E19

and, as a consequence, the average vesicle radius equals

$$⟨r_v⟩=\frac{\underset{1}{\overset{\infty }{\int }}{r}_v^2{e}^{-8\mathit{\pi \lambda }{r}_v^2}d{r}_v}{\underset{1}{\overset{\infty }{\int }}{r}_v{e}^{-8\mathit{\pi \lambda }{r}_v^2}d{r}_v}=\frac{e^{-8\mathit{\pi \lambda }}}{2\pi \sqrt{2\lambda }\left(1-erf\sqrt{8\mathit{\pi \lambda }}\right)}$$E20

In the limit λ → 0 (corresponding to r_v ≫ 1), Eq. (19) can be rearranged so as to relate the dimensionless interfacial tension λ with the volume fraction of vesicles

$$\lambda =\frac{{\xi }^3{e}^{-{\alpha }_v}}{v{\varphi }_{\mathit{ves}}}$$E21

and Eq. (20) becomes simplified to

$$⟨r_v⟩=\frac{1}{2\pi \sqrt{2\lambda }}$$E22

Combining Eqs. (21) and (22) give the following rather simple relation between average reduced vesicle radius, bending energy, and vesicle volume fraction, respectively.

$$⟨r_v⟩=\frac{e^{{\alpha }_v/2}}{2\pi }\sqrt{\frac{v{\varphi }_{\mathit{ves}}}{2{\xi }^3}}$$E23

This means that thermodynamically stable unilamellar vesicles with a finite size are predicted to exist from our theory. The average radius in Eq. (23) may be rationalized as the result of a balance between the entropy of self-assembly, tending to decrease the size of vesicles, and a size-independent positive bending energy, tending to increase the size of vesicles. Notably, the size-independent curvature energy of unilamellar vesicles is crucial for enabling the reversible formation of vesicles with finite size.

In Figure 2, we have plotted the average vesicle radius r_v against the bilayer bending constant k_bi according to Eq. (20) at some given volume fractions of vesicles. As a result of the entropy becoming more important with decreasing ϕ_ves, the size of the vesicles increases with increasing concentration, similar to the growth behaviour of surfactant micelles [3]. The impact of bending properties on the size of equilibrium vesicles is significant. Vesicles in the range r_v = 2–100 in dilute solutions are obtained in a rather narrow span of k_bi around kT, and the vesicle size increases with increasing vesicle bending energy α_v = 4πk_bi/kT.

Figure 2.

Reduced vesicle midplane radius r_v = R_v/ξ plotted against the bilayer bending constant k_bi at volume fractions ϕ_ves = 0.001 (dotted line), 0.01 (solid line), and 0.1 (dashed line) of vesicle bilayers in accordance with Eqs. (19) and (20). Dash-dotted lines are calculated from the approximate equation (23).

5. Comparison with geometrically open bilayer disks

The following expression for the curvature free energy of a circular disk with a semi-toroidal rim (cf.Figure 3) as a function of the dimensionless radius may be derived from Eq. (9)

Figure 3.

Schematic picture of a disk consisting of a planar circular bilayer, with radius Rd and thickness 2ξ, surrounded by a semi-toroidal rim vid radius ξ. The upper image shows the disk when the bilayer is parallel to the paper plane. The lower image is seen from the edge side of the disk when the bilayer is perpendicular to the paper plane.

$${\mathrm{E}}_{\mathit{\text{disk}}}\left(r_d\right)={\alpha }_d+\mathit{\pi \beta }{r}_d+2\mathit{\pi \lambda }{r}_d^2$$E24

The dimensionless disk radius is defined as r_d = R_d/ξ, where R_d is the radius of the central planar bilayer part of the disks and ξ, as before, is half the bilayer thickness (cf.Figure 3). In Eq. (24), we have introduced the two dimensionless parameters

$${\alpha }_d=\frac{2\pi }{\mathit{kT}}\left(3k_c+2{\overline{k}}_c-8\xi k_c{H}_0\right)$$E25

$$\beta =\frac{\pi }{\mathit{kT}}\left(k_c-4\xi k_c{H}_0\right)$$E26

that reflect two terms with different size dependence in the expression for the curvature energy of disks. Both α_d and β depend on the three bending elasticity constants defined in the Helfrich expression in Eq. (8).

According to Eqs. (11) and (26), a transition from micelles to bilayers is expected as β = 0. Negative β-values mean that micelles are more favourable than bilayers and, hence, various bilayer structures, including disks, are expected to form as β > 0.

The total volume fraction of surfactants present as disks can be written as

$${\varphi }_{\mathit{\text{disk}}}=\frac{\pi {\xi }^3}{v}{e}^{-{\alpha }_d}\underset{1}{\overset{\infty }{\int }}\left(4r_d+1\right)e^{-\left(\mathit{\pi \beta }{r}_d+2\mathit{\pi \lambda }{r}_d^2\right)}d{r}_d$$E27

In accordance with Eq. (27), the formation of bilayer disks is limited as a maximum value of the volume fraction of disks is reached in the limit λ → 0 equalling

$${\varphi }_{\mathit{\text{disk}}}=\frac{{\xi }^3}{\mathit{v\beta }}\left(5+\frac{4}{\mathit{\pi \beta }}\right)e^{-{\alpha }_d-\mathit{\pi \beta }}$$E28

Hence, it follows that an appreciable amount of disks is only predicted to form close to the micelle-to-bilayer transition as the parameter β ≈ 0 and Eq. (28) may be simplified to

$$\beta =\frac{2{\xi }^{3/2}}{\sqrt{\mathit{v\pi }{\varphi }_{\mathit{\text{disk}}}}}{e}^{-\pi \left(k_c+2{\overline{k}}_c\right)/\mathit{kT}}$$E29

Furthermore, β may assume low values only as $k_c+2{\overline{k}}_c$ is larger than about unity. Likewise, the average dimensionless disk radius is obtained from the expression

$$⟨r_d⟩=\frac{\underset{1}{\overset{\infty }{\int }}{r}_d\left(4r_d+1\right)e^{-\pi \left(\beta r_d+2\lambda r_d^2\right)}d{r}_d}{\underset{1}{\overset{\infty }{\int }}\left(4r_d+1\right)e^{-\pi \left(\beta r_d+2\lambda r_d^2\right)}d{r}_d}$$E30

In the limit λ → 0, Eq. (30) turns into

$$⟨r_d⟩=\frac{5{\pi }^2{\beta }^2+9\mathit{\pi \beta }+8}{5{\pi }^2{\beta }^2+4\mathit{\pi \beta }}=1+\frac{5\mathit{\pi \beta }+8}{\mathit{\pi \beta }\left(5\mathit{\pi \beta }+4\right)}=1+\frac{2}{\mathit{\pi \beta }}-\frac{5}{5\mathit{\pi \beta }+4}$$E31

It follows from Eq. (31) that appreciably large disk radii, that is $⟨r_d⟩\gg 1$, are only possible close to the micelle-to-bilayer transition (β ≈ 0) as $k_c+2{\overline{k}}_c\gtrsim 1$. ⟨r_d⟩ larger than about unity cannot be attained for positive values of β, that is in the regime where bilayer aggregates predominate over micelles. The term including β in Eq. (24) stems from the curvature of the semi-toroidal rim, and positive values mean that the rim is unfavourable as compared to the central planar part of the disks. Even in the limit λ → 0, the disks are prevented to grow to a finite size of appreciable magnitude due to the unfavourable rim energy.

Both the unilamellar vesicle and the bilayer disk are geometrically heterogeneous aggregates composed of different geometrical parts. However, because of the detailed geometry of the two bilayer structures, they are expected to behave completely differently from a thermodynamic point of view. The two geometrical parts of a vesicle, the positively curved outer layer and the negatively curved inner layer, together give rise to a size-independent bending energy that enables the formation of an equilibrium distribution of unilamellar vesicles of finite size in so far $H_0<1/4\xi$. On the other hand, the geometrically heterogeneous structure of a bilayer disk, with a central planar bilayer part and a half toroidal rim, gives rise to two curvature free energy terms that scale differently with respect to aggregate size. The formation of bilayer disks of reasonable size is only promoted by small β values (corresponding to $H_0\approx 1/4\xi$), but so are micelles over bilayer aggregates since a transition from bilayers to micelles is expected as β turns to negative values. This means that in contrast to unilamellar vesicles, the geometrically composed structure of bilayer disks prevents the formation of large disks in the regime where bilayers are favoured over micelles.

6. Influence of bending elasticity on the size of unilamellar vesicles

According to Eqs (16) and (17), the bilayer bending constant k_bi and bending energy α_v increase with increasing bending rigidity k_c, increasing saddle-splay constant ${\overline{k}}_c,$ and decreasing effective spontaneous curvature k_cH₀. Moreover, according to Eq. (23), the size of equilibrated vesicles follows similar trends with the three bending elasticity constants. This means that a small size of unilamellar vesicles is favoured by small values of k_c and ${\overline{k}}_c$, and high values of k_cH₀.

The spontaneous curvature (H₀) represents the sign and magnitude of the preferential curvature of a single surfactant monolayer. The dependence of k_cH₀ on the size of unilamellar vesicles is the result of the outer vesicle monolayer always being more voluminous than the inner layer with a larger number of surfactant/lipid molecules. This tendency increases in magnitude with decreasing vesicle size. A small vesicle size means larger curvature of the outer vesicle layer and smaller (i.e. more negative) curvature of the inner layer, and since effects in the outer layer trump effects in the inner layer, vesicles decrease in size with increasing k_cH₀.

The bending rigidity (k_c) is a measure of the ability of a monolayer to resist deviations from a uniform mean curvature equal to the spontaneous curvature. k_c must always be a positive quantity in order to realize a minimum of γ as a function of mean curvature in Eq. (8). Large bending rigidities favour geometrically homogeneous aggregates with a uniform curvature, or in the case of geometrically heterogeneous aggregates, smaller deviations in curvature between the different geometrical parts. This means that k_c usually mainly influences the shape, but not the size, of an aggregate, which is true, for instance, for geometrically homogeneous spherical microemulsion droplets. However, in the case of a unilamellar vesicle, with a positively curved outer layer and a negatively curved inner layer, the difference in curvature between the two geometrical parts decreases in magnitude with increasing vesicle size. This means that, in contrast to micelles and microemulsion droplets, the bending rigidity has an explicit influence on the size of unilamellar vesicles. Hence, small vesicles are favoured by low values of k_c.

The third bending elasticity constant, the saddle-splay constant (${\overline{k}}_c$), is related to the Gaussian curvature K. As implied by its name, high positive values of ${\overline{k}}_c$ influence the curvature of an interface to favour a saddle-like structure, with negative Gaussian curvature, that is curvatures with opposite signs in perpendicular directions. According to the Gauss-Bonnet theorem, the last integral in Eq. (9) is always equal

$$\int \mathit{KdA}=4\pi \left(1-g\right)$$E32

where the genus g represents the number of handles or holes present in a surfactant monolayer. From a geometrical point of view, a vesicle consists of two closed interfaces (i.e. the inner and outer monolayers, respectively) giving g = −1. As a result, Eq. (32) equals $8\pi {\overline{k}}_c$, the quantity of which does not depend on the size of the vesicle. Hence, from a mathematical perspective, ${\overline{k}}_c$ has a similar impact as k_cH₀ and k_c, contributing a size-independent term to k_bi and α_v. Consequently, small vesicles are favoured by small (possibly negative) values of ${\overline{k}}_c.$ We may also note that vesicles belong to a different topology (g = −1) than micelles (g = 0). This means that micelles, with a higher genus number, are favoured with respect to vesicles as ${\overline{k}}_c$assumes positive values, whereas the opposite holds (vesicles favoured with respect to micelles) when ${\overline{k}}_c$ is negative. This effect is taken into account in the simple relation in Eq. (11).

7. Effects of molecular structure on the bending elasticity constants

Whether unilamellar vesicles may form spontaneously or not, or the size of equilibrium vesicles, largely depends on the chemical structure of the amphiphilic components making up a bilayer. It also depends on environmental quantities such as ionic strength, pH, and temperature. Hence, the size of spontaneous unilamellar vesicles may be rationalized by means of investigating the molecular interpretation of the various bending elasticity constants.

The main influence from the head group of amphiphilic molecules comes from the molecular charge number, which gives rise to a surface charge in the head group layers in a bilayer. The effect of surface charge density occurring from ionic surfactants and lipids on various bending elasticity constants may be calculated from the Gouy-Chapman theory [24]. Electrostatic effects tend to raise both spontaneous curvature and bending rigidity while contributing a negative value to the saddle-splay constant. Bilayer aggregates are expected to predominate over micelles at lower values of the spontaneous curvature according to Eq. (11). This means that the presence of smaller vesicles is usually promoted by small values of k_c, rather than large H₀, and lower surface charge densities.

The head group of a nonionic surfactant appears to have a similar effect of promoting high and positive values of the effective spontaneous curvature k_cH₀ [18]. The effect on bending rigidity appears to be more complicated and it has been theoretically demonstrated that a maximum k_c appears at a certain hydrophilic-lipophilic balance (HLB) of a nonionic surfactant [18].

It has been demonstrated that additional contributions to all bending elasticity constants, due to the finite thickness of the hydrophobic part, appear for a monolayer composed of surfactant or amphiphilic lipid with flexible hydrocarbon tails [16]. Most interestingly, this contribution is completely absent for molecules with a rigid tail. The hydrophobic finite thickness effects tend to substantially lower the effective spontaneous curvature k_cH₀ as well as increase the bending rigidity k_c. Lowering k_cH₀ tends to favour bilayer aggregates over micelles and a flexible hydrocarbon tail of at least one component seems to be necessary for a bilayer to form at all. However, the finite thickness effects also tend to raise k_bi and the vesicle bending energy, thus promoting larger vesicles rather than smaller vesicles. The flexibility of the hydrophobic part may be reduced by adding one or more double bonds to an aliphatic chain. Most interestingly, conspicuously small (ultrasmall) unilamellar vesicles have been observed to form spontaneously when a drug surfactant is mixed with a phosphatidyl choline phospholipid with unsaturated hydrophobic tail, but not with a phospholipid with a saturated tail [11].

8. Effect of mixing amphiphilic molecules

The spontaneous formation of unilamellar vesicles has virtually exclusively been observed in surfactant/surfactant or surfactant/lipid mixtures. Most interestingly, when investigating the three bending elasticity constants k_cH₀, k_c, and ${\overline{k}}_c$, it turns out that the bending rigidity behaves fundamentally differently than the other two constants in the sense that k_c has an explicit contribution from the effect of mixing two amphiphilic molecules that always brings it down [17, 20, 25, 26]. The effect is enhanced as the asymmetry between two components is increased and k_c assumes a minimum magnitude as the composition in the aggregates is optimized. By means of reducing k_c while leaving k_cH₀ approximately unchanged, it is possible to promote the formation of small unilamellar vesicles without promoting micelle formation.

Figure 4 shows model calculations of the mixing contribution to the bending rigidity k_c for mixtures of an ionic and nonionic surfactant with identical flexible hydrophobic tails [17]. It is seen that pure mixing effects might significantly bring down k_c, and there exists an optimized composition in the bilayer aggregates where k_c is expected to reach a minimum value. It has been demonstrated that an ionic single-tailed/nonionic double-tailed surfactant mixture is more asymmetric than an ionic single-tailed/nonionic single-tailed surfactant mixture, giving rise to an even more negative mixing contribution to k_c [17].

Figure 4.

Explicit mixing contribution to the bending rigidity k_c for a mixture of an ionic and a nonionic surfactant with identical tails [17].

The reason for the reduction of bending rigidity, and as a consequence, the bilayer bending constant as well as the average size of vesicles, is that the composition in a monolayer consisting of two surfactants with different spontaneous curvatures becomes a strong function of curvature. As a result, different equilibrium compositions in the outer and inner monolayers of a vesicle, where the amphiphilic component with a high spontaneous curvature prefers the outer layer and the component with lower H₀ prefers the inner layer, give rise to lower bending rigidity and smaller vesicles (cf.Figure 5).

Figure 5.

Schematic images of a monolayer and a bilayer composed of two amphiphilic components. The component with the red head group prefers to be located in parts with a positive curvature and the component with the blue head group to be located in parts with negative curvature [17].

Spontaneous formation of small unilamellar vesicles has been observed in several aqueous systems of two oppositely charged surfactants, where one of the surfactants is present in excess. In this case, the asymmetry between the surfactants comes from the different head group charge numbers, giving rise to a difference in charge density between the two monolayers. The surfactant in excess prefers the outer layer, implying a higher charge density in the outer positively curved layer and a lower charge density in the inner layer that is preferred by the surfactant in deficit. The difference in charge number between two surfactants with z = +1 and −1 is two, which enhances the reduction in k_c and k_bi as compared with a mixture of a monovalent ionic surfactant and a nonionic surfactant [17]. Moreover, mixing an ionic single-tailed surfactant with a double-tailed oppositely charged surfactant, in excess of the single-tailed surfactant, enhances the asymmetry between the surfactants and, as a consequence, the reduction of k_c is enhanced. As a matter of fact, ultrasmall unilamellar vesicles have been observed in mixtures of the ionic single-tailed surfactant sodium dodecyl sulphate (SDS) with the double-tailed cationic surfactant didodecyldimethyl ammonium bromide (DDAB) dissolved in water in excess of SDS [27].

Ultrasmall vesicles, with a diameter less than 20 nm, have also recently been observed to form spontaneously in mixtures of an ionic surfactant with short and rigid tail and a zwitterionic phospholipid at high ionic strength [11]. In this case, there is a large asymmetry in spontaneous curvature with the small surfactant preferring to be located in the outer layer and the phospholipid preferring the inner layer of the vesicle. In a dilution series, the vesicles are found to reach a minimum size at some optimal surfactant/lipid composition.

Similar to a vesicle, a bilayer disk is a geometrically heterogeneous structure composed of a central planar bilayer surrounded by a more curved semi-toroidal rim at the edge of the disk. In a binary mixture of amphiphilic components, the component with higher spontaneous curvature prefers to be located in the rim, whereas the other component prefers to be located in the planar central part. As for vesicles, the segregation of components has an explicit impact on bending rigidity k_c but not on the other bending elasticity constants. Lowering k_c by means of mixing amphiphilic components will reduce the bending parameter α_d in Eq. (25), but this effect cannot promote the formation of large disks, since the formation of the latter is restricted to the vicinity of the micelle-to-bilayer transition as β ≈ 0.

9. Concentrated vesicle solutions and overpacking

The theoretical arguments presented so far are strictly valid for a dilute solution of vesicles where inter-aggregate interactions have been neglected. Since vesicles, for geometrical reasons, are comparatively voluminous structures, including a substantial amount of aqueous phase in the interior of the closed vesicle bilayer, they may become overpacked above a certain concentration of total amphiphilic component. This limit is significantly reduced as the size of the vesicles increases. Assuming a simple cubic packing of the vesicles, the overpacking limit, in terms of the solute volume fraction where spherical vesicles are closely packed, can be calculated from the following simple relation

$${\varphi }_{\mathit{\text{pack}}}=\frac{8\pi R_v^2\xi }{{\left(2R_v\right)}^3}=\frac{\pi }{r_v}$$E33

where r_v = R_v/ξ. Hence, Eq. (33) reveals the overpacking limit to be inversely proportional to the vesicle radius. This means that comparatively large vesicles might be overpacked at very low surfactant/lipid concentrations. Moreover, inter-aggregate interactions may play a significant role even at substantially lower concentrations than ϕ_pack. A consequence of these interactions is that geometrically closed vesicles open up to form bilayer disks that may pack more densely by means of excluding the water trapped in the vesicle cores. According to Eq. (33), the overpacking limit for substantially large vesicles, that is r_v larger than 100, is lower than a few volumes per cents. This is consistent with the experimental observations that a significant amount of disks are usually present in solutions of comparatively large vesicles and the amount of disks seems to increase with increasing vesicle size [7, 27, 28].

10. Conclusions

Liposomes shaped as unilamellar vesicles may form spontaneously in certain surfactant-phospholipid mixtures and it is possible to manufacture a novel type of thermodynamically stable colloidal solution with similar properties as micellar and microemulsion solutions [11]. This kind of solution consists of rather small vesicles, the size of which is stable over time in a fixed environment and the solution may become reversibly reproduced when returning to its original environmental conditions. We have demonstrated the theoretical principles of colloidal liposome solutions by combining bending elasticity theory and solution thermodynamics. In particular, we have shown that geometrically closed bilayer vesicles may be equilibrium structures with a wide range of finite average sizes. The work of bending a planar bilayer into a geometrically closed vesicle is constant with vesicle size. As a result, equilibrium vesicles with finite size may form as a result of a balance between positive bending energy, favouring larger vesicles, and entropy of self-assembly favouring smaller vesicles. Hence, the average size of equilibrated vesicles increases with increasing bending energy (or bilayer bending constant) as well as increasing vesicle bilayer concentration. Moreover, bilayer aggregates such as vesicles and disks are expected to predominate over micelles in so far the spontaneous curvature falls below H₀ = 1/4ξ, where ξ denotes the monolayer thickness. Small unilamellar vesicles are favoured by small values of the bilayer bending constant $k_{\mathit{bi}}=2\left(2k_c+{\overline{k}}_c-4k_c\xi H_0\right)$. k_bi may be reduced by means of increasing the spontaneous curvature. However, increasing H₀ favours micelles over bilayer aggregates. A more efficient way to reduce k_bi to magnitudes compatible with small unilamellar vesicles, while still being in the region where bilayers predominate over micelles, is to reduce the bending rigidity k_c. Most interestingly, k_c may be reduced by means of mixing two amphiphilic components with different spontaneous curvatures, where one of the components prefers the outer, oppositely curved, monolayer, whereas the other prefers the inner, negatively curved, layer.

In contrast to vesicles, bilayer lipodisks may only be stable under exceptional circumstances, such as near a micelle-bilayer coexistence region or at amphiphile concentrations close to or above the overpacking limit of vesicles. The positively curved rim of disks cannot be stable in a regime where bilayers are favoured over micelles by low spontaneous curvature of its components. Reducing bending rigidity by means of mixing asymmetric components has no impact on the thermodynamic stability of non-interacting disks. However, disks may become more stable at higher concentrations, above the overpacking limit of unilamellar vesicles, because disks are able to pack more densely than vesicles. Since the overpacking limit concentration depends on vesicle size (it is proportional to one over vesicle radius), disks are expected to be favoured by large values of the bilayer bending constant k_bi, large bilayer aggregate sizes, and high concentrations of amphiphilic components.