^{1}

### Abstract

Self-consistent field (SCF) theory is used to find bending moduli of surfactant and lipid bilayers. Recently, we successfully applied low-memory search methods to solve the SCF equations. Using these we are now able to directly evaluate the Gaussian bending modulus for molecularly detailed models of bilayers by evaluating the excess Helmholtz energy of tensionless bilayers in a (part of the) Im3m cubic phase. The result prompted us to reconsider the protocol that has been used thus far to find the mean bending modulus *k* _{ c } and Gaussian bending modulus . With respect to previous predictions, the value of *k* _{ c } is reduced by a factor of two and the Gaussian bending modulus is less negative and much closer to zero. In line with experimental data we now find that can also become positive. In this paper we use the non-ionic surfactants series of the type C_{ n }E_{ m } for illustration.

A large part of this work has been performed during the stay of F.L. at the laboratory of physical chemistry 1 of Lund University (Sweden). The hospitality of its members and especially P. Linse is highly appreciated. Special thanks to H. Wennerström for discussions.

I. INTRODUCTION

II. BILAYERS IN CYLINDRICAL AND SPHERICAL GEOMETRY

A. Cylindrical geometry

B. Spherical geometry

III. THE OLD PROTOCOL BASED ON CURVATURE DEPENDENCE OF GRAND POTENTIAL

A. On the use of the excess Helmholtz energy

IV. RESULTS

A. Preliminary analysis

B. Bending rigidities

C. Membrane in a saddle-shape

V. DISCUSSION

VI. CONCLUSION

### Key Topics

- Surfactants
- 61.0
- Chemical potential
- 25.0
- Vesicles
- 25.0
- Free energy
- 13.0
- Lipids
- 6.0

## Figures

(a) The volume fraction profile of head group (E) and tail segment (C) across the planar tensionless bilayer. The *z* = 0 is positioned at the symmetry plane. (b) The corresponding (planar) grand potential density ω(*z*) and excess Helmholtz density *f* ^{σ}(*z*) profiles. (c) The radial volume fraction profile of a cylindrically (continuous lines) as well as spherically (only points) curved bilayer. The radius of both vesicles was exactly *R* = 60 (from first moment analysis). The *r* = 0 coordinate is the center of the vesicle. (d) The grand potential density and excess Helmholtz density profiles for the cylindrically curved vesicle of panel (c). Membranes are composed of C_{16}E_{4} molecules with the default parameter set.

(a) The volume fraction profile of head group (E) and tail segment (C) across the planar tensionless bilayer. The *z* = 0 is positioned at the symmetry plane. (b) The corresponding (planar) grand potential density ω(*z*) and excess Helmholtz density *f* ^{σ}(*z*) profiles. (c) The radial volume fraction profile of a cylindrically (continuous lines) as well as spherically (only points) curved bilayer. The radius of both vesicles was exactly *R* = 60 (from first moment analysis). The *r* = 0 coordinate is the center of the vesicle. (d) The grand potential density and excess Helmholtz density profiles for the cylindrically curved vesicle of panel (c). Membranes are composed of C_{16}E_{4} molecules with the default parameter set.

(a) The surface tension as a function of the number of C_{16}E_{4} molecules in the bilayer per unit area Γ. (b) The equilibrium volume fraction in the bulk of C_{16}E_{4} molecules that coexists with the bilayer (equivalent to the critical micellisation concentration CMC) as a function of Γ. The vertical dotted line represents the equilibrium tensionless bilayer. The horizontal lines help to evaluate the corresponding CMC.

(a) The surface tension as a function of the number of C_{16}E_{4} molecules in the bilayer per unit area Γ. (b) The equilibrium volume fraction in the bulk of C_{16}E_{4} molecules that coexists with the bilayer (equivalent to the critical micellisation concentration CMC) as a function of Γ. The vertical dotted line represents the equilibrium tensionless bilayer. The horizontal lines help to evaluate the corresponding CMC.

Set of interaction curves wherein Δμ ≡ μ_{ surf }(*d*) − μ_{ surf }(∞) in units of *k* _{ B } *T* is plotted as a function of *d* which is half the distance between two centres of bilayers. The interaction parameter which controls the hydrophobicity of the head groups χ_{ OW } is varied as indicated. Default parameters settings for the surfactant.

Set of interaction curves wherein Δμ ≡ μ_{ surf }(*d*) − μ_{ surf }(∞) in units of *k* _{ B } *T* is plotted as a function of *d* which is half the distance between two centres of bilayers. The interaction parameter which controls the hydrophobicity of the head groups χ_{ OW } is varied as indicated. Default parameters settings for the surfactant.

(a) The chemical potential of the surfactant as a function of the imposed curvature *J* for cylindrically and spherically curved bilayers as indicated. The lines are a fit through the data. (b) The excess number of molecules per unit area of the bilayer Γ as a function of the imposed curvature *J* for cylindrically and spherically curved bilayers as indicated. The lines are a fit through the data (using the low *J* values). Membranes are composed of C_{16}E_{4} molecules with the default parameter set.

(a) The chemical potential of the surfactant as a function of the imposed curvature *J* for cylindrically and spherically curved bilayers as indicated. The lines are a fit through the data. (b) The excess number of molecules per unit area of the bilayer Γ as a function of the imposed curvature *J* for cylindrically and spherically curved bilayers as indicated. The lines are a fit through the data (using the low *J* values). Membranes are composed of C_{16}E_{4} molecules with the default parameter set.

(a) γ × 2π*R* ^{2}, and Δ*f* ^{σ} × 2π*R* ^{2} (in units *k* _{ B } *T*), being measures for *k* _{ c } and κ, respectively, as a function of the curvature of the cylindrical bilayer *J* = 1/*R*. The horizontal lines are a fit to the data for not too high *J* = 1/*R* values. (b) The grand potential Ω and Δ*F* ^{σ} ≡ Δ*f* ^{σ} × 4π*R* ^{2} as a function of the curvature *J* = 2/*R* of the spherical vesicle. Membranes are composed of C_{16}E_{4} molecules with the default parameter set.

(a) γ × 2π*R* ^{2}, and Δ*f* ^{σ} × 2π*R* ^{2} (in units *k* _{ B } *T*), being measures for *k* _{ c } and κ, respectively, as a function of the curvature of the cylindrical bilayer *J* = 1/*R*. The horizontal lines are a fit to the data for not too high *J* = 1/*R* values. (b) The grand potential Ω and Δ*F* ^{σ} ≡ Δ*f* ^{σ} × 4π*R* ^{2} as a function of the curvature *J* = 2/*R* of the spherical vesicle. Membranes are composed of C_{16}E_{4} molecules with the default parameter set.

The mean bending modulus κ (open triangle), the Gaussian bending modulus (open circles), and the effective bending modulus in spherical geometry (solid dots) (all quantities in units of *k* _{ B } *T*) as a function of (a) the surfactant tail length *n* with *m* = 4 and χ_{ OW } = −0.6, (b) the surfactant head length *m* with fixed tail length *m* = 16 and χ_{ OW } = −0.6, (c) the interaction parameter χ_{ OW } = −0.6, for fixed (*n*,*m*) = (16,40). It has been argued before that −χ_{ OW }∝*T* ^{−1}, which means that at high *T* the χ_{ OW } is low and vice versa.

The mean bending modulus κ (open triangle), the Gaussian bending modulus (open circles), and the effective bending modulus in spherical geometry (solid dots) (all quantities in units of *k* _{ B } *T*) as a function of (a) the surfactant tail length *n* with *m* = 4 and χ_{ OW } = −0.6, (b) the surfactant head length *m* with fixed tail length *m* = 16 and χ_{ OW } = −0.6, (c) the interaction parameter χ_{ OW } = −0.6, for fixed (*n*,*m*) = (16,40). It has been argued before that −χ_{ OW }∝*T* ^{−1}, which means that at high *T* the χ_{ OW } is low and vice versa.

A schematic picture of a bicontineous Im3m cubic phase. In this phase the bilayer is curved in a saddle shape. The elementary SCF box, which has reflecting boundary conditions in all directions is pointed at on the left-hand side of the drawing. Two spots with numbers (1) and (2), respectively, are indicated on two corners of this box. These spots reflect the “eye”-positions of the viewgraphes given for a 3D SCF density plot given in Figs. 8(a) and 8(c) , respectively. On the right-hand side of the drawing the molecular information is given of how the surfactant bilayer is expected to be positioned in this case. The inner water phase has a dark gray color. The outer water phase is obviously not shown, but only pointed at. The surfactant layer is given a white filling. The head groups are in intermediate gray color. The viewgraph is adopted from Ref. ^{ 33 } .

A schematic picture of a bicontineous Im3m cubic phase. In this phase the bilayer is curved in a saddle shape. The elementary SCF box, which has reflecting boundary conditions in all directions is pointed at on the left-hand side of the drawing. Two spots with numbers (1) and (2), respectively, are indicated on two corners of this box. These spots reflect the “eye”-positions of the viewgraphes given for a 3D SCF density plot given in Figs. 8(a) and 8(c) , respectively. On the right-hand side of the drawing the molecular information is given of how the surfactant bilayer is expected to be positioned in this case. The inner water phase has a dark gray color. The outer water phase is obviously not shown, but only pointed at. The surfactant layer is given a white filling. The head groups are in intermediate gray color. The viewgraph is adopted from Ref. ^{ 33 } .

Contour plots of the surfactant density in a 3 gradient calculation for C_{16}E_{4} with χ_{0W } = −0.4. In panels (a) and (c), the view positions are indicated in Fig. 7 by (1) and (2), respectively and the contours on the outer edges of the SCF-box are given. In panel (b) three cross-sections at heights *z* = 8, *z* = 17, and *z* = 28 in the *x*-*y* plane. The colors not necessarily match the core-corona composition of the surfactant layer. However, the corona layer is near the white regions, and the core is well withing the red regions. Water is given by the bright blue color. The grand potential in the box is to numerical precision zero, indicated that the optimization of the number of surfactants in the box optimized and the bilayer has a mean average curvature of zero. For more data we refer to Table I .

Contour plots of the surfactant density in a 3 gradient calculation for C_{16}E_{4} with χ_{0W } = −0.4. In panels (a) and (c), the view positions are indicated in Fig. 7 by (1) and (2), respectively and the contours on the outer edges of the SCF-box are given. In panel (b) three cross-sections at heights *z* = 8, *z* = 17, and *z* = 28 in the *x*-*y* plane. The colors not necessarily match the core-corona composition of the surfactant layer. However, the corona layer is near the white regions, and the core is well withing the red regions. Water is given by the bright blue color. The grand potential in the box is to numerical precision zero, indicated that the optimization of the number of surfactants in the box optimized and the bilayer has a mean average curvature of zero. For more data we refer to Table I .

(a) Volume fraction distributions of heads and tails for the default case (i.e., C_{16}E_{4} with χ_{ OW } = −0.4). (b) The corresponding grand potential density distributions. The points are for the tensionless planar bilayer (with the symmetry plane halfway layer *z* = 18. The continuous lines connecting the densities at lattice points in a cross-section through the cubic phase at a location where the bilayer is perpendicular to one of the axis of the unit cell (here we used the *z*-axis, but there are two other possible places where we could have done the same).

(a) Volume fraction distributions of heads and tails for the default case (i.e., C_{16}E_{4} with χ_{ OW } = −0.4). (b) The corresponding grand potential density distributions. The points are for the tensionless planar bilayer (with the symmetry plane halfway layer *z* = 18. The continuous lines connecting the densities at lattice points in a cross-section through the cubic phase at a location where the bilayer is perpendicular to one of the axis of the unit cell (here we used the *z*-axis, but there are two other possible places where we could have done the same).

The difference in volume fraction profile between the δφ(ζ) defined by the difference of the volume fraction profile across the planar and that across the radial cylindrical or spherically curved bilayers both with radius *R* = 60, as indicated. The ζ coordinate is defined with respect to the center of the bilayer (see text). Default surfactant with default parameters. Data already presented in Fig. 1 have been used to compute the differences.

The difference in volume fraction profile between the δφ(ζ) defined by the difference of the volume fraction profile across the planar and that across the radial cylindrical or spherically curved bilayers both with radius *R* = 60, as indicated. The ζ coordinate is defined with respect to the center of the bilayer (see text). Default surfactant with default parameters. Data already presented in Fig. 1 have been used to compute the differences.

## Tables

Collection of thermodynamic data of C_{16}E_{4} surfactants in (i) tensionless bilayers (“*L* _{α}”), or (ii) a relaxed cubic (“Im3m”) phase with box size *M* = 35. The bulk volume fraction of surfactants φ^{ b }, the corresponding chemical potential μ, and the number of surfactants found in the box *n* ^{σ}(Im3m).

Collection of thermodynamic data of C_{16}E_{4} surfactants in (i) tensionless bilayers (“*L* _{α}”), or (ii) a relaxed cubic (“Im3m”) phase with box size *M* = 35. The bulk volume fraction of surfactants φ^{ b }, the corresponding chemical potential μ, and the number of surfactants found in the box *n* ^{σ}(Im3m).

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