^{1}, Gerhard Gompper

^{1}and Daniel M. Kroll

^{2}

### Abstract

Monte Carlo simulations of dynamically triangulated surfaces of variable topology are used to investigate the scattering intensities of bicontinuous microemulsions. The bulk scattering intensity is shown to follow the Teubner-Strey expression. The domain size and the correlation length are extracted from the scattering peaks as a function of the bending rigidity, saddle-splay modulus, and surfactant density. The results are compared to earlier theories based on Ginzburg-Landau and Gaussian random field models. The ratio of the two length scales is shown to be well described by a linear combination of logarithmically renormalized bending rigidity and saddle-splay modulus with universal prefactors. This is in contrast to earlier theoretical predictions in which the scattering intensity is independent of the saddle-splay modulus. The equation of state, and the asymptotics of the bulk and film scattering intensities for high and low wave vectors are determined from simulations and compared with theoretical results.

Discussions with T. Auth, H. Frielinghaus, O. Holderer, M. Monkenbusch, and D. Richter are gratefully acknowledged.

I. INTRODUCTION

II. THEORETICAL BACKGROUND

A. Ginzburg-Landau theory

B. Gaussian random field model and membrane curvature elasticity

C. Bending rigidity renormalization

D. Film scattering

E. Porod laws

III. SIMULATIONS

A. Monte Carlo of dynamically triangulated surfaces

B. Fluctuating topology

IV. RESULTS

A. Equation of state

B. Bulk scattering

C. Film scattering

D. Topology

V. DISCUSSION AND CONCLUSIONS

### Key Topics

- Surfactants
- 38.0
- Microemulsions
- 33.0
- Ginzburg Landau theory
- 28.0
- Topology
- 17.0
- X-ray scattering
- 17.0

## Figures

Monte Carlo step for changing the topology of a triangulated surface. (a) Two surface triangles are removed. (b) The two holes are connected by a prism of six triangles.

Monte Carlo step for changing the topology of a triangulated surface. (a) Two surface triangles are removed. (b) The two holes are connected by a prism of six triangles.

Typical configurations from simulations for *N* = 2207 beads for κ = 3.5, , and *p* = 0.01 (upper panel) as well as for κ = 5.0, , and *p* = 0.03 (lower panel). The two sides of the membrane are colored differently to make the bicontinuous structure more easily visible. A view of the structure for κ = 3.5, , and *p* = 0.01 from different angles to reveal the shape and topology more clearly, and an animation of the temporal evolution of the microemulsion structure are shown (enhanced online) (a), (b). [URL: http://dx.doi.org/10.1063/1.3701265.1] [URL: http://dx.doi.org/10.1063/1.3701265.2]10.1063/1.3701265.110.1063/1.3701265.2

Typical configurations from simulations for *N* = 2207 beads for κ = 3.5, , and *p* = 0.01 (upper panel) as well as for κ = 5.0, , and *p* = 0.03 (lower panel). The two sides of the membrane are colored differently to make the bicontinuous structure more easily visible. A view of the structure for κ = 3.5, , and *p* = 0.01 from different angles to reveal the shape and topology more clearly, and an animation of the temporal evolution of the microemulsion structure are shown (enhanced online) (a), (b). [URL: http://dx.doi.org/10.1063/1.3701265.1] [URL: http://dx.doi.org/10.1063/1.3701265.2]10.1063/1.3701265.110.1063/1.3701265.2

The equation of state [*p*(ϕ)δ^{3}/*T*]ϕ^{−3} vs. ln (ϕ) of sponge phases for several values of κ and as indicated by the legend. All data are for system size *N* = 2207. The symbols show the simulation results, and the thick dotted lines are the fits as discussed in the text.

The equation of state [*p*(ϕ)δ^{3}/*T*]ϕ^{−3} vs. ln (ϕ) of sponge phases for several values of κ and as indicated by the legend. All data are for system size *N* = 2207. The symbols show the simulation results, and the thick dotted lines are the fits as discussed in the text.

Bulk scattering intensity *S* _{ b }(*q*) (symbols) as a function of the wave vector *q* for *p* = 0.1, , *N* = 2207, and several values of the bending rigidity κ (upper panel), and for , *N* = 2207, and several values of the bending rigidity κ and the osmotic pressure *p* (lower panel). In the upper panel, the solid lines are fits corresponding to the Teubner-Strey formula, Eq. (5), with the line colors matching the symbol colors in the corresponding simulation data.

Bulk scattering intensity *S* _{ b }(*q*) (symbols) as a function of the wave vector *q* for *p* = 0.1, , *N* = 2207, and several values of the bending rigidity κ (upper panel), and for , *N* = 2207, and several values of the bending rigidity κ and the osmotic pressure *p* (lower panel). In the upper panel, the solid lines are fits corresponding to the Teubner-Strey formula, Eq. (5), with the line colors matching the symbol colors in the corresponding simulation data.

The amplitude of the large-*q* tail of the bulk scattering intensity *S* _{ b }(*q*) as a function of *S*/*V* for several values of κ and , together with a linear fit forced to go through the origin.

The amplitude of the large-*q* tail of the bulk scattering intensity *S* _{ b }(*q*) as a function of *S*/*V* for several values of κ and , together with a linear fit forced to go through the origin.

The scaled domain size, *dS*/*V*, as a function of *S*/*V* for several values of κ and for system size *N* = 2207. The inset shows the same for *N* = 607.

The scaled domain size, *dS*/*V*, as a function of *S*/*V* for several values of κ and for system size *N* = 2207. The inset shows the same for *N* = 607.

The correlation length ξ*S*/*V* as a function of *S*/*V* for several values of κ and for system size *N* = 2207. The inset shows the same for *N* = 607.

The correlation length ξ*S*/*V* as a function of *S*/*V* for several values of κ and for system size *N* = 2207. The inset shows the same for *N* = 607.

The length scale ratio *k* _{0}ξ as a function of *S*/*V* for several values of κ and for system sizes (a) *N* = 2207 and (b) *N* = 607. The symbols show the simulation results and the dashed-dotted lines are the fits discussed in the text. The color coding indicates which values of κ and the dashed-dotted lines correspond to.

The length scale ratio *k* _{0}ξ as a function of *S*/*V* for several values of κ and for system sizes (a) *N* = 2207 and (b) *N* = 607. The symbols show the simulation results and the dashed-dotted lines are the fits discussed in the text. The color coding indicates which values of κ and the dashed-dotted lines correspond to.

Film scattering intensity *S* _{ f }(*q*) as a function of the wave vector *q* for *p* = 0.01, , *N* _{ p } = 2207, and several values of the bending rigidity κ together with a power law fit to the tail (line shifted upwards for clarity).

Film scattering intensity *S* _{ f }(*q*) as a function of the wave vector *q* for *p* = 0.01, , *N* _{ p } = 2207, and several values of the bending rigidity κ together with a power law fit to the tail (line shifted upwards for clarity).

The amplitude *a* _{ f } of the large-*q* tail of the film scattering intensity *S* _{ f }(*q*) as a function of *S*/*V* for several values of κ an , together with a linear fit forced to go through the origin.

The amplitude *a* _{ f } of the large-*q* tail of the film scattering intensity *S* _{ f }(*q*) as a function of *S*/*V* for several values of κ an , together with a linear fit forced to go through the origin.

The small-*q* behavior of the film scattering intensity for various cases. (a) , *p* = 0.1, *N* = 2207, and κ varies as indicated by the legend. (b) κ = 3.0, *p* = 0.01, *N* = 2207, and varies as indicated by the legend. The main panels show the small-*q* part of *S* _{ f }(*q*) on linear scale, where the full lines are fits to Eq. (19). The insets display the full scattering functions on double-logarithmic scales. The colors of the data sets in the insets match those in the main figures.

The small-*q* behavior of the film scattering intensity for various cases. (a) , *p* = 0.1, *N* = 2207, and κ varies as indicated by the legend. (b) κ = 3.0, *p* = 0.01, *N* = 2207, and varies as indicated by the legend. The main panels show the small-*q* part of *S* _{ f }(*q*) on linear scale, where the full lines are fits to Eq. (19). The insets display the full scattering functions on double-logarithmic scales. The colors of the data sets in the insets match those in the main figures.

The scaled Euler characteristic γ = 〈χ_{ E } *V* ^{2} *S* ^{−3}〉, as a function of the surfactant density δ*S*/*V*. The symbols denote the simulation data for various values of κ and , as indicated. The dashed-dotted lines are a least-squares fit to Eq. (38). The inset shows the constant term γ_{0} in Eq. (38) as a function of κ for different values of . The solid lines are guides to the eye.

The scaled Euler characteristic γ = 〈χ_{ E } *V* ^{2} *S* ^{−3}〉, as a function of the surfactant density δ*S*/*V*. The symbols denote the simulation data for various values of κ and , as indicated. The dashed-dotted lines are a least-squares fit to Eq. (38). The inset shows the constant term γ_{0} in Eq. (38) as a function of κ for different values of . The solid lines are guides to the eye.

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