^{1,a)}, S. A. Safran

^{2}, T. Sottmann

^{3}and R. Strey

^{3}

### Abstract

The equilibrium microstructures in microemulsions and other self-assembled systems show complex, connected shapes such as symmetric bicontinuous spongelike structures and asymmetric bicontinuous networks formed by cylinders interconnected at junctions. In microemulsions, these cylinder network microstructures may mediate the structural transition from a spherical or globular phase to the bicontinuous microstructure. To understand the structural and statistical properties of such cylinder network microstructures as measured by scattering experiments, models are needed to extract the real-space structure from the scattering data. In this paper, we calculate the scattering functions appropriate for cylinder network microstructures. We focus on such networks that contain a high density of network junctions that connect the cylindrical elements. In this limit, the network microstructure can be regarded as an assembly of randomly oriented, closed packed network junctions (i.e., the cylinder scattering contributions are neglected). Accordingly, the scattering spectrum of the network microstructure can be calculated as the product of the junction number density, the junction form factor, which describes the scattering from the surface of a single junction, and a structure factor, which describes the local correlations of different junctions due to junction interactions (including their excluded volume). This approach is applied to analyze the scattering data from a bicontinuous microemulsion with equal volumes of water and oil. In a second approach, we included the cylinder scattering contribution in the junction form factor by calculating the scattering intensity of junctions to which three rods with spherical cross section are attached. The respective theoretical predictions are compared with results of neutron scatteringmeasurements on a water-in-oil microemulsion with a connected microstructure.

The authors thank T. Tlusty and A. Besser for stimulating discussions, and H. Frielinghaus, A. Radulescu, and S. Maccarrone for support with the neutron scattering measurements at the Juelich Research Center. Furthermore they thank Travis Hodgdon (University of Delaware) for his careful reading of the manuscript. Support from the European Union Network of Excellence *SoftComp*, the German-Israeli Foundation, and the Schmidt Minerva Center are gratefully acknowledged.

I. INTRODUCTION

II. JUNCTION SCATTERINGFORM FACTOR

A. Junction surface scattering function

B. -star scattering function

III. RESULTS AND COMPARISON WITH EXPERIMENTS

A. Junction surface scattering

B. -star scattering

IV. CONCLUSION

### Key Topics

- Form factors
- 43.0
- Networks
- 38.0
- Microemulsions
- 35.0
- Surfactants
- 27.0
- Neutron scattering
- 18.0

## Figures

Geometry of the junction surface: (a) Picture of the network junction with three cylinders attached, taken from Ref. 32. (b) Definition of the relevant lengths by a cut through the junction in the plane of the coordinate system: The junction consists of three semitoroidal parts (gray) and six flat pieces (white, in the middle of the three semitoroidal parts; two flat pieces belong to each semitoroidal part, respectively, whereby one flat piece is above the plane, the other one below the plane). (c) The vector describes the points on the junction surface and can be decomposed into the vectors and , where the angles and define the orientation of the vector in space.

Geometry of the junction surface: (a) Picture of the network junction with three cylinders attached, taken from Ref. 32. (b) Definition of the relevant lengths by a cut through the junction in the plane of the coordinate system: The junction consists of three semitoroidal parts (gray) and six flat pieces (white, in the middle of the three semitoroidal parts; two flat pieces belong to each semitoroidal part, respectively, whereby one flat piece is above the plane, the other one below the plane). (c) The vector describes the points on the junction surface and can be decomposed into the vectors and , where the angles and define the orientation of the vector in space.

(a) Geometry of the star, which is a network junction to which three rods are attached. Each star consists of three branches (, , and ) with respective lengths , , and . (b) Definition of the angles and that define the orientation of the vector in space.

(a) Geometry of the star, which is a network junction to which three rods are attached. Each star consists of three branches (, , and ) with respective lengths , , and . (b) Definition of the angles and that define the orientation of the vector in space.

Normalized form factor for junction surface scattering as a function of the dimensionless wave vector [ radius, see Fig. 1(b)], for various values of the structural parameter (which defines the radius relative to ) for (a) fixed junction orientations in space (the lines for different values of were shifted by factors in order to show them in the same plot), and (b) for an ensemble of randomly oriented junctions.

Normalized form factor for junction surface scattering as a function of the dimensionless wave vector [ radius, see Fig. 1(b)], for various values of the structural parameter (which defines the radius relative to ) for (a) fixed junction orientations in space (the lines for different values of were shifted by factors in order to show them in the same plot), and (b) for an ensemble of randomly oriented junctions.

Normalized form factors as a function of the dimensionless wave vector [ radius, see Fig. 1(b)]; form factors are shown for surface scattering from an ensemble of randomly oriented junctions with spherical cross section (black full line), and with rectangular cross section with polydispersity in the tube radius , as the standard deviation, , of the radius distribution function is varied.

Normalized form factors as a function of the dimensionless wave vector [ radius, see Fig. 1(b)]; form factors are shown for surface scattering from an ensemble of randomly oriented junctions with spherical cross section (black full line), and with rectangular cross section with polydispersity in the tube radius , as the standard deviation, , of the radius distribution function is varied.

Comparison of the calculated junction form factor with experimental data: (a) Small angle neutron scattering data (empty symbols, taken from Ref. 52) for a microemulsion with bicontinuous microstructure containing , -octane(d18) and (film contrast). The junction form factor (, , light gray line, overlapped by the black line for large , as indicated by the dashed illustration) with polydispersity in the tube radius, , fits the experimental data at intermediate and high values. Dividing the experimental data by this form factor allows us to deduce the effective experimental structure factor [, crossed symbols]. This experimental structure factor is fitted with a structure factor for particles that interact with a “soft” Gaussian potential [, dark gray line]. Thus the experimental intensity can be quantitatively reproduced (black line) by multiplying the junction form factor with the theoretical structure factor .

Comparison of the calculated junction form factor with experimental data: (a) Small angle neutron scattering data (empty symbols, taken from Ref. 52) for a microemulsion with bicontinuous microstructure containing , -octane(d18) and (film contrast). The junction form factor (, , light gray line, overlapped by the black line for large , as indicated by the dashed illustration) with polydispersity in the tube radius, , fits the experimental data at intermediate and high values. Dividing the experimental data by this form factor allows us to deduce the effective experimental structure factor [, crossed symbols]. This experimental structure factor is fitted with a structure factor for particles that interact with a “soft” Gaussian potential [, dark gray line]. Thus the experimental intensity can be quantitatively reproduced (black line) by multiplying the junction form factor with the theoretical structure factor .

(a) -star scattering intensity as a function of the dimensionless scattering vector for a ratio of the branch length and the cylinder radius without (light gray line) and with cylinder cross-section scattering function (black line). The -star scattering intensity multiplied with cylinder cross-section scattering function is also calculated for (dark gray line). The low- and intermediate- scattering is dominated by the branch-branch cross correlation (dashed line) and the rodlike branch self-correlation (dash-dotted line). (b) Comparison of the calculated -star scattering intensity with neutron scattering data from an oil-rich --decane(d22)- microemulsion (film contrast) with a network microstructure. The -star scattering intensity with polydispersity in the branch lengths (width of the length distribution function: , black line) agrees much better with the experimental data than the two rod-form factors (rod lengths for black dashed line and for gray dashed line). In all fits, a cylinder cross-section radius of is used.

(a) -star scattering intensity as a function of the dimensionless scattering vector for a ratio of the branch length and the cylinder radius without (light gray line) and with cylinder cross-section scattering function (black line). The -star scattering intensity multiplied with cylinder cross-section scattering function is also calculated for (dark gray line). The low- and intermediate- scattering is dominated by the branch-branch cross correlation (dashed line) and the rodlike branch self-correlation (dash-dotted line). (b) Comparison of the calculated -star scattering intensity with neutron scattering data from an oil-rich --decane(d22)- microemulsion (film contrast) with a network microstructure. The -star scattering intensity with polydispersity in the branch lengths (width of the length distribution function: , black line) agrees much better with the experimental data than the two rod-form factors (rod lengths for black dashed line and for gray dashed line). In all fits, a cylinder cross-section radius of is used.

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