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.
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.
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.
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