Small-Angle Scattering. Ian W. Hamley

fluctuations of layers.

Different models have been proposed to describe the thermal fluctuations and hence the structure factor terms Sk. In the thermal disorder model, each layer fluctuates with an amplitude Δ = 〈(dk − d)²〉, where d is the average lamellar spacing. The structure factor is that of an ideal one‐dimensional crystal multiplied by a Debye‐Waller factor:

      (1.47)

      In the second model, the paracrystalline model (this type of model is discussed further in Section 1.6.3), the position of an individual fluctuating layer in a paracrystal is determined solely by its nearest neighbours. Then [21, 22]

      (1.48)

      In a third alternative model, introduced by Caillé [23] and modified to allow for finite lamellar stacks [24, 25], the fluctuations are quantified in terms of the flexibility of the membranes:

      (1.49)

      Here γ is Euler's constant and

      (1.50)

      is the Caillé parameter, which depends on the bulk compression modulus B and the bending rigidity K of the layers.

      1.6.2 Periodic Structures and Bragg Reflections

      For ordered systems such as colloid crystals, liquid crystals or block copolymer mesophases, the structure factor will comprise a series of Bragg reflections (or pseudo‐Bragg reflections, strictly, for layered structures). The amplitude structure factor for a hkl reflection (where h, k and l are Miller indices) of a lattice is given by [12]

(1.51)

Here a*, b*, and c* are reciprocal space axis vectors. In Eq. (1.51), ρ(r) is used rather than Δρ(r) since if Fhkl is determined on an absolute scale, the Fourier transform of Eq. (1.51) permits the determination of ρ(r) in absolute units (e.g. electron Å⁻³ in the case of SAXS data).

The location of the observed Bragg reflections (ratio of peak positions) is characteristic of the symmetry of the structure. Table 1.1 lists the observed reflections for common structures observed for soft materials (and some hard materials). Figure 1.7 shows examples of structures with the lowest indexed planes indicated. The generating equations for allowed reflections for different space groups are available in crystallography textbooks [26] and elsewhere [27].

Figure 1.7 Representative one‐, two‐, and three‐dimensional structures, with lowest indexed diffraction planes indicated. For the lamellar structure, three‐dimensional Miller indices have been employed while for the hexagonal structure two‐dimensional indices are used.

Figure 1.8 shows representative SAXS intensity profiles for some common ordered phases in soft materials, exemplified by data for block copolymer melts. The sequences of observed reflections are consistent with Table 1.1.

Figure 1.8 Compilation of SAXS profiles measured for PEO‐PBO [polyoxyethylene‐b‐polyoxybutylene] diblock copolymer melts. The x‐axis uses a q‐scale normalized to q^*, the position of the first order peak. BCC: body‐centred cubic, Gyr: gyroid, Hex: hexagonal‐packed cylinders, Lam: lamellar, DIS: disordered.

      Source: From Ref. [28].

      The layer spacing d for a lamellar phase is given by

(1.52)

      Here q_l is the position of the lth order Bragg peak.

      For a two‐dimensional hexagonal structure [26, 29]:

(1.53)

      where qhk is the position of the Bragg peak with indices hk and a is the lattice constant.

For a cubic structure [26, 29]

(1.54)