Small-Angle Scattering. Ian W. Hamley

R).

It can be difficult in some cases (in particular, when measurements extend over only a small q range and/or for highly polydisperse systems) to disentangle polydispersity from the effect of particle anisotropy, which can also smear out form factor minima. This is illustrated by the example of Figure 1.20 which compares the calculated form factor for a sphere (R = 30 Å, σ = 1.5) with that of a monodisperse ellipsoid with R₁ = 29 Å and R₂ = 34.8 Å. The form factors are quite similar at low q (and this example has not been optimized for maximal similarity). Especially with lower resolution SANS experiments (see Section 5.5.1 for a discussion of the resolution function for SANS), the two profiles in Figure 1.20 are not distinguishable at low q.

Figure 1.20 Comparison of form factor of a polydisperse sphere (R = 30 Å, σ = 1.5) and a monodisperse ellipsoid with R₁ = 29 Å and R₂ = 34.8 Å.

1.8 FORM AND STRUCTURE FACTORS FOR POLYMERS

      Polymers adopt coiled conformations. In the simplest picture, these are described as ideal Gaussian coils. Polymers in the melt or in solution (under theta conditions) adopt this conformation which is the most basic model of polymer conformation. The form factor for Gaussian coils can easily be calculated (as follows) to yield the Debye function. By analogy with Eq. (1.18), but integrating over a Gaussian distribution the intensity (form factor) is [61]

(1.102)

      The Gaussian function for a random coil depends on the end‐to‐end distance between points i and j, R_ij and |i – j|, which is the number of links between these points:

      (1.103)

      The integral over Rij can be evaluated using

      (1.104)

where x = (2|i − j|b²)/3. Substituting in Eq. (1.102) we obtain

      (1.105)

For a large chain (N large), the summations can be replaced by integrals

      (1.106)

      This can be evaluated [61] to give

(1.107)

Here (b denotes a statistical segment length). Eq. (1.107) is the Debye form factor function for a polymer. Figure 1.21 shows an example of a computed Debye function. This function has a Guinier‐like asymptote at low q, but at high q (qR_g ≫ 1), , i.e. it shows a scaling:

(1.108)

Figure 1.21 Debye form factor for a polymer with Rg = 50 Å.

The expressions in Eqs. (1.107) (1.108) are valid for dilute polymer solutions under theta (ideal solution) conditions. However, this form factor and scaling behaviour is most commonly observed for polymers in the melt, which can adopt an ideal conformation since the local density within a polymer in the melt is the same as that at the macroscopic scale meaning that individual polymer chains are not perturbed by inter‐chain interactions [20, 62]

      The Debye form factor can be generalized to the case of expanded/collapsed coils for which Rg = bN^ν, where v is the Flory exponent (v = 1/2 for Gaussian coils such as polymer chains in a theta solvent, v ≈ 3/5 for polymers in good solvents, v = 1/3 in a poor solvent).