Ian W. Hamley - Small-Angle Scattering

The Fourier transform of the OZ equation is written as

( 1.75)

Where H ( q ) is the Fourier transform of h ( r ) and C ( q ) is the transform of c ( r ).

The structure factor is related to the pair correlation function g ( r ) via

( 1.76)

It may be noted that this equation differs from that given in Eq. (1.32)since the spherical average is not employed, and the normalization of g ( r ) differs.

Equation 1.76along with Eq. (1.75)lead to the OZ expression for the structure factor:

( 1.77)

In the limit ρ → 0 the integral in Eq. (1.74)vanishes and the direct correlation function is [50]

( 1.78)

Here β = 1/ kB T . If βv ( r ) ≪ 1 Eq. (1.78)gives

(1.79)

Which is known as the mean spherical approximation (MSA), used to evaluate the OZ equation.

The OZ equation can be solved using closure relationships for c ( r ) such as the hypernetted chain (HNC) approximation and the Percus‐Yevick (PY) approximation. The HNC equation is [50]

(1.80)

or

(1.81)

This simplifies to the MSA in the limit r → ∞ since h ( r ) → 0 in this limit, but generalized for any density.

The PY closure equation is given by

(1.82)

This is obtained from an expansion of the HNC closure [50].

Exact solutions of these closure relationships are known, for instance using the PY approximation for hard spheres [50] or the Carnahan‐Starling approximation gives the expression in Eq. (1.38).

Further information on integral equation theories for liquids is available elsewhere [51].

Analytical expressions are available for the form factor of particles of many geometries, as well as polymers discussed in Section 1.8. Some examples of more commonly used form factors are listed in Table 1.2. Extensive compilations of form factors are available [7, 17, 18, 21, 55].

Table 1.2 Expressions for common form factors.

Form factor	Equation	Parameters
Homogeneous sphere		Radius, R ( ), Scattering contrast, Δ ρ
Spherical shell	P ( q ) = [ Ks ( q , R 1, Δρ ) − Ks ( q , R 2, Δρ (1 − μ ))] 2 Ks ( q , R , Δ ρ ) as for a homogeneous sphere as above (other parameterizations are possible)	Outer radius R 1with shell scattering contrast Δ ρ Inner radius R 2with core scattering contrast Δ ρ (1− μ )
Gaussian polymer chain –Debye function (see Section 1.8for derivation)		I 0, forward scattering intensity Rg , radius of gyration ( b , statistical segment length)
Sphere with attached Gaussian chains (used for block copolymer micelles) [52, 53]	where and with	Nc , aggregation number R , core radius Rg , radius of gyration of attached chains d , displacement of chains (for non‐penetration into core, d ≈ 1) Δ ρs , scattering contrast of spherical core Δ ρc , scattering contrast of attached chains
Ellipsoid of revolution (spheroid)	Ks ( q , R , Δ ρ ) as for a homogeneous sphere as above (other parameterizations are possible)	Radius in polar direction Rp Equatorial radius Re = νRp Scattering contrast, Δ ρ
Tri‐axial ellipsoid	P ( q ) = ∫ [ Ks ( q , R ( a , b , c , α , β , Δρ )] 2sin α d α d β R ( a , b , c , α , β ) =[( a 2sin 2 β + b 2cos 2 β )sin 2 α + c 2cos 2 α ] 1/2 Ks ( q , R , Δ ρ ) as for a homogeneous sphere as above (other parameterizations are possible)	Semi axis lengths a , b , c Scattering contrast, Δ ρ
Rectangular parallelepipedons (including cubes)	Where	Edge lengths a , b , c Scattering contrast, Δ ρ
Circular cylinder		Radius, R . length, L . Scattering contrast, Δ ρ
Infinitely thin circular disc		Radius, R Scattering contrast, Δ ρ
Lipid bilayer with Gaussian scattering density profile (within a disc)	P ( q ) = [ Pdisc ( q )/(Δ ρ ) 2] Pcs ( q ) where the cross‐section form factor Pcs ( q ) is	Cross‐section parameters illustrated in Figure 1.16
Helices and helical tapes (Pringle–Schmidt form factor [54])	Here and ɛ 0= 1 and ɛn = 2 for n ≥ 1	H helix length, P helix pitch, other parameters defined in Figure 1.16

Figure 1.16 Parameters for complex form factors. (a) Gaussian bilayer, (b) Projection of a helical structure showing definitions of angles and radii in the corresponding Pringle–Schemidt form factor ( Table 1.2).