alt="StartFraction 1 Over upper S left-parenthesis 0 right-parenthesis EndFraction equals StartFraction 1 Over phi upper N Subscript upper A Baseline EndFraction plus StartFraction 1 Over left-parenthesis 1 minus phi right-parenthesis upper N Subscript upper B Baseline EndFraction minus 2 chi"/>
This is generalized for non‐zero wavenumbers using the random‐phase approximation (RPA) to give [12, 49, 61, 64]
(1.116)
Here P(q, N) is the corresponding form factor of an ideal chain, i.e. the Debye function (Eq. (1.107)). These expressions apply to the case of polymers in solution, the subscripts A and B referring to solvent and solute. These expressions are also useful for the case of blends of protonated and deuterated polymers in SANS studies (see Section 5.8), where A and B label the respective unlabelled and labelled chains. The RPA is a mean field method, widely employed in polymer science to calculate the structure factor in terms of the form factor of single chains [62].
SAS data from blends is often fitted using the Ornstein‐Zernike function:
(1.117)
This expression can be derived from Eq. (1.116) in the limit of small q, [62, 65] with
(1.118)
The correlation length in Eq. (1.117) can be shown [61] to be
(1.119)
Analogous equations to Eq. (1.116) have been obtained for block copolymer melts, also using the random‐phase approximation. The result for a diblock copolymer (degree of polymerization N, Flory‐Huggins interaction parameter χ and volume fraction of one component f) is [28, 66]
(1.120)
Here F(X) is a combination of Debye functions (cf. Eq. (1.107), X is defined after this equation) as follows:
(1.121)
and
(1.122)
Figure 1.23 shows an example of structure factors for a diblock copolymer in a disordered melt at several χN values, calculated using Eq. (1.120). Since χ is inversely proportional to temperature, the S(q) functions become less intense and broader as temperature increases, as expected.
Figure 1.23 Structure factor for a diblock copolymer melt with f = 0.25 at three values of χN indicated [66]. The order‐disorder transition within this model occurs at χN = 17.6 at this composition.
Source: From Leibler [66]. © 1980, American Chemical Society.
Further information about scattering from block copolymers can be found elsewhere [28, 67, 68].
REFERENCES
1 1. van de Hulst, H.C. (1957). Light Scattering by Small Particles. New York: Dover.
2 2. Brown, W. (ed.) (1996). Light Scattering ‐ Principles and Development. Oxford: Oxford University Press.
3 3. Lindner, P. and Zemb, T. (eds.) (2002). Neutrons, X‐Rays and Light. Scattering Methods Applied to Soft Condensed Matter. Amsterdam: Elsevier.
4 4. Glatter, O. (2018). Scattering Methods and their Application in Colloid and Interface Science. Amsterdam: Elsevier.
5 5. Guinier, A. and Fournet, G. (1955). Small Angle Scattering of X‐Rays. New York: Wiley.
6 6. Glatter, O. and Kratky, O. (eds.) (1982). Small Angle X‐Ray Scattering. London: Academic.
7 7. Feigin, L.A. and Svergun, D.I. (1987). Structure Analysis by Small‐Angle X‐Ray and Neutron Scattering. New York: Plenum.
8 8. Hamley, I.W. (1991). Scattering from uniform, cylindrically symmetric particles in liquid crystal phases. The Journal of Chemical Physics 95: 9376–9383.
9 9. Svergun, D.I. and Koch, M.H.J. (2003). Small‐angle scattering studies of biological macromolecules in solution. Reports on Progress in Physics 66: 1735–1782.
10 10. Glatter, O. (2002). The inverse scattering problem in small‐angle scattering. In: Neutrons, X‐Rays and Light Scattering Methods Applied to Soft Condensed
