Guiniers Law

Scattering experiments at low angles on dilute colloidal systems, polymer solutions included, can be applied for a determination of the molar mass and the size of colloids or polymers. The relation may be explained as follows. 

We start from Eq. (A. 13) and consider a dilute system of polymers or, more general, colloids, each one composed of N monomeric units. For J fp colloids in the sample we have 

For a dilute system we neglect all interferences between different colloids. Equation (A. 13) can then be rewritten as 

In the low angle range close to the origin we may use a series expansion up 

For isotropic systems the linear term vanishes and the quadratic term transforms into 

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Sample compositions and scattering length densities are known quantities, hence the adjustable parameters are and a / and if used and 15(0). Uncertainties in a / Rf and may be taken as 2 A, 0.02 and 10 A respectively. We use the multi-model FISH program for our SANS analysis (see reference 13). For dilute non-interacting systems (absence of any obvious S(Q)) estimates for the micelle radii R can also be obtained using the Guinier law which is valid at low Q (QR[Pg.305]

For a homogeneous sphere with radius R, Rg=Rg, oo which may be calculated by the relation R =3I5 R. Thus, a homogenous sphere of 100 nm diameter is characterized by Rg=38.73 nm. The validity of the Guinier-law Equation (17) requires that Rgqlatex systems with sufficient accuracy. 

Guinier law (equation (8-15)), in practice, applies to the scattering intensity produced by a dilute set of polydisperse nano-objects, provided the size distribution has a moderate... 

Guinier law for a two electron density system becomes (Guinier, 1955) h qD)=HPi-pofV c- < -... 

Other methods of analysis based on Guinier law [74,97] invariants [97], and Porod law [93] have also been used. 

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