1. Chapter 31 – STRUCTURE FACTORS FOR POLYMER SYSTEMS 31:5. THE ZERO CONTRAST METHOD 31:6. THE RANDOM PHASE APPROXIMATION

2. 31:5. THE ZERO CONTRAST METHOD Scattering cross section: Contrasts: Partial scattering factors: -- Consider deuterated and hydrogenated polymers of same M w -- Mix them in solution -- Keep polymer volume fraction constant, but vary deuterated fraction -- Mix deuterated and hydrogenated solvent fraction to obtain zero contast dPEO hPEO D2OD2O H2OH2O

3. Defined standard deviation: Defined average squared: Scattering cross section (cm -1 ): single-chain inter-chain When contrast is zero, the inter-chain term does not contribute and the scattering cross section is proportional to single-chain form factor only SOME DETAILS

4. Derivations: Cross section: Zero contrast condition: MORE DETAILS

5. 31:6. THE RANDOM PHASE APPROXIMATION Derivations: Flory-Huggins interaction parameter: Single-chain scattering factor: Single-chain form factor: Incompressibility condition: Cross section: Spinodal condition:

6. COMMENTS -- The zero contrast method and the Random Phase Approximation (RPA) method are useful methods for analyzing SANS data from polymers. -- The zero contrast method measures single-chain properties even in concentrated polymer mixtures. -- The RPA is a single method to calculate the structure factor for polymer mixtures in the mixed phase (homogeneous) region.