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Analysis Techniques for Polymer Systems: Zero Contrast Method and Random Phase Approximation

This chapter discusses the Zero Contrast Method and the Random Phase Approximation (RPA) as analytical techniques for small-angle neutron scattering (SANS) data from polymer systems. It details the process of mixing deuterated and hydrogenated polymers to achieve zero contrast scattering cross-sections, focusing on isolating single-chain properties. The chapter also introduces the RPA, which helps calculate structure factors for polymer mixtures in homogeneous phases. Together, these methods provide crucial insights into the behavior of concentrated polymer solutions.

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Analysis Techniques for Polymer Systems: Zero Contrast Method and Random Phase Approximation

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

  2. dPEO D2O hPEO H2O 31:5. THE ZERO CONTRAST METHOD -- Consider deuterated and hydrogenated polymers of same Mw -- Mix them in solution -- Keep polymer volume fraction constant, but vary deuterated fraction -- Mix deuterated and hydrogenated solvent fraction to obtain zero contast Scattering cross section: Contrasts: Partial scattering factors:

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

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

  5. 31:6. THE RANDOM PHASE APPROXIMATION Derivations: Single-chain scattering factor: Flory-Huggins interaction parameter: 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.

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