Chapter 34 THE MULTICOMPONENT RANDOM PHASE APPROXIMATION 34

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Chapter 34 – THE MULTI-COMPONENT RANDOM PHASE APPROXIMATION 34: 2. INCOMPRESSIBLE POLYMER MIXTURE 34:

Chapter 34 – THE MULTI-COMPONENT RANDOM PHASE APPROXIMATION 34: 2. INCOMPRESSIBLE POLYMER MIXTURE 34: 3. THE SINGLE-CHAIN FORM FACTORS 34: 4. BINARY HOMOPOLYMER BLEND MIXTURE 34: 5. TERNARY HOMOPOLYMER BLEND MIXTURE 34: 7. THE DIBLOCK COPOLYMER CASE

34: 2. INCOMPRESSIBLE POLYMER MIXTURE Scattering cross section: Matrix notation: Non-interacting system scattering factors:

34: 2. INCOMPRESSIBLE POLYMER MIXTURE Scattering cross section: Matrix notation: Non-interacting system scattering factors: Excluded volume terms: Scattering length densities: 2 3 1 2 1 3

MORE ON THIS Incompressibility condition: This implies: Spinodal condition:

MORE ON THIS Incompressibility condition: This implies: Spinodal condition:

34: 3. THE SINGLE-CHAIN FORM FACTORS “Bare” homopolymer scattering factor: “Bare” copopolymer scattering factor:

34: 3. THE SINGLE-CHAIN FORM FACTORS “Bare” homopolymer scattering factor: “Bare” copopolymer scattering factor: Homopolymer form factor: 1 4 2 Complex architectures: 3 10 0 5 6 7 12 0 8 11 0 9

34: 4. BINARY HOMOPOLYMER BLEND MIXTURE Bare scattering factor for component 1: Excluded volume

34: 4. BINARY HOMOPOLYMER BLEND MIXTURE Bare scattering factor for component 1: Excluded volume factor for component 1: Fully interacting system scattering factor: Scattering cross section: This is referred to as the de Gennes formula

34: 5. TERNARY HOMOPOLYMER BLEND MIXTURE Bare scattering factors: Excluded volume factors: Scattering cross

34: 5. TERNARY HOMOPOLYMER BLEND MIXTURE Bare scattering factors: Excluded volume factors: Scattering cross section:

34: 7. THE DIBLOCK COPOLYMER CASE Homopolymer blend mixture Fully interacting system scattering factor:

34: 7. THE DIBLOCK COPOLYMER CASE Homopolymer blend mixture Fully interacting system scattering factor: Scattering cross section: Diblock copolymer

COMMENTS -- The Random Phase Approximation (RPA) model applies to homogeneous (single-phase) polymer mixtures.

COMMENTS -- The Random Phase Approximation (RPA) model applies to homogeneous (single-phase) polymer mixtures. -- It does not apply to the demixed phase (phase separated) region. -- It can handle homopolymers and other architectures (copolymers, comb polymers, star polymers, etc). -- It can predict the scattering cross section as well as the spinodal temperature. -- It can work for Lower Critical Spinodal Temperature (LCST) and Upper CST blend mixtures.