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Phase Transitions: Liquid-Liquid Unmixing– Equilibrium Phase Diagram

Phase Transitions: Liquid-Liquid Unmixing– Equilibrium Phase Diagram. Soft-Condensed Matter Department of Physics,Tunghai-University. Phase Transition and Order Parameters.

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Phase Transitions: Liquid-Liquid Unmixing– Equilibrium Phase Diagram

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  1. Phase Transitions: Liquid-Liquid Unmixing– Equilibrium Phase Diagram Soft-Condensed Matter Department of Physics,Tunghai-University

  2. Phase Transition and Order Parameters • Order parameter: change from a more ordered state to a less ordered state, and vice versa → order parameters are necessary to describe the change of the states • First order transition: order parameter changes discontinuously between zero and finite values • Second order transition: order parameter changes continuously between zero and finite values

  3. Phase Transition in Soft Matter • The self-assembled process • The states of soft matters are usually very complex • A transition means the atoms of the system to rearrange themselves → usually takes longer time to reach the equilibrium • If the time scale for the rearrangement is too long, we may observe the non-equilibrium states

  4. Liquid-Liquid Unmixing Problem A B A+B

  5. Regular Solution Model: A Mean-Field Approach • Change of free energy of mixing: Fmix = FA+B – (FA+FB) • A and B can mix if Fmix < 0, phase separation for Fmix > 0 • Assume the liquids are incompressible • Assume the molecules are located at lattice points with coordinate number = z • Ф: volume fractions

  6. Regular Solution Model (Conti.): Entropy part • Mean-field approximation: the neighboring sites are independent of each other • Boltzmann formula: • In this case:

  7. Regular Solution Model (Conti.): Energy part • Assume only n.n. interactions • Assume the interactions are pairwise additive • Mean-field approximation: there are zФA A molecules and zФB B molecules at the neighbors of each site (no matter the site is occupied by A or B) • єAA, BB, AB are the contact energies for AA, BB, and AB n.n. contacts

  8. Regular Solution Model (Conti.): Energy part

  9. Free Energy for mixing

  10. Stable and Unstable Cases

  11. Phase Separation • For Fig.3.3 (b), the mixed state is unstable and the system will become a phase-separated state

  12. Metastable State Unstable Metastable

  13. Phase Diagram

  14. Interface between Phases and Interfacial Tension • For phase separated liquids, there is an interface • The interface costs free energy → Surface tension • The force needed to keep the interface: F=γL

  15. Interfacial Tension • The definition is performed under the constant temperature condition, i. e., isothermalrather than adiabatic • The interfacial tension is an interfacial free energy rather than internal energy • For ideal sharp interface:

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