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: F mix = F A+B – (F A +F B ) A and B can mix if F mix 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 zФ A A molecules and zФ B B molecules at the neighbors of each site (no matter the site is occupied by A or B) є AA, 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., isothermal rather than adiabatic The interfacial tension is an interfacial free energy rather than internal energy For ideal sharp interface: