Phase Transitions: Liquid-Liquid Unmixing– Equilibrium Phase Diagram

Soft-Condensed MatterDepartment of Physics,Tunghai-University

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

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

Liquid-Liquid Unmixing Problem

A B

A+B

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 F

mix > 0 Assume the liquids are incompressible Assume the molecules are located at lattice points

with coordinate number = z Ф: volume fractions

BA

BABA VV

V

,

,

Regular Solution Model (Conti.): Entropy part Mean-field approximation: the neighboring

sites are independent of each other Boltzmann formula:

In this case:

i iiB ppkS ln

BBAABmix kS lnln

Regular Solution Model (Conti.): Energy part Assume only n.n. interactions Assume the interactions are pairwise additive Mean-field approximation: there are zФA A molec

ules 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

Regular Solution Model (Conti.): Energy part

Free Energy for mixing

Stable and Unstable Cases

Phase Separation

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

Metastable State

Unstable

Metastable

Phase Diagram

Interface between Phases and Interfacial Tension For phase separated liqui

ds, there is an interface The interface costs free e

nergy → Surface tension The force needed to kee

p the interface: F=γL

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: