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Subsections

# Free Energy Minima and Phases

## Phase Coexistence

Homogeneous state:
All part of my system are alike
Inhomogeneous state:
Some parts are different:

Definition:
Macroscopic states, coexisting at different conditions, are called phases. Each phase exists in the thermodynamic limit
Question:
Is microphase separation in co-polymers a true phase equilibrium?

No, we do not have separate independent macroscopic phases! This is just one microscopically inhomogeneous macroscopic phase.

## Free Energy Minimum

### Problem

Suppose our system is at constant temperature. Total free energy is

 A(N1,N2,V1,V2,T) = A1(N1,V1,T) + A2(N2,V2,T) (1)

with
 (2)
We want to minimize (1) under conditions (2).

### Method of Lagrange Multipliers

We want to minimize f(x,y) under conditions g(x,y)=0. Construct a new function

and minimize this as function of independent variables x, y, . Minimizing by we obtain g(x,y)=0, i.e.

Example:
I want to fence a rectangular lot of the given area A. How can I save money on the fence?

Solution:
For an lot the area is A=xy, the fence is f=2x+2y long. We want to minimize

We minimize:

and obtain

The result is --I'd better make a square lot.

### Minimization

To minimize free energy (1) under conditions (2), we minimize a new function
 (3)

#### Minimization with respect to V1 and V2:

The derivatives here are just (minus) pressures! We obtained:
1.
In equilibrium both phases have equal pressures (mechanic equilibrium) P1=P2=P
2.
Lagrange multiplier is minus pressure of the system:

#### Minimization with respect to N1 and N2:

1.
Both phases have equal chemical potentials
2.
Lagrange multiplier is the chemical potential of the system:

#### Conclusion

In coexisting phase P, T, are equal (as well as other thermodynamic fields).

#### Slightly Different View:

Let's forget about phase 2. The condition for N1 for phase 1 can be obtained from minimization a function

with external pressure and chemical potential P0 and .In equilibrium is -potential (we already derived this in previous lectures ).

#### Stability Condition:

The extremum should be a minimum second derivatives are positive!

### General Situation

Suppose we have an extensive variable xi. It has conjugated field

In equilibrium Xi is constant in all phases, and the matrix with

is positive (in equilibrium S has maximum, and A minimum!). We can obtain this by minimizing

with external value Xi(0)

Next: Geometric Interpretation and Phase Up: Phase Equilibria Previous: Phase Equilibria

© 1997 Boris Veytsman and Michael Kotelyanskii
Thu Oct 2 21:02:12 EDT 1997