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Soft Matter Physics$

Masao Doi

Print publication date: 2013

Print ISBN-13: 9780199652952

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199652952.001.0001

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(p.230) B Restricted free energy

(p.230) B Restricted free energy

Source:
Soft Matter Physics
Publisher:
Oxford University Press

According to statistical mechanics, the free energy of a system at equilibrium can be calculated if the Hamiltonian of the system is known. The Hamiltonian is a function of the set of generalized coordinates ( q 1 , q 2 , , q f ) and momenta ( p 1 , p 2 , , p f ) . For simplicity we shall denote the 2 f set of variables by Γ,

(B.1)
Γ = ( q 1 , q 2 , , q f , p 1 , p 2 , , p f )

Then the Hamiltonian is written as

(B.2)
H ( Γ ) = H ( q 1 , q 2 , , q f , p 1 , p 2 , , p f )

Given H ( Γ ) , the free energy of the system at temperature T is calculated by

(B.3)
A ( T ) = 1 β ln [ d Γ e β H ( Γ ) ]

where β = 1 / k B T .

Equation ( B.3 ) represents the free energy of the system which is at true equilibrium under a given temperature T.

In thermodynamics, we often consider the free energy of a system which is at equilibrium under some hypothetical constraints. Such a free energy is called a restricted free energy. For example, in discussing the phase separation of solutions, we consider the system consisting of two homogeneous solutions (each having concentration ϕ 1 , ϕ 2 and volume V 1 , V 2 ), and then minimize the total free energy F t o t ( ϕ 1 , ϕ 2 , V 1 , V 2 ; T ) with respect to ϕ i and V i to obtain the true equilibrium state. The free energy F t o t ( ϕ 1 , ϕ 2 , V 1 , V 2 ; T ) is an example of the restricted free energy.

The restricted free energy appears many times in this book. For example, in Section 3.2.3 , we considered the free energy U ( r ) of a polymer chain whose end-to-end vector is fixed at r . In Section 5.2.3 , we considered the free energy F ( S ; T ) of a liquid crystalline system in which the order parameter is fixed at S. In Section 7.2.1 , we considered the free energy A ( x ) for the non-equilibrium state specified by variable x. In this appendix, we discuss the restricted free energy from the general viewpoint of statistical mechanics.

(p.231) Let us consider a system in which the values of certain physical quantities x ˆ i ( Γ ) ( i = 1 , 2 , , n ) are fixed at x i , i.e.,

(B.4)
x i = x ˆ i ( Γ ) ( i = 1 , 2 , , n )

The free energy of such a system is given by

(B.5)
A ( x , T ) = 1 β ln d Γ   Π i = 1 n δ ( x i x ˆ i ( Γ ) )   e β H ( Γ )

where x stands for ( x 1 , x 2 , , x n ) . The constraints can be accounted for by introducing the hypothetical potential

(B.6)
U c ( Γ , x ) = 1 2 k i ( x ˆ i ( Γ ) x i ) 2

where k i is a large positive constant which gives a penalty for the deviation of the quantity x ˆ i ( Γ ) from the given value x i . Using the formula

(B.7)
δ ( x ) = lim k k 2 π 1 / 2 e k x 2

the free energy of the constrained system can be written as

(B.8)
A ( x , T ) = 1 β ln d Γ     e β [ H ( Γ ) + U c ( Γ , x ) ]

where the coefficient Π i ( k i β / 2 π ) 1 / 2 has been omitted since it only gives terms independent of x i .

From eq. ( B.5 ), it is easy to derive the following properties of the restricted free energy.

  1. (1) If the system is at equilibrium without the constraints, the probability that the physical quantity x ˆ i ( Γ ) has the value x i is given by

    (B.9)
    P ( x )   e β A ( x , T )

  2. (2) The free energy of the system at equilibrium is given by

    (B.10)
    A ( T ) = 1 β ln d x   e β A ( x , T )

Equations ( B.9 ) and ( B.10 ) indicate that A ( x , T ) is playing the role of the Hamiltonian H ( Γ ) in the new phase space specified by x = ( x 1 , x 2 , , x n ) .

According to eq. ( B.9 ), the most probable state is the state x which minimizes A ( x , T ) . This corresponds to the thermodynamic principle that the true equilibrium state is the state which minimizes the restricted free energy A ( x , T ) .

(p.232) To fix the values of the physical quantities at x, external forces need to be applied to the system. For example, to keep the end-to-end vector of a polymer chain at the fixed value r , a force f ˆ needs to be applied at the chain ends (see the discussion in Section 3.2.3 ). The force needed to keep x ˆ i ( Γ ) at x i is given by

(B.11)
F ˆ i ( Γ , x ) = U c ( Γ , x ) x i

The average of F ˆ i ( Γ , x ) is given by

(B.12)
F i ( x ) = F ˆ i ( Γ , x ) = d Γ U c ( Γ , x ) x i e β [ H ( Γ ) + U c ( Γ , x ) ] d Γ e β [ H ( Γ ) + U c ( Γ , x ) ] = A ( x , T ) x i

This is the generalization of eq. ( 3.30 ).

The restricted free energy A ( x , T ) can be calculated by eqs. ( B.5 ) or ( B.8 ). There is another way of calculating A ( x , T ) . This method has already been used in Section 3.2.3 to calculate the restricted free energy U ( r ) of a polymer chain whose end-to-end vector is fixed at r . There, instead of fixing the end-to-end vector at r , we assumed that a constant force f is applied at the chain end. If f is suitably chosen, the average end-to-end vector becomes equal to the given value r . 1 If the relation between f and r is known, the restricted free energy U ( r ) is obtained by eq. ( 3.30 ).

The above method can be generalized as follows. We consider the Hamiltonian

(B.13)
H ˜ ( Γ , h ) = H ( Γ ) i h i x ˆ i ( Γ )

where h i represents the hypothetical external force conjugate to x ˆ i ( Γ ) . We calculate the free energy A ˜ ( h i , T ) for the Hamiltonian ( B.13 )

(B.14)
A ˜ ( h , T ) = 1 β ln d Γ   e β H ˜ ( Γ , h )

The average of x ˆ i ( Γ ) for given forces ( h 1 , h 2 , ) is calculated as

(B.15)
x i = d Γ   x ˆ i ( Γ ) e β H ˜ ( Γ , h ) d Γ e β H ˜ ( Γ , h ) = 1 β d Γ   H ˜ ( Γ , h ) h i e β H ˜ ( Γ , h ) d Γ e β H ˜ ( Γ , h ) = A ˜ ( h , T ) h i

(p.233) On the other hand, the free energy A ˜ ( h , T ) is equal to the minimum of A ( x , T ) i h i x i with respect to x i . Therefore

(B.16)
x i A ( x , T ) i h i x i = 0

Hence

(B.17)
h i = A ( x , T ) x i

and A ( x , T ) is written as

(B.18)
A ( x , T ) = A ˜ ( h , T ) + i h i x i ,

Equations ( B.16 ) and ( B.15 ) are the Legendre transform. Therefore, the restricted free energy A ( x , T ) is obtained either by eq. ( B.16 ), or by ( B.15 ). We now consider two examples of the restricted free energy.

First we consider a two-component solution. Let r p i ( i = 1 , 2 , , N p ) and r s i ( i = 1 , 2 , , N s ) be the position vectors of solute molecules and solvent molecules. The potential energy of the system is written as

(B.19)
U t o t ( r p , r s ) = i j u p p ( r p i r p j ) + i , j u p s ( r p i r s j ) + i j u s s ( r s i r s j )

where u p p ( r ) , u p s ( r ) , and u s s ( r ) are the potentials for solute–solute, solute–solvent, and solvent–solvent interactions.

The effective interaction potential between solute molecules is given by the restricted free energy:

(B.20)
A p p ( r p ) = 1 β ln Π i d r s i   e β U t o t ( r p , r s )

This represents the effective interaction between the solute molecules with the effect of the solvent included. The effective interaction Δ ϵ introduced in eq. ( 2.49 ) is an example of such a potential for the lattice model. The interaction potential U ( h ) between the colloidal particles introduced in Section 2.5.1 is another example of such a potential.

In the theory of solutions, A p p ( r p ) is called the potential of the mean force since the average of the force acting on solute i is given by the partial derivative of A p p ( r p ) :

(B.21)
f i ( r p ) = A p p ( r p ) r p i

This is a special case of eq. ( B.10 ). (The minus sign arises from the difference in the definition of the force.) Notice that even though the original potential U t o t is written as a sum of two-body potentials such (p.234) as u p p ( r ) , u p s ( r ) , and u s s ( r ) , the potential A p p is generally not written as a sum of the two-body potentials.

As the next example, let us consider the free energy introduced in Section 5.2.3 .

We consider N nematic forming molecules, each pointing along the unit vector u i ( i = 1 , 2 , , N ). We assume that the scalar order parameter of the system is S, i.e., u i satisfy the constraint of eq. ( 5.28 ). To take into account this constraint, we assume that the Hamiltonian of the system is written as

(B.22)
H ˜ ( u i , h ) = h N i 3 2 u i z 2 1 3

where we have assumed (for simplicity) that the molecules are not interacting with each other (i.e., U = 0 ).

The free energy A ˜ ( h ) can be written as

(B.23)
A ˜ ( h ) = 1 β ln [ d u e β h ( 3 / 2 N ) ( u z 2 1 / 3 ) ] N = A ˜ ( 0 ) N β ln e β h ( 3 / 2 N ) ( u z 2 1 / 3 ) 0

where A ˜ ( 0 ) is the free energy of the system with no constraint, and 0 is the average for u for an isotropic distribution of molecules. In the following, we set A ˜ ( 0 ) equal to zero.

The right-hand side of eq. ( B.21 ) can be calculated as a power series with respect to h:

(B.24)
e β h ( 3 / 2 N ) ( u z 2 1 / 3 ) 0   = [ 1 + 3 β h 2 N ( u z 2 1 / 3 ) + 9 β 2 h 2 8 N 2 ( u z 2 1 / 3 ) 2 ] 0   = 1 + ( β h ) 2 10 N 2 +

Hence

(B.25)
A ˜ ( h ) = β 10 N h 2

Therefore the order parameter S of the system is given by

(B.26)
S = A ˜ ( h ) h = β 5 N h

Hence

(B.27)
A ( S ) = A ˜ ( h ) + h S = β 10 N h 2 = 5 N k B T 2 S 2

(p.235) If there is an interaction between the particles, the interaction energy is written as (see eq. ( 5.11 ))

(B.28)
E ( S ) = N U 2 ( u u ) 2 = N U 3 S 2

Hence the free energy is given by

(B.29)
A ( S ) = 5 N k B T 2 S 2 N U 3 S 2 = 5 N k B 2 ( T T c ) S 2

where T c = 2 U / 15 k B . Equation ( B.29 ) is the first term in the expansion of eq. ( 5.35 ).