## 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

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)
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Then the Hamiltonian is written as

(B.2)
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Given $H ( Γ )$, the free energy of the system at temperature T is calculated by

(B.3)
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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)
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The free energy of such a system is given by

(B.5)
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where x stands for $( x 1 , x 2 , … , x n )$. The constraints can be accounted for by introducing the hypothetical potential

(B.6)
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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)
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the free energy of the constrained system can be written as

(B.8)
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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)
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2. (2) The free energy of the system at equilibrium is given by

(B.10)
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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)
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The average of $F ˆ i ( Γ , x )$ is given by

(B.12)
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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)
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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)
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The average of $x ˆ i ( Γ )$ for given forces $( h 1 , h 2 , … )$ is calculated as

(B.15)
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(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)
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Hence

(B.17)
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and $A ( x , T )$ is written as

(B.18)
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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)
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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)
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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)
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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)
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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)
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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)
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Hence

(B.25)
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Therefore the order parameter S of the system is given by

(B.26)
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Hence

(B.27)
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(p.235) If there is an interaction between the particles, the interaction energy is written as (see eq. ( 5.11 ))

(B.28)
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Hence the free energy is given by

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