Flory Huggins Theory for Polymer Blends

Flory Huggins Theory

The Flory Huggins mean-field theory introduces entropic and enthalpic contributions of mixing of two different chemical species. This theory considers interactions between molecules are due to the interaction of a given molecule and an average interaction force of all the other molecules in the system. The mixture is imagined as a lattice and all the molecules have occupied lattice sites.

In Flory Huggins theory, there are two assumptions are considered,

1. When mixing two components, the overall volume does not change.
2. After mixing, a homogeneous mixture is obtained.

When mixing two species of A and B, if the volume of species A is V_A and the volume of species B is V_B, the total volume after mixing is given by V_A+V_B. It considers the components are randomly mixed to fill the entire lattice according to the Flory Huggins theory. The volume fractions of both components A and B are given by,

Lattice site

The lattice site is defined as the position of a molecule. The volume of the lattice site is similar to the volume of a single molecule. Here it is assumed that all the simple molecules have the same volume. One simple molecule occupies a single lattice site while large molecules such as polymers occupy multiple connected lattice sites. Generally, a repeating unit of a polymer occupies a single lattice site. If the volume of a single lattice site is V₀, the volume of components A and B can be represented as,

V_A = N_AV₀

V_B = N_BV₀

N_A and N_B are the number of lattice sites occupied by each molecule. The total number of lattice sites are given by,

Number of lattice sites occupied by species A

Number of lattice sites occupied by species B

Regular Solutions:

Mixtures of low molar mass species with N_A = N_B = 1

Polymer Solutions:

Mixtures of macromolecule (N>> 1) with the low molar mass solvent (N=1)

Polymer Blends:

Mixtures of macromolecules of different chemical species having N_A >>1 and N_B>>1

The entropy of mixing ΔS

Entropy is a measure of disorderness. By definition

S = k ln Ω

• S= entropy
• k = Boltzmann constant
• Ω = number of ways a molecule can arrange on the lattice site

In a homogeneous mixture of A and B, each molecule has

Ω_AB = n

Possible states, where n is the total number of lattice sites of the combined system

Ω_A = n Φ_A

Ω_B = n Φ_B

For a single molecule of species, A, the entropy change on mixing is

ΔS_A = k ln Ω_AB - k ln Ω_A

ΔS_A = k ln (Ω_AB/ Ω_A)

Where, Ω_AB = n and Ω_A = n Φ_A,

ΔS_A = k ln (n_/ n Φ_A)

ΔS_A = k ln (1_/ Φ_A)

ΔS_A = - k ln (Φ_A)

ΔS_B = - k ln (Φ_B)

Since volume fraction is always < 1, the Entropy change of mixing is always positive

ΔS= - k ln (Φ) > 0

To calculate the total entropy of mixing, the entropy contributions from each molecule in the system are summed.

ΔS _mix = n_A ΔS_A + n_B ΔS_B

ΔS _mix = - k (n_A Φ_{A +} n_B Φ_B)

Where n_A and n_B are the numbers of molecules per species A and B.

n_A = (n Φ_A)/ N_A

n_B = (n Φ_B)/ N_B

For a mixture of two polymers A and B; Entropy of mixing per lattice site is given by,

• Φ_A – volume fraction of species A
• Φ_B – volume fraction of species B
• N_A – number of sites occupied by a molecule A
• N_B – number of sites occupied by a molecule B
• k – Boltzmann's constant

Free energy / Enthalpy of mixing ΔH

Gibbs free energy (ΔG) is defined under constant temperature and pressure. When it is defined under constant temperature and volume, it is called Helmholtz free energy (ΔF). Under constant pressure and temperature, the internal energy is called enthalpy (ΔH).

ΔG = ΔH - T ΔS

ΔF = ΔU - T ΔS

Where,

• ΔG – Gibbs free energy
• ΔH – Enthalpy
• ΔF – Helmholtz free energy
• ΔU – Internal energy
•  T – Absolute temperature.

F-H model considers the constant volume condition. Thus, the energy of interactions will be explained in terms of Helmholtz's free energy of mixing. There are some assumptions considered,

• The interactions between monomers are assumed to be small enough that they do not affect the random positioning of the polymer chains on the lattice.
• Monomer volumes of each species are identical.

Interactions can be explained in terms of pairwise interaction energies between adjacent lattice sites. In a binary mixture following interactions are occurred.

• u_AA = interactions between species A
• u_BB = interactions between species B
• u_AB = interactions between species A and B

The probability of finding adjacent cells filled by components A and B is given by assuming the probability that a given cell is occupied by particular species is equal to the volume fraction of that species. Therefore, average pairwise interactions of a monomer species A with its neighbor can be given

U_A = u_AA Φ_A + u_AB Φ_B

Likewise,

U_B = u_BB Φ_B + u_AB Φ_A

Each lattice site of a regular lattice has “z” nearest neighbors (z-coordination number of the lattice)

• For 2D lattice z = 4
• For 3D lattice z = 6

Therefore, the average interaction energy of an A monomer with all of its z neighbors is zH_A. The average energy per monomer is half of this energy (zH_A/2). The total number of monomers of species A and B are n Φ_A and n Φ_B respectively. Therefore, the total interaction energy of the mixture is given by,

U₂ = zn/2 [H_A Φ_A + H_A Φ_A]

U₂ = zn/2 {[ Φ_A (u_AA Φ_A + u_AB Φ_B)]+ [Φ_B (u_BB Φ_B + u_AB Φ_A)]}

U₂ = zn/2 (u_AA Φ_A²+ u_BB Φ_B²+ 2u_AB Φ_AΦ_B)

Energy before mixing

Interaction energy per site in a pure A component before mixing is (zu_AA /2) Because before mixing, A molecules are surrounded by only A molecules. The total number of monomers of species A and B are nΦ_A and nΦ_B respectively. Therefore, the total energy of species A and B before mixing is

U₁ = zn/2 (u_AA Φ_A + u_BB Φ_B)

Therefore, the enthalpy change of mixing is

ΔU = H₂ - H₁

ΔU = zn/2 (u_AA Φ_A²+ u_BB Φ_B²+ 2u_AB Φ_AΦ_B) _- zn/2 (u_AA Φ_A + u_BB Φ_B)

ΔU = zn/2 (u_AA Φ_A²- u_AA Φ_A + u_BB Φ_B² - u_BB Φ_B + 2u_AB Φ_AΦ_B)

ΔU = zn/2 (u_AA Φ_A (Φ_A - 1) + u_BB Φ_B (Φ_B -1) + 2u_AB Φ_AΦ_B)

Where, Φ_B = 1 – Φ_A

ΔU = zn/2 [u_AA Φ_A (Φ_A - 1) + u_BB (1- Φ_A) (1 - Φ_A -1) + 2u_AB Φ_A(1-Φ_A)]

ΔU = zn/2 [u_AA Φ_A (Φ_A - 1) + u_BB (1- Φ_A) (1 - Φ_A -1) + 2u_AB Φ_A(1-Φ_A)]

ΔU = zn/2 [u_AA Φ_A (Φ_A - 1) - u_BB Φ_A (1 - Φ_A) + 2u_AB Φ_A(1-Φ_A)]

ΔU = zn/2 (Φ_A Φ_B) (u_AB -u_AA - u_BB)

Helmholtz free energy for mixing per lattice site,

ΔU̅ = ΔU/n

ΔU̅ = z/2 (Φ_A Φ_B) (u_AB -u_AA - u_BB)

Flory Huggins interaction parameter, χ

χ is a dimensionless measure of the differences in the strength of pairwise interaction energies between species in a mixture. χ value depends on the temperature.

Where,

• χ – Flory Huggins interaction parameter
• Z – coordination number
• k – Boltzmann's constant
• T – Absolute temperature

Helmholtz free energy for mixing per lattice site,

ΔF̅ _mix = ΔU̅ _mix - T ΔS̅ _mix