Flory Huggins Theory for Polymer Blends

What is the Flory Huggins Theory?

The Flory-Huggins mean-field theory provides a foundational model for understanding the thermodynamics of mixing two different chemical components. At its core, the theory analyzes the process by breaking it down into two fundamental contributions: the change in molecular disorder, known as entropy (ΔS), and the change in interaction energy, known as enthalpy (ΔH).

To accomplish this, the theory imagines the mixture on a conceptual lattice, where each molecule occupies a set number of sites. By calculating the entropic and enthalpic changes within this framework, we can determine the overall Gibbs free energy of mixing (ΔG), which ultimately governs the behavior of regular solutions, polymer solutions, and polymer blends.

The Flory-Huggins mean-field theory introduces entropic and enthalpic contributions to the mixing of two different chemical species. This theory considers that interactions between molecules are due to the interaction of a given molecule and the 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.

Mixing of two components

In the Flory-Huggins theory, there are two assumptions 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 that 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 and the volume of the 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

a. Number of lattice sites occupied by each molecule in a regular solution

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

b. Number of lattice sites occupied by each molecule in a polymer solution

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

c. Number of lattice sites occupied by each molecule in a polymer blend

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 disorder. 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, the 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

Gibbs free energy ΔG, Helmholtz free energy (ΔF), and 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 (ΔU) 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.

The Flory-Huggins model considers the constant volume condition. Thus, the energy of interactions will be explained in terms of Helmholtz's free energy of mixing (ΔF_mix). 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.

Deriving the Enthalpy of Mixing/ change in internal energy (ΔU)

In the Flory-Huggins model, the enthalpy of mixing (ΔH_mix) represents the net change in interaction energy when two components are mixed. Because the theory assumes constant volume, we calculate this as the change in internal energy (ΔU). The derivation follows these logical steps.

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

• 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 a particular species is equal to the volume fraction of that species. Therefore, the 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 is 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 is 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. The χ value depends on the temperature.

In simple terms, the Flory-Huggins interaction parameter (χ) is a single number that measures the energy of repulsion or attraction between two different types of molecules in a mixture.

Where,

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

Flory-Huggins equation for the free energy of mixing per lattice site

Helmholtz free energy for mixing per lattice site,

ΔF̅ _mix = ΔU̅ _mix - T ΔS̅ _mix

Limitations of the Flory-Huggins Theory

While foundational, the Flory-Huggins theory is a simplified model and has several important limitations that cause it to deviate from real-world experimental results.

1. The Interaction Parameter (χ) is Not a True Constant

The Flory-Huggins theory treats χ as a constant that only represents the enthalpy of interaction. In reality, χ is a more complex parameter that:

• Depends on Concentration: The interaction parameter often changes as the polymer concentration changes.
• Has an Entropic Component: It isn't purely enthalpic; it also includes non-combinatorial entropic effects that the simple lattice model ignores.
• It is temperature-dependent: While the basic form (χ ∝ 1/T) captures general trends, the actual relationship can be more complex.

2. Assumption of Zero Volume Change (ΔV_mix = 0)

The model assumes that the volume of the polymer and the solvent is perfectly additive, meaning there is no expansion or contraction when they are mixed. For many real systems, a small but significant volume change occurs, which the theory does not account for.

3. Ignores Specific Molecular Architecture

The simple lattice model treats the polymer as a chain of identical segments without any specific structure. This means it overlooks important factors like:

• Chain Stiffness: Real polymer chains have varying degrees of flexibility, which affects their arrangement and entropy.
• Specific Interactions: The theory averages out all molecular forces into the single χ parameter, failing to properly model strong, directional interactions like hydrogen bonding.

4. Inability to Predict Lower Critical Solution Temperature (LCST)

The classic Flory-Huggins theory successfully predicts that a polymer solution will phase-separate upon cooling (an Upper Critical Solution Temperature, or UCST). However, it cannot explain the phenomenon where some polymer solutions mix at low temperatures but phase-separate upon heating. This is known as the Lower Critical Solution Temperature (LCST), and explaining it requires more advanced theories that account for free volume differences and equation-of-state effects.