# Flory Huggins Theory for Polymer Blends

B.Sc (Hons, USJ) (Polymer Science and Technology, Chemistry, Physics)
Categories :ย
Last Updated Onย :ย June 24, 2022
Published Date : January 29, 2022

## 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 of two components, 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 VA and the volume of species B is VB, the total volume after mixing is given by VA+VB. 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

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,

VA = NAV0

VB = NBV0

NA and NB are the numbers 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 NA = NB = 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 NA >>1 and NB>>1

## Entropy of mixing ฮS

Entropy is a measure of disorderness. By definition

S = k ln ฮฉ

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

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

ฮฉAB = n

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

ฮฉA = n ฮฆA

ฮฉB = n ฮฆB

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

ฮSA = k ln ฮฉAB - k ln ฮฉA

ฮSA = k ln (ฮฉAB/ ฮฉA)

Where, ฮฉAB = n and ฮฉA = n ฮฆA,

ฮSA = k ln (n/ n ฮฆA)

ฮSA = k ln (1/ ฮฆA)

ฮSA = - k ln (ฮฆA)

ฮSB = - k ln (ฮฆB)

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

ฮS= - k ln (ฮฆ) > 0

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

ฮS mix = nA ฮSA + nB ฮSB

ฮS mix = - k (nA ฮฆA + nB ฮฆB)

Where nA and nB are the numbers of molecules per species A and B.

nA = (n ฮฆA)/ NA

nB = (n ฮฆB)/ NB

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

• ฮฆA โ volume fraction of species A
• ฮฆB โ volume fraction of species B
• NA โ number of sites occupied by a molecule A
• NB โ number of sites occupied by a molecule B
• k โ Boltzmann constant

## Free energy / Enthalpy of mixing ฮH

Gibbs free energy (ฮG) is defined under constant temperature and pressure. When it is defined under constant temperature and volume, it is called Helmholtz free energy (ฮF). Under constant pressure and temperature, the internal energy is called enthalpy (ฮH).

ฮG = ฮH - T ฮS

ฮF = ฮU - T ฮS

Where,

• ฮG โ Gibbs free energy
• ฮH โ Enthalpy
• ฮF โ Helmholtz free energy
• ฮU โ Internal energy
• ย T โ 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.

• uAA = interactions between species A
• uBB = interactions between species B
• uAB = 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

UA = uAA ฮฆA + uAB ฮฆB

Likewise,

UB = uBB ฮฆB + uAB ฮฆA

Each lattice site of a regular lattice has โzโ 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 zHA. The average energy per monomer is half of this energy (zHA/2). The total number of monomers of species A and B are n ฮฆA and n ฮฆB respectively. Therefore, the total interaction energy of the mixture is given by,

U2 = zn/2 [HA ฮฆA + HA ฮฆA]

U2 = zn/2 {[ ฮฆA (uAA ฮฆA + uAB ฮฆB)]+ [ฮฆB (uBB ฮฆB + uAB ฮฆA)]}

U2 = zn/2 (uAA ฮฆA2+ uBB ฮฆB2+ 2uAB ฮฆAฮฆB)

Energy before mixing

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

U1 = zn/2 (uAA ฮฆA + uBB ฮฆB)

Therefore, the enthalpy change of mixing is

ฮU = H2 - H1

ฮU = zn/2 (uAA ฮฆA2+ uBB ฮฆB2+ 2uAB ฮฆAฮฆB) - zn/2 (uAA ฮฆA + uBB ฮฆB)

ฮU = zn/2 (uAA ฮฆA2- uAA ฮฆA + uBB ฮฆB2 - uBB ฮฆB + 2uAB ฮฆAฮฆB)

ฮU = zn/2 (uAA ฮฆA (ฮฆA - 1) + uBB ฮฆB (ฮฆB -1) + 2uAB ฮฆAฮฆB)

Where, ฮฆB = 1 โ ฮฆA

ฮU = zn/2 [uAA ฮฆA (ฮฆA - 1) + uBB (1- ฮฆA) (1 - ฮฆA -1) + 2uAB ฮฆA(1-ฮฆA)]

ฮU = zn/2 [uAA ฮฆA (ฮฆA - 1) + uBB (1- ฮฆA) (1 - ฮฆA -1) + 2uAB ฮฆA(1-ฮฆA)]

ฮU = zn/2 [uAA ฮฆA (ฮฆA - 1) - uBB ฮฆA (1 - ฮฆA) + 2uAB ฮฆA(1-ฮฆA)]

ฮU = zn/2 (ฮฆA ฮฆB) (uAB -uAA - uBB)

Helmholtz free energy for mixing per lattice site,

ฮUฬ = ฮU/n

ฮUฬ = z/2 (ฮฆA ฮฆB) (uAB -uAA - uBB)

## Flory Huggins interaction parameter, ฯ

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

Where,

• ฯ โ Flory Huggins interaction parameter
• Z โ coordination number
• k โ Boltzmann's constant
• T โ Absolute temperature

Helmholtz free energy for mixing per lattice site,

ฮFฬ mix = ฮUฬ mix - T ฮSฬ mix

## References and Attributes

### Figures:

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