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Flory Huggins Theory for Polymer Blends

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.

Molecular distribution in different lattices
Figure 01: Molecular distribution in different lattices

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.

Mixing of two components
Figure 02: Mixing of two components

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,

Flory Huggins Theory for Polymer Blends eq1

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,

Flory Huggins Theory for Polymer Blends eq2

Number of lattice sites occupied by species A

Flory Huggins Theory for Polymer Blends eq3

Number of lattice sites occupied by species B

Flory Huggins Theory for Polymer Blends eq4

MixtureNANB
Regular solution11
Polymer solutionDegree of polymerization1
Polymer blendDegree of polymerizationDegree of polymerization
Table 01: Table 1 - The number of lattice sites occupies per molecule

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 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

ΩAB = n

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

ΩA = n ΦA

ΩB = n ΦB

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

ΔSA = k ln ΩAB - k ln ΩA

ΔSA = k lnAB/ ΩA)

Where, ΩAB = n and ΩA = n ΦA,

ΔSA = k ln (n/ n ΦA)

ΔSA = k ln (1/ ΦA)

ΔSA = - k lnA)

ΔSB = - k lnB)

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 = nA ΔSA + nB ΔSB

ΔS mix = - k (nA ΦA + nB ΦB)

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

nA = (n ΦA)/ NA

nB = (n ΦB)/ NB

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

Flory Huggins Theory for Polymer Blends eq5
  • ΦA – volume fraction of species A
  • ΦB – volume fraction of species B
  • NA – number of sites occupied by a molecule A
  • NB – number of sites occupied by a molecule B
  • k – Boltzmann 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.

  • 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 ΦA + uAB ΦB

Likewise,

UB = uBB ΦB + uAB ΦA

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 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 ΦA and n ΦB respectively. Therefore, the total interaction energy of the mixture is given by,

U2 = zn/2 [HA ΦA + HA ΦA]

U2 = zn/2 {[ ΦA (uAA ΦA + uAB ΦB)]+ [ΦB (uBB ΦB + uAB ΦA)]}

U2 = zn/2 (uAA ΦA2+ uBB ΦB2+ 2uAB ΦAΦB)

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 nΦA and nΦB respectively. Therefore, the total energy of species A and B before mixing is

U1 = zn/2 (uAA ΦA + uBB ΦB)

Therefore, the enthalpy change of mixing is

ΔU = H2 - H1

ΔU = zn/2 (uAA ΦA2+ uBB ΦB2+ 2uAB ΦAΦB) - zn/2 (uAA ΦA + uBB ΦB)

ΔU = zn/2 (uAA ΦA2- uAA ΦA + uBB ΦB2 - uBB ΦB + 2uAB ΦAΦB)

ΔU = zn/2 (uAA ΦAA - 1) + uBB ΦBB -1) + 2uAB ΦAΦB)

Where, ΦB = 1 – ΦA

ΔU = zn/2 [uAA ΦAA - 1) + uBB (1- ΦA) (1 - ΦA -1) + 2uAB ΦA(1-ΦA)]

ΔU = zn/2 [uAA ΦAA - 1) + uBB (1- ΦA) (1 - ΦA -1) + 2uAB ΦA(1-ΦA)]

ΔU = zn/2 [uAA ΦAA - 1) - uBB ΦA (1 - ΦA) + 2uAB ΦA(1-ΦA)]

ΔU = zn/2 (ΦA ΦB) (uAB -uAA - uBB)

Helmholtz free energy for mixing per lattice site,

ΔU̅ = ΔU/n

ΔU̅ = z/2 (ΦA ΦB) (uAB -uAA - uBB)

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.

Flory Huggins Theory for Polymer Blends eq6 2

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

Flory Huggins Theory for Polymer Blends eq7

χ valueSolubility
+Not favorable for mixing
-Favorable for mixing always
0Favorable for mixing. Ideal situation
Table 02: Solubility depends on the χ value

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References and Attributes

Figures:

The cover image was designed by using an image by frakir, licensed under CC BY-SA 3.0, via Wikimedia Commons


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