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B.Sc (Hons, USJ) (Polymer Science and Technology, Chemistry, Physics)

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,

- When mixing of two components, overall volume does not change.
- 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,

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 V_{0}, the volume of components A and B can be represented as,

**V _{A }= N_{A}V_{0}**

**V _{B} = N_{B}V_{0}**

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

Mixture | N_{A} | N_{B} |

Regular solution | 1 | 1 |

Polymer solution | Degree of polymerization | 1 |

Polymer blend | Degree of polymerization | Degree of polymerization |

**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**

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 **

**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**– volume fraction of species A_{A}**Φ**– volume fraction of species B_{B}**N**– number of sites occupied by a molecule A_{A}**N**– number of sites occupied by a molecule B_{B}**k**– Boltzmann's constant

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**= interactions between species A_{AA}**u**= interactions between species B_{BB}**u**= interactions between species A and B_{AB}

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,

**U _{2} = zn/2 [H_{A} Φ_{A }+ H_{A} Φ_{A}]**

**U _{2} = zn/2 {[ Φ_{A} (u_{AA }Φ_{A }+ u_{AB }Φ_{B})]+ [Φ_{B} (u_{BB }Φ_{B }+ u_{AB }Φ_{A})]}**

**U _{2} = zn/2 (u_{AA }Φ_{A}^{2}+ u_{BB }Φ_{B}^{2}+ 2u_{AB }Φ_{A}Φ_{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

**U _{1} = zn/2 (u_{AA }Φ_{A }+ u_{BB }Φ_{B})**

Therefore, the enthalpy change of mixing is

**ΔU = H _{2 }- H_{1}**

**ΔU = zn/2 (u _{AA }Φ_{A}^{2}+ u_{BB }Φ_{B}^{2}+ 2u_{AB }Φ_{A}Φ_{B})_{ - }zn/2 (u_{AA }Φ_{A }+ u_{BB }Φ_{B})**

**ΔU = zn/2 (u _{AA }Φ_{A}^{2}- u_{AA }Φ_{A }+ u_{BB }Φ_{B}^{2} - u_{BB }Φ_{B} + 2u_{AB }Φ_{A}Φ_{B})**

**ΔU = zn/2 (u _{AA }Φ_{A} (Φ_{A} - 1) + u_{BB }Φ_{B} (Φ_{B }-1) + 2u_{AB }Φ_{A}Φ_{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})**

* χ* is a dimensionless measure of the differences in the strength of pairwise interaction energies between species in a mixture.

Where,

– Flory Huggins interaction parameter*χ***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}**

χ value | Solubility |

+ | Not favorable for mixing |

- | Favorable for mixing always |

0 | Favorable for mixing. Ideal situation |

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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