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The '''Flory-Huggins theory''' | The '''Flory-Huggins theory''' for [[solutions]] of [[polymers]] was developed by [[Maurice L. Huggins ]] (Ref. 1) and [[Paul J. Flory]] (Ref. 2). The Flory-Huggins theory defines the volume of a [[polymers |polymer system]] as a lattice which is divided into small subspaces (called sites) of the same volume. In the case of polymer solutions, the solvent is assumed to occupy single sites, while each polymer chain occupies <math>n</math> sites. The repulsive forces in the system are modelled by requiring each lattice site to be occupied by only a single segment. Attractive interactions between non-bonded sites are included at the lattice neighbour level. Assuming random and ideal mixing of two polymers, i.e. mixing volume <math>\Delta V_m = 0</math>, it is possible to obtain the well-known expression for the combinatorial [[entropy]] of mixing <math>\Delta S_m</math> per site of the Flory Huggins theory, | ||
:<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math> | :<math>\Delta S_m = -R \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B \right]</math> | ||
where <math>\phi_i</math> is the [[volume fraction]] of the component <math>i</math>, and <math>R</math> is the [[molar gas constant]]. Applying the concept of regular solutions and assuming all pair interactions in the framework of a mean-field theory yields for the [[enthalpy]] of mixing | where <math>\phi_i</math> is the [[volume fraction]] of the component <math>i</math> and <math>n_i</math> is the number of segments in each type of polymer chain, and <math>R</math> is the [[molar gas constant]]. Applying the concept of regular solutions and assuming all pair interactions in the framework of a mean-field theory yields for the [[enthalpy]] of mixing | ||
:<math>\Delta H_m = RT \chi \phi_A \phi_B</math> | :<math>\Delta H_m = RT \chi \phi_A \phi_B</math> | ||
where <math>T</math> is the absolute [[temperature]]. | where <math>T</math> is the absolute [[temperature]]. | ||
According to the preceding equations, the Flory-Huggins equation can be expressed in terms of the [[Gibbs energy function]] of mixing for a binary system | According to the preceding equations, the Flory-Huggins equation can be expressed in terms of the [[Gibbs energy function]] of mixing for a binary system | ||
:<math>\Delta G_m = RT \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B + \chi \phi_A \phi_B\right]</math> | :<math>\Delta G_m = -RT \left[ \frac{\phi_A}{n_A} \ln \phi_A + \frac{\phi_B}{n_B} \ln \phi_B + \chi \phi_A \phi_B\right]</math> | ||
where <math>\chi</math> is the | where <math>\chi</math> is the Flory-Huggins binary interaction parameter, defined as: | ||
:<math>\chi | :<math>\chi \approx \epsilon_{ij} - \frac{(\epsilon_{ii}-\epsilon_{jj})}{2} </math> | ||
where | where <math>\epsilon_{ij}</math> is the net energy associated with two neighbouring lattice sites of the different | ||
polymer segments for the same type or for the different types of polymer chains. Although the theory considers <math>\chi</math> as a fixed parameter, experimental data reveal that actually <math>\chi</math> depends on such quantities as temperature, concentration, [[pressure]], molar mass, molar mass distribution. From a theoretical point of view it may also depend on model parameters as the coordination number of the lattice and segment length. | polymer segments for the same type or for the different types of polymer chains. Although the theory considers <math>\chi</math> as a fixed parameter, experimental data reveal that actually <math>\chi</math> depends on such quantities as temperature, concentration, [[pressure]], molar mass, molar mass distribution. From a theoretical point of view it may also depend on model parameters as the coordination number of the lattice and segment length. | ||
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Therefore: | Therefore: | ||
*Positive values of <math>\chi</math> necessarily lead to | *Positive values of <math>\chi</math> necessarily lead to incompatibility for polymer mixtures of high molecular weight. | ||
*Polymer mixing always take place if the <math>\chi</math> parameter is negative. | *Polymer mixing always take place if the <math>\chi</math> parameter is negative. Compatible polymer mixtures with negative <math>\chi</math> are due to specific interactions between given polymer segments. Compatibility can be induced by introducing opposite charges in the different polymers or by the presence of a copolymer containing A and B segments. | ||
*For polymer | *For a polymer solution, the critical Flory-Huggins parameter is close to <math>1/2</math>. | ||
The <math>\chi</math> parameter is somewhat similar to a [[second virial coefficient]] expressing binary interactions between molecules and, therefore, it usually shows a linear dependence of <math>1/T</math> | The <math>\chi</math> parameter is somewhat similar to a [[second virial coefficient]] expressing binary interactions between molecules and, therefore, it usually shows a linear dependence of <math>1/T</math> | ||
:<math>\chi(T) = | :<math>\chi(T) = A + \frac{B}{T}</math> | ||
where <math> | where <math>A</math> and <math>B</math> are assumed to be constants, but can actually depend on density, | ||
concentration, molecular weight, etc. A usual interpretation is that <math> | concentration, molecular weight, etc. A usual interpretation is that <math>A</math> represents an [[enthalpy |enthalpic]] quantity and <math>B</math> an [[entropy | entropic]] contribution, although both <math>A</math> and <math>B</math> are actually empirical parameters. According to this description, the systems should show an upper critical temperature. Many polymer systems and solutions, however, show an increase of <math>\chi</math> for increasing temperatures (negative entropic contribution) what implies the existence of a lower critical temperature. | ||
==References== | ==References== | ||
#[http://dx.doi.org/10.1063/1.1750930 Maurice L. Huggins "Solutions of Long Chain Compounds", Journal of Chemical Physics '''9''' p. 440 (1941)] | |||
#[http://dx.doi.org/10.1063/1.1723621 Paul J. Flory "Thermodynamics of High Polymer Solutions", Journal of Chemical Physics '''10''' pp. 51-61 (1942)] | |||
[[Category: Polymers]] | [[Category: Polymers]] |