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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 descrition can be easily generalized to the case of polymer mixtures. The Flory-Huggins theory defines the volume of a [[polymers |polymer system]] as a lattice which is divided into microscopic subspaces (called sites) of the same volume. In the case of polymer solutions, the solvent molecules are assumed to occupy single sites, while a polymer chain of a given type, i, occupies <math>n_i</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, 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 for the general case of a mixture of two components, A (polymer or solvent) and polymer B, | ||
:<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>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, which can be expressed as: | ||
:<math>\chi | :<math>\chi \propto \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 several methods. For instance, introducing opposite charges in the different polymers or adding a copolymer containing A and B segments. | ||
*For polymer | *For a polymer solution, <math>n_A</math>=1, the critical Flory-Huggins parameter is close to <math>1/2</math>. The temperature corresponding to this value <math>\chi</math>=<math>1/2</math> would be the critical temperature if the polymer is infinitely long and defines the [[theta solvent | theta temperature]] of the polymer-solvent system. | ||
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> | ||
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==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]] |