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The '''Flory-Huggins theory''' ( | The '''Flory-Huggins theory''' (perhaps chronologically speaking it should be known as the Huggins-Flory theory) for [[solutions]] of [[polymers]] was developed by [[Maurice L. Huggins ]] <ref>[http://dx.doi.org/10.1063/1.1750930 Maurice L. Huggins "Solutions of Long Chain Compounds", Journal of Chemical Physics '''9''' p. 440 (1941)]</ref> and [[Paul J. Flory]] <ref>[http://dx.doi.org/10.1063/1.1750971 Paul J. Flory "Thermodynamics of High Polymer Solutions", Journal of Chemical Physics '''9''' pp. 660-661 (1941)]</ref><ref>[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)]</ref>, following the work by Kurt H. Meyer <ref>[http://dx.doi.org/10.1002/hlca.194002301130 Kurt H. Meyer "Propriétés de polymères en solution XVI. Interprétation statistique des propriétés thermodynamiques de systèmes binaires liquides", Helvetica Chimica Acta '''23''' pp. pp. 1063-1070 (1940)]</ref>. The description 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, <math>i</math>, 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 (Eq. 10.1 of Ref. 3): | ||
:<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> | ||
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where <math>\chi</math> is the dimensionless Flory-Huggins binary interaction parameter (similar to the [[Johannis Jacobus van Laar |van Laar]] [[heat of mixing]] <ref>[http://www.dwc.knaw.nl/DL/publications/PU00013947.pdf J. J. van Laar "On the latent heat of mixing for associating solvents", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''7''' pp. 174-177 (1905)]</ref>), which can be expressed as (Eq. 21 of Chapter XII): | where <math>\chi</math> is the dimensionless Flory-Huggins binary interaction parameter (similar to the [[Johannis Jacobus van Laar |van Laar]] [[heat of mixing]] <ref>[http://www.dwc.knaw.nl/DL/publications/PU00013947.pdf J. J. van Laar "On the latent heat of mixing for associating solvents", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''7''' pp. 174-177 (1905)]</ref>), which can be expressed as (Eq. 21 of Chapter XII): | ||
:<math>\chi = \frac{z\Delta w_{ | :<math>\chi = \frac{z\Delta w_{12}}{RT}</math> | ||
where <math>z</math> is the coordination number and (Eq. 17 of Chapter XII) | where <math>z</math> is the coordination number and (Eq. 17 of Chapter XII) | ||
:<math>\Delta w_{ | :<math>\Delta w_{12} = w_{12} - \frac{(w_{11}+w_{22})}{2} </math> | ||
where <math>w_{ | where <math>w_{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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*Positive values of <math>\chi</math> necessarily lead to immiscibility for polymer mixtures of high molecular weight. | *Positive values of <math>\chi</math> necessarily lead to immiscibility for polymer mixtures of high molecular weight. | ||
*Polymer mixing always take place if the <math>\chi</math> parameter is negative. Miscible polymer mixtures with negative <math>\chi</math> exist due to specific interactions between given polymer segments. Miscibility or 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. | *Polymer mixing always take place if the <math>\chi</math> parameter is negative. Miscible polymer mixtures with negative <math>\chi</math> exist due to specific interactions between given polymer segments. Miscibility or 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== | ||
<references/> | <references/> | ||
[[Category: Polymers]] | [[Category: Polymers]] |