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| '''Flory–Huggins solution theory''' is a [[mathematical model]] of the [[thermodynamics]] of [[polymer]] [[solution]]s which takes account of the great dissimilarity in [[molecule|molecular]] sizes in adapting the usual [[expression (mathematics)|expression]] for the [[entropy of mixing]]. The result is an equation for the [[Gibbs free energy]] change <math>\Delta G_m</math> for mixing a polymer with a [[solvent]]. Although it makes simplifying assumptions, it generates useful results for interpreting experiments.
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| The [[thermodynamic potentials|thermodynamic equation]] for the [[Gibbs free energy]] change accompanying mixing at constant [[temperature]] and (external) [[pressure]] is
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| :<math>\Delta G_m = \Delta H_m - T\Delta S_m \,</math>
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| A change, denoted by <math>\Delta</math>, is the [[number|value]] of a [[Variable (mathematics)|variable]] for a [[solution]] or [[mixture]] minus the values for the pure [[Component (thermodynamics)|components]] considered separately. The objective is to find explicit [[formula]]s for <math>\Delta H_m</math> and <math>\Delta S_m</math>, the [[enthalpy]] and [[entropy]] increments associated with the mixing [[Process (science)#Processes in Science|process]].
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| The result obtained by [[Paul Flory|Flory]]{{Ref|1}} and [[Maurice Loyal Huggins|Huggins]]{{Ref|2}} is
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| :<math>\Delta G_m = RT[\,n_1\ln\phi_1 + n_2\ln\phi_2 + n_1\phi_2\chi_{12}\,] \,</math> | |
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| The right-hand side is a [[function (mathematics)|function]] of the number of [[mole (unit)|moles]] <math>n_1</math> and volume fraction <math>\phi_1</math> of [[solvent]] ([[Component (thermodynamics)|component]] <math>1</math>), the number of moles <math> n_2 </math> and volume fraction <math>\phi_2 </math> of polymer (component <math>2</math>), with the introduction of a parameter [[chi (letter)|chi]] <math>\chi</math> to take account of the [[energy]] of interdispersing polymer and solvent molecules. <math>R</math> is the [[gas constant]] and <math>T</math> is the [[thermodynamic temperature|absolute temperature]]. The volume fraction is analogous to the [[mole fraction]], but is weighted to take account of the relative sizes of the molecules. For a small solute, the mole fractions would appear instead, and this modification is the innovation due to Flory and Huggins.
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| == Derivation ==
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| We first calculate the [[entropy of mixing|''entropy'' of mixing]], the increase in the [[Entropy#Mathematical description|uncertainty]] about the locations of the molecules when they are interspersed. In the pure condensed [[phase (matter)|phases]] — [[solvent]] and polymer — everywhere we look we find a molecule.{{Ref|3}} Of course, any notion of "finding" a molecule in a given location is a [[thought experiment]] since we can't actually examine [[space|spatial]] locations the size of molecules. The [[expression (mathematics)|expression]] for the [[entropy of mixing]] of small molecules in terms of [[mole fraction]]s is no longer reasonable when the [[solution|solute]] is a [[macromolecule|macromolecular]] [[Chain (sequence)|chain]]. We take account of this dis[[symmetry]] in molecular sizes by assuming that individual polymer segments and individual solvent molecules occupy sites on a [[lattice (group)#Lattices in two dimensions: detailed discussion|lattice]]. Each site is occupied by exactly one molecule of the solvent or by one [[monomer]] of the polymer chain, so the total number of sites is
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| :<math>N = N_1 + xN_2\,</math> | |
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| <math>N_1</math> is the number of solvent molecules and <math>N_2</math> is the number of polymer molecules, each of which has <math>x</math> segments.{{Ref|4}}
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| From [[statistical mechanics]] we can calculate the [[entropy]] change, the increase in [[space|spatial]] [[information entropy#Formal definitions|uncertainty]], as a result of mixing solute and solvent.
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| :<math>\Delta S_m = -k[\,N_1\ln(N_1/N) + N_2\ln(xN_2/N)\,]\,</math>
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| where <math>k</math> is [[Boltzmann constant|Boltzmann's constant]]. Define the lattice ''volume fractions'' <math>\phi_1</math> and <math>\phi_2</math>
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| :<math>\phi_1 = N_1/N\,</math>
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| :<math>\phi_2 = xN_2/N\,</math>
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| These are also the probabilities that a given lattice site, chosen at [[randomness|random]], is occupied by a solvent molecule or a polymer segment, respectively. Thus
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| :<math>\Delta S_m = -k[\,N_1\ln\phi_1 + N_2\ln\phi_2\,]\,</math> | |
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| For a small solute whose molecules occupy just one lattice site, <math>x</math> equals one, the volume fractions reduce to [[mole fraction|molecular or mole fractions]], and we recover the usual equation from [[ideal solution|ideal mixing]] theory.
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| In addition to the entropic effect, we can expect an ''enthalpy'' change.{{Ref|5}} There are three molecular interactions to consider: solvent-solvent <math>w_{11}</math>, monomer-monomer <math>w_{22}</math> (not the [[covalent bond]]ing, but between different chain sections), and monomer-solvent <math>w_{12}</math>. Each of the last occurs at the expense of the average of the other two, so the energy increment per monomer-solvent contact is
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| :<math>\Delta w = w_{12} - \begin{matrix} \frac{1}{2} \end{matrix} (w_{22} + w_{11})\,</math>
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| The total number of such contacts is
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| :<math>xN_2z\phi_1 = N_1\phi_2z\,</math>
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| where <math>z</math> is the coordination number, the number of nearest neighbors for a lattice site, each one occupied either by one chain segment or a solvent molecule. That is, <math>xN_2</math> is the total number of polymer segments (monomers) in the solution, so <math>xN_2z</math> is the number of nearest-neighbor sites to ''all'' the polymer segments. Multiplying by the probability <math>\phi_1</math> that any such site is occupied by a solvent molecule,{{Ref|6}} we obtain the total number of polymer-solvent molecular interactions. An approximation following [[mean field theory]] is made by following this procedure, thereby reducing the complex problem of many interactions to a simpler problem of one interaction.
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| The enthalpy change is equal to the energy change per polymer monomer-solvent interaction multiplied by the number of such interactions
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| :<math>\Delta H_m = N_1\phi_2z\Delta w\,</math>
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| The polymer-solvent interaction parameter [[chi (letter)|''chi'']] is defined as
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| :<math>\chi_{12} = z\Delta w/kT \,</math> | |
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| It depends on the nature of both the solvent and the solute, and is the only ''material-specific'' parameter in the model. The enthalpy change becomes
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| :<math>\Delta H_m = k T N_1\phi_2\chi_{12} \,</math>
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| Assembling terms, the total free energy change is
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| :<math>\Delta G_m = RT[\,n_1\ln\phi_1 + n_2\ln\phi_2 + n_1\phi_2\chi_{12}\,] \,</math>
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| where we have converted the expression from molecules <math>N_1</math> and <math>N_2</math> to moles <math>n_1</math> and <math>n_2</math> by transferring [[Avogadro's number]] <math>N_A</math> to the [[gas constant]] <math>R = kN_A</math>.
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| The value of the interaction parameter can be estimated from the [[Hildebrand solubility parameter]]s <math>\delta_a</math> and <math>\delta_b</math>
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| :<math>\chi_{12} = V_{seg}(\delta_a - \delta_b)^2/RT \,</math>
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| where <math>V_{seg}</math> is the actual volume of a polymer segment.
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| This treatment does not attempt to calculate the [[Chemical structure|conformational]] entropy of folding for polymer chains. (See the [[random coil#Random walk model|random coil]] discussion.) The conformations of even [[amorphous solid|amorphous]] polymers will change when they go into solution, and most [[thermoplastic]] polymers also have [[lamellae (materials)|lamellar]] crystalline regions which do not persist in solution as the chains separate. These events are accompanied by additional entropy and energy changes.
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| More advanced models exist, such as the [[Flory-Krigbaum theory]].
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| == External links ==
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| * [http://www.informit.com/content/images/chap3_0130181684/elementLinks/chap3_0130181684.pdf "Conformations, Solutions and Molecular Weight" (book chapter)], Chapter 3 of Book Title: Polymer Science and Technology; by Joel R. Fried; 2nd Edition, 2003
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| == References and footnotes ==
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| # {{Note|1}} "[[Thermodynamics]] of High [[Polymer]] [[Solution]]s," [[Paul Flory|Paul J. Flory]] ''Journal of Chemical Physics,'' August 1941, Volume 9, Issue 8, p. 660 [http://link.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000009000008000660000002&idtype=cvips&gifs=Yes Abstract]. Flory suggested that Huggins' name ought to be first since he had published several months earlier: Flory, P.J., "Thermodynamics of high polymer solutions," ''J. Chem. Phys.'' '''10''':51-61 (1942) [http://www.garfield.library.upenn.edu/classics1985/A1985AFW3100001.pdf ''Citation Classic'' No. 18, May 6, 1985]
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| # {{Note|2}} "Solutions of Long Chain [[Chemical compound|Compound]]s," [[Maurice Loyal Huggins|Maurice L. Huggins]] ''Journal of Chemical Physics,'' May 1941 Volume 9, Issue 5, p. 440 [http://link.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000009000005000440000001&idtype=cvips&gifs=yes Abstract]
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| # {{Note|3}} We are ignoring the ''free volume'' due to molecular disorder in liquids and amorphous solids as compared to [[crystal]]s. This, and the assumption that [[monomer]]s and solute molecules are really the same size, are the main ''geometric'' approximations in this model.
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| # {{Note|4}} For a real [[chemical synthesis|synthetic]] polymer, there is a [[statistics|statistical]] [[random variable|distribution]] of [[Chain (sequence)|chain]] lengths, so <math>x</math> would be an [[average]].
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| # {{Note|5}} The [[enthalpy]] is the [[internal energy]] corrected for any [[pressure]]-[[volume]] [[mechanical work|work]] at constant (external) <math>P</math>. We are not making any distinction here. This allows the approximation of [[Helmholtz free energy]], which is the natural form of free energy from the Flory-Huggins lattice theory, to Gibbs free energy.
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| # {{Note|6}} In fact, two of the sites adjacent to a polymer segment are occupied by other polymer segments since it is part of a [[Chain (sequence)|chain]]; and one more, making three, for [[branching (chemistry)|branching]] sites, but only one for [[Polymer|terminal]]s.
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| {{Chemical solutions}}
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| {{DEFAULTSORT:Flory-Huggins solution theory}}
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| [[Category:Polymer chemistry]]
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| [[Category:Solutions]]
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| [[Category:Thermodynamic free energy]]
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| [[Category:Statistical mechanics]]
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