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Flory–Huggins solution theory is alattice model of thethermodynamics ofpolymer solutions which takes account of the great dissimilarity inmolecular sizes in adapting the usualexpression for theentropy of mixing. The result is an equation for theGibbs free energy change${\displaystyle \mathrm{\Delta }{G}_{\mathrm{m}\mathrm{i}\mathrm{x}}}$ for mixing a polymer with asolvent. Although it makes simplifying assumptions, it generates useful results for interpreting experiments.

Thethermodynamic equation for theGibbs energy change accompanying mixing at constanttemperature and (external)pressure is

${\displaystyle \mathrm{\Delta }{G}_{\mathrm{m}\mathrm{i}\mathrm{x}}=\mathrm{\Delta }{H}_{\mathrm{m}\mathrm{i}\mathrm{x}}-T\mathrm{\Delta }{S}_{\mathrm{m}\mathrm{i}\mathrm{x}}}$

A change, denoted by${\displaystyle \mathrm{\Delta }}$, is thevalue of avariable for asolution ormixture minus the values for the purecomponents considered separately. The objective is to find explicitformulas for${\displaystyle \mathrm{\Delta }{H}_{\mathrm{m}\mathrm{i}\mathrm{x}}}$ and${\displaystyle \mathrm{\Delta }{S}_{\mathrm{m}\mathrm{i}\mathrm{x}}}$, theenthalpy andentropy increments associated with the mixingprocess.

The result obtained byFlory^[1] andHuggins^[2] is

${\displaystyle \mathrm{\Delta }{G}_{\mathrm{m}\mathrm{i}\mathrm{x}}=RT[n_1\ln {\varphi }_1+n_2\ln {\varphi }_2+n_1{\varphi }_2{\chi }_{12}]}$

The right-hand side is afunction of the number ofmoles${\displaystyle n_1}$ and volume fraction${\displaystyle {\varphi }_1}$ ofsolvent (component ${\displaystyle 1}$), the number of moles${\displaystyle n_2}$ and volume fraction${\displaystyle {\varphi }_2}$ of polymer (component ${\displaystyle 2}$), with the introduction of a parameter${\displaystyle \chi }$ to take account of theenergy of interdispersing polymer and solvent molecules.${\displaystyle R}$ is thegas constant and${\displaystyle T}$ is theabsolute temperature. The volume fraction is analogous to themole 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. In the most general case the mixing parameter,${\displaystyle \chi }$, is a free energy parameter, thus including an entropic component.^[1][2]

We first calculate theentropy of mixing, the increase in theuncertainty about the locations of the molecules when they are interspersed. In the pure condensedphases –solvent and polymer – a molecule exists for any arbitrarily small volume element.^[3] Theexpression for theentropy of mixing of small molecules in terms ofmole fractions is no longer reasonable when thesolute is amacromolecularchain. We take account of this dissymmetry in molecular sizes by assuming that individual polymer segments and individual solvent molecules occupy sites on alattice. Each site is occupied by exactly one molecule of the solvent or by onemonomer of the polymer chain, so the total number of sites is

${\displaystyle N=N_1+x{N}_2}$

where${\displaystyle N_1}$ is the number of solvent molecules and${\displaystyle N_2}$ is the number of polymer molecules, each of which has${\displaystyle x}$ segments.^[4]

For arandom walk on a lattice^[3] we can calculate theentropy change (the increase inspatialuncertainty) as a result of mixing solute and solvent.

${\displaystyle \mathrm{\Delta }{S}_{\mathrm{m}\mathrm{i}\mathrm{x}}=-k_{\mathrm{B}}\left[N_1\ln \frac{N_1}{N}+N_2\ln \frac{x{N}_2}{N}\right]}$

where${\displaystyle k_{\mathrm{B}}}$ is theBoltzmann constant. Define the latticevolume fractions${\displaystyle {\varphi }_1}$ and${\displaystyle {\varphi }_2}$

${\displaystyle {\varphi }_1=\frac{N_1}{N},{\varphi }_2=\frac{x{N}_2}{N}}$

These are also the probabilities that a given lattice site, chosen atrandom, is occupied by a solvent molecule or a polymer segment, respectively. Thus

${\displaystyle \mathrm{\Delta }{S}_{\mathrm{m}\mathrm{i}\mathrm{x}}=-k_{\mathrm{B}}[N_1\ln {\varphi }_1+N_2\ln {\varphi }_2]}$

For a small solute whose molecules occupy just one lattice site,${\displaystyle x}$ equals one, the volume fractions reduce tomolecular or mole fractions, and we recover the usualentropy of mixing.

In addition to the entropic effect, we can expect anenthalpy change.^[5] There are three molecular interactions to consider: solvent-solvent${\displaystyle w_{11}}$, monomer-monomer${\displaystyle w_{22}}$ (not thecovalent bonding, but between different chain sections), and monomer-solvent${\displaystyle w_{12}}$. Each of the last occurs at the expense of the average of the other two, so the energy increment per monomer-solvent contact is

${\displaystyle \mathrm{\Delta }w=w_{12}-\frac{1}{2}(w_{22}+w_{11})}$

The total number of such contacts is

${\displaystyle x{N}_{2}z{\varphi }_1=N_1{\varphi }_{2}z}$

where${\displaystyle z}$ 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,${\displaystyle x{N}_2}$ is the total number of polymer segments (monomers) in the solution, so${\displaystyle x{N}_{2}z}$ is the number of nearest-neighbor sites toall the polymer segments. Multiplying by the probability${\displaystyle {\varphi }_1}$ that any such site is occupied by a solvent molecule,^[6] we obtain the total number of polymer-solvent molecular interactions. An approximation followingmean field theory is made by following this procedure, thereby reducing the complex problem of many interactions to a simpler problem of one interaction.

The enthalpy change is equal to the energy change per polymer monomer-solvent interaction multiplied by the number of such interactions

${\displaystyle \mathrm{\Delta }{H}_{\mathrm{m}\mathrm{i}\mathrm{x}}=N_1{\varphi }_{2}z\mathrm{\Delta }w}$

The polymer-solvent interaction parameterchi is defined as

${\displaystyle {\chi }_{12}=\frac{z\mathrm{\Delta }w}{k_{\mathrm{B}}T}}$

It depends on the nature of both the solvent and the solute, and is the onlymaterial-specific parameter in the model. The enthalpy change becomes

${\displaystyle \mathrm{\Delta }{H}_{\mathrm{m}\mathrm{i}\mathrm{x}}=k_{\mathrm{B}}T{N}_1{\varphi }_2{\chi }_{12}}$

Assembling terms, the total free energy change is

${\displaystyle \mathrm{\Delta }{G}_{\mathrm{m}\mathrm{i}\mathrm{x}}=RT[n_1\ln {\varphi }_1+n_2\ln {\varphi }_2+n_1{\varphi }_2{\chi }_{12}]}$

where we have converted the expression from molecules${\displaystyle N_1}$ and${\displaystyle N_2}$ to moles${\displaystyle n_1}$ and${\displaystyle n_2}$ by transferring theAvogadro constant${\displaystyle N_{\text{A}}}$ to thegas constant${\displaystyle R=k_{\mathrm{B}}{N}_{\text{A}}}$.

The value of the interaction parameter can be estimated from theHildebrand solubility parameters${\displaystyle {\delta }_a}$ and${\displaystyle {\delta }_b}$

${\displaystyle {\chi }_{12}=\frac{V_{\mathrm{s}\mathrm{e}\mathrm{g}}({\delta }_a-{\delta }_b{)}^2}{RT}}$

where${\displaystyle V_{\mathrm{s}\mathrm{e}\mathrm{g}}}$ is the actual volume of a polymer segment.

In the most general case the interaction${\displaystyle \mathrm{\Delta }w}$ and the ensuing mixing parameter,${\displaystyle \chi }$, is a free energy parameter, thus including an entropic component.^[1][2] This means that aside to the regular mixing entropy there is another entropic contribution from the interaction between solvent and monomer. This contribution is sometimes very important in order to make quantitative predictions of thermodynamic properties.

More advanced solution theories exist, such as theFlory–Krigbaum theory.

Polymers can separate out from the solvent, and do so in a characteristic way.^[4] The Flory–Huggins free energy per unit volume, for a polymer with${\displaystyle N}$ monomers, can be written in a simple dimensionless form

${\displaystyle f=\frac{\varphi }{N}\ln \varphi +(1-\varphi )\ln (1-\varphi )+\chi \varphi (1-\varphi )}$

for${\displaystyle \varphi }$ the volume fraction of monomers, and${\displaystyle N\gg 1}$. Theosmotic pressure (in reduced units) is

${\displaystyle \mathrm{\Pi }=\frac{\varphi }{N}-\ln (1-\varphi )-\varphi -\chi {\varphi }^2}$.

The polymer solution is stable with respect to small fluctuations when the second derivative of this free energy is positive. This second derivative is

${\displaystyle f^{″}=\frac{1}{N\varphi }+\frac{1}{1-\varphi }-2\chi }$

and the solution first becomes unstable when this and the third derivative

${\displaystyle f^{‴}=-\frac{1}{N{\varphi }^2}+\frac{1}{(1-\varphi {)}^2}}$

are both equal to zero. A little algebra then shows that the polymer solution first becomes unstable at a critical point at

${\displaystyle {\chi }_{\text{cp}}\simeq 1/2+N^{-1/2}+\cdots {\varphi }_{\text{cp}}\simeq N^{-1/2}-N^{-1}+\cdots }$

This means that for all values of the monomer-solvent effective interaction is weakly repulsive, but this is too weak to cause liquid/liquid separation. However, when${\displaystyle \chi >1/2}$, there is separation into two coexisting phases, one richer in polymer but poorer in solvent, than the other.

The unusual feature of the liquid/liquid phase separation is that it is highly asymmetric: the volume fraction of monomers at the critical point is approximately${\displaystyle N^{-1/2}}$, which is very small for large polymers. The amount of polymer in the solvent-rich/polymer-poor coexisting phase is extremely small for long polymers. The solvent-rich phase is close to pure solvent. This is peculiar to polymers, a mixture of small molecules can be approximated using the Flory–Huggins expression with${\displaystyle N=1}$, and then${\displaystyle {\varphi }_{\text{cp}}=1/2}$ and both coexisting phases are far from pure.

Synthetic polymers rarely consist of chains of uniform length in solvent. The Flory–Huggins free energy density can be generalized^[5] to an N-component mixture of polymers with lengths${\displaystyle r_i}$ by

${\displaystyle f(\{{\varphi }_i,r_i\})=\sum _{i=1}^N\frac{{\varphi }_i}{r_i}\ln {\varphi }_i+\frac{1}{2}\sum _{i,j=1}^N{\varphi }_i{\varphi }_j{\chi }_{ij}}$

For a binarypolymer blend, where one species consists of${\displaystyle N_A}$ monomers and the other${\displaystyle N_B}$ monomers this simplifies to

${\displaystyle f(\varphi )=\frac{\varphi }{N_A}\ln \varphi +\frac{1-\varphi }{N_B}\ln (1-\varphi )+\chi \varphi (1-\varphi )}$

As in the case for dilute polymer solutions, the first two terms on the right-hand side represent the entropy of mixing. For large polymers of${\displaystyle N_A\gg 1}$ and${\displaystyle N_B\gg 1}$ these terms are negligibly small. This implies that for a stable mixture to exist, so for polymers A and B to blend their segments must attract one another.^[6]

Flory–Huggins theory tends to agree well with experiments in the semi-dilute concentration regime and can be used to fit data for even more complicated blends with higher concentrations. The theory qualitatively predicts phase separation, the tendency for high molecular weight species to be immiscible, the${\displaystyle \chi \propto T^{-1}}$ interaction-temperature dependence and other features commonly observed in polymer mixtures. However, unmodified Flory–Huggins theory fails to predict thelower critical solution temperature observed in some polymer blends and the lack of dependence of the critical temperature${\displaystyle T_{\text{c}}}$ on chain length${\displaystyle r_i}$.^[7] Additionally, it can be shown that for a binary blend of polymer species with equal chain lengths${\displaystyle (N_A=N_B)}$ the critical concentration should be${\displaystyle {\psi }_{\text{c}}=1/2}$; however, polymers blends have been observed where this parameter is highly asymmetric. In certain blends, mixing entropy can dominate over monomer interaction. By adopting the mean-field approximation,${\displaystyle \chi }$ parameter complex dependence ontemperature, blend composition, and chain length was discarded. Specifically, interactions beyond the nearest neighbor may be highly relevant to the behavior of the blend and the distribution of polymer segments is not necessarily uniform, so certain lattice sites may experience interaction energies disparate from that approximated by the mean-field theory.

One well-studied^[4][6] effect on interaction energies neglected by unmodified Flory–Huggins theory is chain correlation. In dilute polymer mixtures, where chains are well separated, intramolecular forces between monomers of the polymer chain dominate and drive demixing leading to regions where polymer concentration is high. As the polymer concentration increases, chains tend to overlap and the effect becomes less important. In fact, the demarcation between dilute and semi-dilute solutions is commonly defined by the concentration where polymers begin to overlap${\displaystyle c^{\ast }}$ which can be estimated as

${\displaystyle c^{\ast }=\frac{m}{\frac{4}{3}\pi R_{\text{g}}^3}}$

Here,m is the mass of a single polymer chain, and${\displaystyle R_{\text{g}}}$ is the chain'sradius of gyration.

• "Conformations, Solutions and Molecular Weight" (book chapter), Chapter 3 of Book Title: Polymer Science and Technology; by Joel R. Fried; 2nd Edition, 2003

• Polymer chemistry
• Solutions
• Thermodynamic free energy
• Statistical mechanics

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