Flory-Huggins solution theory is a
mathematical modelA mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
of the
thermodynamicsThermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
of
polymerA polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
solutionIn chemistry, a solution is a homogeneous mixture composed of only one phase. In such a mixture, a solute is dissolved in another substance, known as a solvent. The solvent does the dissolving.- Types of solutions :...
s which takes account of the great dissimilarity in
molecularA molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
sizes in adapting the usual
expressionIn mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers , variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic...
for the
entropy of mixingIn thermodynamics the entropy of mixing is the increase in the total entropy of a compound system, when different and chemically non-reacting chemical substances or material components are mixed by removing partition between the system's initially separate volumes...
. The result is an equation for the
Gibbs free energyIn thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...
change
for mixing a polymer with a
solventA solvent is a liquid, solid, or gas that dissolves another solid, liquid, or gaseous solute, resulting in a solution that is soluble in a certain volume of solvent at a specified temperature...
. Although it makes simplifying assumptions, it generates useful results for interpreting experiments.
The
thermodynamic equationA thermodynamic potential is a scalar function used to represent the thermodynamic state of a system. The concept of thermodynamic potentials was introduced by Pierre Duhem in 1886. Josiah Willard Gibbs in his papers used the term fundamental functions. One main thermodynamic potential that has a...
for the
Gibbs free energyIn thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...
change accompanying mixing at constant
temperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
and (external)
pressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
is
A change, denoted by
, is the
valueA number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
of a
variableIn mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
for a
solutionIn chemistry, a solution is a homogeneous mixture composed of only one phase. In such a mixture, a solute is dissolved in another substance, known as a solvent. The solvent does the dissolving.- Types of solutions :...
or
mixtureIn chemistry, a mixture is a material system made up by two or more different substances which are mixed together but are not combined chemically...
minus the values for the pure
componentsIn thermodynamics, a component is a chemically distinct constituent ofa system. Calculating the number of components in a system is necessary, for example, when applying Gibbs phase rule in determination of the number of degrees of freedom of a system....
considered separately. The objective is to find explicit
formulaIn mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....
s for
and
, the
enthalpyEnthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
and
entropyEntropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
increments associated with the mixing process.
The result obtained by
FloryPaul John Flory was an American chemist and Nobel laureate who was known for his prodigious volume of work in the field of polymers, or macromolecules...
and
HugginsMaurice Loyal Huggins was a scientist who independently conceived the idea of hydrogen bonding and who was an early advocate for their role in stabilizing protein secondary structure...
is
The right-hand side is a
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of the number of
molesThe mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...
and volume fraction
of
solventA solvent is a liquid, solid, or gas that dissolves another solid, liquid, or gaseous solute, resulting in a solution that is soluble in a certain volume of solvent at a specified temperature...
(
componentIn thermodynamics, a component is a chemically distinct constituent ofa system. Calculating the number of components in a system is necessary, for example, when applying Gibbs phase rule in determination of the number of degrees of freedom of a system....
), the number of moles
and volume fraction
of polymer (component
), with the introduction of a parameter
chiChi is the 22nd letter of the Greek alphabet, pronounced as in English.-Greek:-Ancient Greek:Its value in Ancient Greek was an aspirated velar stop .-Koine Greek:...
to take account of the
energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
of interdispersing polymer and solvent molecules.
is the
gas constantThe gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...
and
is the
absolute temperatureThermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...
. 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.
Derivation
We first calculate the
entropy of mixingIn thermodynamics the entropy of mixing is the increase in the total entropy of a compound system, when different and chemically non-reacting chemical substances or material components are mixed by removing partition between the system's initially separate volumes...
, the increase in the uncertainty about the locations of the molecules when they are interspersed. In the pure condensed
phasesIn the physical sciences, a phase is a region of space , throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, and chemical composition...
—
solventA solvent is a liquid, solid, or gas that dissolves another solid, liquid, or gaseous solute, resulting in a solution that is soluble in a certain volume of solvent at a specified temperature...
and polymer — everywhere we look we find a molecule. Of course, any notion of "finding" a molecule in a given location is a
thought experimentA thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
since we can't actually examine
spatialSpace is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
locations the size of molecules. The
expressionIn mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers , variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic...
for the
entropy of mixingIn thermodynamics the entropy of mixing is the increase in the total entropy of a compound system, when different and chemically non-reacting chemical substances or material components are mixed by removing partition between the system's initially separate volumes...
of small molecules in terms of
mole fractions is no longer reasonable when the
soluteIn chemistry, a solution is a homogeneous mixture composed of only one phase. In such a mixture, a solute is dissolved in another substance, known as a solvent. The solvent does the dissolving.- Types of solutions :...
is a
macromolecularA macromolecule is a very large molecule commonly created by some form of polymerization. In biochemistry, the term is applied to the four conventional biopolymers , as well as non-polymeric molecules with large molecular mass such as macrocycles...
chain. We take account of this dis
symmetrySymmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
in molecular sizes by assuming that individual polymer segments and individual solvent molecules occupy sites on a lattice. Each site is occupied by exactly one molecule of the solvent or by one
monomerA monomer is an atom or a small molecule that may bind chemically to other monomers to form a polymer; the term "monomeric protein" may also be used to describe one of the proteins making up a multiprotein complex...
of the polymer chain, so the total number of sites is
is the number of solvent molecules and
is the number of polymer molecules, each of which has
segments.
From
statistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
we can calculate the
entropyEntropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
change, the increase in
spatialSpace is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
uncertainty, as a result of mixing solute and solvent.
where
is
Boltzmann's constant. Define the lattice
volume fractions 
and
These are also the probabilities that a given lattice site, chosen at
randomRandomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....
, is occupied by a solvent molecule or a polymer segment, respectively. Thus
For a small solute whose molecules occupy just one lattice site,
equals one, the volume fractions reduce to
molecular or mole fractions, and we recover the usual equation from
ideal mixingIn chemistry, an ideal solution or ideal mixture is a solution with thermodynamic properties analogous to those of a mixture of ideal gases. The enthalpy of solution is zero as is the volume change on mixing; the closer to zero the enthalpy of solution is, the more "ideal" the behavior of the...
theory.
In addition to the entropic effect, we can expect an
enthalpy change. There are three molecular interactions to consider: solvent-solvent
, monomer-monomer
(not the
covalent bondA covalent bond is a form of chemical bonding that is characterized by the sharing of pairs of electrons between atoms. The stable balance of attractive and repulsive forces between atoms when they share electrons is known as covalent bonding....
ing, but between different chain sections), and monomer-solvent
. Each of the last occurs at the expense of the average of the other two, so the energy increment per monomer-solvent contact is
The total number of such contacts is
where
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,
is the total number of polymer segments (monomers) in the solution, so
is the number of nearest-neighbor sites to
all the polymer segments. Multiplying by the probability
that any such site is occupied by a solvent molecule, we obtain the total number of polymer-solvent molecular interactions. An approximation following
mean field theoryMean field theory is a method to analyse physical systems with multiple bodies. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases . The n-body system is replaced by a 1-body problem with a chosen good external field...
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
The polymer-solvent interaction parameter
chiChi is the 22nd letter of the Greek alphabet, pronounced as in English.-Greek:-Ancient Greek:Its value in Ancient Greek was an aspirated velar stop .-Koine Greek:...
is defined as
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
Assembling terms, the total free energy change is
where we have converted the expression from molecules
and
to moles
and
by transferring
Avogadro's numberIn chemistry and physics, the Avogadro constant is defined as the ratio of the number of constituent particles N in a sample to the amount of substance n through the relationship NA = N/n. Thus, it is the proportionality factor that relates the molar mass of an entity, i.e...
to the
gas constantThe gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...
.
The value of the interaction parameter can be estimated from the
Hildebrand solubility parameterThe Hildebrand solubility parameter provides a numerical estimate of the degree of interaction between materials, and can be a good indication of solubility, particularly for non polar materials such as many polymers...
s
and
where
is the actual volume of a polymer segment.
This treatment does not attempt to calculate the
conformationalA chemical structure includes molecular geometry, electronic structure and crystal structure of molecules. Molecular geometry refers to the spatial arrangement of atoms in a molecule and the chemical bonds that hold the atoms together. Molecular geometry can range from the very simple, such as...
entropy of folding for polymer chains. (See the random coil discussion.) The conformations of even
amorphousIn condensed matter physics, an amorphous or non-crystalline solid is a solid that lacks the long-range order characteristic of a crystal....
polymers will change when they go into solution, and most
thermoplasticThermoplastic, also known as a thermosoftening plastic, is a polymer that turns to a liquid when heated and freezes to a very glassy state when cooled sufficiently...
polymers also have
lamellarA lamella is a gill-shaped structure: fine sheets of material held adjacent to one another, often with fluid in between though sometimes simply a set of 'welded' plates. They appear in biological and engineering contexts, such as filters and heat exchangers...
crystalline regions which do not persist in solution as the chains separate. These events are accompanied by additional entropy and energy changes.
More advanced models exist, such as the Flory-Krigbaum theory.
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