Flory–Huggins solution theory
   HOME

TheInfoList



OR:

Flory–Huggins solution theory is a lattice model of the
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
of polymer solutions which takes account of the great dissimilarity in
molecular A molecule is a group of two or more atoms that are held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemistry, ...
sizes in adapting the usual expression for the
entropy of mixing In thermodynamics, the entropy of mixing is the increase in the total entropy when several initially separate systems of different composition, each in a thermodynamic state of internal equilibrium, are mixed without chemical reaction by the ther ...
. The result is an equation for the
Gibbs free energy In thermodynamics, the Gibbs free energy (or Gibbs energy as the recommended name; symbol is a thermodynamic potential that can be used to calculate the maximum amount of Work (thermodynamics), work, other than Work (thermodynamics)#Pressure–v ...
change \Delta G_ for mixing a polymer with a
solvent A solvent (from the Latin language, Latin ''wikt:solvo#Latin, solvō'', "loosen, untie, solve") is a substance that dissolves a solute, resulting in a Solution (chemistry), solution. A solvent is usually a liquid but can also be a solid, a gas ...
. Although it makes simplifying assumptions, it generates useful results for interpreting experiments.


Theory

The thermodynamic equation for the
Gibbs energy In thermodynamics, the Gibbs free energy (or Gibbs energy as the recommended name; symbol is a thermodynamic potential that can be used to calculate the maximum amount of work, other than pressure–volume work, that may be performed by a ther ...
change accompanying mixing at constant
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
and (external)
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
is : \Delta G_ = \Delta H_ - T\Delta S_ A change, denoted by \Delta, is the value of a variable for a
solution Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solu ...
or
mixture In chemistry, a mixture is a material made up of two or more different chemical substances which can be separated by physical method. It is an impure substance made up of 2 or more elements or compounds mechanically mixed together in any proporti ...
minus the values for the pure
components Component may refer to: In engineering, science, and technology Generic systems *System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis * Lumped e ...
considered separately. The objective is to find explicit
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
s for \Delta H_ and \Delta S_, the
enthalpy Enthalpy () is the sum of a thermodynamic system's internal energy and the product of its pressure and volume. It is a state function in thermodynamics used in many measurements in chemical, biological, and physical systems at a constant extern ...
and
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
increments associated with the mixing
process A process is a series or set of activities that interact to produce a result; it may occur once-only or be recurrent or periodic. Things called a process include: Business and management * Business process, activities that produce a specific s ...
. The result obtained by Flory and Huggins is : \Delta G_ = RT ,n_1\ln\phi_1 + n_2\ln\phi_2 + n_1\phi_2\chi_\, The right-hand side is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
of the number of moles n_1 and volume fraction \phi_1 of
solvent A solvent (from the Latin language, Latin ''wikt:solvo#Latin, solvō'', "loosen, untie, solve") is a substance that dissolves a solute, resulting in a Solution (chemistry), solution. A solvent is usually a liquid but can also be a solid, a gas ...
(
component Component may refer to: In engineering, science, and technology Generic systems *System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis * Lumped e ...
 1), the number of moles n_2 and volume fraction \phi_2 of polymer (component 2), with the introduction of a parameter \chi to take account of the
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
of interdispersing polymer and solvent molecules. R is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment p ...
and T is the
absolute temperature Thermodynamic temperature, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion. Thermodynamic temperature is typically expres ...
. The volume fraction is analogous to the
mole fraction In chemistry, the mole fraction or molar fraction, also called mole proportion or molar proportion, is a quantity defined as the ratio between the amount of a constituent substance, ''ni'' (expressed in unit of moles, symbol mol), and the to ...
, 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, \chi, is a free energy parameter, thus including an entropic component.


Derivation

We first calculate the ''entropy'' of mixing, the increase in the
uncertainty Uncertainty or incertitude refers to situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown, and is particularly relevant for decision ...
about the locations of the molecules when they are interspersed. In the pure condensed phases
solvent A solvent (from the Latin language, Latin ''wikt:solvo#Latin, solvō'', "loosen, untie, solve") is a substance that dissolves a solute, resulting in a Solution (chemistry), solution. A solvent is usually a liquid but can also be a solid, a gas ...
and polymer – a molecule exists for any arbitrarily small volume element. The expression for the
entropy of mixing In thermodynamics, the entropy of mixing is the increase in the total entropy when several initially separate systems of different composition, each in a thermodynamic state of internal equilibrium, are mixed without chemical reaction by the ther ...
of small molecules in terms of
mole fraction In chemistry, the mole fraction or molar fraction, also called mole proportion or molar proportion, is a quantity defined as the ratio between the amount of a constituent substance, ''ni'' (expressed in unit of moles, symbol mol), and the to ...
s is no longer reasonable when the
solute In chemistry, a solution is defined by IUPAC as "A liquid or solid phase containing more than one substance, when for convenience one (or more) substance, which is called the solvent, is treated differently from the other substances, which are ...
is a
macromolecular A macromolecule is a "molecule of high relative molecular mass, the structure of which essentially comprises the multiple repetition of units derived, actually or conceptually, from molecules of low relative molecular mass." Polymers are physi ...
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
. We take account of this dis
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
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
monomer A monomer ( ; ''mono-'', "one" + '' -mer'', "part") is a molecule that can react together with other monomer molecules to form a larger polymer chain or two- or three-dimensional network in a process called polymerization. Classification Chemis ...
of the polymer chain, so the total number of sites is : N = N_1 + xN_2 where N_1 is the number of solvent molecules and N_2 is the number of polymer molecules, each of which has x segments. For a
random walk In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space. An elementary example of a rand ...
on a lattice we can calculate the
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
change (the increase in spatial
uncertainty Uncertainty or incertitude refers to situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown, and is particularly relevant for decision ...
) as a result of mixing solute and solvent. : \Delta S_ = -k_\left N_1\ln\tfrac + N_2\ln\tfrac \right/math> where k_ is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
. Define the lattice ''volume fractions'' \phi_1 and \phi_2 : \phi_1 = \frac, \quad \phi_2 = \frac These are also the probabilities that a given lattice site, chosen at
random In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. ...
, is occupied by a solvent molecule or a polymer segment, respectively. Thus : \Delta S_ = -k_ ,N_1\ln\phi_1 + N_2\ln\phi_2\,/math> For a small solute whose molecules occupy just one lattice site, x equals one, the volume fractions reduce to molecular or mole fractions, and we recover the usual
entropy of mixing In thermodynamics, the entropy of mixing is the increase in the total entropy when several initially separate systems of different composition, each in a thermodynamic state of internal equilibrium, are mixed without chemical reaction by the ther ...
. In addition to the entropic effect, we can expect an ''enthalpy'' change. There are three molecular interactions to consider: solvent-solvent w_, monomer-monomer w_ (not the
covalent bond A covalent bond is a chemical bond that involves the sharing of electrons to form electron pairs between atoms. These electron pairs are known as shared pairs or bonding pairs. The stable balance of attractive and repulsive forces between atom ...
ing, but between different chain sections), and monomer-solvent w_. Each of the last occurs at the expense of the average of the other two, so the energy increment per monomer-solvent contact is : \Delta w = w_ - \tfrac(w_ + w_) The total number of such contacts is : xN_2z\phi_1 = N_1\phi_2z where 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, xN_2 is the total number of polymer segments (monomers) in the solution, so xN_2z is the number of nearest-neighbor sites to ''all'' the polymer segments. Multiplying by the probability \phi_1 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 theory In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of ...
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 : \Delta H_ = N_1\phi_2z\Delta w The polymer-solvent interaction parameter ''chi'' is defined as : \chi_ = \frac 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 : \Delta H_ = k_ T N_1\phi_2\chi_ Assembling terms, the total free energy change is : \Delta G_ = RT ,n_1\ln\phi_1 + n_2\ln\phi_2 + n_1\phi_2\chi_\, where we have converted the expression from molecules N_1 and N_2 to moles n_1 and n_2 by transferring the
Avogadro constant The Avogadro constant, commonly denoted or , is an SI defining constant with an exact value of when expressed in reciprocal moles. It defines the ratio of the number of constituent particles to the amount of substance in a sample, where th ...
N_\text to the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment p ...
R = k_N_\text. The value of the interaction parameter can be estimated from the
Hildebrand solubility parameter The Hildebrand solubility parameter (δ) provides a numerical estimate of the degree of interaction between materials and can be a good indication of solubility, particularly for nonpolar materials such as many polymers. Materials with similar val ...
s \delta_a and \delta_b : \chi_ = \frac where V_ is the actual volume of a polymer segment. In the most general case the interaction \Delta w and the ensuing mixing parameter, \chi, is a free energy parameter, thus including an entropic component. 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 the Flory–Krigbaum theory.


Liquid-liquid phase separation

Polymers can separate out from the solvent, and do so in a characteristic way. The Flory–Huggins free energy per unit volume, for a polymer with N monomers, can be written in a simple dimensionless form : f=\frac\ln\phi+(1-\phi)\ln(1-\phi)+\chi\phi(1-\phi) for \phi the volume fraction of monomers, and N\gg 1. The
osmotic pressure Osmotic pressure is the minimum pressure which needs to be applied to a Solution (chemistry), solution to prevent the inward flow of its pure solvent across a semipermeable membrane. It is also defined as the measure of the tendency of a soluti ...
(in reduced units) is : \Pi=\frac -\ln(1-\phi)-\phi-\chi\phi^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 : f'' = \frac+\frac-2\chi and the solution first becomes unstable when this and the third derivative : f = -\frac + \frac are both equal to zero. A little algebra then shows that the polymer solution first becomes unstable at a critical point at : \chi_\text\simeq 1/2+N^+\cdots \qquad \phi_\text\simeq N^-N^+\cdots This means that for all values of 0<\chi\lesssim 1/2 the monomer-solvent effective interaction is weakly repulsive, but this is too weak to cause liquid/liquid separation. However, when \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 N^, 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 N=1, and then \phi_\text=1/2 and both coexisting phases are far from pure.


Polymer blends

Synthetic polymers rarely consist of chains of uniform length in solvent. The Flory–Huggins free energy density can be generalized to an N-component mixture of polymers with lengths r_ by : f\Bigl(\\Bigr) = \sum_^N \frac\ln\phi_i + \frac\sum_^N \phi_i \phi_j \chi_ For a binary
polymer blend In materials science, a polymer blend, or polymer mixture, is a member of a class of materials analogous to metal alloys, in which at least two polymers are blended together to create a new material with different physical properties. History Du ...
, where one species consists of N_ monomers and the other N_ monomers this simplifies to : f(\phi) = \frac\ln\phi + \frac\ln(1-\phi) + \chi\phi(1-\phi) 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 N_ \gg 1 and N_ \gg 1 these terms are negligibly small. This implies that for a stable mixture to exist \chi < 0 , so for polymers A and B to blend their segments must attract one another.


Limitations

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 \chi \propto T^ interaction-temperature dependence and other features commonly observed in polymer mixtures. However, unmodified Flory–Huggins theory fails to predict the lower critical solution temperature observed in some polymer blends and the lack of dependence of the critical temperature T_\text on chain length r_. Additionally, it can be shown that for a binary blend of polymer species with equal chain lengths (N_ = N_) the critical concentration should be \psi_\text = 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, \chi parameter complex dependence on
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
, 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 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 c^ which can be estimated as : c^ = \frac Here, ''m'' is the mass of a single polymer chain, and R_\text is the chain's
radius of gyration The radius of gyration or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concent ...
.


Footnotes

# "
Thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
of High
Polymer A polymer () is a chemical substance, substance or material that consists of very large molecules, or macromolecules, that are constituted by many repeat unit, repeating subunits derived from one or more species of monomers. Due to their br ...
Solution Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solu ...
s", Paul J. Flory ''Journal of Chemical Physics'', August 1941, Volume 9, Issue 8, p. 66
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
''Citation Classic'' No. 18, May 6, 1985
# "Solutions of Long Chain Compounds", Maurice L. Huggins ''Journal of Chemical Physics'', May 1941 Volume 9, Issue 5, p. 44
Abstract
# We are ignoring the ''free volume'' due to molecular disorder in liquids and amorphous solids as compared to
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
s. This, and the assumption that
monomer A monomer ( ; ''mono-'', "one" + '' -mer'', "part") is a molecule that can react together with other monomer molecules to form a larger polymer chain or two- or three-dimensional network in a process called polymerization. Classification Chemis ...
s and solute molecules are really the same size, are the main ''geometric'' approximations in this model. # For a real
synthetic Synthetic may refer to: Science * Synthetic biology * Synthetic chemical or compound, produced by the process of chemical synthesis * Synthetic elements, chemical elements that are not naturally found on Earth and therefore have to be created in ...
polymer, there is a
statistical Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a varia ...
of
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
lengths, so x would be an
average In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by ...
. # The
enthalpy Enthalpy () is the sum of a thermodynamic system's internal energy and the product of its pressure and volume. It is a state function in thermodynamics used in many measurements in chemical, biological, and physical systems at a constant extern ...
is the
internal energy The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accoun ...
corrected for any
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
-
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an ani ...
at constant (external) P. We are not making any distinction here. This allows the approximation of
Helmholtz free energy In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature ( isothermal). The change in the Helmholtz ene ...
, which is the natural form of free energy from the Flory–Huggins lattice theory, to Gibbs free energy. # In fact, two of the sites adjacent to a polymer segment are occupied by other polymer segments since it is part of a
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
; and one more, making three, for branching sites, but only one for terminals.


References


External links


"Conformations, Solutions and Molecular Weight" (book chapter)
Chapter 3 of Book Title: Polymer Science and Technology; by Joel R. Fried; 2nd Edition, 2003 {{DEFAULTSORT:Flory-Huggins solution theory Polymer chemistry Solutions Thermodynamic free energy Statistical mechanics