Lec-10 Theories of Liquid Mixtures………………………………………………..…Eng. Auda Jabbar Ms.C.
Theories of Liquid Mixtures
The theory makes it possible to:
1- Interpret and predict different material’s behavior in a wide range of
states and independent variables, viz. gases, low molecular weight
organic liquids, metals, polymers in a glassy or molten state
2-Determine miscibility of gases or liquids in polymers
3- Compute the phase diagrams of polymer blends, etc.
1- Lattice, Cell, and Hole Theories
The statistical mechanics methods that use a pseudo-crystalline model
of regularly placed elements on a “lattice” are known as lattice theories.
Many theories, known under the names of free volume, cell-hole,
tunnel, Monte Carlo, or molecular dynamics belong here.
Of these, only two will be mentioned:
1- The first, and the best known, was originally developed by Huggins
[1941] and by Flory [1941], then extended by many authors [e.g.
Utracki, 1962; Koningsveld, 1967].
2-The second, is the cell-hole Simha-Somcynsky [1969] theory that
has been incessantly evolving during the intervening years.
1.1 Huggins-Flory theory
For binary systems that contain an ingredient i = 1 or 2 (traditionally,
for polymer solutions the subscript 1 indicates solvent, and 2 polymer)
the Huggins-Flory, H-F, relation has been expressed in several equivalent
forms
The drawback of the H-F theory was the initial assumption that all lattice
cells are occupied by either solvent molecules or polymeric segments
that are of equal size. As a consequence the free volume contribution was
neglected.
1.2 Equation of State Theories
Equation of State (EoS) or PVT Relationships .In his Ph.D. thesis of
1873, van der Waals proposed the first EoS. The relation is frequently
written in terms of reduced variables, indicating expected observance of
the corresponding states principle:
Eq 2.16 also introduced the free volume concept — note that as T 0,
V b. Van der Waals considered that molecules move in “cells” made
by the surrounding molecules with a uniform potential. The volume
within which the center of a molecule can freely move, is what defines
the free volume. Thus, one may distinguish:
Lec-10 Theories of Liquid Mixtures………………………………………………..…Eng. Auda Jabbar Ms.C.
Thermodynamically, the free volume is expressed in terms of the entropy
of vaporization:
1.3 Gas-lattice Model
The gas-lattice model considers liquid to be a binary mixture of
randomly distributed, occupied and vacant sites. P and T can change the
concentration of holes, but not their size. A molecule may occupy m sites.
According to the gas-lattice theory, four factors determine the
polymer/polymer miscibility:
1. Interacting surface areas of segments.
2. Coil dimensions (dependent on T, , and MW).
3. Molecular weight polydispersity.
4. Free volume fraction.
2. Off-lattice Theories
2.1 Strong Interactions Model
For incompressible systems having strong interactions, e.g., acid-base
type, the directional-specific model of segmental interactions may be
used [Walker and Vause, 1982; ten Brinke and Karasz, 1984].
The following expression was derived:
where U1 and U2 are the attractive and repulsive energies, respectively,
and q is the degeneracy number. Depending on the relative magnitude of
U1 and U2, Eq 2.49 predicts either UCST or LCST.
Lec-10 Theories of Liquid Mixtures………………………………………………..…Eng. Auda Jabbar Ms.C.
2.2 Heat of Mixing Approach
When the configurational entropy of a polymer blend is vanishingly
small, the enthalpic effects dominate, hence the adiabatic calorimetry
should be able to predict polymer/polymer miscibility
[Cruz et al., 1979)].
After experimentally confirming validity of this idea, the principal
authors attempted to use this approach for explanation of the so call
“miscibility windows” [Paul and Barlow, 1984]. The latter term refers to
either polymer/copolymer blends that show miscibility only within a
limited rage of the copolymer composition
2.3 Solubility Parameter Approach
The concept of the solubility parameter originates from Hildebrand’s
work on enthalpy of regular solutions . Accordingly, in a strict sense, the
molecular interactions should be nonspecific, without forming
associations or orientation, hence not of the hydrogen or polar type.
Another fundamental assumption was that the intermolecular
interactions 1-2, are geometric mean of the intramolecular interactions,
1-1 and 2-2:
For molecules without polar groups the solubility parameter, i, may
be determined from:
1. The definition :
2. An empirical correlation with the surface tension coefficient, Vi :
3. Knowing the experimental values of Hm for material 1 in a series
of solvents with known value of i.
For small molecules without strong interactions the values of the
solubility parameter vary from 5.9 (for C6F14) to 14.1 (for I2) and 23.8
(for H2O). The standard error of these estimates is ± 0.2.
The solubility parameter of a polymer is usually determined by
measuring its behavior in a solvent, whose solubility parameter is known.
Lec-10 Theories of Liquid Mixtures………………………………………………..…Eng. Auda Jabbar Ms.C.
The biggest drawback of the solubility parameter approach is omission
of the entropic and specific interactions’ effects. Furthermore, the
fundamental dependencies do not take into account either the structural
(isomeric), orientation, or the neighboring group effects.
However, since the contributions that are included in the solubility
parameters are indeed detrimental to miscibility, minimizing their values
must but help the miscibility.
►Hansen has suggested one way of modifying the solubility parameter
to account for the presence of specific interactions between the polymer
and solvent.
In this approach, the solubility parameter is considered to be a vector
made up of three components: one due to hydrogen-bonding, another due
to dipole interactions, and a third due to dispersive forces.
Values of each of the three components for different polymers and
different solvents have been determined based on experimental
observations as well as on theoretical modeling, and these have been
tabulated in books .
A polymer is found to be soluble in a liquid when the magnitude of the
vector difference between the two vectors representing the Hansen
solubility parameters of the polymer and the liquid is less than a certain
amount. This method has found widespread application in the paint
industry.
Flory–Huggins solution theory
Flory–Huggins solution theory is a mathematical model of the
thermodynamics of polymer solutions which takes account of the great
dissimilarity in molecular sizes in adapting the usual expression for the
entropy of mixing. The result is an equation for the Gibbs free energy
change
for mixing a polymer with a solvent. Although it makes
simplifying assumptions, it generates useful results for interpreting
experiments. The thermodynamic equation for the Gibbs free energy
change accompanying mixing at constant temperature and (external)
pressure is
A change, denoted by , is the value of a variable for a solution or
mixture minus the values for the pure components considered separately.
The objective is to find explicit formulas for
and
, the
enthalpy and entropy increments associated with the mixing process.
The result obtained by Flory and Huggins is
Lec-10 Theories of Liquid Mixtures………………………………………………..…Eng. Auda Jabbar Ms.C.
The right-hand side is a function of the number of moles
and volume
fraction
of solvent (component ), the number of moles
and
volume fraction
of polymer (component ), with the introduction of a
parameter chi to take account of the energy of interdispersing polymer
and solvent molecules. is the gas constant and
is the 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.
Derivation:
We first calculate the entropy of mixing, the increase in the uncertainty
about the locations of the molecules when they are interspersed. In the
pure condensed phases — solvent and polymer — everywhere we look
we find a molecule. Of course, any notion of "finding" a molecule in a
given location is a thought experiment since we can't actually examine
spatial locations the size of molecules. The expression for the entropy of
mixing of small molecules in terms of mole fractions is no longer
reasonable when the solute is a macromolecular chain. We take account
of this dissymmetry 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 of the polymer chain, so the total number of sites is
is the number of solvent molecules and
molecules, each of which has segments.
is the number of polymer
From statistical mechanics we can calculate the entropy change, the
increase in spatial uncertainty, as a result of mixing solute and solvent.
where
and
is Boltzmann's constant. Define the lattice volume fractions
Lec-10 Theories of Liquid Mixtures………………………………………………..…Eng. Auda Jabbar Ms.C.
These are also the probabilities that a given lattice site, chosen at
random, 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 mixing theory.
In addition to the entropic effect, we can expect an enthalpy change.
There are three molecular interactions to consider:
1-Solvent-Solvent
2-Monomer-Monomer
(not the covalent bonding, but between
different chain sections)
3- 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,[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.
The enthalpy change is equal to the energy change per polymer
monomer-solvent interaction multiplied by the number of such
interactions
Lec-10 Theories of Liquid Mixtures………………………………………………..…Eng. Auda Jabbar Ms.C.
The polymer-solvent interaction parameter chi 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
moles
and
by transferring Avogadro's number
constant
.
and
to
to the gas
The value of the interaction parameter can be estimated from the
Hildebrand solubility parameters and
where
is the actual volume of a polymer segment.
This treatment does not attempt to calculate the conformational entropy
of folding for polymer chains. (See the random coil discussion.) The
conformations of even amorphous polymers will change when they go
into solution, and most thermoplastic polymers also have lamellar
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.