http://www.ccl.net/cca/software/MS-WIN95-NT/miscibility_maps/mm_tutorial.tex.shtml
CCL mm_tutorial.tex
\documentclass[12pt]{article}
\sloppy

\begin{document}

%\title{}

%\author{}

%\date{}

%\maketitle

The problem of the compatibility in polymer blends
has long been a subject of interest for both
theorists and experimentalists.
Unlike homopolymers the blends comprising random copolymers
often show windows'' of miscibility on their phase diagram
even in the absence of specific interaction between monomer units.
Such windows'' were observed in systems
where mutual repulsion between dissimilar units
in the copolymer macromolecule would suffice
to overcome the repulsion between them and
units in other components of the blend in hand.
This repulsion effect'' is conveniently elucidated
in terms of a simple Flory-Huggins type Theory
enabling one to predict the location of the boundaries
of miscibility regions for an arbitrary polymer blend
with known values of the degree of polymerization of the components
and parameters $\chi _{\alpha \beta}$ of interaction
between all pairs $\mathrm{M} _{\alpha}$ and $\mathrm{M} _{\beta}$
of their monomer units.
Such attractive theoretical approach,
in view of its simplicity,
when treating experimental data on the investigation
of the phase states of polymer blends
and especially for the search of new advanced materials on their basis.

Typical problem arising for thermodynamic examination
of blends involving copolymers consists in determining
under given temperature those regions of values of their composition,
at which the system remains homophase.
The size and shape of such miscibility regions
of particular polymer specimen
have been suggested to term miscibility maps'' (MM).
An exhaustive classification was put forward of all
topologically conceivable kinds of such MMs
depending on the Flory-Huggins parameters
for a mixture of two binary statistical copolymers
having a common monomer unit.
Later the principles were formulated of such a classification
for arbitrary blends consisting of any number of components,
each comprising any kinds of units.
In the mathematical model underlaying this version
of the Miscibility Maps'' software
the consideration was restricted for simplicity sake
to the case of rather high-molecular polymers
where the dependence of the MM shape on their molecular weight
may be neglected.
Such an approximation is widely used to estimate
the compatibility regions of real polymer blends.
It corresponds to ignoring the combinatorial entropy contribution
to the free energy of mixing
that for such blends is normally small
as compared to the enthalpy contribution.
This simple approach has an indisputable advantage
when establishing the correlation
between the topology of the boundary,
which separates the regions of compatibility and incompatibility
of the blend.
Within the framework of such an approximation
it is sufficient to have an information
only on the values of the Flory-Huggins parameters
$\chi _{\alpha \beta}$ for all pairs $\mathrm{M} _{\alpha}$
and $\mathrm{M} _{\beta}$ of monomer units
of polymer blend under examination.
To each given set of these parameters there corresponds
the particular MM in the space of compositions of blend components.
Once the MM has been obtained, one can predict theoretically
the limits of the variation of these compositions
where the appearance of miscibility windows'' can be expected
in the blends comprising statistical copolymers.

Our software product permits a user
to construct the MM of systems of different types,
proceeding from given values of the Flory-Huggins parameters.
Hence this product offers a possibility
not only to reveal the topology of the MM of a polymer blend,
containing copolymers prepared by a polymerization of given monomers
but also to determine quantitatively
the boundaries of compatibility regions
of any blend depending on composition of the copolymers involved.

The program has highly intuitive
user-friendly interface.
To try the Miscibility Maps'' download
the self-extracting archive, unzip it and run mmdemo.exe.
To get an idea of the potentialities
of the program load one of the predefined
base variants (by selecting {\bf Set Base Variant} option
and perform calculations by clicking on the {\bf Compute!}
the Miscibility Maps'' or any other