Polymers with COSMO-RS(-SAC)
****************************

Many types of polymers can be described with COSMO-RS theory.
However, the typical procedure for calculating \sigma-profiles (a
geometry optimization with ADF, a single point COSMO calculation,
processing the surface points to make a \sigma-profile) is far too
expensive for generating the \sigma-profiles of polymers.  Instead,
the \sigma-profile of a polymer can be represented as the \sigma-
profile of a monomer scaled to the size of the polymer.  This means
that given a \sigma-profile of an "average monomer," we can generate a
\sigma-profile for the polymer by multiplying the monomer \sigma-
profile by a factor equal to the number of repeat units in the
polymer.  This procedure, though not without some shortcomings, makes
the treatment of polymers with COSMO-RS computationally tractable.

   [image]COSMO surface of the Polyether ether ketone (PEEK) monomer
   within a trimer.  The "average monomer" \sigma-profile is
   calculated from the central monomer unit of the trimer.

   [image]COSMO surface of the isolated Polyether ether ketone (PEEK)
   monomer.

There are many possible approaches to generating the \sigma-profile
for an "average monomer."  The approach used to generate the monomers
in the 2019 ADFCRS-Polymer Database is the following:

   (1) Construct a trimer from 3 units of the monomer
   (2) Cap the ends of the outer two monomers with methyl groups
   (3) Optimize the geometry of the trimer with ADF
   (4) Do a single point COSMO calculation
   (5) Generate the \sigma-profile for *only the central unit* of the trimer

This method provides the \sigma-profile of a monomer surrounded by two
copies of itself on either side.  Though perhaps longer polymer chains
could be considered in the determination of the "average monomer," the
trimer representation was found to be sufficiently accurate for
calculating various thermodynamic properties while not being
exorbitantly expensive in the ADF step.  In cases where this monomer
representation fails to capture the behavior of the polymer, the
following alternative procedures are possible:

* Generating a structure from a longer polymer chain and taking the
  "average monomer" \sigma-profile as an average of the \sigma-
  profiles of all of the interior monomers

* Sampling the conformational space of the polymer and using different
  structures for different problems (e.g., a polymer with a possible
  intramolecular H-bond may exhibit this internal H-bonding in a
  neutral solvent but not in a more polar one)


Additional properties/units for polymer systems
===============================================


Average molecular weight
------------------------

As mentioned above, polymers in COSMO-RS are calculated using a scaled
version of a monomer's \sigma-profile.  However, because polymers come
in many different lengths, we must be able to adjust the length of the
polymer and accordingly adjust the scaling of the monomer \sigma-
profile.  This is done with an average molecular weight parameter,
which represents the Number Average Molecular Weight M^N, defined as
the following:

   M^{N} = \frac{\sum_j n_j w_j}{\sum_j n_j}

where *j* is in index over all different molecules (different-length
polymers are different molecules) in solution, w_j is the weight of
each molecule, and n_j is the number of molecules of type *j* in
solution.  Because this parameter is adjustable, the same monomer
\sigma-profile can be used for calculations with polymers of different
lengths.  Note that a default value of 10000 g/mol is used if no input
is given.


Mole fractions of the monomers and polymers
-------------------------------------------

Because a typical polymer consists of many polymer chains of different
lengths, the mole fraction of the polymer is well-defined.  In the
COSMO-RS program, there are two mole fractions defined:


x(monomer)
~~~~~~~~~~

x(monomer) is simply the mole fraction of a monomer in the solution,
treating all polymeric species as a number of monomers equal to the
length of the polymer chain.  For non-polymeric components, x(monomer)
is the mole fraction in relation to a solution of monomers.


x(polymer)
~~~~~~~~~~

x(polymer) defines a mol of each polymer chain using the average
molecular weight parameter to determine the average chain length.
Specifically, we can define a term R_i as the number of repeat units
in the average polymer of species *i*.  For non-polymeric components,
R_i=1.  R_i is shown below:

   R_i = \frac{M_{i}^N}{M(monomer)_i}

where M(monomer)_i refers to the molar mass of the monomer.  Using
this R_i, we can define x(polymer)_i:

   x(polymer)_i = \frac{ x(monomer)_i / R_i }{ \sum_j x(monomer)_j /
   R_j }

where now *j* is an index over all distinct components.  All polymers
of the same type are assumed to be of length R_i.


Weight- and volume-fraction activity coefficients
-------------------------------------------------

As a further consequence of the ill-defined mole fraction for polymer
solutions, activity coefficients are often reported in the literature
in terms of weight- or volume-fraction.  These are also reported for
calculations in the COSMO-RS program in which at least one species is
a polymer.  They are defined from the activity of each species:

   a_i = \gamma_i x(polymer)_i

where \gamma_i is the activity coefficient in relation to the polymer
mole fractions.  Note: this is the default value for \gamma_i in
polymer calculations.  The weight-fraction activity coefficient
(\Omega_i) can then be calculated as:

   \Omega_i = \frac{a_i}{w_i}

where w_i is the weight fraction of component *i*.  Finally, the
volume fraction activity coefficient (\gamma^{\phi}_i) can be defined
as follows:

   \gamma^{\phi}_i = \frac{a_i}{\phi_i}

where \phi_i is the volume fraction of component *i*.


Flory-Huggins parameter
-----------------------

The Flory-Huggins model is used extensively for binary polymer/solvent
and polymer/polymer mixtures.  In the Flory-Huggins model, the \chi
parameter is the single system-specific parameter and is intended to
quantify the enthalpic interactions between system components.  Users
of the Flory-Huggins model can deduce important system behavior (phase
stability, solubility, etc.) from the value of the \chi parameter.

The \chi parameter was originally intended to be composition-and
temperature- independent, although it is now known to vary
significantly across compositions and temperatures for many systems.
Fortunately, system changes related to composition and temperature are
captured in the COSMO-RS program and are reflected in the calculation
of the Flory-Huggins \chi parameter.  This is due to the calculation
of the \chi parameter from the free energy of mixing of two species:

   \frac{G_{mix}}{RTV} = \frac{\phi_1}{v_1} ln(\phi_1) +
   \frac{\phi_2}{v_2} ln(\phi_2) + \frac{ \phi_1 \phi_2 \chi_{12}
   }{v_r}

where G_{mix} is the free energy of mixing, *R* is the gas constant,
*T* is the absolute temperature, *V* is the system volume, \phi_i is
the volume fraction of component *i*, v_i is the molar volume of
species *i*, and v_r is a reference volume.  Note that the difference
in v_1 and v_2 can be significant in the case of a polymer/solvent
mixture.  Since G_{mix} does vary with composition and temperature,
the \chi parameter will also exhibit this variance.

Important:

  **Reference volumes (** v_r **) used in calculating the Flory-
  Huggins parameter**

  * polymer/solvent systems:  v_r is equal to the molar volume of the
    solvent

  * polymer/polymer systems: v_r is set to a value of 0.6022140857\
    L/mol, which corresponds to a site size of 1\ nm^{3}

The COSMO-RS program can calculate thermodynamic properties for multi-
component systems, including polymer-containing systems.  However, the
Flory-Huggins \chi parameter is traditionally defined for binary
mixtures (although some extensions to multi-component mixtures exist).
In the case of multi-component mixtures containing polymers, we
calculate a \chi parameter for each species.  This is summarized in
the following:

Note:

  **Calculating the Flory-Huggins parameter for systems with more than
  2 components**For systems with more than 2 components, a \chi
  parameter is defined for each component.  This \chi_i parameter is
  calculated as the \chi parameter for a species *i* as if all the
  other species in solution were combined into one component.  This
  means that for a ternary mixture of solvent/polymer A/polymer B, the
  \chi parameter of the solvent would correspond to the \chi parameter
  of the system defined by the solvent and the *co-polymer* of A and
  B.  Similarly, the \chi parameter of polymer A would correspond to
  the \chi parameter of the binary system of polymer A and the meta-
  solvent of a combination of polymer B and the solvent.


Modified combinatorial term
===========================

The entropy of polymer solutions cannot be calculated in the same way
as the entropy of small molecule solutions.  For this reason, the
normal combinatorial term in COSMO-RS is replaced with a polymer-
specific combinatorial term.  Specifically, the combinatorial term
used is that of Elbro (1990) [1], which has previously been applied
successfully in the context of COSMO-RS [2].  The Elbro combinatorial
requires the free volume fraction (\phi^{fv}_i) of each component:

   \phi^{fv}_i = \frac{x_i (v_i - v^*_i) }{ \sum_j x_j (v_j - v^*_j )}

where x_i is the (polymer) mole fraction of compound *i*, v_i is the
molar volume of compound *i*, v^*_i is the molar hard-core volume of
compound *i*, and *j* is also an index over the compounds.  Using the
values of \phi^{fv}_i for each component, the combinatorial
contribution to the activity coefficient can be defined as follows:

   ln(\gamma^{fv}_i) = ln \left( \frac{ \phi^{fv}_i }{x_i} \right) + 1
   - \frac{ \phi^{fv}_i }{x_i}

Finally, this combinatorial term is used to calculate the final
activity coefficient:

   ln(\gamma_i) = ln(\gamma^{crs}_i) + ln(\gamma^{fv}_i)

where \gamma^{crs}_i is the activity coefficient calculated with
COSMO-RS without any combinatorial corrections.


General application guidelines/warnings
=======================================

At present, the quality of the predictions of the COSMO-RS program is
*unreliable* or *untested* for the following types of systems:

* **Cross-linked polymers.**  The structures of cross-linked polymers
  are not perfectly compatible with the assumptions in the COSMO-RS
  model for polymers.  These types of systems are not recommended to
  be used with COSMO-RS.

* **Polymers with significant swelling behavior.**  Polymer swelling
  is not easily captured with COSMO-RS.  It is likely that some of the
  error due to swelling can be offset by using correct molar volume
  values for the polymers in different solvents/at different
  temperatures.


Downloading the ADFCRS-POLYMER-2019 database
============================================

See the section on the ADFCRS-POLYMER-2019 database on how to download
this database.


Tutorial on polymer calculations
================================

There is a tutorial on COSMO-RS calculations with polymers that
demonstrates basic the basic functionality of the COSMO-RS GUI with
polymers.  This will cover calculations using the polymer database as
well as *inputting your own polymer structures.*

[1] H.S. Elbro, A. Fredenslund, and P. Rasmussen, *A new simple
    equation for the prediction of solvent activities in polymer
    solutions*,  Macromolecules 23, 4707 (1990)

[2] C. Loschen and A. Klamt, *Prediction of solubilities and partition
    coefficients in polymers using COSMO-RS*, Industrial & Engineering
    Chemistry Research 53, 11478 (2014)
