Theory
=======================

Here, we give some basic algorithms of ABPolymer.

Growing Algorithm
-------------------------

TODO

Properties
-----------------

In ABPolymer, many properties are calculated to assist users get deeper insight into the growth of amorphous materials.

.. option:: Radius of Gyration 

The radius of gyration :math:`R_\text{g}` quantifies the spatial extent of a material structure, which is defined as:

.. centered::
    :math:`R_\text{g} = \sqrt{\frac{\sum_i m_i\left(\mathbf{r}_i-\mathbf{r}_\text{M}\right)^2}{\sum_i{m_i}}}`

where, :math:`\mathbf{r}_\text{M}` is the center of mass of the material, :math:`m_i` and :math:`\mathbf{r}_i` is the mass and coordinate of atom :math:`i`, respectively.

In ABPolymer, you can find gyration radius like this:

.. code-block:: 
    :linenos:

    - Radius of gyration:    39.58137592 Angstrom

.. option:: Fractal Dimension

The fractal dimension :math:`D` of a material is formally defined as (`Phys. Rev. E 2005, 71, 011912 <https://journals.aps.org/pre/abstract/10.1103/PhysRevE.71.011912>`_)

.. centered::
    :math:`D = \lim_{N\to \infty} \frac{\ln m(N)}{\ln R(N)}`

where, :math:`m(N)` and :math:`R(N)` is the total mass and a length of the material containing :math:`N` units, respectively.

For an amorphous material, if :math:`D` is like ``2.32``, this means that it is a 3D material but with a lot of pores.

In ABPolymer, you can find fractal dimension like this:

.. code-block:: 
    :linenos:

    - Fractal dimension:     2.11696273 (R^2 = 0.9748)
    
In ABGrow, fractal dimension is calculated using some linear fitting, so ``R^2`` is a linear correlation coefficient. The larger ``R^2`` is, the more reliable the fractal dimension is.

.. option:: Accessible Surface Area (ASA)

The accessible surface area (ASA) quantifies the surface area accessible by a probe molecule, see Figure below. With a given probe radius (``r_probe``), ABGrow calculates ASA with a Monte Carlo algorithm. 

.. figure:: _static/p5.png
   :align: center

In ABPolymer, you can find ASA like this:

.. code-block:: 
    :linenos:

    - Absolute ASA:          107558.89743315 Angstrom^2
    - Material ASA:          6562.65203354 m^2/g

An important observation is that for many materials (zeolites, MOFs, etc) **ASA can be compared with the experimentally determined BET (Brunauer-Emmett-Teller) area** as long as the BET analysis is performed under an appropriate pressure range (`Langmuir 1998, 14, 2097 <https://pubs.acs.org/doi/10.1021/la9710379>`_; `J. Am. Chem. Soc. 2007, 129, 8552 <https://pubs.acs.org/doi/10.1021/ja071174k>`_, `Langmuir 2010, 26, 5475 <https://pubs.acs.org/doi/10.1021/la100449z>`_).

.. option:: Network Assortativity

The `network assortativity <https://en.wikipedia.org/wiki/Assortativity>`_ of a material is defined as the tendency for units to connect to other units with similar connectivity properties within a network. For positive and negative assortativity, the units tend to connect to other units with similar and dissimilar connectivities, respectively.