2. Theory
Here, we give some basic algorithms of ABPolymer.
2.1. Growing Algorithm
TODO
2.2. Properties
In ABPolymer, many properties are calculated to assist users get deeper insight into the growth of amorphous materials.
- Radius of Gyration
The radius of gyration \(R_\text{g}\) quantifies the spatial extent of a material structure, which is defined as:
\(R_\text{g} = \sqrt{\frac{\sum_i m_i\left(\mathbf{r}_i-\mathbf{r}_\text{M}\right)^2}{\sum_i{m_i}}}\)
where, \(\mathbf{r}_\text{M}\) is the center of mass of the material, \(m_i\) and \(\mathbf{r}_i\) is the mass and coordinate of atom \(i\), respectively.
In ABPolymer, you can find gyration radius like this:
1- Radius of gyration: 39.58137592 Angstrom
- Fractal Dimension
The fractal dimension \(D\) of a material is formally defined as (Phys. Rev. E 2005, 71, 011912)
\(D = \lim_{N\to \infty} \frac{\ln m(N)}{\ln R(N)}\)
where, \(m(N)\) and \(R(N)\) is the total mass and a length of the material containing \(N\) units, respectively.
For an amorphous material, if \(D\) 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:
1- 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.
- 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.
In ABPolymer, you can find ASA like this:
1- Absolute ASA: 107558.89743315 Angstrom^2
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; J. Am. Chem. Soc. 2007, 129, 8552, Langmuir 2010, 26, 5475).
- Network Assortativity
The network 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.