Читать книгу Encyclopedia of Glass Science, Technology, History, and Culture - Группа авторов - Страница 314

In atomistic simulations the positional correlation of atoms is easily investigated within a radius of half of simulation cell size (~25 Å). The most widely derived results are the PDF, the RDF, or total correlation function, T(r), which can be readily compared with those obtained in diffraction studies.

In simulations, the PDF and the RDF are derived as follows. The single and pair (two‐body) probability density P_N ⁽¹⁾, P_N ⁽²⁾ are defined as

(16)

(17)

where N is again the number of atoms and r_i is the coordinate of atom i.

The value of P_N ⁽¹⁾(r) in homogeneous system turns out to be the number density ρ, which is defined as N/V, where V is the volume.

Moreover, P_N ⁽²⁾(r) is expressed in terms of the PDF, g(r, r′), and number density ρ as:

(18)

As to the RDF, J(r), it is defined as the number of atoms between r and r + dr from the center of an arbitrary origin atom:

(19)

An alternative function called total distribution function, T(r), is calculated as:

(20)

The information directly obtained from diffraction experiments is the intensity I(Q), which is related to J(r) by

(21)

Finally, the frequently used structure factor, S(Q), is the Fourier transform of the number density ρ_Q first calculated in atomistic simulation:

(22)

Then, S(Q) is calculated from ρ_Q:

(23)

It is quite important to reproduce the experimental J(r) in the real space domain or I(Q) in the wave‐number domain to validate the calculated three‐dimensional structure.

Depending on the atoms considered, X‐ray and neutron diffraction experiments can yield different profiles so that both kinds of profiles should be calculated and compared with the relevant data as done in Figures 4 and 5 for MD‐simulated B₂O₃ glass [6]. Once the total correlation and interference functions have been validated, more detailed analysis based on PDF functions can be performed, as shown in Figure 6, and important insight into structural order be obtained. Although the experimental peak positions are reproduced reasonably well by the calculated T(r) and Q I(Q), there are some discrepancies for the peak values. The position and width of the first peak represent the average length and the length distribution of B─O bonds, respectively. The second peak position and peak curve are mostly affected bond angles of O─B─O and B─O─B and size distributions. As indicated by a detailed analysis of the data, most of the discrepancy is due to a simulated fraction of only 30–50% for the so‐called boroxol B₃O₆ rings (cf. Chapter 7.6) compared to the 60–80% range of the experimental values [13].

Figure 4 Comparisons between the experimental [13] and simulated [6] X‐ray (a) and neutron (b) total correlation functions of B₂O₃ glass.