Encyclopedia of Glass Science, Technology, History, and Culture

(17)

where N is again the number of atoms and r iis 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 + d r 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 Qfirst 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 4and 5for MD‐simulated B 2O 3glass [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 B 3O 6rings (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 B 2O 3glass.

Of particular interest for characterizing short‐range order are CN. In atomistic simulation, this parameter is well defined as the number of atoms falling within a given distance from an arbitrary atom. For each atomic pair this cutoff radius is typically estimated either from the corresponding bond lengths in crystal structures or from the position of minimum between the first and second peaks of the pair‐distribution function. Alternatively, the CN can be estimated experimentally from the height of the first peak observed in the X‐ray diffraction or neutron diffraction spectra or from the chemical and isomer shifts in NMR or Mössbauer spectra, respectively. In MD studies the oxygen CN of network‐forming cations (Si, B, P, Ge, etc.) are generally calculated to be within 5% of the experimental data even for the changes with varying pressure or concentration of network‐modifier alkali or alkaline earth cations. Such coordination changes from 3 to 4 for B atoms in borate glasses and from 4 to 6 for Si and Ge in silicate and germanate glasses have been well documented in this way (e.g. [12]).

Figure 5 Comparisons between the experimental [13] and simulated [6] X‐ray (a) and neutron (b) interference functions of B 2O 3glass.

On the other hand, the oxygen CN is not well defined for intermediate network‐forming cation (Al, Fe, Zr, etc.) or network‐modifier cations when the distance distribution between cation and oxygen atom is broad. A slight change in the definition of cutoff radius then translates in a large change in CN. In MD simulations on sodium aluminosilicate glasses, the switch of Al from a network‐forming to a network‐modifying role has nonetheless been evidenced by a CN increase from six in crystal to four and five in glass (e.g. [14]) whereas the existence of fivefold coordinated aluminum and threefold coordinated oxygens has also been evidenced [15].

In silicate glasses another fundamental feature to describe variations of structure and properties is the “ Q ndistribution,” where the subscript n indicates the number of bridging oxygen (BO) in an SiO 4tetrahedra ( Chapter 2.3). In MD simulations it is easy to identify nonbridging oxygens (NBO) on the basis of the cutoff radius. The calculated Q ndistributions for sodium‐silicate glasses have been compared with that determined by MAS‐NMR experiments [16]. The MD calculations reproduce the experiments reasonably, although the extremely rapid quenching rates prevailing in the MD simulation may broaden the distribution, which does depend on actual T , P conditions. The analysis of Q nis also important for phosphate glasses, because Q ndistribution reflects their polymer‐like structure that results from the existence of doubly bonded oxygen atoms. In the case of phosphate glass not many MD studies have been published and more validated potential models need to be developed. Recent MD calculations on iron phosphate glasses have demonstrated that the network connectivity is indeed dominated by the expected Q n[11].

Figure 6 PDF functions in simulated B 2O 3glass and their structural assignments [6]. The peaks labeled in (a) refer to the specific distances indicated in the elementary structural units (b).