\(Q^n\) Distribution & Network Connectivity¶

The \(Q^n\) distribution characterizes the connectivity of network-forming polyhedra (typically tetrahedra) in oxide glasses by counting bridging oxygens (BOs).

Theory¶

\(Q^n\) Species¶

Each network-forming cation (e.g., Si, Al, B in 4-fold coordination) is bonded to a certain number of oxygen atoms. Some of these oxygens are bridging (shared with another network former) and some are non-bridging (NBOs, bonded to only one former).

The \(Q^n\) label indicates the number of bridging oxygens:

Species	Bridging O	Non-bridging O	Network role
\(Q^0\)	0	4	Isolated tetrahedron
\(Q^1\)	1	3	Chain end
\(Q^2\)	2	2	Chain middle
\(Q^3\)	3	1	Sheet/branching
\(Q^4\)	4	0	Fully cross-linked (vitreous silica)

Bridging Oxygen Definition¶

An oxygen is classified as bridging if it is bonded to \(\geq 2\) network-forming cations (within the specified cutoff distance).

Network Connectivity¶

The network connectivity (NC) is the average number of bridging oxygens per network former:

\[ \text{NC} = \frac{\sum_{n=0}^{4} n \cdot f(Q^n)}{\sum_{n=0}^{4} f(Q^n)} = \sum_{n=0}^{4} n \cdot x(Q^n) \]

where \(f(Q^n)\) is the fraction of formers with \(n\) bridging oxygens and \(x(Q^n)\) is the normalized \(Q^n\) distribution.

Typical values: - Pure SiO₂ glass: NC ≈ 4.0 (all \(Q^4\)) - 75SiO₂-25Na₂O: NC ≈ 3.0 (mainly \(Q^3\) and \(Q^4\)) - 50SiO₂-50Na₂O: NC ≈ 2.0 (mainly \(Q^2\) and \(Q^3\))

The NC provides a single number that correlates with many glass properties including viscosity, fragility, glass transition temperature, and elastic moduli.

Usage¶

`compute_qn(structure, cutoff, former_types, o_type)`¶

from amorphouspy import compute_qn, compute_network_connectivity

# Compute Qn distribution
qn = compute_qn(
    structure=glass_structure,
    cutoff=2.0,                  # Si-O bond cutoff (Å)
    former_types=["Si"],         # Network formers to analyze
    o_type="O",                  # Oxygen symbol
)

# Returns: {'Q0': 0.01, 'Q1': 0.05, 'Q2': 0.15, 'Q3': 0.45, 'Q4': 0.34}

# Compute network connectivity from Qn distribution
nc = compute_network_connectivity(qn)
print(f"Network connectivity: {nc:.2f}")

Parameters:

Parameter	Type	Default	Description
`structure`	`Atoms`	—	ASE Atoms object
`cutoff`	`float`	—	Bond cutoff distance for former-O bonds (Å)
`former_types`	`list[str]`	—	Element symbols of network formers (e.g., `["Si", "Al"]`)
`o_type`	`str`	`"O"`	Element symbol for oxygen

Returns: A dictionary mapping "Q0" through "Q4" to fractions (summing to 1.0).

Multi-former analysis¶

When multiple network formers are present, the \(Q^n\) distribution is computed separately for each former species:

qn = compute_qn(
    structure=glass_structure,
    cutoff=2.2,
    former_types=["Si", "Al"],
    o_type="O",
)

# Returns separate distributions:
# {'Si': {'Q0': 0.00, 'Q1': 0.02, 'Q2': 0.10, 'Q3': 0.48, 'Q4': 0.40},
#  'Al': {'Q0': 0.00, 'Q1': 0.05, 'Q2': 0.15, 'Q3': 0.55, 'Q4': 0.25}}

Note: The cutoff should be chosen to match the first minimum in the former-O RDF. For Si-O this is typically ~2.0 Å; for Al-O it is ~2.2 Å. Using a single cutoff for multiple formers is an approximation — choose a value that works reasonably for all.

Relationship to Composition¶

For binary alkali silicate glasses \(x\text{M}_2\text{O} \cdot (1-x)\text{SiO}_2\), the theoretical \(Q^n\) distribution can be predicted from composition using the binary model:

\[ \text{NC} = 4 - \frac{2x}{1-x} \]

This assumes each M₂O adds one non-bridging oxygen, converting one \(Q^n\) → \(Q^{n-1}\).

Deviations from this model (measured by the actual \(Q^n\) distribution) reveal important structural features: - Disproportionation: \(2Q^3 \to Q^2 + Q^4\) indicates local unmixing - Aluminum avoidance: Al prefers \(Q^4\) to satisfy Loewenstein's rule