Melting Criterion Based on Configuron Percolation

1. Introduction

Phase transitions in condensed matter systems are traditionally classified according to the Ehrenfest scheme into first-order and higher-order transitions. Crystalline melting is the archetypal first-order transition, characterized by latent heat, a discontinuity in entropy, and coexistence of phases at a sharply defined melting point Tₘ. In contrast, the glass transition lacks these discontinuities and is often described as a dual-nature having both kinetic and second-order-like transition character which occurs over a temperature interval around T_g. Despite decades of research, a unified microscopic theory describing both transformations within a single coherent framework remains elusive.

Classical approaches to melting and vitrification include: (i) the Lindemann criterion for melting, based on the thermal vibration amplitude of lattice atoms [1]; (ii) the free-volume theory for the glass transition, relating molecular mobility to available void space [2]; (iii) mode-coupling theory for dynamic arrest, describing the slowing of structural relaxation near T_g [3]; and (iv) the energy landscape paradigm, interpreting the glass transition as kinetic trapping in a multidimensional potential surface [4]. However, none of these frameworks directly explains why melting of crystals is thermodynamically discontinuous while the glass transition is continuous, nor do they provide a common thermodynamic quantity governing both phenomena.

An alternative viewpoint considers the topological connectivity of atomic bond networks [5,6,7,8]. The mechanical rigidity of a solid depends on the strength and integrity of interatomic bonds. Thermal excitation leads to progressive bond breaking, degrading the network connectivity. When connectivity falls below a critical percolation threshold, the material loses long-range rigidity and the solid-to-liquid transition occurs [9,10]. This picture is formalized by treating broken bonds as discrete thermodynamic excitations—termed configurons—whose statistical mechanics governs the transition [5,11].

This work develops a unified theory of melting and glass transition based on three pillars: (a) the statistical thermodynamics of configurons as broken-bond excitations; (b) percolation theory of network connectivity and its relation to mechanical rigidity; and (c) quantum mechanical localization effects that differentiate the behavior of configurons in ordered crystals from that in structurally disordered glasses. The central result is that both transitions correspond to a percolation of broken bonds, but the order of the resulting transition—first-order or continuous—is determined by whether configurons are delocalized (crystals) or Anderson-localized (glasses).

2. Statistical Thermodynamics of Configurons

A configuron is defined as an elementary excitation of a condensed matter system corresponding to a thermally broken chemical bond [5,11]. Formally, it is an atom (or atomic group) deprived of one or more of its nearest-neighbour bonds, accompanied by the local structural rearrangement required to accommodate the broken-bond state. As bond breaking occurs due to thermal fluctuations, the formation of a configuron can be represented as a reaction when a lattice phonon (ℏω) is absorbed by an intact bond (•) resulting in in formation of a quasiparticle termed a configuron (∘):

ℏω + • →∘

(1)

Configuron quasiparticles were introduced first by Angell [11] based on the idea that there are groups of atoms in a disordered material that at high anharmonicity have simultaneously moved far enough from their original positions so that it becomes energetically unfavourable for them to return. The formation of a configuron is characterized by two thermodynamic parameters: the formation enthalpy H_d and the formation entropy S_d. These quantities encapsulate the energetics and the change in local configurational disorder associated with breaking a bond in the condensed phase [5,6,10].

Treating configuron formation as a two-state equilibrium between intact-bond and broken-bond states, and applying standard statistical mechanics the equilibrium mole fraction of configurons at absolute temperature T is [5,6,11,12]:

$$c={\displaystyle \frac{1}{1+exp\left(\frac{H_d-T·S_d}{k_{B}T}\right)}},$$

(2)

where k_ʙ is the Boltzmann constant. This expression is the configuron analogue of the point-defect equilibrium in crystalline solids. In the dilute limit (c ≪ 1), Equation (2) reduces to the Arrhenius form:

$$c\approx A·exp\left(-{\displaystyle \frac{H_d}{k_{B}T}}\right),$$

(3)

where A = exp(S_d/k_B) is a pre-exponential factor determined by the formation entropy. For vitreous silica (SiO₂), representative values are H_d ≈ 5.5 eV and S_d ≈ 15 k_ʙ, yielding T_g ≈ 1446 K, in close agreement with experiment [13]. The increase of c with temperature drives the system toward the percolation threshold, which defines the transition temperature.

Let cₚ denote the critical configuron concentration at which the percolating broken-bond cluster first spans the system. The transition temperature T* (equal to Tₘ for crystals or T_g for glasses) is implicitly defined by the condition:

where A = exp(S_d/k_B) is a pre-exponential factor determined by the formation entropy. For vitreous silica (SiO₂), representative values are H_d ≈ 5.5 eV and S_d ≈ 15 k_ʙ, yielding T_g ≈ 1446 K, in close agreement with experiment [13]. The increase of c with temperature drives the system toward the percolation threshold, which defines the transition temperature.

Let cₚ denote the critical configuron concentration at which the percolating broken-bond cluster first spans the system. The transition temperature T* (equal to Tₘ for crystals or T_g for glasses) is implicitly defined by the condition:

$$c\left(T^{\mathrm{*}}\right)=c_p,$$

(4)

where cₚ depends on lattice geometry and dimensionality. For three-dimensional bond percolation, cₚ ranges from approximately 0.179 (face-centered cubic) to 0.249 (simple cubic) [14]. The value appropriate for a given material reflects its local coordination environment. Near the threshold, classical percolation theory [14] predicts that the mean cluster size S diverges as:

$$S\sim {|c-c_p|}^{\gamma },$$

(5)

with exponent γ ≈ 1.80 in three dimensions [14], and the correlation length characterizing the homogeneity diverges as ξ ∼ |c − cₚ|^-ν with critical exponent ν ≈ 0.88. The correlation length has different meanings above and below the percolation threshold: it characterises the average size of clusters made of broken bonds below it and gives the average size of pores in the percolation cluster formed above the percolation threshold [6]. At the threshold, the infinite percolating cluster is a fractal object, characterized by its Hausdorff–Besicovitch dimension D_H.

At the percolation threshold, the spanning configuron cluster is not a homogeneous three-dimensional object but a fractal with a well-defined Hausdorff–Besicovitch dimension. In three-dimensional space, percolation theory gives [14] $D_H\approx 2.523$. Exceeding the condition for the spanning cluster to permeate three-dimensional space and remove long-range rigidity serves thus as melting criterion [15]. The melting or glass transition condition in 3D is therefore:

$$D_H>2.55\pm 0.05$$

(6)

This geometric criterion is universal: it applies equally to crystals and glasses and depends neither on the specific chemical system nor on the particular transition mechanism.

3. Rigidity Percolation and Mechanical Stability

The theoretical foundation linking mechanical rigidity to network connectivity was established by Kantor and Webman [9], who proved rigorously that in elastic networks the rigidity threshold coincides exactly with the percolation threshold. In physical terms: a network of bonds loses macroscopic shear rigidity precisely at the bond-percolation threshold, not before and not after. This theorem provides the direct bridge between the statistical mechanics of configuron formation (Section 2) and the observable mechanical transition.

Complementary insight comes from the constraint-counting theory of Phillips [7] and Thorpe [8]. In a covalent network, each bond contributes constraints to the rigidity. The mean number of constraints per atom is determined by the mean coordination number ⟨r⟩. The rigidity threshold occurs when the number of constraints equals the number of degrees of freedom, giving the critical mean coordination:

$${⟨r⟩}_c=2.4,$$

(7)

Networks with ⟨r⟩ > 2.4 are rigid (over-constrained), while those with ⟨r⟩ < 2.4 are floppy (under-constrained). Each broken bond (configuron) reduces the actual network ⟨r⟩ by removing constraints. The transition occurs when sufficient configurons have formed to drive ⟨r⟩ below 2.4, consistent with the percolation picture. Both approaches—the Kantor–Webman theorem and the Phillips–Thorpe constraint theory—converge on the same physical conclusion: rigidity is lost at a well-defined connectivity threshold, and configuron formation is the microscopic mechanism driving the system to that threshold.

Configurons effectively remove bond constraints from the network. Their progressive accumulation with increasing temperature reduces both the mean coordination number and the bond-percolation connectivity, driving the system toward the rigidity percolation threshold at T*. The transition from a rigid solid to a structurally fluid state is thus a geometric, connectivity-driven event, not merely an energetic one.

4. Quantum Nature of Configurons

In a perfect crystal, translational symmetry ensures that all bond sites are equivalent. A configuron formed at any site can propagate through the lattice without energy cost, since every equivalent site offers an identical environment. By the Bloch theorem [16,17], the quantum mechanical wavefunction of such a mobile excitation extends coherently over the entire lattice, forming Bloch-like quasiparticle states. The configuron in a crystal is therefore fundamentally delocalized: it is not pinned to a specific lattice position but participates in band-like propagation through the periodic potential.

This delocalization has a critical thermodynamic consequence: delocalized configurons can migrate, cluster, and condense at nucleation sites (grain boundaries, dislocations, surfaces). On condensation the above-described reaction (1) proceeds in the opposite direction, a configuron recombines creating a joining bond and releasing the thermal energy in form of a phonon ℏω in the lattice:

∘ → ℏω + •

(8)

The condensation of configurons is exothermic—it corresponds to the annihilation of broken bonds through the formation of a new, higher-entropy liquid phase. The total energy released in this collective condensation event in molar units is precisely the latent heat of fusion L_m observed at Tₘ.

In a structurally disordered solid (glass), the spatial randomness of the network introduces a random potential landscape for configuron propagation. Bonds of different strengths and geometries create a distribution of site energies. In such a system, Anderson’s seminal 1958 analysis [18] demonstrated that for sufficiently strong disorder, all quantum states become spatially localized: the wavefunction of each excitation decays exponentially from its formation site with a characteristic localization length ξ_A:

$$\left|\psi \right(r\left)\right|\sim exp\left(-{\displaystyle \frac{r}{{\xi }_A}}\right),$$

(9)

where r is the distance from the configuron’s formation site. Anderson localization suppresses long-range configuron transport [18,19,20,21]: hence each configuron is effectively pinned to its local structural environment and cannot migrate to nucleation sites. As a result:

• No collective condensation of configurons occurs.
• No latent heat is released.
• The transition from solid-like to liquid-like behaviour is spatially gradual and thermodynamically continuous.

This localization is the fundamental quantum mechanical reason why the glass transition resembles a second-order transition, in contrast to the sharp, latent-heat-accompanied first-order melting of crystals. The structural disorder of the glass is not merely a geometrical complication; it fundamentally alters the quantum statistics of the broken-bond excitations.

5. Consequences of Configuron Localization

In crystals, delocalized configurons accumulate with rising temperature until their concentration reaches cₚ at T = Tₘ. At this point, the macroscopic percolation of broken bonds corresponds to loss of lattice rigidity. Simultaneously, mobile configurons condense at pre-existing defects and grain boundaries, releasing the formation enthalpy H_d per bond. Summing over all bonds involved in the percolation event yields the macroscopic latent heat:

$$L_m=N\mathrm{ₚ}·H_d,$$

(10)

where Nₚ is the number of bonds per mole involved in the percolation event. This energy release creates the characteristic temperature plateau at Tₘ, the coexistence of solid and liquid phases, and the entropy discontinuity ΔSₘ = Lₘ/Tₘ, all hallmarks of a first-order transition consistent with Gibbs thermodynamics [17,22]. The transition is sharp and well-defined because the percolation event is cooperative and the energy release is rapid compared to the thermal equilibration time.

In glasses, Anderson-localized configurons cannot migrate or condense. Their concentration nevertheless increases with temperature according to Equation (1). At high temperatures, when the concentration of configurons becomes sufficiently high, they are inevitably in the vicinity of each other so that they form clusters:

∘ + ∘ ↔ ∘∘; … ∘ + ∘∘…∘ ↔ ∘∘∘…∘

(11)

Clusters formed as a result of these reactions are dynamic structures in thermal equilibrium with surrounding [6].

At T = T_g the percolation threshold is reached in the same geometric sense as for crystals (Eq. (4)). This gives for the glass transition temperature

$$T_g={\displaystyle \frac{H_d}{S_d+k_{B}ln\left[\right(1-{\mathsf{\varphi }}_c)/{\mathsf{\varphi }}_c]}}$$

(12)

where ${\mathsf{\varphi }}_c$ is the percolation threshold, and H_d and S_d are the formation enthalpy and entropy of configurons respectively [5,12]. T_g is hence dependent on quasi-equilibrium thermodynamic parameters of bonds i.e. on enthalpy and entropy of formation configurons at given conditions which are not necessarily for an equilibrium state. However, because no condensation occurs, the energy associated with configuron formation is not released abruptly. Instead, the enthalpy changes smoothly, producing the characteristic step in heat capacity ΔCₚ observed at T_g, rather than a latent heat spike [15,23,24,25,26]. The continuous nature of the transition reflects the absence of a collective many-body event; configurons simply percolate through the disordered network without condensing, and the solid-to-liquid crossover is spatially and temporally distributed. Worth noting recent work which shows that fully crystalline, low-molecular-weight solids can display heat-capacity changes during melting that closely mimic those seen at the glass transition [27].

The nature of glass transition is dual – formation of glass exhibits both thermodynamic and kinetic features [28]. The kinetic aspect of the glass transition—its dependence on cooling rate—also emerges naturally: at faster cooling rates, the system is driven out of equilibrium before the configuron concentration can track Eq. (2), shifting the apparent T_g to higher values. This is consistent with the empirical Vogel–Fulcher–Tammann equation and the concept of fragility introduced by Angell [29].

6. Role of Defects and Processing History

Both Tₘ and T_g depend on configuron statistics, which are sensitive to the pre-existing structural state of the material. Several factors modulate the effective cₚ and the configuron formation parameters H_d and S_d:

• Cooling rate: Faster quenching freezes a higher-than-equilibrium configuron concentration, effectively raising T_g up to its maximum possible level [5,30].
• Structural relaxation (annealing): Sub-T_g annealing allows the system to reduce c toward its equilibrium value, downing T_g toward its minimal possible level [5,30].
• Intrinsic defects and impurities: Pre-existing broken bonds or network-disrupting dopants modify the effective bond enthalpy H_d and the percolation threshold cₚ.
• Grain boundaries and interfaces: In polycrystalline materials, high-angle grain boundaries act as configuron sinks, modifying the condensation dynamics at Tₘ.

Recent ab initio analysis by Shirai [31] demonstrates that intrinsic defect formation shifts the effective melting conditions in covalent crystals, in quantitative agreement with the configuron picture. This confirms that the framework is robust to the presence of structural imperfections and can account for the well-known variation of Tₘ with sample purity and processing history.

7. Discussion

Table 1 compares the configuron-percolation theory (CPT) framework with the principal established theories of melting and glass transition.

The configuron-percolation framework offers several conceptually new insights beyond those available from existing theories:

• Unified origin: Melting and glass transition share an identical microscopic origin—the percolation of broken chemical bonds. They differ only in the quantum mechanical character of the broken-bond excitations, not in the underlying thermodynamic driving force.
• Order determined by localization: The first-order versus continuous character of the transition is not an empirical distinction but a consequence of Anderson localization. Any disordered solid must exhibit a continuous transition, and any sufficiently ordered solid must exhibit a first-order transition—a fundamental prediction of the framework.
• Universal geometric criterion: The condition D_H > 2.5 (Eq. (6)) is a universal, material-independent criterion for the solid-to-liquid transition in 3D, derived from percolation geometry rather than from specific interatomic potentials. Notable that for 2D materials the condition is D_H > 1.896 [32].
• Role of excitations: High concentrations of configurons as elementary excitations lead to phase transitions hence the primary role is shifted from atomic motions to excitation dynamic with expected anticorrelation between glass forming ability and electrical conductivity.

The universal melting criterion naturally connects condensed matter thermodynamics, percolation topology, fractal geometry, and quantum mechanics within a single, coherent framework and makes several specific, testable predictions. The universal critical configuron concentration: The ratio c(T*)/cₚ should equal unity at both Tₘ and T_g for any material. Neutron and X-ray scattering studies of bond-length distributions near T_g should reveal a universal threshold in broken-bond fraction. Near Tₘ or T_g, the configuron cluster should exhibit fractal correlations with D_H ≈ 2.52 observable in small-angle scattering experiments as a power-law scattering intensity I(q) ∼ q⁻ᴰ_ᴴ [10,33]. Below the T_m either a supercooled melt or a crystalline phase exists while below the T_g only the solid phase exists, either as a thermodynamically stable crystal or a thermodynamically metastable glass [34] and that glass is not a melt [35]. For crystals with controlled defect concentrations, Tₘ should shift predictably according to the modified configuron statistics, providing a quantitative test via precision calorimetry. More fragile glass formers (in Angell’s sense [29]) should correspond to shorter Anderson localization lengths ξ_A, reflecting stronger network disorder. This connection could be tested by correlating fragility indices with structural disorder parameters from pair-distribution-function analysis.

8. Conclusions

A unified theoretical framework for melting of crystalline solids and the glass transition of amorphous materials has been developed based on the statistical thermodynamics of configurons—thermally broken chemical bonds—and their percolation through the atomic bond network. The principal conclusions are:

• Both transitions from solid to melt correspond to percolation of broken bonds through the condensed matter network, occurring when the Hausdorff–Besicovitch dimension of the configuron cluster exceeds D_H = 2.5, the rigidity threshold in three-dimensional space.
• Rigidity loss at the transition coincides exactly with the bond-percolation threshold, consistent with the Kantor–Webman theorem for elastic networks and the Phillips–Thorpe constraint theory.
• The transition condition is governed by the universal fractal dimension D_H ≈ 2.523, providing a material-independent geometric criterion for the solid-to-liquid transition in 3D systems.
• The first-order character of crystal melting arises from the delocalization (Bloch-like propagation) of configurons in periodic lattices, enabling collective condensation and latent heat release at Tₘ.
• The continuous, second-order-like character of the glass transition is a direct consequence of Anderson localization of configurons in structurally disordered networks, which suppresses condensation, eliminates latent heat, and yields a smooth solid-to-liquid crossover at T_g.
• The framework accommodates the influence of defects, cooling rate, and processing history through modifications of the equilibrium configuron concentration and the effective percolation threshold, consistent with recent ab initio results.

This approach provides a consistent, physically transparent, and experimentally testable description of phase transitions in ordered and disordered solids within a single unified framework.