Understanding why and how crystalline solids melt remains a central problem in condensed-matter physics. Dislocation loops are fundamental topological excitations that control the thermodynamic stability of crystals, yet their role in setting universal aspects of melting has remained unclear. Here we show, within dislocation-mediated melting theory, that the free-energy condition for loop proliferation leads to a universal ratio between the energy of a minimal dislocation loop and the thermal energy at melting. For minimal dislocation loops that begin to proliferate at the onset of melting, this ratio takes the purely geometric value $\mathcal{E}_* = E_{\rm loop}/(k_B T_m) \approx 25.1$, independent of elastic moduli and chemistry-dependent details. This result provides a microscopic explanation for recent empirical findings by Lunkenheimer \emph{et al.}, who identified a closely related universal energy scale $\approx 24.6$ from viscosity data. The same framework also rationalizes the empirical $2/3$ rule relating the glass-transition and melting temperatures.
The microscopic mechanism by which crystalline solids melt remains a long-standing challenge in condensedmatter physics [1]. Early theoretical approaches described melting as a mechanical instability of the lattice, associated with phonon softening or the collapse of elastic constants near the melting temperature [2][3][4][5][6]. While successful in specific contexts, such quasi-harmonic descriptions fail to capture the essential role of topological defects, large-amplitude collective rearrangements [7], and disorder [8], which accompany melting in real materials [9].
A more fundamental perspective interprets melting as a defect-unbinding transition driven by the proliferation of dislocations. This viewpoint was developed in a unified form by Kleinert through his gauge-field theory of stresses and defects in solids [10], in which dislocations emerge as dynamical degrees of freedom and melting occurs when the configurational entropy of defect loops compensates their elastic and core energies. Subsequent analytical work by Burakovsky, Preston and collaborators demonstrated that dislocation-mediated models can quantitatively reproduce melting temperatures and latent heats of crystalline metals with different lattice symmetries [11,12], building on classical dislocation elasticity [13,14].
The notion of melting as a defect-proliferation transition echoes the unbinding of vortices in two-dimensional superfluids [15] and the Halperin-Nelson theory of twodimensional crystal melting [16]. In three dimensions, however, the relevant excitations are closed dislocation loops, whose energetics and entropy jointly determine the stability of the crystalline phase. Despite decades of work, the extent to which this picture implies universal features of three-dimensional melting has remained largely unexplored.
Molecular-dynamics simulations provide strong support for a dislocation-mediated scenario, directly reveal-ing the nucleation and proliferation of dislocation segments and loops as melting is approached [17][18][19][20][21]. These studies show that melting is preceded by intense defect activity, shear localisation, and the emergence of configurational excitations [22][23][24], consistent with a topological interpretation of the transition.
We demonstrate that three-dimensional crystal melting is governed by a previously unrecognized universal energy scale. Specifically, we show that at the melting temperature the ratio between the energy of a minimal dislocation loop and the thermal energy k B T m becomes a geometry-controlled constant, E * ≈ 25.1. Remarkably, this ratio is independent of elastic constants, core energies, and chemical details. We further show that this universal constant naturally connects to recent work by Lunkenheimer, Samwer and Loidl [25], who identified a universal cooperativity-free activation energy at melting corresponding to a closely related value ≈ 24.6.
The universal energy ratio 24.6 at 3D melting follows directly from the Arrhenius construction shown in Fig. 1 and does not involve any adjustable parameter. In Fig. 1(a), besides the non-Arrhenius Vogel-Fulcher-Tammann (VFT) curve, a hypothetical Arrhenius line is drawn to represent the dynamics of the liquid in the absence of cooperativity. This line is written as
where E is an effective activation barrier expressed in Kelvin. Here y(T ) denotes a generic dynamical quantity characterizing molecular mobility in the liquid. Depending on context, y represents either the structural relaxation time τ (T ) or the viscosity η(T ). Both quantities are treated on equal footing because they exhibit very similar temperature dependencies over many decades and are approximately proportional to each other in glass-forming liquids. The idealized melting temperature T id m is defined as the temperature at which this Arrhenius line reaches the universal relaxation time identified earlier in Ref. [25],
indicated by the star in Fig. 1 (
Using the standard assumptions employed throughout the analysis, log 10 y 0 = -14 for relaxation times, one obtains
which leads directly to,
Thus, the factor 24.6 is simply the dimensionless slope of the Arrhenius line in Fig. 1, corresponding to the logarithmic separation between the microscopic time scale y 0 and the universal ideal melting point. For illustration purposes, this construction is shown in Fig. 1(b) for a specific material, polyethylene oxide. In Ref. [25], the authors have shown that the same construction applied to a variety of different materials always yields the same result, and the universal, constant energy ratio of 24.6 in Eq. (6).
In what follows, we shall present a mathematical derivation of how this constant energy ratio arises from a dislocation-loop theory of 3D melting.
Within dislocation-mediated melting theory, melting occurs when the free energy of a dislocation loop vanishes due to the compensation between elastic energy and configurational entropy. As shown in the Appendix, for
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