Ring Statistics in 2D Silica: Effective Temperatures in Equilibrium

The thermodynamic properties of subsystems in strong interaction with the neighborhood can largely differ from the standard behavior. Here we study the thermodynamic properties of rings and triplets in equilibrated disordered 2D silica. Their statistics follows a Boltzmann behavior, albeit with a strongly reduced temperature. This effective temperature strongly depends on the length scale of the chosen subsystem. From a systematic analysis of the 1D Ising model and an analytically solvable model, we suggest that these observations reflect the presence of strong local positive energy correlations.

Figure 1

Sketch of the definition of the ring energies for a defect-free configuration, using periodic boundary conditions. ($\mathrm{Red}=\mathrm{O}$, $\mathrm{Green}=\text{Si}$). (a) All O-particle energies are equally added to the connected Si particles. (b) All redefined Si-particle energies are equally redistributed to the connected rings. The combination of three rings in (b), marked with gray shade, is an example of a “triplet of rings” as they share a common Si particle at the center. The image is partially reproduced from Ref. [16] by permission of the PCCP Owner Societies.

Figure 2

Comparison of the logarithm of (a) complementary ring probabilities and (b) complementary triplet probabilities at $T=0.015$ ($\beta =66.7$) of the 2D-silica ring network, with their respective average energies. The correlation coefficients are shown in the graph. For (a) and (b), the inverse effective temperatures are ${\beta }_{\mathrm{ring}}^{\mathrm{eff}}=344.5$ and ${\beta }_{\text{triplet}}^{\mathrm{eff}}=182.5$, respectively. For (b), the data are shown whenever $P_t{P}_t^{*}>{10}^{-6}$. In (c) the probabilities of the different states of spin triplets and quadruplets for the 1D Ising model are compared with the respective average energies at $T=1.5$ ($\beta =0.67$). The permutation factors for the substructures are represented by “f.” The respective interaction range (2 or 6 neighbors) is listed in the figure. The resulting inverse effective temperatures ${\beta }_{\text{Ising}}^{\mathrm{eff}}$ from the Boltzmann fit are 0.67 ($\approx \beta$) (triplets, 2 neighbors), 0.90 (triplets, 6 neighbors), and 0.78 (quadruplets, 6 neighbors).

Figure 3

(a) Density of states for six member rings where the reweighting was performed at ${\lambda }_6\beta$. The value of ${\lambda }_6=8.0$ is determined by minimizing the overlap error ${\chi }^2$ between the resulting density of states curves for $T=0.013$ and $T=0.019$. $c$ represents a simple shift parameter. We fit the density of states curves with a Gaussian function (see SI.VII for details about the Gaussian properties). All data are plotted where probabilities for both temperatures are larger than 0.002 (thus excluding the high-energy regime where anharmonic effects become relevant [30]). (b) The dependence of ${\chi }^2/N_{\mathrm{dof}}$ where the normalization factor denotes the number of data points minus the number of fitted parameters [32]. The optimum ${\lambda }_{\mathrm{tot}}$ value for the total defect-free system is very close to its theoretical value of 1.0. (c) The size dependence of the optimum ${\lambda }_r$ values, scaled with the trivial factor of 3, are shown. For $r=9$ the noise is very high [see (b)] so that ${\lambda }_9$ should be taken with caution.