Response to “Comment on ‘Dynamic aspects of the liquid-liquid phase transformation in silicon’ ” [J. Chem. Phys. 130, 247102 (2009)]

The dynamics of undercooled liquids remains one of the issues in condensed matter physics which is not completely solved at the present time and is the subject of intense activities.¹ In this research area, liquids can be classified as fragile or strong, depending on the temperature dependence of dynamical quantities or structural relaxation times, in the undercooled regime near the glass transition temperature $Tg$⁠. Beside these quantitative differences, strong liquids are characterized by the existence even above $Tg$ of a so-called boson peak (BP), a characteristic feature in the short time dynamics.² For fragile systems the BP is always weak and is rapidly hidden by diffusive modes as the temperature rises above $Tg$⁠. The BP is revealed experimentally as excess modes in the vibrational density of states at low frequencies (typically of the order of 1 THz) which are also manifested by an excess of the heat capacity with respect to the harmonic solid value at low temperatures. The origin of the BP is still under debate, which has therefore stimulated many recent experimental, theoretical, and simulation studies. Especially, computer simulations are of great interest to gain additional insight into the microscopic nature of the BP.

For the case of $SiO2$⁠, which is recognized as a typical strong glass former at normal density with one of the strongest BP, several simulation studies have been performed in order to elucidate the BP. It has been shown that the latter is manifested in dynamical quantities such as the Debye–Waller factor,³ the mean-square displacement,^3,4 the longitudinal and transverse current-correlation function,⁵ and the intermediate scattering function.⁴ For the latter, it is associated with a dip that appears between the short time dynamics and the plateau characteristic of the cage effect preceding the $α$-relaxation (see also Ref. 1 where this feature was given some emphasis in the last section on short time processes). Upon compression to high density $(ρhigh/ρlow>1.5)$⁠, liquid silica becomes fragile^3,6 as the silicon coordination number increases and the intermediate-range order decreases. In this high density state, the boson dip no longer persists above $Tg$⁠, either in the intermediate scattering function⁴ or in the Debye–Waller factor.³ Therefore the disappearance of the “boson dip” in the intermediate scattering function is clearly associated with the change from strong-to-fragile behavior, contrary to the claim in the comment of Yannopoulos.⁷ 

For silicon, computer simulations,^8–11 using the Stillinger–Weber empirical potential, have given evidence of a transition from a high density liquid (HDL) to a low-density liquid (LDL), as suggested long ago by the Harvard–Bell Laboratories group¹² and supported by recent experiments.¹³ Such a liquid-liquid (LL) transition was recently confirmed using a hybrid simulation technique that combines classical MD and first-principle MD.¹⁴ The transition between the HDL and the LDL is accompanied by a significant increase in the local tetrahedral arrangement as well as an enhancement of the intermediate-range order.¹⁵ The establishment of a tetrahedral network in the LDL suggests examining the possibility that the LL transition of silicon is accompanied by a fragile-to-strong crossover, as in the case of silica and also other systems such as the yttria-alumina compositions of Ref. 15 and triphenyl-phosphite.¹⁶ 

In classical,⁹ as well as first-principles¹⁷ MD simulations of undercooled silicon, the intermediate scattering function shows the appearance of a boson dip in the LDL which is not present in the HDL. Even if finite size effects exist in the LDL, for two system sizes,⁹ qualitatively the same result is obtained. As in the case of silica, such a feature is in favor of the strong character of the LDL phase. Quite independent support for the strong character of the LDL of silicon is provided in Fig. 1, which displays the heat capacity at constant pressure, $CP$⁠, as a function of temperature in the undercooled liquid regime. $CP$ can be calculated at each temperature from the enthalpy fluctuations observed in the classical MD simulations, and the rapid increase in $CP$ as the LDL spinodal underlying the LL transition is approached has already been reported (see Ref. 9). Figure 1 not only contains those data but also includes a value obtained from the 20 ns trajectory in the LDL phase from which the diffusivity reported in Ref. 9 was obtained. $CP$ is seen¹⁸ to have dropped abruptly to a value of only $3.6NkB$⁠, about the same excess over the harmonic solid value of $3NkB$ as seen¹⁹ in the archetypal strong liquid, $SiO2$⁠.

In summary, the strong character of the LDL phase of undercooled liquid silicon that was suggested originally by the boson dip in the intermediate scattering function of a classical MD simulation and recently supported by the comparable behavior seen in ab initio studies, is confirmed by both dynamic and thermodynamic routes: (i) by demonstrating the same boson dip behavior in well-characterized fragile/strong systems and (ii) by establishing the presence of a sudden decrease in the specific heat at constant pressure as the HDL to LDL transition occurs, as known for other fragile/strong systems.

We acknowledge the CINES and IDRIS under Project No. INP2227/72914 as well as PHYNUM CIMENT for computational resources. The ANR is gratefully acknowledged for financial support under Grant No. ANR:BLAN06-3_138079.