Study of sulfur α-S8 crystals with an anisotropic inter-molecular potential model
Introduction
In previous works we studied the α, β and γ crystalline phases of the S8 molecular compound, as given by a simple and flexible molecular model [1], [2]. These calculations were a first attempt to obtain the crystalline phase diagram at zero pressure, via a series of classical molecular dynamics (MD) simulations and using a constant temperature T and constant pressure P algorithm. In the literature, a lot of attention has been usually paid to the liquid phase of elemental sulfur [3], [4], [5], because of its liquid–liquid phase transition, correlated to large changes in its molecular composition as a function of the thermodynamic parameters T and P. This is not the case for the crystalline phases of elemental sulfur, that still have been not systematically identified [6], [7]. Moreover, recent experiments have shown an unexpected superconductor phase of elemental sulfur under high pressure [8], motivating the study of its high pressure phase diagram.
Over a wide variety of sulfur allotropes [6], [7], S8 is the most stable at ambient pressure and temperature (STP) in solid, liquid and gas phases. This cyclic crown-shaped molecule packs in an orthorhombic structure to form the most stable STP phase of sulfur (α-S8). Although α-S8 is the natural form in which sulfur is found, there are, at least, two other known crystalline phases of the S8 compound: β- and γ-S8. The orthorhombic α-S8 phase belongs to the Fd (D242h) spatial group, with 16 molecules in the unit cell [9]. In an inset of Fig. 1 there is a sketch of the unit cell in the [1 1 0] direction. α-S8 has a metastable melting point at 385.5 K, but when it is slowly heated it can go through a phase transition towards β-S8 at 369 K, this phase melts at 393 K [6].
The α-S8 crystal structure and its Raman and infrared frequencies have been intensely studied using lattice sums methods [10] and with MD simulations in the constant volume–constant energy ensemble [11].
In Refs. [1], [2] we performed a series of constant T–constant P MD simulations in the 50–500 K temperature interval. The employed MD algorithm was useful in the study of the lattice stability and structural phase transitions as a function of the thermodynamic parameters P and T. Our simulations [1], [2] were performed using a simple and isotropic atom–atom Lennard-Jones (LJ) inter-molecular potential, with its potential parameters (εiso and σiso) fitted to reproduce the experimental data on lattice parameters and sublimation heat of α-S8 at STP. This potential, without further modifications, was afterwards used through all calculations of all crystalline phases at all studied temperatures. In this way we could reproduce the experimental volume per molecule, configurational energy, and lattice cell parameters of α-, β- and γ-S8 in a wide range of temperatures, including the high temperature orientationally disordered phase of β-S8 [1], [2].
In spite of the amount of experimental data that this simple isotropic atom–atom reproduces, it fails to maintain the α-S8 structure below 200 K. The calculated low temperature structure turned out to be monoclinic, with a molecular arrangement quite similar to the orthorhombic phase, and was labeled as α′-S8. Both structures are plotted in insets in Fig. 1. According to the experimental data [12], the orthorhombic structure does not change at low temperatures and maintains the same spatial group down to 100 K.
If we assume that the monoclinic low temperature phase [12] does not correspond to a metastable state of α-S8, we can correlate the failure of the calculations in Refs. [1], [2] to the extreme simplicity of the isotropic atom–atom inter-molecular potential used. In the next sections we describe an improved anisotropic atom–atom potential model, the present MD calculations, our new results and conclusions.
Section snippets
The intra- and inter-molecular potential model
The intra-molecular potential model for this flexible molecule is the same as that used in Refs. [1], [2], except that the potential wells are lowered now by ∼10%, in order to improve the fit of the calculated intra-molecular frequencies to their experimental values. This model [1], [2] was, in turn, an adaptation of that proposed in Ref. [11], and takes into account 16 internal degrees of freedom per molecule. As the S–S stretching modes have a thermal activation energy of about 700 K, well
Calculations
The numerical MD algorithms and MD samples used in this paper are equal to those used in our calculations of Refs. [1], [2]. In this way, the comparison between results obtained with both, isotropic and anisotropic atom–atom models, is possible.
The numerical simulations are performed using a classical constant T–constant P algorithm, that allows changes in the MD box volume and shape in order to balance the external pressure with the strongly temperature dependent internal stresses [16], [17],
Results
The overall results of the present simulations, using the anisotropic model, are very similar to those obtained with the isotropic potential model in the 200–400 K temperature interval [1], [2]. This comparison is valid for all calculated values of configurational energies, volumes per molecule, lattice parameters, pair distribution functions, several correlation functions and vibrational density of states.
The main difference between the behavior of both potentials is found below 200 K. Using
Conclusions
Due to the unexpected results of Refs. [1], [2], where MD simulations of α-S8 were performed using a simple isotropic atom–atom model, we repeated the MD calculations using an improved inter-molecular potential that takes into account the lone pair charge distribution of the S atoms. The overall results and conclusions obtained with both models are similar, being the isotropic model more sensitive to changes in the thermodynamic parameters T and P than the more `realistic' anisotropic one.
Due
Acknowledgements
The authors thank CONICET for the grant PIP 0859/98.
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