This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Elastic stiffness coefficients of thenardite and their pressure and temperature dependence Dirk Arbeck I , Eiken Haussu ¨hl * ,I , Victor L. Vinograd I , Bjo ¨rn Winkler I , Natalia Paulsen I , Siegfried Haussu ¨hl II , Victor Milman III and Julian D. Gale IV I Institut fu ¨r Geowissenschaften, Abt. Kristallographie, Goethe Universita ¨t Frankfurt, Altenho ¨ferallee 1, 60438 Frankfurt am Main, Germany II Institut fu ¨r Kristallographie, Universita ¨t zu Ko ¨ln, Zu ¨lpicher Str. 49b, 50674 Ko ¨ln, Germany III Accelrys Inc., 334 Cambridge Science Park, Cambridge CB4 0WN, United Kingdom IV Nanochemistry Research Institute, Department of Chemistry, Curtin University, GPO Box U1987, Perth, WA 6845 Australia In memoriam Professor Friedrich Liebau Received November 7, 2011; accepted January 12, 2012 Published online: July 9, 2012 Thenardite / Elasticity / Piezoelastic constants / DFT / Na 2 SO 4 Abstract. The elastic stiffness coefficients, c ij , of orthor- hombic Na 2 SO 4 thenardite (space group Fddd) were meas- ured with an ultrasonic plane wave technique at ambient temperature as a function of hydrostatic pressure in the range of 0.1–70 MPa. The variation of the c ij in the range of 1– 5000 MPa was studied with density functional theory (DFT) based calculations. The experimental results and the DFT calculations were used to derive a force-field model, which was then employed to compute lattice parameters and elastic stiffness tensors of thenardite and of two other Na 2 SO 4 poly- morphs as functions of the temperature based on quasi-har- monic lattice dynamics. The structural parameters of the three polymorphs measured at high temperatures are repro- duced to within 1.7% by the present calculations. Phases II (space group Pbnm) and III (Cmcm) appear to have signifi- cantly higher entropies than thenardite in agreement with their metastable formation at higher temperatures. 1. Introduction Sodium sulphate, Na 2 SO 4 , exists in four different poly- morphs that have been structurally characterised, often la- belled as I (space group P6 3 /mmc), II (Pbnm), III (Cmcm) and V (Fddd) [1]. Thenardite (phase V), which is stable at ambient conditions, is a major commodity chemical [2]. Its principal use is in processing wood pulp for paper pro- duction. The knowledge of the phase equilibria of thenar- dite, particularly on its transformation into mirabilite (Na 2 SO 4  10 H 2 O), are important for understanding me- chanisms of weathering and damage of buildings and monuments of cultural heritage [3, 4]. Phase relations be- tween the four polymorphs have been studied by Rasmus- sen et al. [1], Brodale and Giauque [5], Tanaka et al. [6]. Vickers hardness, abrasive properties, thermal expansion, and elastic and thermoelastic coefficients at ambient pres- sure and temperature have been reported by Bayh and Haus- su ¨hl [7]. The present study is concerned with the elastic behaviour of thenardite and the other polymorphs at higher pressures and temperatures. Here we report the results of measurements of the piezoelastic coefficients of thenardite at moderately high pressures using a technique based on measuring the speed of an ultrasonic wave traveling across plane parallel plates. These studies are complemented with resonant ultrasound spectroscopic experiments and with DFT-based and empirical force-field atomistic calculations. The development of a force-field model for a material like thenardite is not straightforward. The conventional static fitting procedure, in which the force-field parameters are adjusted in order to reproduce structural data and elastic stiffness coefficients, may not be appropriate given that the- nardite is a compressible material. Hence, thermal expansion and zero point motion should not be ignored and the struc- tural data collected at ambient temperature cannot be used as constraints in a static lattice energy minimisation procedure. Therefore, the force-field fitting performed here was based on a different strategy that explicitly accounts for lattice dy- namical effects. We show that the use of a free energy mini- misation derived set of input parameters allows the struc- tures of the three phases to be modelled in a correct manner. The paper is organised as follows. First, we introduce the theory behind the ultrasonic measurements performed here. Then we describe further details of the experimental setup. Next, we determine the same properties by means of atomistic calculations and compare the experimental re- sults with the theoretically obtained values. 2. Crystal structure First analyses of the crystal structure of Na 2 SO 4 (V) were performed by Gossner and Mussgnug [8] and Colby [9]. Z. Kristallogr. 2012, 227, 503–513 / DOI 10.1524/zkri.2012.1476 503 # by Oldenbourg Wissenschaftsverlag, Mu ¨nchen * Correspondence author (e-mail: haussuehl@kristall.uni-frankfurt.de)