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Title:
First-principles investigation of the lattice vibrations in the alkali feldspar solid solution

Paper sections:
Introduction
<p>The heat capacity of solid solutions when plotted as a function of composition often deviates from linear behaviour at low temperatures. The resulting excess heat capacities are of vibrational and/or non-lattice (e.g. magnetic and electronic) origin. If the non-lattice contributions are not present or can be neglected, as it is the case with the alkali feldspars, only excess vibrational entropies are produced by the excess heat capacities. Excess vibrational entropies may be in the order of the configurational entropy and may have thus large effects on phase stability calculations (for a review of the vibrational entropy in solid solutions, see e.g. Van de Walle and Ceder 2002; Fultz 2010).</p><p>According to the "bond length versus bond stiffness interpretation" of the excess vibrational entropy, changes of the stiffness of a chemical bond are due to accompanied changes of the bond length, which occur during compositional variations (e.g. Van de Walle and Ceder 2002; Wu et al. 2003; Burton and van de Walle 2006; Van de Walle 2009). Assuming a typical value of 2 for the Grüneisen parameter Fultz (2010) estimated that a bond length increase of only 1 % decreases the force constant by 12 %, which is a large softening effect. A bond length increase of 10 % drops the force constant to zero. Bond softening increases the vibrational entropy because frequencies of lattice vibrations are shifted to lower values causing an increase in heat capacity and thus entropy, whereas bond stiffening has the opposite effect (Van de Walle and Ceder 2002). Mutual substitution of atoms of different size generally results in an increase in bond lengths around the smaller atom (bond softening) and a corresponding decrease around the larger atom (bond stiffening) compared to the bond lengths of the respective end-member structures. One of the both effects, however, may dominate the vibrational behaviour in the solid solution resulting in positive or negative excess vibrational entropies. Assuming a binary solid solution with end-members A and B, which have different volumes and elastic constants, it is to be expected that with compositional variation the bond length change of the stiffer end-member is less pronounced compared to that of the softer (Benisek and Dachs 2011). Under the assumption that the larger end-member is elastically stiffer, the atoms of the smaller end-member have to enlarge their bond lengths to a high degree, producing positive excess vibrational entropies as a net effect. The opposite situation may produce negative excess vibrational entropies, if the difference in elasticity is large enough. These considerations lead to an empirical relationship (Benisek and Dachs 2011, 2012) allowing the estimation of the maximum extent of the excess vibrational entropy, Δmax
S
exc, from the differences of end-member volumes, ΔV = V
A–V
B, and bulk moduli, ΔK = K
A–K
B:1\documentclass[12pt]{minimal}
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				egin{document}$$\Delta_{\hbox{max} } S^{	ext{exc}} = (\Delta V + {	ext{m}}\;\Delta K){	ext{f}}$$\end{document}ΔmaxSexc=(ΔV+mΔK)f</p><p>The value of ΔV is defined to be positive so that ΔK may be positive or negative depending on which end-member is elastically stiffer (smaller or larger) distinguishing the above-mentioned cases. The parameters m and f were determined based on data of some silicate solid solutions and binary alloys (Benisek and Dachs 2012). In the following, this ΔV versus ΔK approach was applied to other binary solid solution systems as well (Benisek and Dachs 2013, Eremin et al. 2013, Benisek et al. 2013, 2014a, b; Dachs et al. 2014a, b).</p><p>First-principles methods enable nowadays to calculate the effects of the vibrational modes in solid solutions producing more and more realistic phase stability diagrams (e.g. Turchi et al. 1991; Tepesch et al. 1996; Burton and van de Walle 2006; Urusov et al. 2007; Vinograd and Winkler 2010). Using the density functional theory (DFT), the lattice vibrations of the low albite—low microcline series, i.e. Al, Si ordered alkali feldspars in the system NaAlSi3O8–KAlSi3O8 abbreviated with Ab and Or are modelled in this study. The frequencies of the acoustic and optical modes are plotted as a function of composition in order to identify those vibrational modes that are most affected within the solid solution. Low-frequency modes are to be expected to be strongly softened in samples with intermediate compositions since alkali feldspars are characterised by distinct positive excess heat capacities at about 80 K (Benisek et al. 2014a). Such behaviour, however, is not reflected by the measured infrared (IR) spectra of the alkali feldspars (Zhang et al. 1996). Although the spectra in the low-wavenumber region are not easy to interpret, the lowest frequency mode measured experimentally is shifted to slightly higher frequencies in samples with intermediate composition, which is at variance with expectations from the heat capacity data.</p>


<p>DFT parameter settings</p><p>aConvergence was investigated using an increasingly dense k-point mesh</p>

Structural models
<p>All investigated cells are characterised by a fully ordered Al, Si distribution, i.e. Al occupies the T1O-site and Si the other three tetrahedral sites. The end-member structures of albite and microcline have a triclinic symmetry (\documentclass[12pt]{minimal}
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				egin{document}$$ar{1}$$\end{document}1¯), whereas in cells with intermediate compositions, the Na–K substitution destroys the symmetry elements. This is different to a real alkali feldspar crystal, where the Na and K sites are thermodynamically averaged. To investigate different Na, K configurations, four cells with Ab50Or50 composition were constructed: two ordered configurations using the single unit cell (cell 1 and 2) containing 4 A-sites (occupied either by Na or K) and two supercells containing 16 A-sites. For one of the supercells, a random number generator of Mathematica® constructed a quasi-random Na, K distribution (cell 3) and the other supercell was used to simulate a clustered Na, K distribution (cell 4). The clusters represent thin (up to 2 atoms) irregularly shaped lamellae in the (100)-plane. For the other compositions (Ab75Or25 and Ab25Or75), single unit cells were used, where only one A-site was substituted by the other atom. Appendix A (available as electronic supplementary material) contains the crystal symmetry, the lattice parameters, the fractional coordinates, the enthalpies and plots of the relaxed structures of all investigated cells. The calculated volumes are smaller by ~5 % compared to measured volumes. The calculated excess volumes of mixing, however, agree well with measured ones, i.e. they lie in the middle of the range defined by the results of Kroll et al. (1986), Hovis and Peckins (1978), Hovis (1988) as shown in Appendix A.</p>


<p>Heat capacities of low albite plotted against temperature. Points measured isobaric heat capacity (C
P) from Benisek et al. (2014a); Broken line calculated isochoric heat capacity (C
V); in the temperature region between 270 and 350 K, C
V was transformed into C
P (solid line)</p><p>Phonon dispersion relation (frequency, ν, versus wave vector) and phonon DOS of low albite. Partial phonon DOS of the Na atoms is given by the shaded area. Γ represents the centre, whereas Y, M and Z are points at the edges of the Brillouin zone with the coordinates (0, ½, 0), (0, ½, ½) and (0, 0, ½), respectively</p><p>Excess heat capacities (C
P, Vexc) plotted against temperature (T). Points measurements from Benisek et al. (2014a), lines represent the DFT results. a Data of Ab50Or50 of cell 1 (solid), cell 2 (dashed-dotted), cell 3 (dotted) and cell 4 (dashed). b Measured data of Ab70Or30, calculated ones of Ab75Or25. c Data of Ab25Or75. Error bars as shown in a represent 1 SD</p><p>The excess vibrational entropy (S
exc) of Ab50Or50 plotted against temperature. Data points represent the calorimetric results of Benisek et al. (2014a) and the lines those of the DFT calculations of cell 1 (solid), cell 2 (dashed-dotted), cell 3 (dotted) and cell 4 (dashed)</p><p>Low-frequency part of the phonon DOS for low albite (dotted line), low microcline (dashed line) and low structural state Ab50Or50 of cell 1 (solid line)</p><p>Mean frequencies (ν) of vibrational modes in the low-frequency region plotted against composition (X
Or). Solid circles Acoustic modes at the edge of the Brillouin zone (Z); Open squares the five lowest optical modes at the centre of the Brillouin zone (Γ); Open diamonds the same five optical modes at point Z; Open triangles three optical modes at the centre of the Brillouin zone at ~3 THz; Open circles an optical mode at ~8 THz. The lines represent the behaviour of a linear combination of the end-member frequencies</p><p>Mean bond lengths (R) of Na–O (closed symbols) and K–O (open symbols) as a function of composition. Solid lines represent a fit through the data, whereas the broken lines show the bond lengths without changes across the solid solution</p><p>Mean DFT calculated Na, K–O bond lengths (R) in the respective end-members and their relative changes (Δ) due to compositional changes (from end-member to Ab50Or50 composition). Na and K are coordinated fivefold and sevenfold, respectively, which is in accordance with Downs et al. (1996)</p><p>Square root of the force constants in x , y and z direction (k
x, k
y and k
z) of sodium and potassium as a function of composition (X
Or). The curves are based on a linear fit of k versus X
Or</p><p>Volumes (V) and bulk moduli (K) of low albite (Ab) and low microcline (Or)</p><p>aKroll and Ribbe (1983)</p><p>bAllan and Angel (1997)</p><p>cDowns et al. (1994)</p><p>IR spectra in the low-frequency region. The abscissa is shown in units of wavenumber (w) as well as frequency (ν). Solid line measured by Zhang et al. (1996); Vertical bars DFT calculations</p>

Discussion
<p>As shown in Figs. 3 and 4, the DFT results obtained for Na, K ordered or clustered distributions yield better agreement with the calorimetric results than the cell with the disordered Na, K distribution. This observation is consistent with investigations on Al, Si ordered alkali feldspars by means of 23Na NMR spectroscopy (Phillips et al. 1988). The respective samples were synthesised in the same way and at the same conditions as the samples used for determining the excess heat capacities calorimetrically. The NMR data of these samples indicate short-range clustering, i.e. the Na, K environment of a Na atom is skewed towards a relative enrichment of Na atoms (Phillips et al. 1988) supporting the DFT results.</p>

Conclusions
<p>Calculations of the phonon spectra and the heat capacity by density functional methods supply a consistent picture combining findings from different studies, viz. from calorimetric and NMR data, as well as from a macroscopic description of the excess vibrational entropy (ΔV vs. ΔK approach). Furthermore, the calculations succeeded in resolving contradicting results derived from calorimetry and IR spectroscopy.</p>


<p>Supplementary material 1 (PDF 582 kb)</p><p>Supplementary material 2 (PDF 1476 kb)</p><p>Supplementary material 3 (PDF 11 kb)</p>
The heat capacities of Al, Si ordered alkali feldspars of different Na, K compositions were calculated using the density functional theory. The effect of the Na, K distribution, if random, ordered or clustered, on the resulting heat capacity was investigated on different cells with Ab50Or50 composition. For all compositions and distributions studied, the excess heat capacity of mixing is positive at low temperatures with a maximum at ~60 K. This produces an increasing excess vibrational entropy that reaches a constant value above ~200 K. The amount of the excess heat capacity of Ab50Or50, however, depends on the Na, K distribution. Best agreement with measured excess heat capacities is achieved, if the distribution of Na and K is either ordered or clustered. The positive excess heat capacities can be attributed to a strong softening of the acoustic and the lowest optical modes related to a strong increase of Na–O bond lengths in samples with intermediate compositions. The softening of the lowest optical mode is, however, not reflected by thoroughly measured literature IR data. Comparing calculated and measured IR spectra suggests that the resolution of the measured spectra was insufficient for detecting the lowest IR-active modes.Electronic supplementary materialThe online version of this article (doi:10.1007/s00269-014-0715-8) contains supplementary material, which is available to authorized users.