On the Tammann Rule Rotraut Merkle und Joachim Maier* Stuttgart, Max Planck Institut für Festkörperforschung Received December 28th , 2004. Dedicated to Professor Rüdiger Kniep on the Occasion of his 60th Birthday Abstract. The validity of Tammann's rule is related to the fact that the unavoidable thermal generation of point defects leads to defect- defect interactions and finally to a breakdown of the structure. It is shown that the onset of this defect avalanche, which can be esti- mated by a cube root law, roughly corresponds to the Tammann temperature. The investigation of simple compounds corroborates Introduction In the early days of solid state chemistry the so called Tammann rule [1] was formulated stating that at tempera- tures higher than about two thirds of the melting point Tm solid state materials become reactive [2] (some authors also propose 1/2 Tm, depending on the properties examined, see e. g. [3]). In the light of a modern mechanistic understand- ing which is based on the pioneering work of Wagner and Schottky [4], it became clear that the occurrence of solid state reactions in ionic systems being very frequently trans- port controlled, presupposes the presence of point defects, i.e. the presence of ionic charge carriers. For mass transport to proceed, conductivities of at least two carriers (two ionic species or one ionic and one electronic species) are neces- sary. So evidently the product of mobility and concen- tration is decisive. While a correlation between mobility and melting point is difficult to achieve, in this contribution evi- dence for a correlation between defect concentration and melting point will be given, which is based on a simple model that takes account of Coulombic interaction of point defects [5]. In Ref. [6] empirical arguments have been re- ported for a critical defect concentration at the melting point. Notwithstanding the fact that such a relation is not directly connected with Tammann's rule its validity is inves- tigated, too. In view of the fact that the mobilities are not so different in the high temperature zone of interest, we can formulate the Tammann rule as: At about 2/3 of the melt- ing point, the charge carrier concentration in solids be- comes substantial. * Prof. Dr. J. Maier Max-Planck-Institut für Festkörperforschung Heisenbergstraße 1 D-70569 Stuttgart Fax.: 49 (0)711 689 1722 email: s.weiglein@fkf.mpg.de Z. Anorg. Allg. Chem. 2005, 631, 1163 1166 DOI: 10.1002/zaac.200400540 2005 WILEY-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim 1163 this picture and also the observation of a critical defect concen- tration. Examples are given that Tammann's rule can be used to systematically search for new solid electrolytes. Keywords: Phase transitions; Thermodynamics; Ion conduction; Defect interaction; Superionic state Thermal destiny of ionic crystals At zero Kelvin the equilibrium charge carrier concentration is zero. (In reality a nonzero frozen in concentration will be realized, not to mention defects induced by effectively charged impurities which we will neglect in the following). Thermal equilibrium necessarily requires the formation of point defects. In primarily Frenkel disordered materials such as AgX (X Cl, Br, I) this is primarily a finite concen- tration of interstitial silver ions and silver vacancies; in primarily Schottky disordered materials such as alkaline halides, both cation and anion vacancies prevail [7]. As long as the temperature is low, the concentration is small and the defects will be randomly distributed. Then ideal mass action laws hold with the consequence that the defect con- centration (x) follows a van't Hoff law (as long as the defect formation parameters H°, S° can be considered as temperature independent) x,ideal exp S0 2R exp H0 2RT (1) Hence x increases steeply with temperature according to Eq. (1) until the defect concentration is so high that the defects perceive each other. The interaction of the two op- positely charged defects is primarily an attractive Coulomb interaction. This attractive interaction reduces the effective formation enthalpy H° (we ignore effects on S°) to H0 H (x) [5, 8]. It hence becomes increasingly easier for the next defects to be formed. As a consequence more defects are formed than expected according to the ideal mass action law (i.e. according to the van't Hoff relation). In a ln(x) vs 1/T plot an upward bending of the graph occurs. Such deviations are also observed in the tempera- ture graph of the ionic conductivity (see e. g. [9, 10] and references cited in [5]) and in an anomalous increase in specific heat (see e. g. [11]). As there is a positive feedback (more defects lead to even more defects), an avalanche of charge carriers occurs which eventually leads to a phase R. Merkle, J. Maier transformation which can be of first or higher order. This process describes the transition to a superionic state within the same structure. In the case of a first order transform- ation realistically a transition into another solid structure or into the totally molten state occurs. In particular in the case of Schottky disorder, the molten state represents the naturally expected superionic phase. In such cases the tran- sition will occur even at lower temperatures (by T) as the free enthalpy (G) of the real superionic phase is smaller than the free enthalpy of the virtual superionic phase (no structural modification). If this G difference is large and hence T substantial, it can be that the premelting regime is completely suppressed. The just described process consti- tutes a universal behavior for simple compounds and pre- dicts at least an upper limit of the disorder temperature whenever the phase does not undergo a structural change before it melts [13]. Of course, in the case of complex crystal structures, these considerations may not be sufficient to completely describe their thermal evolution. Cube root model as a simple means for the description of the thermal destiny It was shown in previous papers [5, 12, 13] that a cube root law in x well describes the defect interaction and leads to a satisfactory description of the disorder in simple crystals including the premelting zone; in the case of a higher order transition it may also describe the disorder in the superionic state, while in the case of first order transition only an up- per limit for the transition into the superionic state is ob- tained. The validity of this model was demonstrated for AgI which undergoes a transition to the AgI phase, for AgCl and AgBr which undergo a transition to the molten state, as well as for PbF2 which undergoes a higher order transition within the fluorite structure [5, 12 15]. In the latter case the conductivity behavior could even be pre- dicted for the superionic state. The basis of the cube root law is the assumption that the interaction between defects that are more or less randomly distributed can be approximately mimicked by the interac- tion that a system of the same carrier concentrations per- ceives in which they all have the same distances. Then it is just necessary to calculate the Madelung energy of a period- ically ordered defect structure, the lattice of which is super- imposed to the perfect (host material) lattice. In order to avoid misunderstandings we simply refer to this as the "de- fect lattice" (spanned by the defects) in the following. This directly leads to the implicit formula, Eq. (2), the solution of which yields the x(T) relation for the whole temperature range (including the superionic state provided the structure is maintained): H0 TS0 Jx1/3 kT ln x2 gigv( i x)( v x) (2) (gi, gv, i, v, denote degeneracy and number of available crystallographic sites). The defect interaction parameter 2005 WILEY-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim zaac.wiley-vch.de Z. Anorg. Allg. Chem. 2005, 631, 1163 11661164 Figure 1 Molar fractions of Frenkel defects calculated according to eq. (2). Dotted straight lines indicate ideal defect concentrations in the absence of defect interactions. J can be traced back to those parameters that determine the electrostatic interaction energy of the defect lattice [12]: UM (Madelung energy of a perfect lattice with same cation/ anion stoichiometry as the defect lattice), dielectric con- stant r and fd/f (ratio of the Madelung constants of a crys- talline phase and the value for the corresponding melt, fd/f 0.9 [16]) J 4UMfd 3rf (3) Figure 1 displays the defect concentrations x(T) calculated from Eqs. (2), (3) for the silver halides (for first- or higher order phase transitions, a vertical step at Tc has to replace the unphysical S-shaped solution of Eq. (2); this new line corresponds to the solution of Eq. (2) with lowest Gibbs energy). If the transition at Tc is of first order as is the case when the modification is altered or the phase melts, the calculated transition temperature is, as already men- tioned, to be taken as the upper limit. The fact that in AgCl, AgBr and AgI, Tc is close to the experimental value means that the difference G (virtual high temperature phase) G (real high temperature phase) is small. In Ref. [5] also a quantitative criterion has been derived based on Eq. (2) which decides upon whether the transition is of first or higher order. The correlation betwen defect concentration evolution and melting point forms the basis of our interpretation of Tammann's rule. Figure 2 shows the increase in x relative to xideal (calculated without defect interactions according to Eq. (1)). Owing to the steep self amplified augmentation it does not matter whether we chose x / xideal 1.1 or x/xideal 1.01 as the beginning of the anomalous increase. Moreover, since in this temperature zone all the materials exhibit similar xideal values, we may even chose an absolute value (e.g. x 10 4 ) as a criterion for the onset tempera- ture (it is already qualitatively clear that the onset of the interaction avalanche requires a similar defect concen- tration, i. e. a similar mean distance, given the small varia- On the Tammann Rule Figure 2 Ratio of defect molar fraction x / xideal with and with- out defect interactions. The dotted line marks the chosen criterion x / xideal 1.1 for determination of the onset temperature. tions in r and in fd/f). Table 1 gives Tc as well as Tonset together with Tm, and Figure 3 demonstrates graphically the correlation of the latter quantities. Indeed it is seen that Tonset is proportional to Tm with the proportionality con- stant between 0.5 and 2/3 as proposed by the modified Tam- mann relation. A better agreement is not to be expected because of the qualitative character of Tammann's rule. Interestingly, the calculated defect concentrations at the predicted critial temperatures Tc shown in Table 1 are simi- lar and fall in the range of 210 3 710 2 . Here the "ther- mal destiny" outlined above explains the observation of a "critical" defect concentration. The finding that the correlation between Tm and x di- rectly translates into a correlation between Tm and ionic conductivity is due to the fact that the mobilitites of simple compounds are not so different close to the critical tem- perature. This was shown for fluorites in Ref. [17] with mo- Table 1 Melting temperature Tm(* or superionic phase transition for AgI, PbF2), onset temperature Tonset (x/xideal 1.1), critical temperature Tc, defect molar fraction x at T Tc, and references for defect formation parameters of selected halides. ** Tc for the fluorite materials exhibiting only a diffuse phase transformation is estimated graphically from plots of x versus 1/T. Please note: for the fluorite materials, the defect lattice (anion va- cancies and interstitials) has 1:1 stoichiometry and is thus approxi- mated by a NaCl structure with appropriate lattice constant Tm/K T(x 10 4 ) T(x 10 6 ) T(x/xid 1.1) Tc /K x (Tc) Ref. NaF 1265 1030 791 699 [24] NaCl 1074 971 751 710 1347 0.063 [24] NaBr 1020 764 587 525 942 0.0025 [24] NaI 933 756 580 550 1083 0.075 [24] AgCl 728 576 448 422 727 0.0062 [5] AgBr 703 475 362 343 687 0.056 [5] AgI 419* 300 240 230 419 0.071 [5] CaF2 1665 1268 937 951 2000** [25] SrF2 1673 1196 869 879 2000** [25] BaF2 1593 940 686 661 1550** [25] PbF2 700* 471 358 339 680** [5] Z. Anorg. Allg. Chem. 2005, 631, 1163 1166 zaac.wiley-vch.de 2005 WILEY-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim 1165 Figure 3 Plot of onset temperature of defect interaction versus experimental melting temperature (for AgI and PbF2: versus expe- rimental transformation temperature to superionic phase). Solid symbols: relative criterion x / xideal 1.1; open symbols: absolute criterion x 10 4 . The dotted lines correspond to 2/3 Tm and 1/ 2 Tm, respectively. bilities of 310 3 cm2 V 1 s 1 ; it holds also for the silver halides [18] and the alkali halides [19, 20] exhibiting com- parable mobilities close to Tc. This fact is accepted as an empirical finding in this paper but of course also relies on energetic and entropic reasons. As we face a relation be- tween thermodynamic and kinetic quantities this is more difficult to explain. The most important quantities for high defect concen- trations are of course the defect formation parameters H0 and S0 . UM and fd/f of the defect superlattice will not vary much because for Schottky as well as for Frenkel disorder (to mention the most frequent defect types) cationic and anionic defects are formed in a 1:1 stoichiometric ratio. The influence of r essentially becomes effective through H0 . For a given structure a larger r usually implies a lower defect formation enthalpy H0 (because the host structure becomes dielectrically softer), which outweighs the decrease in J. A weaker influence of r appears via J, there the effect is opposite as a high r weakens the defect interactions. Turning around Tammann's rule: Search strategy for good ion conductors If we assume that the molten state is the superionic state of interest (i.e. no superionic solid phase exists, as in the case of Schottky disordered solids) we can state that high defect concentrations imply low melting points. The reversal is not as strict because of the discrepancy between virtual high temperature structure and real structure. Nonetheless, searching for low melting ionic compounds is a powerful search strategy for materials that easily form defects and hence offer the possibility to be good ion conductors. This qualitative tendency is obvious e. g. when we consider the alkali halides AX (variation of the cation in the series LiCl, R. Merkle, J. Maier NaCl, KCl, RbCl, CsCl or of the anion LiCl, LiBr, LiI). It is also striking that soft, low melting solids such as silver halides, stoichiometric lithium halide-alcohol adducts [21] or alkali triflates [22] exhibit high ionic conductivities. A spectacular example are ionic liquids. Ionic materials such as quartenary amines or imidazolium salts possess a melt- ing point around room temperature and some of them have been recognized to exhibit high conductivities in the solid state at moderate temperatures, see e. g. [23]. Conclusions Owing to interactions between defects an overexponential increase of defect formation starts typically at a tempera- ture that can be identified with Tammann's temperature. This defect avalanche unavoidably leads to a molten state (if a superionic phase or a phase with different structure is not thermodynamically available), thus connecting melting point and defect concentrations. This behavior forms an ex- planation of Tammann's rule. As the latter refers to reactiv- ities, i. e. in the diffusion controlled case to conductivities, the validity of the approach presupposes comparable mo- bilities in the premelting zone, which is indeed the case for many simple materials. If a solid phase undergoes a modifi- cation change before it melts, we have to refer to the solid high temperature phase and the picture may change. Fi- nally, we gave examples that, in the case of simple struc- tures, Tammann's rule can be used as a guideline to search for new solid electrolytes. References [1] G. Tammann, Lehrbuch der Metallkunde, 4. ed., Verlag Voss, Berlin 1929. [2] K. 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