1. Introduction

Van den Berg et al. (1973) discovered crystal structure modulation in isomorphic crystals of K₂MoO₄ and K₂WO₄, samples of which were obtained in the manner described by Kools et al. (1970). Warczewski (1979) has studied powder samples of K₂MoO₄ and K₂WO₄ with the aid of a Guinier-Lennè camera and observed the hysteresis of their phase transitions. Their low-temperature monoclinic non-modulated phase was first observed by Gatehouse & Leverett (1969). It turned out that for K₂MoO₄ one can distinguish two regions in the temperature dependence of the modulation parameter p in both the first and the second heating curve (see Fig. 4a of Warczewski, 1979): (a) a region with p changing with temperature (this is the incommensurately modulated phase); (b) a region with p constant as a function of temperature [this is the lock-in commensurately modulated (superstructure) phase]. So far, van den Berg et al. (1983) have studied the average structure of K₂MoO₄ in the temperature range of the incommensurate phase.

On the other hand, for K₂WO₄, only one region was observed with p constant as a function of temperature in both the first and the second heating curve (see Fig. 4b in Warczewski, 1979), which means that it is a commensurately modulated (superstructure) phase.

The present work is a study of the binary system K₂Mo_xW_1-xO₄ and the dependence in this system of the modulation parameter p on the concentration x at a given temperature in order to find evidence for incommensurately modulated structures within the interval 0 x 1.

2. Crystal growth and synthesis of powder samples in the system K₂Mo_xW_1-xO₄

The compounds of the series K₂Mo_xW_1-xO₄ were obtained by conversion (crystallization from H₂O) of an appropriate mixture of the oxides MoO₃ and WO₃ with excess of KOH (in aqueous solution at room temperature in nitrogen atmosphere) according to the reaction (Bzowski et al., 1998; Bzowski, 1999):

Samples with five different concentrations, namely x = 0.00, 0.25, 0.50, 0.75, 1.00, were prepared for investigation.

The reactions were carried out in the protective atmosphere of nitrogen. Some water was added after mixing the dry oxides. After about one week, hygroscopic single crystals of the compounds under study were grown, the majority of them being twinned. Then the single crystals were washed to remove the excess of hygroscopic KOH with the aid of anhydrous ethanol and finally kept in a dryer at 393  K, in the presence of NaOH to absorb water. Atomic absorption spectroscopy (Siemens 300S spectrometer) proved with an accuracy of 1% the assumed concentrations x. The powder patterns of the end members, i.e. K₂MoO₄ and K₂WO₄, were checked and confirmed by comparison with the powder patterns of the database PDF-1 using the computer program XRAYAN. The powder patterns of the remaining intermediate compounds are qualitatively identical. A second set of powder samples was prepared by heating appropriate mixtures of K₂CO₃, MoO₃ and WO₃ at different temperatures: 1133, 1173 and 1213  K. In addition, samples with an excess of K₂CO₃ of 2%, 5%, 10% and 20% were synthesized. These experiments had to be performed because of inconsistencies in the differential scanning calorimetry (DSC) and high-temperature X-ray powder results (see below).

3. Phase transitions in the system K₂Mo_xW_1-xO₄

DSC measurements were made with a Netsch STA 409 thermal analyser. For all samples, three heating and cooling runs were performed in the temperature range between room temperature and 973  K at rates of 20  K  min^-1 (heating) and 10  K  min^-1 (cooling). All authors who have performed X-ray diffraction experiments report two phase transitions, whereas in differential thermal analysis (DTA) experiments mostly three peaks were observed (see Table 2 of van den Akker et al., 1970).

Figs. 1 and 2 represent the heating and cooling DSC runs for a sample of K₂MoO₄ composition synthesized by sintering at 1213  K. Figs. 3 and 4 show analogous heating and cooling runs for a K₂MoO₄ sample with 10 wt% excess of K₂CO₃. The heating and cooling diagrams for the other intermediate compositions in the solid-solution series are very similar and are therefore not reproduced here.

The interpretation of the phase transitions is given in Fig. 5 for the heating and in Fig. 6 for the cooling experiments. On heating, the low-temperature stable monoclinic phase (M) changes into the hexagonal modulated (H_mod) structure at 623  K for K₂MoO₄ and 663  K for K₂WO₄. This phase transforms to the hexagonal non-modulated phase (H) at 733  K. This value is constant over the whole series. The transformation is always reversible, and shows no hysteresis. The third, rather large, DSC signal between about 773 and 873  K arises from the melting of a K₂(Mo,W)₂O₇ phase, which crystallizes because of evaporation of K₂O. With addition of at least 10% K₂CO₃ the formation of this phase is suppressed and, hence, the highest DSC peak disappears. This peak is thus not related to the phase transitions of K₂Mo_xW_1-xO₄.

The phase transition sequence is significantly different on cooling the samples (Fig. 6): the curve of crystallization of the K₂Mo₂O₇ phase now intersects the curve of the K₂MoO₄ phase transition H to H_mod, resulting in a reversal of the sequence of the two DSC peaks in Fig. 2. Furthermore, in contrast to the heating sequence in Fig. 5, the hexagonal modulated phase H_mod now changes below 573  K to an orthorhombic phase (O) of the low-K₂SO₄ type, which also becomes modulated with decreasing temperature; the relations between the cell parameters are a = 2d_100(hex), b = b_(hex), c = 2c_(hex) (for individual cell parameters see Table 1). No DSC signal is observed for the phase transition hexagonal to orthorhombic. This modulated orthorhombic phase was first observed for K₂MoO₄ and K₂WO₄ by Warczewski (1979), but with c_(ortho) = c_(hex).

The stable monoclinic phase is obtained on further cooling below 423  K and accompanied by a very intense DSC peak (see Fig. 2). This peak is not observed in the samples with at least 10% excess of K₂CO₃ (see Fig. 4), but the monoclinic low-temperature phase is always observed after cooling to room temperature.

The powder patterns and the lattice parameters of this monoclinic phase show no significant deviations between the samples with and without excess K₂CO₃. Lattice parameters for the pure end members are given in Table 2.

4. Modulation parameter p of the solid solutions K₂Mo_xW_1-xO₄

The two series of samples were studied at higher temperatures with a Siemens D500 powder diffractometer with Cu K radiation ( = 1.54059  Å) and a step width of 0.02° in the 2 range 10-50°. The heating device used was constructed by Behruzi et al. (1991).

The compounds used for both the first and the second series are described in §2; for the second series the pure end members K₂MoO₄ and K₂WO₄ synthesized at 1213  K without any excess of K₂CO₃ were used. In the diffractometer the samples of the first series were heated to 833  K and for the second series to 923  K. The sample temperatures and compositions used for the following investigations are shown for heating and cooling runs in Figs. 5 and 6. The most important temperature range is just above and below 733  K, where the hexagonal modulated phase H_mod changes to the non-modulated phase H. The variation of the cell parameters and the modulation parameter p with composition x was measured at 693  K, both on heating and cooling. The cell parameters of the average structure were calculated by means of the main reflections using normal least-square programs, whereas the parameter p was calculated by indexing the satellite reflections by trial and error with the following formula, used earlier by Warczewski (1979):

where h, k, l, r, s are integers, the r and s values being equal to 0, +1 and -1.

Fig. 7 presents as an example the powder pattern at 693  K of K₂MoO₄, showing the main and satellite reflections. Table 3 shows the cell parameters and the modulation parameter p as a function of the composition x for the hexagonal modulated phase H_mod at 693  K. In Figs. 8, 9 and 10 the cell parameters a, c and V versus composition x are plotted for 693 and 833  K. Note that at the phase transition H_mod to H (which takes place at 733  K for all compositions x; see above) the a axis changes only by about 0.02  Å, whereas the c axis increases drastically by about 0.15  Å. Furthermore, the relative variation of the cell volume versus composition is much smaller for the monoclinic room-temperature phase (Table 1) than for the hexagonal phase (Table 3 and Fig. 10). The modulation parameter p covers the range from 0.25 to 0.30 and changes nearly linearly with composition x (Fig. 11). Fig. 11 shows that either at x = 0 exactly (i.e. for K₂WO₄) or close to this value (i.e. within the interval x = 0-0.25) a lock-in phase transition takes place, locking in the value of the modulation parameter to p = 0.25.

5. Discussion

From the latter result it can be concluded that the phenomena of incommensurate crystal structures and lock-in phase transitions occur not only for the case of temperature changes but also for the case of composition changes. There are at least two effects observed that deserve further investigation. The first one is the constant temperature (733  K) of the H_mod to H phase transition for all compositions x (Figs. 5 and 6), suggesting that this phase transition is independent of the Mo/W ratio, i.e. it is dependent only on the K and/or O atoms. This implies that only the positions of the K and/or O atoms are being displacively modulated, but that there is no significant substitutional modulation of Mo and W. The second effect is the variation of the modulation parameter p over the entire Mo/W composition range; this variation is nearly linear, possibly with a small non-linearity occurring at x_Mo = 0.25, as can be seen in Fig. 11. In order to explain these two apparently inconsistent effects, one could suppose that the phase transition mechanism involves only the K and O atoms, whereas the modulation wavelength (for a given temperature) depends also on the Mo and W atoms. Near x_Mo = 0, the predominance of W atoms leads to a lock-in phase transition with a rational value of the modulation parameter. These hypotheses, however, require further studies, which will be the subject of future work.