Experimental test

Numerous experiments on the ordering-disordering of different cations in various minerals with two nonequivalent sites have been reported in the literature, but most of these experimental data are equilibrium site occupancies. Orthopyroxenes, however, are an exception. Several researchers have investigated the kinetic site occupancies of Fe2+ and Mg2+ at the Mj and M2 sites in orthopyroxenes using Mössbauer spectrometry (Virgo and Hafner, 1969; Besancon, 1981; Anovitz et al., 1988; Skogby, 1992) and single-crystal X-ray structure refinements (Saxena et al., 1987; Saxena et al., 1989; Sykes-Nord and Molin, 1993). To test our model, it may be useful to consider the following three cases for orthopyroxene:

(1) Two-cation ordering-disordering in pure binary solid solutions. Only Fe2+ and M g2+ are assumed to take over the Mj and M2 sites, and other trace or minor cations are neglected. As a first approximation, the system is treated as a two-cation order- disorder process in an assumed pure binary solid solution o f Fe and Mg at each ot the two sites. Reported experimental data include those o f Besancon (1981), Anovitz et al. (1988) and Skogby (1992).

(2) Two-cation ordering-disordering in multicomponent solid solutions. Both Fe2+ and M g2+ as w ell as other cations such as Ca2+, Mn2+, Fe3+, 161A13+ and Ti4+ are considered to occupy the M\ and M2 sites, but the site occupancies o f these cations except for Fe2+ and Mg2+ are assumed to be constant, therefore only Fe2+ and Mg2+ are supposed to participate in the order-disorder processes. Experiments o f this sort include those o f Saxena et al. (1987) and Sykes-Nord and Molin (1993).

0.26 - 0.24 ■ 0.22 - 0.20 - sample TZ T = 873 K A experimental • theoretical x \ e = 0.246 - 0.077Exp(-0.00007060 0.18 ■ 0.16 ■ X = 0.754 + 0.077Exp(-0.00007060 0.84 ■ 0.82 - 0.80 ■ 0.78 - 0.76 ■ 0.74 - 150000 200000 50000 100000 t (s)

Fig. 2-3. Theoretical curves showing that two-site order-disorder processes are either a monotonously increasing function (A) or a monotonously decreasing function (B), as illustrated by the disordering of Fe2+ and Mg2+ at the Mj site in an orthopyroxene. The parameters are estimated from the experimental data (solid triangles) o f sample TZ given by Besancon (1981).

0.050 0.045 - Fe 0.040 - 0.035 - sample AV77 T= 923 K 0.030 - 0.025 ; 0.105 - 0.100 - 0.095 0.090 - 0.085 - 0.080 0.975 ■ Mg 0.970 - 0.965 0.960 - 0.955 - 0.950 0.905 - 0.900 - 0.895 - 0.890 • 0.885 0.880 200000 300000 100000

Fig. 2-4. Two-cation ordering-disordering in a pure binary solid solution. Calculated values (curves) are compared with experimental site occupancies (solid circles) o f Fe2+ and M g2+ at the M\ and M2 sites in an orthopyroxene sample AV77 at 923 K. (Skogby, 1992). Site occupancies are: (A) Fe2+ at Mj; (B) Fe2+ at M2; (C) Mg2+ at Mi; (D) Mg2+ at M2.

0.25 - Fe 0.20 - 0.15 ■ sample OPXS95-13 T= 1023 K 0.10 ■ 0.05 ■ 0.85 - Fe 0.80 - 0.75 • 0.70 ■ 0.38 - Fe 0.36 ■ 0.34 - 0.32 - 0.30 ■ 0.88 - Fe 0.86 ■ 0.84 ■ 0.82 ■ 0.80 - 10000

Fig. 2-5. Two-cation ordering-disordering in a multicomponent solid solution. Calculated results (curves) based on the model are compared with the experimental site occupancies (solid circles) o f Fe2+ at the Mj and M2 sites in the orthopyroxenes. (A) Fe2+ at Mj, (B) F e 2+ at M2, sample O PXS95-13; (C) Fe2+ at M i, (D) Fe2+ at M2, sample OPX4. Experimental data are from Saxena et al. (1987) and Sykes-Nord and Molin (1993).

0.009 0.008 - 0.007 - 0.006 - 0.005 ■ _{sample OPXIO} 7= 873 K 0.004 O 0.003 0.86 - 0.84 - 0.82 - 0.80 - 0.18 - 0.16 - 0.14 - 0.12 - 0.10

, ,

Fig. 2-6. Multi-cation ordering-disordering in a multicomponent solid solution, as illustrated by Mn2+, Mg2+ and Fe2+ disordering in an orthopyroxene sample OPXIO. (A)

Mn2+ at Mi; (B) Mg2+ at Mi; (C) Fe2+ at M\. The experimental data (solid circles) (Saxena

et al., 1989) are in agreement with the theoretical calculations (curves).

(3) Multi-cation ordering-disordering in multicomponent solid solutions. More than two cations, such as Fe2+, Mg2+ and Mn2+, are considered to be involved in the ordering-disordering, but the site occupancies of other cations as mentioned above are still assumed to be constant. The only available kinetic data of this kind have been published by Saxena et al. (1989).

Figures 2-3 A and 2-3 B show the experimental results of Besancon (1981) on the orthopyroxene sample TZ at 873 K. The site occupancy of Fe2+at M\ increases with

time because = -0 .0 7 7 ±0.009 < 0 , and finally approaches its asymptote

c £ = 0.246 ± 0.004; whereas the concentration of Mg2^ at M\ which is estimated from

the site occupancies of Fe2+ at M\ by assuming that x f e + = 1 (pure binary solution), decreases with time since c* =0.0770 > 0 , and gradually approaches another asymptote c^g=0.754 ±0.004. The two asymptotes are the equilibrium concentrations of Fe2+ and Mg2+ at M i, respectively. These features are consistent with the characteristics that have been predicted from the theoretical Equation 2-26.

Figures 2-4A to 2-4D demonstrate the experimental data of Skogby (1992) on the ordering-disordering in an orthopyroxene (sample AV77) at 923 K, which was treated as a binary solid solution.

Saxena et al. (1987) and Sykes-Nord and Molin (1993) have studied the order- disorder kinetics in orthopyroxenes by considering the site occupancies of not only Fe2+ and Mg2+ but of other cations as well. They assigned Ca2+ and Mn2+ to the M2 site, and l6lAl, Fe3+, Cr3+ as well as Ti4+ to the M] site, and the concentrations of these cations are assumed to be constant at both sites. The experimental results on samples OPXS95-13 at 1023 K (Saxena et al., 1987) and OPX4 at 898 K (Sykes-Nord and Molin, 1993) are illustrated in Figs 2-5A to 2-5D. Similar results on the Fe-Mg ordering-disordering in orthopyroxenes have been given by other investigators (Virgo and Flafner, 1969; Anovitz et al., 1988), but for brevity, they are not replotted here.

Saxena et al. (1989) have experimentally investigated the order-disorder kinetics of three cations Mn2+, Fe2+ and Mg2+ in orthopyroxenes. Their results on sample OPX10 at 873 K are shown in Figs 2-6A to 2-6C. Nonlinear estimations give the following kinetic coefficients:

kg” = (1.47±0.27)xl0“V 1, k£n = (4.14±1.32)xlO "V ; k£* = (4.59± 0.79)x 10' V 1, k%* = (1.77± 0.25)x lO ^ V 1;

=(1.81 ± 0.27) x IO-4*'1, k[{ = (4.43 ± 0.75) x 10"5s~l. From these diagrams, the following conclusions can be made:

(1) The experimental results can be fitted well with model Equations 2-16 and 2-17 in all the three cases: Fe-Mg ordering-disordering in pure binary (Figs 2-3 and 2-4) and multicomponent (Fig. 2-5) solid solutions as well as multi-cation (Mn2+, Fe2+ and Mg2+) ordering-disordering in multicomponent systems (Fig. 2-6).

(2) The kinetic characteristics of experimental data are in good agreement with the theoretical features which are predictable from our model. These include:

(a) Variation trends of the site occupancies of a cation with time (kinetic curves). Given sample AV77 as an example (Figs 2-4A to 2-4D), the initial x\ / x,° for Fe2+ and Mg2+ in the unheated natural sample is 4.47 and 0.909, respectively (Skogby, 1992). The kinetic coefficients, which are estimated from the experimental data, are listed as

follows:

= (4.59±1.29)xK T V \ k% = (2.15±0.65)xICTV1; kg* = (3.28 ± 0.86) x 1(T55"1, kg8 = (3.47 ± 0.91) x 10"5 s~l.

Since KFf = kg / = 2.14 < (jcJ / = 4.47, we can expect that the site occupancies of Fe2+ will increase at Mi but decrease at M2 with increasing time according to Inequality

2-27. Similarly, because K j 8 = kg8 / k2[8 = 0.945 > (jc2 / x°)Mk = 0.909, the concentrations of Mg2+ should decrease at Mi and increase at M2 according to Inequality 2-28. These

theoretical predictions are consistent with the experimental results shown in Figs 2-4A to 2-4D.

(b) According to Equation 2-14, theoretically, the two coefficients cn and c21 for a given cation in site occupancy Functions 2-16 and 2-17 should have equal absolute values but opposite signs at the two sites; and both sites should have the same X value.

Therefore, if there were no experimental errors, the above parameters estimated from the experimental site occupancies of a given cation at the two sites should satisfy these requirements. Given sample AV77 (Skogby, 1992) as an example of a binary system, nonlinear estimations from the site occupancies of Fe2+ at Mi give the following coefficients:

eg =(-1.69±0.18)xl0"2 Xx = (-6.74±1.76)xl0'5 The site occupancies of Fe2^ at M2 give

c[[ = (+1.69± 0.18)x 10“2, X2 = (-6.74± 1.76)x 10"5. Therefore, c/7 =-cf{ and Xt =X2=X.

Similarly, Mn2+ in the multi-cation ordering-disordering in the orthopyroxene sample OPXIO of Saxena et al. (1989) is another example. At the M\ site, we have

= (-4.19± 0.21)xl0'\ X{ =(-1.89±0.26)xl0-4; at the M2 site, we get

c%n =(4.19±0.21)xl0~3, X2 =(-1.89±0.26)xl0“4.

Within experimental uncertainties, the value of cu in each example is considered to be identical to that of c21 for a given cation, and this is consistent with the theoretical requirements.