Phase diagram studies in n-CoSb 3 skutterudites
5.4 Stable compositions and Vegard’s law
5.4.4 Vegard’s law in ternary phase diagram system
Figure 5.22 Illustration of typical eutectic phase diagram of binary A-B system. The green line represents the isothermal line. The orange section represents the solubility region of B in A. The purple dot represents the solubility limit of B in A at the selected temperature.
Figure 5.23 Illustration of isothermal section of ternary In-Co-Sb phase diagram. The green triangle represents the isothermal section. The orange area represents the solubility region of R-CoSb3. The purple boundary represents the solubility limit of R in CoSb3 at the selected temperature.
However, in a ternary system, e.g., R-Co-Sb, it is more complicated with an additional degree of freedom due to the thickness of the solubility region of RxCo4Sb12, as shown in Figure 5.23. In a ternary system, the isothermal section is a surface rather than a line in a binary system, so is the solubility region. The solubility limit also changes from a single value (represented by purple dot in Figure 5.22) in a binary system to a boundary represented by the purple contour of the orange solubility region in Figure 5.23. The experimentally determined solubility limit is thus largely dependent on the phase regions adjacent to skutterudite phase where the nominal compositions are, as illustrated in Figure 5.24. For the Vegard’s law to be applicable in a ternary system, the nominal compositions should be along the solubility direction, e.g., (InVF)2x/3Co4Sb12-x/3(InSb)x/3 – CoSb3,so it
can be considered as in a pseudo-binary system. If nominal compositions are off the solubility direction, e.g., (InVF)xCo4Sb12 – CoSb3 for In-Co-Sb system (purple solid line in Figure 5.14b) and (YbVF)xCo4Sb12.2 – Co4Sb12 for Yb-Co-Sb system (yellow dashed line in Figure 5.25a), then deviations from the Vegard’s law can result from nominal compositions entering either two-phase (2-p in Figure 5.24) or three-phase region (3-p’ in Figure 5.24) other than the one that will give the maximum solubility limit (xmax in3-p, Figure 5.24). As a result, the solubility limit observed will be smaller than the maximum solubility limit, because its determination is misled by slope change or false plateau in the relationship between lattice parameter and nominal filler content x.
Figure 5.24 Dependence of solubility limit on adjacent phase regions in a ternary phase diagram system. Triangles enclosed in yellow lines are three-phase regions (on the Co-rich side: 3-p; on the Sb-rich side: 3-p’), with x3-p and x3-p’ indicating respectively the solubility limit of R in equilibrium skutterudite phase when nominal compositions are in these phase regions. A two-phase region (2-p) is located right above the Sb-rich three-phase region. The solubility limit of R in equilibrium skutterudite phase when nominal compositions are in this two-phase region is represented by the purple dot marked at x2-p. The maximum solubility limit xmax is represented by the purple dot that is the furthest point along the elongated direction (solubility direction).
Figure 5.25 serves as a good example to illustrate the inapplicability of Vegard’s law when the nominal compositions are off the solubility direction in a ternary Yb-Co-Sb system. It demonstrates that reaching a plateau of lattice constant could be because of the existence of stable compositions and thus does not necessarily imply reaching the solubility limit.
Figure 5.25 Samples with different nominal Yb content x in YbxCo4Sb12.2 (marked as empty orange rectangles) but the same Sb excess lead to a nonlinear dependence of lattice constant (b) due to the sample traversing different two- and three- phase regions of the phase diagram (a).
Excess of Sb is often added when synthesizing CoSb3 skutterudites due to the high volatility of Sb.
In samples with even a slight Sb excess (depicted in Figure 5.25(a)), increasing the nominal Yb content will lead to samples in the Sb-rich three-phase region of YbxCo4Sb12, YbSb2, and liquid Sb first, then samples in the two-phase region of YbxCo4Sb12 and YbSb2 before reaching samples in the Co-rich three-phase region of YbxCo4Sb12, YbSb2, and CoSb2. Thus the actual Yb content in Yb filled CoSb3 as measured by the lattice constant of the skutterudite phase will stop increasing and be constant for a period (first plateau in Figure 5.25(b), or blue dashed line in Figure 5.20(b)) while the nominal composition is in the Sb-rich three-phase region and the resulting skutterudite phase has stable composition (blue point in Figure 5.25(a)). This Yb content at which lattice constant stops increasing can be easily mistaken as the Yb solubility limit139. However, as seen from Figure
0.72 0.74 0.76
0.22
0.24
0.26
0.28
0.02 0.04 0.06
CoSb+3Yb Sb
2
Co Sb
CoSb3+CoSb2
CoSb3 +YbSb2 + liquid Sb
CoSb3+ liquid Sb CoS
b+C3 oS
b+2Yb Sb2
Yb
CoSb3
(a) T=973K
0.2 0.3 0.4 0.5 0.6 0.7
9.05 9.06 9.07
Lattice constant (a(A))
x in nominalYbxCo4Sb12.2 Easily mistaken as
solubility limit (b)
Stable composition 1 (Red, Co-rich) Stable composition 2
(Blue, Sb-rich)
5.25(b), further Yb addition can change the Yb actual content in the skutterudite phase dramatically as the nominal composition moves away from the Sb-rich three-phase region and into the two-phase region, and consequently the lattice constant would resume the increasing trend. As the nominal composition moves into the Co-rich three-phase region, the actual Yb content reaches the maximum solubility (red point in Figure 5.25(a)) and thus the lattice constant reaches a plateau again (second plateau in Figure 5.25(b) or red dashed line plateau in Figure 5.20(b)). An overall investigation of the phase relations near the targeted phase is thus needed in order to make a conclusion about solubility limit because: observing a plateau in lattice constant is insufficient in determining the solubility limit in a ternary system, nor is observing impurity phases sufficient (the nominal composition might just be in a two- or three-phase region).