Structure, Bonding, and Properties
Rule 4: Linking of polyhedra having different cations
1.12. STRUCTURE TRANSFORMATION AND STABILITY 1. PHASE DIAGRAM
A phase means a matter having certain definite composition and structure. Phases in equilibrium with one another obey Gibbs' phase law:
(1.20) where F is the number of degrees of freedom (i.e., the number of variables of states such as temperature and pressure which can be varied independently), P is the number of phases, and C is the number of components. Components are to be understood as independent, pure substances (elements or compounds) from which the other compounds that eventually occur in the system can be formed.
For a single-component system, a phase diagram in which pressure is plotted versus temperature exhibits the existence ranges for the different phases of a system comprising only one component. For a two-component system, the chemical composition is plotted versus one of the variables (pressure or temperature) with the other unchanged. A phase diagram is usually a plot of the composition versus temperature at normal pressure.
The phase diagram of iron and neodymium (Fig. 1.20) is an example of a system with two components that form a compound, Fe17Nd2, in the middle component. In the Nd-rich side it is a typical example of two components: one pure element, Nd, and the other a compound, Fe2Nd, and they form neither solid solution (except for very low concentrations) nor a compound, but are miscible in the liquid state. As a special feature, an acute minimum is observed in the diagram, called the eutectic point. It marks the melting point of the eutectic mixture, which has a lower melting point than either pure components or any mixtures. The eutectic line is a horizontal line that passes through the eutectic point (E in the diagram). The area underneath is a region in which both components (neodymium and Fe2Nd) coexist as solids in two phases. A liquid solution of
0 u
Q) I.. :l
...-~ I-. Q)
0..
6
Q)
E-<
neodymium and iron containing a molar fraction of 40% iron and having a temperature of 1100D
e
corresponds to point A in Fig. 1.20. Upon cooling the liquid as shown in the diagram. At the moment the liquidus line is reached, pure Fe2Nd begins to crystallize. As a consequence, the composition of the liquid changes, since it now contains an increasing fraction of neodymium; this corresponds to a rightward movement in the diagram, where the crystallization temperature for Fe2Nd is lower. According to the amount of crystallizing Fe2Nd, the temperature for the crystallization of Fe2Nd continuously decreases until the eutectic point is reached, where both neodymium and Fe2Nd solidify.The term "incongruent solidification" serves to express the continuous change of the solidification temperature.
In the Fe-rich side, there is no eutectic point between pure iron and Fe17Nd2. We can only discern two kinks in the liquidus line, and the expected maximum of the melting point of the Fe17Nd2 is "covered." The horizontal line running through the kink is called the peritectic line. When Fe17Nd2 is heated to 12lODC, it decomposes into a solid y-Fe phase and a liquid phase with a higher Nd content (30%). As the temperature rises, the composition of the liquid contains less neodymium and the solid (y-Fe) reduces until 1538DC is reached, which is the melting point of iron, except the polymorphism transition between y-Fe and 8-Fe at 1394DC.
A phase diagram gives valuable information about the compounds that can form in a system with several components. Figure 1.21 gives the phase diagram of a
1500
1300
1100
900
700
0 10 20 30 40 50 1~
, , ,
~(-yFe) ,
, , ' ~oC
, 912"c
~(ClFe) ,
'tl Z
&:
1l~"c
_________ !t3.P~C
Weight Percent of Neodymium
'tl Z
60 70 80
L
A
90 100
1021"c ,
, , , ,
,/(PNd)i
/,/,..:_!l§Q~G._!
863°C,''' (aNd)--+
, , ,
~---~~---~~---~
&:
500~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
o 10 20 30 40 50 60 70 80 90 100
Fe Atomic Percent of Neodymium Nd
Figure 1.20. Phase diagram of the Fe-Nd binary system, where E indicates the eutectic point.
53
STRUCTURE, BONDING, AND PROPERTIES
54
CHAPTER 1
Weight Change, mg
o
10 20 30 40 50 60 70 80 90 1001000
t) 800
0 Q)
'/ \1\ , " I
I a
I I
I I
...
::Ico
...
600Q) c.
E 400
... Q)
200
I I
I I 1
----
I I ,----- ....
I }-:':::-,.1 '\ \
I
f
___ I
~... "' I
I
"/ \\ b
r \ \
I
I I ~
I \K' I ~ c 13
I 1\
I \
1.50 1.60 1.70 1.80 1.90 2.00 Compositon x, in PrO X
Figure 1.21. Phase diagram of PrOx system,
praseodymium-oxygen system, which is a very interesting system for functional materials. For oxygen content between 1.7 and 2.0 there are a series of phases that have linear composition at lower temperatures (lower than 200°C) but a narrow range of composition at higher temperature (about 300-600°C). Phases with linear compositions are called linear stoichiometric phases, and phases with a narrow range of composition are called nonstoichiometric phases. These phases have special structural relationships among them and special physical properties (Chapter 4). We would like to emphasize that the composition variation with temperature and environment can cause a tremendous change in property. For example, NaxW03 is an insulator if x < 0.3 and a conductor for x > 0.3. LaxMn03 is an insulator and antiferromagnetic if x
=
1, but a conductor and ferromagnetic if x < 1. This is a basic feature of oxide functional materials.The phase diagram is determined experimentally using the method described in the following. In differential thermal analysis (DT A), a sample of a given composition is heated or cooled slowly together with a thermal different reference substance, and the temperatures of both substances are monitored continuously. When a phase transition occurs in the sample, the enthalpy of conversion is released or absorbed, and therefore a temperature difference shows up between the sample and the reference, thus indicating a phase transition.
DTA is a very useful technique for phase determination, but its accuracy depends on the precision of temperature measurement and rate of heating or cooling. In fact, phase transition automatically introduces a change in crystal structure. Therefore, x-ray diffraction is usually used to identify the phase. For minor phases, especially nonstoichiometric compounds, TEM is most powerful for phase determination. This will be a key topic in future chapters.
1.12.2. THERMODYNAMIC STABILITY
When the free enthalpy of reaction f!G for the transformation of the structure of a compound to any other structure is positive, this structure is thermodynamically stable.
The term f!G depends on the transition enthalpy W and the transition entropy f!S:
f!G
=
W - Tf!S (1.21)and W and f!S, in tum, depend on pressure p and temperature T:
(1.22) A structure is stable only within a certain pressure and temperature. By variation of the pressure and/or temperature f!G eventually becomes negative relative to some other structures and a phase transition will occur. The following rules are given for the temperature and pressure dependence of thermodynamic stable structures in general:
(a) With increasing temperature T structures with a low degree of order will be favored. Their formation involves a positive transition entropy f!S, and the value of f!G depends mainly on T f!S.
(b) Higher pressure p favors structures that occupy smaller volumes (or higher densities). For example, diamond (3.51gcm-3) is more stable than graphite (2.26 g cm - 3) at very high pressure.
On the other hand, a thermodynamically unstable structure can exist when its conversion to some other structure proceeds at a negligible rate. In this case the structure is called metastable, inert or kinetically stable. Glasses, or non-crystalline materials in a broad sense, are metastable phases, which have short-range ordering but lack of long range ordering.