The Effect of Zr-Doping and Crystallite Size on the Mechanical ...

II. Methods <strong>and</strong> Instrumentation
4. Compressing Materials: Equation <strong>of</strong> State
<strong>The</strong> change in volume <strong>of</strong> a material with changing pressure <strong>and</strong> temperature
depends on the isobaric thermal expansion α as well as the isothermal compressibility β
such that:
VT,P = V1, 298 [1+ α(T-298) – β(P-1)] (3)
where α <strong>and</strong> β are not constant but approximate to a certain value, resulting in finite
strain. <strong>The</strong> isothermal volume change upon compression can be described by a number
<strong>of</strong> Equations <strong>of</strong> State (EoS). Here, the Birch-Murnaghan EoS for finite strain was used.
It is based upon the assumption that the strain energy <strong>of</strong> a solid undergoing compression
can be expressed as a Taylor series in the finite strain, f. <strong>The</strong> Birch-Murnaghan EoS
[190] is based upon the Eulerian strain:
fE =0.5 [(V0/V) 2/3 -1]/2 (4)
<strong>and</strong> the expansion to third order in the strain yields the Birch-Munraghan EoS:
P = 1.5 K0 [(V0/V) 7/3 - (V0/V) 5/3 ]·{1 - 0.75 (4- K0’)·[(V0/V) 2/3 -1]} (5)
with K0 = isothermal bulk modulus (K0 = 1/β (dP/dV)T) <strong>and</strong> K0’ = dK0/dP. In this study,
pressure-volume data upon compression <strong>of</strong> the samples were gained during diamond
anvil cell experiments. <strong>The</strong> data were then fitted to a third order Birch Murnaghan EoS
<strong>and</strong> for some instances to a second order EoS with K0’ fixed to 4, which is typical for
most materials. In order to yield estimates for the EoS parameters <strong>and</strong> their uncertainties
in a least square refinement, application <strong>of</strong> a weighing scheme was necessary. For each
data point i, the weight wi was assigned such that:
wi = σ -2 (6)
where σ² is the true variance <strong>of</strong> the data point, comprised <strong>of</strong> the contribution from
uncertainties in the pressure <strong>and</strong> volume measurements.
In order to express the relations <strong>of</strong> P <strong>and</strong> V linearly, the F-f plot was used. Here, Birch’s
normalized pressure F was plotted versus the Eulerian strain fE with:
F = P/(3 fE (2 fE +1)) 5/2 (7)
<strong>The</strong> data are described by a third order truncation <strong>of</strong> the EoS <strong>and</strong> as an
advantage, K0 <strong>and</strong> K0’ can be fitted as parameters <strong>of</strong> a linear function. <strong>The</strong> slope <strong>of</strong> such
a fit equals 3K0(K0’−4)/2. To calculate fE (equation 4), V0 was gained from
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