It is of interest to examine if the precipitation energy alters with the pressure state of the material. Although these simulations have created a description of the energy response of a crystal with respect to the hydrostatic pressure imposed on that pressure, a knowledge of what pressure state the dierent components of the precipitation reaction is in are required. To clarify, there is the pressure state in the initial solid solution and the pressure in the precipitated hydride, both of which must be known. If the energy of a crystal is considered as a function of the pressure imposed on the crystal, then the precipitation equation used so far becomes:
∆EP = [E(ZrxHy, (PHydride) + (Ry)E(Zr, (PZr)] − [yE(ZrRH, (PZr) + xE(Zr, (PZr)] (7.3)
where P is the hydrostatic pressure. The previous thermodynamic calculations investigated concentrations where R was equal to 16, 36 or 96, however, for this series of simulations, only the case where R = 36 will be used, as the necessity of repeating simulations at dierent pres- sures made the addition of another dimension impractical. This concentration is low enough to maintain physical relevance, without imposing an unreasonable burden of having large numbers of atoms to simulate. The solid solution is taken to be the most favourable interstitial site oc- cupancy, as determined by Fig. 7.2. This means that the tetrahedral site is used when the solid solution pressure state is above -4 GPa and octahedral is used below this value.
When one of these reaction equations is formulated, the case is being considered whereby both sides of the equation coexist and the energy of the equation states which side the reaction will be driven towards. Thus, both the reactants and the products must be thought of having pressure states that are linked to each-other. The simplest model of this would be to assume that both the hydride and the solid solution are both experiencing the same hydrostatic pressure. However, this is most likely an overly simplistic model, as Fig. 7.1 shows that hydrides and Zr
have dierent responses to pressure.
Total Hydrostatic Pressure (PTotal)
ΔVZr
ΔVHydride
PZr
PHydride
Figure 7.10: A simple diagram of a hydride embedded in a volume of Zr Metal.
In a real system, hydrides will form embedded in the metal. Their shape and orientation is such that they minimise the lattice strains. An example diagram is given in Fig. 7.10. In this case, the system can be thought of as a quantity of "hard" material surrounded by a volume of "soft" material. Intuitively, if this system is squeezed, the expectation is that the soft material would deform more than the harder material. The pressure states in each would be determined by the ratio of how sti the materials are to each-other, assuming a perfectly bonded interface. Thus, an estimation can be made in which the stress in the Zr and the hydride is partitioned by the ratio of their bulk moduli:
KHydride KZr
= PHydride PZr
. (7.4)
In this work, the values for K are taken from Fig. 7.1. Although this model is slightly more sophisticated, there are still further complications. The orientation of the hydride will alter its stiness due to the needle-like shape of hydrides. More complex models would take into account the idea that the hydrides would appear stier down their long axis than down their short axis.
Also, if the pressure state in the Zr or the hydride increases beyond the yield point, then the elastic stress as used in this model will reach a maximum value and increase no further, as additional stress is relieved by plastic deformation, although the atomic scale stresses will still be larger than the macroscopic yielded stress as discussed previously. It is di cult to account for yielding with DFT, however experimental observations or nite elements models can assist in giving a sense of the magnitude of the pressures involved.
Experimental results cannot directly monitor the stress state in both the hydride and Zr phases. However, X-ray diractometery is capable of measuring the change in the spacing of inter-atomic planes, via observing the shift in diraction peaks as the sample is pressurised. Recent work by Allen et. al. [96], measured the strain in both hydrides and Zr around crack
tips. They report maximum strains of 7.2×10−3 - 8×10−3 in the hydride and 4.10×10−3 -
5.10×10−3 in the metal. Much of the focus of their work involved a study of creep, however
these values were extracted from their data on samples without creep, to avoid adding even more complexity. These provided a third ratio relationship between the stresses in hydrides and Zr metal.
These three models give three possible stress ratios which can reasonably be expected to have some bearing on the actual system. With the two variables of the system (pressure of hydride and pressure of Zr) and the output of the energy of precipitation, the parameter space of the system can be represented as a surface. This is presented in Fig. 7.11.
-40 -36 -32 -28 -24 -20 -16 -12 -8 -4 0 4 8 -40 -36 -32 -28 -24 -20 -16 -12 -8 -4 0 4 8 H yd ro sta tic S tre ss in S ol id S ol ut io n (G Pa )
Hydrostatic Stress in Hydride (GPa)
0.0 eV -0.5 eV -1.0 eV 0.5 eV Favours Precipitat ion Favours Solution
Increasing Tensile Stress
1 eV 0.5 eV 0.0 eV -0.5 eV -1.0 eV
Identical Stress Partitioned Stress
Extrapolation from Allen 2012
Figure 7.11: The energy of precipitation, as dependent upon the pressure states of the reactants and products. Three dierent models describing the relationship between pressure states are shown. The line "Allen 2012" is extrapolated from data found in reference [96].
The coloured contours represent the energy of precipitation of a δ-hydride from a solid solution of 307 wtppm. In this case, the energies are calculated for a hydride with a stoichiometry of 1.625, as this is the closest to the 1.666 ratio available with the cell sizes used. The con guration of H atoms was selected as the most favourable con guration for that stoichiometry, as found from the disordered series investigation in chapter 5. On this contour plot, the three dierent stress state models represent lines, which can be thought of as vertically orientated slices through this energy landscape.
Fig. 7.11 shows that the precipitation enthalpy has a tendency to become more negative as the solid solution stress becomes more compressive. The surface is less sensitive to increasing compression in the hydride phases. There is a small discontinuity around -4 GPa in solid solution pressure as the site occupancy changes from fully tetrahedral to fully octahedral. In reality, if this site occupancy preference is correct, it would be reasonable to expect that this change would be gradual, as more H atoms shift from one type of site to the other. All three of the models show an increase in precipitation energy with system tension and a decrease with system compression,
although the partitioned stress and the Allen extrapolation roughly follow the contours of the surface at pressures below -15 GPa in hydride stress. All models show that precipitation is initially unfavourable from this concentration (in agreement with the previous study), but it becomes more favourable with increasing hydrostatic compression, with a switch between -4 and -8 GPa. The data of the surface is interpolated along the three lines to generate Fig. 7.12.
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 Pr ec ip ta tio n En er gy (e V /A to m )
External Stress (GPa) Identical Stress
Partitioned Stress Extrapolation from Allen 2012
Figure 7.12: The energy of precipitation, as dependent upon the pressure state in the precipitated hydride. The lines eectively correspond to slices taken from Fig. 7.11 along the three dierent lines. Negative energies favour hydride precipitation.
This plot more clearly shows the expected behaviour as theorised from the suggested models. They are all consistent in a qualitative sense and the key features remain - a linear increase above -3 GPa, a sudden negative shift between -3 and -10 GPa, followed by the data remaining negative at pressures below this point. The partitioned model and the extrapolation from experimental data suggest that below -10 GPa, the precipitation energy stabilises and is no longer particularly sensitive to pressure. The sudden jump is related to the sudden discontinuity on the solution side of the reaction, as shown in Fig. 7.2. It should be noted that the magnitude of the pressures required to induce these changes is relatively high.