**6. Pore formation model: nucleation and growth**

It is not well established by what mechanism cavities nucleate [31]; however, the linear relationship between cavity nucleation rate, first observed by Needham et al [32], still holds true to this day [33]. Grain boundary sliding, as necessitated to maintain contact between the grains when they elongate during diffusional creep, is one proposed mechanism [34]. This sliding generates cavities at ledges that are pulled apart at the grain boundary.

We propose a model based on the physics of diffusion and fluctuational theory, known as classical nucleation theory (CNT). CNT was formulated at the beginning of the twentieth-century by the works of Volmer and Weber [35], Becker and Döring [36], Frenkel [37], and Zeldovich [38]. It has been prominent and successful in modeling the nucleation of new phases, precipitates, and similar phenomena.

Balluffi [39] was the first to explain the nucleation of holes (cavities) by vacancy supersaturation. However, Raj and Ashby [40] were the first to consider the mechanical stress as the driving force for nucleation, a theory which was further developed by Hirth and Nix [41] and Riedel [28] and forms the basis for our nucleation model.

While CNT generally speaks of nuclei, which may form new phases, we specify these as clusters of vacancies which may form cavities.

The free energy change on formation of such a cluster in the bulk encompasses the pressure-volume work done by the external stress, *σ*, and the energy required by the newly formed surface between the cluster and the matrix, excluding the elastic energy [42]. Eq. (22) shows the relation between the free energy change, *ΔF*, and the cluster radius, *r*, with the specific surface energy, *γ*:

$$
\Delta F = -\frac{4}{3}\pi r^3 \,\sigma + 4\pi r^2 \,\chi\tag{22}
$$

Plotting this relation over the cluster radius, as shown in **Figure 7**, we see that the free energy reaches a maximum at a certain cluster size, *r\**, designated as the critical radius. This energy barrier needs to be overcome for a cluster to become a stable cavity which will then continue to grow as its total free energy decreases.

From Eq. (22), we derive the critical radius and critical (maximum) free energy as follows:

$$r^\* = \frac{2\gamma}{\sigma} \tag{23}$$

$$
\Delta F^\* = \frac{16}{3}\pi \frac{\gamma^3}{\sigma^2} \tag{24}
$$

Clusters below the critical size are naturally/always present in the microstructure due to thermal fluctuations [35, 43]. Their concentration is determined by the number

**Figure 7.** *Free energy change vs. radius for a spherical cluster of vacancies.*

*Microstructurally Based Modeling of Creep Deformation and Damage in Martensitic Steels DOI: http://dx.doi.org/10.5772/intechopen.104381*

of possible nucleation sites, *NS*, and an Arrhenius term comprised of the free energy change over the atomic thermal energy, *kT*. These are the first two terms in Eq. (25) for the nucleation rate, and they quantify the number of critical cavities available in quasi-static equilibrium. The next term, *β\** , is the vacancy attachment rate and *Z*, is the Zeldovich factor:

$$I = N\_s \exp\left(-\frac{\Delta F^\*}{kT}\right) \beta^\* \ Z \tag{25}$$

Some vacancies, which exist throughout the microstructure and are more prevalent at higher temperatures, may find themselves on the surface of a critical cluster and only one atomic jump away from joining it. The number of these vacancies jumping toward the critical cluster per unit time is described by the vacancy attachment rate, *β\** . In Eq. (26), this is shown to depend on the diffusion coefficient, *D*, the concentration of vacancies, *XV*, and the surface area of a critical cluster, *A\** , with *a* being the average interatomic distance:

$$
\boldsymbol{\beta}^\* = \frac{D}{\boldsymbol{a}^2} \mathbf{X}\_v \frac{\boldsymbol{A}^\*}{\boldsymbol{a}^2} \tag{26}
$$

The Zeldovich factor is explained by its namesake [38] and other literature [28, 44] to reduce the nucleation rate, since steady-state nucleation artificially removes supercritical clusters and because slightly supercritical clusters are still more likely to dissolve rather than grow. It is defined in Eq. (27) with *n\** signifying the number of vacancies in a critical cluster:

$$Z = \frac{1}{n^\*} \sqrt{\frac{\Delta F^\*}{3\pi kT}}\tag{27}$$

While the nucleation rate is only directly proportional to most of the physical parameters in Eq. (25), the Arrhenius term dominates. Small changes in the height of

**Figure 8.** *Cluster formed between two grains.*

the nucleation barrier lead to large variations in the equilibrium number of critical cavities available for nucleation and therefore the final nucleation rate. Smaller critical clusters are more likely to nucleate, such as in the case for clusters formed on grain boundaries as shown in **Figure 8**. The dihedral angle, *δ*, formed between the surface and the grain boundary as a result of the competing grain boundary surface energy, *γgb*, and the cluster surface energy, *γ*, reduce the volume and surface area [45] of a critical cluster, even though the curvature, *r \** , is unchanged. Removing the previous grain boundary area (dashed gray line in **Figure 8**) also reduces the critical free energy:

$$\delta = \arccos\left(\frac{\chi\_{\underline{g}b}}{2\,\chi}\right) \tag{28}$$

Nucleation is further boosted by the quicker diffusion of vacancies along the grain boundaries and the effect of converting the multiaxial stress state to an average stress on the grain boundary [46]. Also, real defects, such as dislocations, interacting with the grain boundary supply additional vacancies which can effectively increase the driving force by several gigapascals [47]. These effects predict cavity nucleation almost exclusively at grain boundaries and do not require extreme threshold stresses for nucleation, which is an enduring criticism of classical nucleation of cavities [48].

Finally, a theory based on generalized broken bonds (GNBBs) [49] is used to calculate the free surface energy from the energy of vacancy formation and a correction is applied [50] when dealing with nanosized critical clusters.

Diffusional cavity growth is less controversial and commonly assumed to follow the rate of radial growth in Eq. (29) by Hull and Rimmer [51]. Its resemblance to the Svoboda, Fischer, Fratzl, Kozeschnik (SFFK) model [20] used in precipitate growth simulations further strengthens its prestige:

$$\dot{r} = \frac{D}{kT} \frac{X\_v a^3}{r} \left(\sigma - \frac{2\gamma}{r}\right) \tag{29}$$

The sintering stress, *2γ/r*, that opposes the driving stress, *σ*, also explains the shrinking of subcritical clusters predicted by CNT.


**Table 5.** *Model parameters and physical constants.* *Microstructurally Based Modeling of Creep Deformation and Damage in Martensitic Steels DOI: http://dx.doi.org/10.5772/intechopen.104381*

## **7. Pore formation model: model implementation**

The equations for nucleation and growth are integrated into a Kampmann-Wagner framework [52] at a constant temperature and external stress state. At regularly spaced time intervals, a class of newly formed cavities with a population derived from Eq. (25) is formed, each with a radius slightly (20%) above the critical radius from Eq. (23).

**Figure 9.**

*Comparison between simulated (red) and measured (blue) histograms of cavities in Nickel-based alloy 625 after 5500 h at 700°C and 183 MPa.*

**Figure 10.**

*Comparison between simulated (red) and measured (blue) histograms of cavities in P23 steel after 9000 h at 600°C and 90 MPa.*

#### *Failure Analysis – Structural Health Monitoring of Structure and Infrastructure Components*


*Microstructurally Based Modeling of Creep Deformation and Damage in Martensitic Steels DOI: http://dx.doi.org/10.5772/intechopen.104381*


#### **Table 6.**

*List of symbols and variables used within the creep model.*

During the intervals, the respective radii of all classes grow according to Eq. (29). As the available nucleation sites are used, the nucleation rate diminishes. The simulation ends when all nucleation sites are consumed and there is no more uncavitated grain boundary area. The number of nucleation sites at grain boundaries is calculated [53] from average grain diameters, assuming all grains to be tetrakaidekahedral (**Table 5**).

## **8. Pore formation model: results and comparison with experiments**

**Figures 9** and **10** compare simulated results of nucleated cavities with experimental results obtained from secondary electron microscopy and density measurements. Case studies comparing our model to experimental investigations have been published [42, 58].

## **9. Conclusion**

We have introduced a complex physically based creep model and demonstrated its capabilities in the case of the martensitic steel P91. The model is capable of simulating the creep deformation as well as the microstructural evolution during creep. As soon as some final parameters have been set, based on a single-creep experiment, those parameters can be used for simulating the creep behavior over a wide range of stresses allowing for extrapolating the creep behavior. Current results suggest an extrapolation of the creep lifetime by a factor of at least 6 over a reference experiment.

Furthermore, we have introduced a physically based model for the formation of creep pores due to vacancy diffusion, which is also showing very good agreement with experiments.
