**3. Cavity nucleation**

#### **3.1. Thermodynamic considerations**

The mechanisms for cavitation nucleation have been a puzzle for a long time as explained in the introduction. However part of the explanation came from studies on copper. Pure copper can show extensive cavitation during creep [39], but the number of particles present is so low that they cannot explain the large number of cavities. Lim suggested that it was the substructure of the dislocations that could nucleate the cavities [40]. He also presented a model that can be used to demonstrate whether a nucleation mechanism is thermodynamically feasible or not. He assumed that pile ups of grain boundary dislocations generate the necessary high stresses for the nucleation. Since these high stresses are stationary as a result of the creep process, it avoids the problem of fast stress relaxation in many models. Lim's model is fairly complex and details in the model will not be given here. When a cavity is formed the free energy Δ*G* is changed in a number of ways that are represented by the terms in the following equation [19]

$$
\Delta G = -r^3 F\_{\text{v}} \sigma\_{\text{approx}} + r^2 F\_{\text{s}} \gamma\_{\text{s}} - r^2 F\_{\text{GB}} \gamma\_{\text{GB}} - \{\Delta G\_1 + \Delta G\_2 + \Delta G\_3\}.\tag{7}
$$

γs and γGB are the surface and grain boundary energies per unit area. *F*<sup>v</sup> = 2π/3 (2–3cos α + cos3 α), *F*<sup>s</sup> = 4π(1−cos α), *F*<sup>b</sup> = πsin<sup>2</sup> α and *F*<sup>v</sup> ′ = 1.5 *F*<sup>v</sup> , where α is half the tip angle of the cavity. The first term in Eq. (7) is the work done by the applied stress. The second and third terms represent the modification in the surface and grain boundary energies. The fourth term is the decrease in the strain energy. Δ*G*<sup>1</sup> is the change in the line energy of the grain boundary dislocations (GBD). Δ*G*<sup>2</sup> is the interaction energy between the remaining and the consumed GBD. The strain energy Δ*G*<sup>3</sup> is the reduction of the strain energy of GBDs outside the cavity. Full details can be found in [19, 40].

Lim's model has been applied to copper and austenitic stainless. As long as energy is gained when a cavity is formed, that is, Δ*G* in Eq. (7) is negative, cavitation is possible. From Eq. (7) Δ*G* is reduced when the applied stress σappl is raised, that is, cavitation becomes more likely which is natural. On the other hand when σappl is reduced cavitation is more difficult. There is minimum stress where cavitation is no longer possible because Δ*G* becomes positive. This minimum stress is plotted as a function of temperature for copper in **Figure 5**.

These minimum stresses are compared with design stresses during creep in copper. It is clear that the stresses required for nucleation are well below the stresses that typically appear in the material. This demonstrates that nucleation based on the substructure is a viable process.

From a technical point of view it is well established that the creep ductility of oxygen free pure copper Cu-OF can be very much lower than for phosphorus alloyed copper Cu-OFP. As a consequence the latter material should be used in creep exposed components [28, 41]. It is evident from **Figure 5** that much lower stresses are needed in Cu-OF than in Cu-OFP, which makes the cavitation in the former material much more abundant. This is believed to be the main reason for the low creep ductility of Cu-OF.

**Figure 5.** Minimum stress to form cavities at cell boundaries versus temperature for oxygen free pure Cu-OF and phosphorus alloyed copper Cu-OFP. For comparison the stress that gives creep rupture after one year (10,000 h) is included. After Ref. [19].

It has also been verified that the minimum nucleation stresses are below typical design stresses for the common stainless steels 304H (18Cr10Ni), 316 (17Cr12Ni2Mo), 321 (18Cr12NiTi) and 347 (18Cr12NiNb). For example this is illustrated for 347 (18Cr12NiNb) in **Figure 6**. The design stresses are 10,000 h rupture data. The minimum cavitation stress lies in the interval 35–50 MPa in the interval from 500°C to 750°C. The minimum cavitation stresses are again below the design stresses. The temperature dependence of Lim's model is probably not fully correct. In general it is thought that the amount of cavitation will increase with temperature. However the temperature dependence of the minimum cavitation stress is weaker than that of the design stress, which suggests the opposite behaviour.

#### **3.2. Strain dependence**

**3. Cavity nucleation**

26 Study of Grain Boundary Character

equation [19]

γs

cos3

**3.1. Thermodynamic considerations**

α), *F*<sup>s</sup> = 4π(1−cos α), *F*<sup>b</sup> = πsin<sup>2</sup>

the decrease in the strain energy. Δ*G*<sup>1</sup>

Full details can be found in [19, 40].

main reason for the low creep ductility of Cu-OF.

dislocations (GBD). Δ*G*<sup>2</sup>

viable process.

GBD. The strain energy Δ*G*<sup>3</sup>

The mechanisms for cavitation nucleation have been a puzzle for a long time as explained in the introduction. However part of the explanation came from studies on copper. Pure copper can show extensive cavitation during creep [39], but the number of particles present is so low that they cannot explain the large number of cavities. Lim suggested that it was the substructure of the dislocations that could nucleate the cavities [40]. He also presented a model that can be used to demonstrate whether a nucleation mechanism is thermodynamically feasible or not. He assumed that pile ups of grain boundary dislocations generate the necessary high stresses for the nucleation. Since these high stresses are stationary as a result of the creep process, it avoids the problem of fast stress relaxation in many models. Lim's model is fairly complex and details in the model will not be given here. When a cavity is formed the free energy Δ*G* is changed in a number of ways that are represented by the terms in the following

*ΔG* = −*r* <sup>3</sup> *F*<sup>v</sup> σappl + *r* <sup>2</sup> *F*<sup>s</sup> γ<sup>s</sup> − *r* <sup>2</sup> FGB γGB − (*Δ G*<sup>1</sup> + *Δ G*<sup>2</sup> + *Δ G*3). (7)

α and *F*<sup>v</sup>

minimum stress is plotted as a function of temperature for copper in **Figure 5**.

and γGB are the surface and grain boundary energies per unit area. *F*<sup>v</sup> = 2π/3 (2–3cos α +

, where α is half the tip angle of the cavity.

is the change in the line energy of the grain boundary

is the interaction energy between the remaining and the consumed

is the reduction of the strain energy of GBDs outside the cavity.

′ = 1.5 *F*<sup>v</sup>

The first term in Eq. (7) is the work done by the applied stress. The second and third terms represent the modification in the surface and grain boundary energies. The fourth term is

Lim's model has been applied to copper and austenitic stainless. As long as energy is gained when a cavity is formed, that is, Δ*G* in Eq. (7) is negative, cavitation is possible. From Eq. (7) Δ*G* is reduced when the applied stress σappl is raised, that is, cavitation becomes more likely which is natural. On the other hand when σappl is reduced cavitation is more difficult. There is minimum stress where cavitation is no longer possible because Δ*G* becomes positive. This

These minimum stresses are compared with design stresses during creep in copper. It is clear that the stresses required for nucleation are well below the stresses that typically appear in the material. This demonstrates that nucleation based on the substructure is a

From a technical point of view it is well established that the creep ductility of oxygen free pure copper Cu-OF can be very much lower than for phosphorus alloyed copper Cu-OFP. As a consequence the latter material should be used in creep exposed components [28, 41]. It is evident from **Figure 5** that much lower stresses are needed in Cu-OF than in Cu-OFP, which makes the cavitation in the former material much more abundant. This is believed to be the Experimentally it has been found many times that the number of cavities is proportional to the creep strain, cf. Eq. (2). To explain this strain dependence, Sandstrom and Wu introduced the *double ledge model* [43]. They considered a sliding grain boundary with dislocation substructures on both sides of the boundary that moved along with the grains. The substructures consist of subgrains that contain fairly few dislocations in their interior but with well-developed subgrain walls. The positions where the subgrain walls meet at the grain boundary are referred to as subgrain corners. Nucleation was assumed to take place when a subboundary on one side of the boundary hits a subgrain corner on the other side. The nucleation rate can be expressed as

$$\frac{dn}{dt} = \frac{u\_{\rm cas}}{d\_{\rm sub}} \frac{1}{d\_{\rm sub}^2} \,' \tag{8}$$

**Figure 6.** Minimum cavitation stress versus temperature for TP347H austenitic stainless steel. 10,000 h rupture data from ECCC [42] for TP347 are shown for comparison.

where *d*sub is the subgrain diameter. The last factor takes into account that one nucleus can be formed in each subgrain on the boundary. The subgrain size is directly related to the applied stress [19]

$$d\_{\rm sub} = K\_{\rm sub} \, \text{GB/} \sigma\_{\rm app}. \tag{9}$$

The constant *K*sub is about 20 for austenitic stainless steels and about 11 for copper. The same model can be applied to particles in the grain boundary that are known to contribute to the nucleation. In the model the subgrain corners are replaced by the particles in the grain boundaries with an interparticle distance of λ. Taking both subgrain corners and particles into account, the resulting expression for the nucleation rate is [44]

$$\frac{dn}{dt} = \frac{0.9 \text{ C}\_s}{d\_{sub}} (\frac{1}{d\_{sub}^2} + \frac{1}{\lambda^2}) \text{ } \dot{\varepsilon} = B\dot{\varepsilon}. \tag{10}$$

In Eq. (10), Eq. (1) has been used. The factor 0.9 in Eq. (10) takes into account the averaging of different orientations [44].

The model in Eq. (10) is compared with experimental data for austenitic stainless steels in **Figure 7**. For three of the experimental data sets TP347 at 550°C and 650°C and TP304 at 727°C, the model gives quite an acceptable description. For TP304XX at 750°C the deviation between model and experiment is larger.

**Figure 7.** Modelling and experimental number of cavities per unit grain boundary area as a function of creep strain [44]. Experimental data from Hong and Nam [45] for TP304 steel, Laha et al. [46] for three different types of austenitic stainless steels and Needham and Gladman [27] for TP347 steel.

#### **3.3. Particle size**

where *d*sub is the subgrain diameter. The last factor takes into account that one nucleus can be formed in each subgrain on the boundary. The subgrain size is directly related to the applied

**Figure 6.** Minimum cavitation stress versus temperature for TP347H austenitic stainless steel. 10,000 h rupture data from

*d*sub = *K*sub GB/*σ*appl. (9)

The constant *K*sub is about 20 for austenitic stainless steels and about 11 for copper. The same model can be applied to particles in the grain boundary that are known to contribute to the nucleation. In the model the subgrain corners are replaced by the particles in the grain boundaries with an interparticle distance of λ. Taking both subgrain corners and particles into

In Eq. (10), Eq. (1) has been used. The factor 0.9 in Eq. (10) takes into account the averaging of

The model in Eq. (10) is compared with experimental data for austenitic stainless steels in **Figure 7**. For three of the experimental data sets TP347 at 550°C and 650°C and TP304 at 727°C, the model gives quite an acceptable description. For TP304XX at 750°C the deviation

*<sup>λ</sup>*<sup>2</sup> ) *<sup>ε</sup>*˙ <sup>=</sup> *<sup>B</sup>ε*˙. (10)

account, the resulting expression for the nucleation rate is [44]

*dt* <sup>=</sup> 0.9 *<sup>C</sup>* \_\_\_\_\_*<sup>s</sup> d*sub ( \_\_\_1 *d*sub <sup>2</sup> <sup>+</sup> \_\_1

\_\_\_ *dn*

ECCC [42] for TP347 are shown for comparison.

28 Study of Grain Boundary Character

between model and experiment is larger.

different orientations [44].

stress [19]

It has been proposed that a critical particle radius exists for nucleation [45, 46]. The radius must exceed a minimum value in order for nucleation to take place. Harris developed a model that related the critical particle size to the GBS velocity [47, 48]. His basic assumption was that particles are not able to stop GBS if the diffusion is fast enough. According to Harris this critical particle radius also represented the minimum radius that could nucleate cavities. Harris gave the following relation between the GBS velocity *u* . GBS and the critical particle radius *r*<sup>c</sup>

$$\dot{\mu}\_{\rm cons} = \frac{\delta \, D\_{\rm cs}}{r\_e^{\prime} \ln \frac{\lambda}{2r\_e}} \left( \exp \frac{2}{k\_b} \frac{\gamma\_\* \, \Omega}{T r\_e} - 1 \right) , \tag{11}$$

**Figure 8.** Comparison of experimental and modelling GBS velocity as a function of particle radius [24]. Experimental data for 304, 321 and 347 austenitic stainless steels from Refs. [45, 46, 49, 50]. Modelling results from Eq. (11).

where λ is the interpaticle spacing, γ<sup>s</sup> the surface energy, δ the grain boundary width, *D*GB the grain boundary self-diffusion coefficient, Ω the atomic volume, *k*B Boltzmann's constant and *T* the absolute temperature. The application of Eq. (11) is illustrated in **Figure 8** for different types of austenitic stainless steels [24].

In **Figure 8** the particle parameters are taken from the experimental references. The minimum particles that nucleated cavities in the experiments are chosen for the critical particle radius. The experimental data clearly support Harris' model.

From the particle size distributions [24] the number of nucleated cavities can be estimated if the critical particle size is known. The computed number of nuclei is compared with the observed ones for austenitic stainless steels in **Figure 9.**
