To simulate the nucleation and growth of cavities, CNT will be used which is based on the work done by Volmer and Weber,[21] Becker and Döring,[22] Frenkel[23] and Zeldovich.[24] CNT has achieved great success over the past 80 years modeling crystallization,[25] phase transformations,[26] condensation[27] and precipitate nucleation in different aluminum alloys, superalloys and steels.[28]
The subsequent growth of these nanoscopic nuclei into micro and macroscopic cavities is simulated using a model proposed by Svoboda et al.,[29] which is used in the materials calculator software Matcalc dealing with precipitates and phases.[30]
Classical Nucleation Theory
Classical Nucleation Theory uses a thermodynamic approach to model nucleation kinetics. Considering the change in free energy as the driving force for changes in the microstructure, the nucleation rate is dependent on the free energy of the newly nucleated phase. The fundamental equation is given below:
$$I={N}_{\text{s}}{\beta }^{\ast}Z{\text{exp}}\left(-\frac{\Delta G\; \text{or}\; \Delta F }{RT}\right)$$
(2)
This provides us with an estimate for the number of nucleation events per unit volume per unit time I, whereby Ns is the number of sites per unit volume where nucleation could occur, β* is the attachment frequency of particles of the new phase to the growing nucleus, Z is the so-called Zeldovich factor, which accounts for the fact that not all nuclei which grow to a stable size will also grow further and the exponential term which gives us the probability of finding a nucleus with the needed Gibbs free energy ΔG for nucleation of new precipitates in the microstructure as it is described in Reference 28 or the Helmholtz free energy ΔF for nucleation of cavities as described in References 31 and 32. The exponential term has a decisive role on the nucleation rate. As a nucleus forms, an interface is created between it and the surrounding bulk material, the creation of which increases the total free energy of the system. Nucleation of a new phase or cavity is favorable for the system if it has a lower free energy than the material in which it nucleates. Its effect on the total free energy is directly proportional to the volume of the new phase (in this case, cavity). As an example, the free energy change vs the radius of a cavity in hydrostatic tension under 100 MPa and a surface energy of 1.6 J m−1 is demonstrated in Figure 1.
The free energy change reaches its maximum, ΔF* at the critical radius r* which is given by:
$${r}^{\ast}=\frac{2{\gamma }_{\text{s}}}{\sigma }$$
(3)
where γs and σ represent the surface energy and applied stress, respectively.
Nucleation of Cavities
Early applications of CNT to the nucleation of cavities in metals in creep conditions were done by Raj and Ashby.[13] Other authors have developed this model for the nucleation of cavities over the years.[12,32] This model takes the mechanical stress σ, which has the unit of force per area or energy per volume and uses it as the driving force of free energy change due to the nucleation of cavities.[33,34] The internal stress (back-stress) attributed to the hindrance of dislocation motion by precipitates does not reduce the driving force in this model.
The interfacial energy is taken to be the free surface energy with some modifications which are described in Reference 35. The number of possible nucleation sites is equal to the number of atomic sites, in the bulk. As derived in Reference 28 the Zeldovich factor is
$$Z= \sqrt{\frac{{\Omega }^{2}{\sigma }^{4}}{64\pi RT{\gamma }_{m}^{3}}}=\sqrt{\frac{{\Delta F}^{\ast}}{3\pi RT}}\frac{1}{{n}^{\ast}}$$
(4)
Cavities that have attained the critical size are still more likely to dissolve than they are to continue growing. This was considered in the Zeldovich factor by its namesake.24 The graphical derivation of the Zeldovich factor is shown in Figure 2, whereby the x axis displays the number of vacancies in the cavity. In this figure, a region is defined by the intersections of a certain amount of energy below the critical free energy (ΔF*–RT) with the curve of the total free energy, the rightmost being defined as n*’. The width of this region is the reciprocal of the Zeldovich factor. Cavities must pass through this region during nucleation while they are still susceptible to dissolution. Any supercritical cavities larger than n*’ are more than one quantum of energy (RT) away from dissolving and will continue to grow as they reduce the overall free energy.
Another important term introduced in Eq. [5] is the atomic attachment rate β*, which is defined as,
$${\beta }^{\ast}=\frac{\sqrt[3]{4\pi {\left(3\Omega {n}^{\ast}\right)}^{2}}}{\sqrt[3]{{\Omega }^{4}}}{n}_{\text{v}}{D}_{\text{v}}=\frac{{A}^{\ast}}{\sqrt[3]{{\Omega }^{4}}}{n}_{\text{v}}{D}_{\text{v}}$$
(5)
whereby A* is the surface area of a critical cavity, nv is the concentration of vacancies in the matrix, Dv is the diffusion coefficient of those vacancies and Ω is the atomic volume. This describes the frequency with which new vacancies attach to the cavity.
Heterogeneous Nucleation
All the formulas above are derived for spherical cavities forming in the matrix of the material, which is known as homogeneous nucleation. However, as typical for crystalline metals, most of the interesting phenomena occur at the grain boundaries. This leads to what is called heterogeneous nucleation. The shapes of the cavities formed can be seen in Figure 3, with a solid line showing the grain boundary and a dashed line showing the grain boundary dissolved by the cavity.
The contact angle δ is given below in Eq. [6] by balancing the forces of the free surface energy, γm, and the grain boundary energy, γgb.
$$\delta =a{\text{cos}}\left(\frac{{\gamma }_{\text{gb}}}{2{\gamma }_{\text{m}}}\right)$$
(6)
This so-called “wetting” effect reduces the volume and surface area of a cavity at the grain boundary, for a given fixed critical radius, and gives these cavities their characteristic lens shape. The cavities’ size is also further reduced at triple boundary lines, where three grains meet and at quadruple grain boundary points, where four grains meet. The equations for volume and surface area were derived in previous works.[36,37] As a result of the reduced volume the required free energy for nucleation of cavities lessens and so they form more readily. The three factors which favor heterogeneous over homogeneous nucleation are the accelerated diffusion along grain boundaries, the lower required free energy for nucleation and the abundant supply of vacancies. The lower number of nucleation sites, especially in the case of triple grain boundary lines and quadruple grain boundary points, compared to the bulk, reduce the nucleation at these sites.
The same “wetting” of cavities occurs on precipitates and inclusions, with the angle θ, defined by the three interfacial energies between the cavity, the matrix, and the precipitate (Figure 4). This relation is shown in Eq. [7], where γm is the free surface energy of the matrix, γp is the free surface energy of the precipitate and γp_m is the interfacial energy between the matrix and the precipitate. The dissolution of the interface between the matrix and precipitate also lessens the critical free energy.
$$\theta =a{\text{cos}}\left(\frac{{\gamma }_{\text{p\_m}}-{\gamma }_{\text{p}}}{{\gamma }_{\text{m}}}\right)$$
(7)
Nucleation Rate for Grain Boundary Cavitation
Inserting Eqs. [3] through [6] and the specific terms for the number of nucleation sites at the grain boundaries, NGB, the grain boundary diffusion coefficient, DGB, and the Arrhenius relationship for the equilibrium concentration of vacancies into Eq. [2], the nucleation rate at grain boundary surfaces becomes
$$I={N}_{\text{GB}} \frac{{A}^{\ast}}{\sqrt[3]{{\Omega }^{4}}}{{\text{exp}}\left(\frac{-{Q}_{v}}{kT}\right)D}_{\text{GB}} \sqrt{\frac{{\Delta F}^{\ast}}{3\pi RT}}\frac{1}{{n}^{\ast}}\text{ exp}\left(-\frac{{\Delta F}^{\ast} }{RT}\right)$$
(8)
With the properties of the critical cavity; V*, n*, A* and ΔF* representing the volume of the critical cavity, the number of vacancies in the critical cavity, the surface area of the critical cavity and the free energy needed to nucleate the critical cavity, respectively. The critical radius is calculated with Eq. [3].
$${V}^{\ast}= \frac{4}{3}\pi {r}^{{\ast}3} \left(\frac{\left(2+{\text{cos}}(\delta )\right)\times {(1-{\text{cos}}(\delta ))}^{2}}{2}\right)$$
(9)
$${n}^{\ast}= \frac{{V}^{\ast}}{\Omega }$$
(10)
$${A}^{\ast}= 4 \pi {r}^{{\ast}2} \left( 1-{\text{cos}}\left(\delta \right) \right)$$
(11)
$${\Delta F}^{\ast}= -\sigma {V}^{\ast}+ {\gamma }_{\text{m}}{A}^{\ast}+ {\gamma }_{\text{gb}}\pi {r}^{{\ast}2}\left({\text{ sin}(\delta )}^{2} \right)$$
(12)
Supercritical Cavity Growth
In the previous section, the nucleation of cavities in the microstructure based on CNT was discussed. This model gives us steady state nucleation rates and number densities of cavities in the bulk, at the grain boundaries and on the surface of hard particles. However, the existing CNT model cannot predict the growth rate of cavities which have reached a critical size. We have modified the SFFK (Svoboda, Fischer, Fratzl, Kozeschnik) model[29] for use in our simulations of cavity growth. The original equation for growth is,
$$\dot{r}= \frac{\sigma -\left(2\gamma /r\right)}{r RT}{D}_{0}{u}_{0}\Omega$$
(13)
This equation arises from Onsager’s principle of maximum entropy production[38] which states that as a system reduces its total Gibbs’ free energy on its way to thermodynamic equilibrium, the entropy increases at its maximum rate. Equation [13] is adapted as follows to generate Eq. [14] for the growth rate of the radius \(\dot{r}.\) σ now represents the hydrostatic stress instead of the chemical driving force (analogous to our modifications to CNT), the diffusion coefficient D0 is replaced with that for vacancy diffusion Dv, the mean site fraction u0 is replaced by the vacancy concentration NV and the sintering stress term (2γ/r) now incorporates the free surface energy γm. Ω is redefined from the molar volume to the atomic volume and correspondingly the Boltzmann constant kb is used in place of the gas constant R.
$$\dot{r}= \frac{\sigma -\left(2{\gamma }_{\text{m}}/r\right)}{{r k}_{\text{b}}T}{D}_{\text{V}}{N}_{\text{V}}\Omega =\frac{{D}_{\text{V}}{N}_{\text{V}}\Omega }{{k}_{\text{b}}T} \left( {\frac{\sigma }{r} - \frac{{2\gamma _{\rm m} }}{{r^{2} }}} \right) $$
(14)
The quotient of γm and r describes the compression that the surface energy exerts on a curved surface which resists growth. Assuming a constant stress σ, a constant temperature T, a constant diffusion coefficient Dv, and a constant supply of excess vacancies Nv, the growth rate is completely deterministic and identical for all cavities. The more complex stress state at a strongly necked portion of a sample cannot currently be represented by this model.
Kampmann–Wagner Framework
To keep track of the nucleation and growth of all cavities in the microstructure we use a modified Kampmann–Wagner framework.[39] During each timestep of the simulation, a new class of cavities is created with a population equal to the number of cavities nucleated in the last time interval. The starting radius of all cavities in this class is taken to be slightly overcritical (20 pct over r*). For all subsequent timesteps, this radius evolves according to Eq. [14].
This framework provides a distribution of cavity sizes which is more useful and informative and can be compared to the histograms in Figure 5.
Simplifications in CNT Model
Our implementation of CNT uses the capillarity approximation which means that nanoscale clusters are considered to have the same properties, such as free surface energy, as the bulk material at the macro scale. We also consider cavities to be spherical or lenticular. Future investigations into the lowest energy equilibrium shapes of cavities (Wulff shapes or similar) may decrease the required free energy to nucleate cavities.
Another valid point of criticism in CNT model is the application of global conditions, such as the stress on the entire sample, to the process of nucleation of nanosized particles (in our case, cavities). We have applied a correction to the stress, proposed by Nix[40] to simplify the multiaxial stress acting on the grains to a single value that is used in our calculations.
Typically, literature values for the free surface energy are specified at room temperature, with a general decrease of surface energy at higher temperatures reported in Reference 41. Benson and Shuttleworth[42] postulate that for the extreme case of a “droplet” consisting of a close-packed cluster of thirteen atoms, surface energies should only be reduced by about 15 pct. We are following the work of Sonderegger and Kozeschnik[43] using generalized nearest broken bond theory to calculate a theoretical surface energy according to Eq. [15].
$$\gamma =\frac{{n}_{\text{s}}{ z}_{\text{S}}}{{N}_{\text{A}} {z}_{\text{L}}}\Updelta {E}_{\text{sol}}$$
(15)
This calculation uses ns as the number of atoms per unit of free surface area, the factor zS as the number of bonds broken at the surface, zL as the total number of bonds and the energy of solution ΔEsol (=Qv in our case). The calculated value of the free surface energy γm = 1.432 J m−1 is lower than other values found in literature.[41,44]
All simulations were run at constant temperature and stress for the duration of the creep time. The calculated nucleation rate is constant. This assumes that the number of possible nucleation sites does not decrease as new cavities are nucleated, which is reasonable at such low nucleation rates.
The mechanism of Ostwald ripening is not expected in this model because of the constant positive driving force, which predicts only the further growth of the stable cavities. Coalescence of cavities is also not considered since the cavity phase fraction is low for even the longest crept samples. This assumption is validated by the lack of coalescing cavities in the SEM images. This would only be needed to accurately model tertiary creep at the necked parts of samples which experience the most deformation.