A novel model of third phase inclusions on two phase boundaries

Consider a 2D surface embedded in 3D space, labelled X. We introduce a second surface, S, a non-negative distance away from X. Figure 1 shows a cross-section of such a scenario, where surfaces S and X represent phase boundaries between phases α, β, and γ. In this schematic, where S is apart from X, the surface S demarks the boundary between α and β and X demarks the boundary between β and γ. Where S coincides with X, the surfaces represent the same boundary between the α and γ phases.

The volume enclosed between S and X represents an included phase on X, as shown in Fig. 2. The intent is to calculate the evolution of the parameterised surface S on the computational domain X, thereby capturing the 3D interface between the three phase boundaries on a single surface. The surface S can be represented as a parametrisation on the surface X. This is accomplished by transforming S onto X following a projection vector, \( \overrightarrow{r} \), as depicted in Fig. 3. This projection vector is required to be smoothly varying everywhere on X, including any sharp edges and vertices and should be close to the local normal as discussed below.

It should be noted that h may be positive or negative, representing surfaces above or below the surface of X relative to \( \overrightarrow{f} \). In principle, it is possible to represent separate surfaces above and below surface X simultaneously leading to two values of h. However, in this work, a single surface S is considered that can be mirrored assuming symmetry above and below X.

In order to perform the integral over the whole domain while distinguishing between “S ≠ X” and “S = X”, a phase function, p(h), is defined that varies between 0 and 1, corresponding to where S is in contact with X, as determined by the local value of h. This scheme is illustrated in Fig. 4 which shows the range of h in which p varies from 1 to 0, and the resulting diffuse interface of width d. This interface is treated as being spatially diffuse, similar to standard phase-field models that demonstrate robustness and versatility of handling complex interface morphology (Cahn and Hilliard 1958). Due to the spatial variation of h as the dependant variable, this implies p(h) be diffuse in h, between h = 0, representing the coincidence of S and X, and h > h′, representing the separation of S and X. The value of h′ is user-defined transition size and linked to the choice of the form of p(h). The region 0 > h >h′ therefore denotes the diffuse triple junction interface between the three phases.

It should be noted that the model of the included phase is sharp across the surface S which has no associated thickness. The diffuse interface occurs where S meets X, over a user-defined transition size in h and therefore an interval in space. This region also defines the α − β − γ triple junction which is therefore partially diffuse. Since it is here that the contact angle is defined, the user-defined transition size impacts the contact angle observed in the model. In the “Stationary point” section, we demonstrate that in the limit of zero transition size, the sharp interface contact angle is recovered. In the “Contact angle” section, we further demonstrate that the results of the numerical model converge towards the sharp interface when the transition size is decreased.

Triple junction energy can be added to the model, which only occurs within the diffuse interface. A function g(p) is introduced, with the requirement that g(p = 0) = g(p = 1) = 0 and \( \frac{\partial g}{\partial p}\left( p=0\right)=\frac{\partial g}{\partial p}\left( p=1\right)=0 \). Once more, the form of this function is flexible, but a standard double well potential, g ∝ p ²(1 − p)², may be used.

This can be used to evaluate the chemical potential of the included phase by Eq. 11. Note that the term \( \frac{\overrightarrow{s_1}\times \overrightarrow{s_2}}{\left|\overrightarrow{s_1}\times \overrightarrow{s_2}\right|}={\widehat{n}}_S, \) the unit normal to the interface.

The gradient in the chemical potential is the driving force for the mass flux, J. The mass flux may apply to a layer along the interface S, in the case of surface diffusion or as an approximation to bulk diffusion, or the enclosed volume, such as vapour phase transport. In general, surface, bulk, and volume diffusion occur, but one of them usually dominates for any specific situation (Welland 2012).

The model therefore requires the solution of the partial differential equation for Eq. 24 with the chemical potential from Eq. 11, the variation of volume with h from Eq. 18, and the variation in total energy with h from Eq. 20. When combined, this is a fourth order non-linear partial differential equation for h, defined on the 2D surface X. Alternately, Eq. 11 may be solved simultaneously with Eq. 24 for μ and h as two coupled second order equations on X. This allows it to be solved using the standard C ₀ continuous Lagrange elements available in most finite-element method codes. Note that although the equations are in terms of the distances between surfaces h, the conserved quantity is still the concentration of the β phase c. This is also equivalent to conservation of the volume of the β phase since density has been assumed constant in this version of the model.

The suitability criteria for the vector field \( \overrightarrow{r} \) and the selection of the focal point \( \overrightarrow{f} \) can now specified. Considering Fig. 3, it is clear that in order for S to be uniquely defined, there must be a one-to-one mapping from X to S. Specifically, \( {\widehat{n}}_S\cdot \widehat{r}\ne 0 \), for all points on X. From this requirement, the contact angle of the included phase with respect to surface X must be less than the minimum value of \( { \sin}^{-1}\left({\widehat{n}}_X\cdot \widehat{r}\right) \) over the domain. This is the reason that contact angles equal to or greater than 90° are not representable in this work in the ideal case where \( \widehat{r}={\widehat{n}}_X \). For practical geometries where \( {\widehat{n}}_X\cdot \widehat{r}<1 \), such as along edges between facets, this limits the contact angle to less than half the internal angle. By careful definition of \( \widehat{r} \), the limitations imposed by this requirement can be minimised. Note that this does not preclude concave surfaces nor does it preclude complex 2D manifolds.

In order to demonstrate and analyse the performance of the model, it is compared to an analytic sharp interface solution representing a 1D azimuthally symmetric geometry in polar coordinates of a precipitate on an interface reflected on surface X. In this case, the segment in Fig. 1 where S is apart from X, corresponding to the β − γ interface is the computational bisection of the precipitate and does not constitute a real interface and therefore σ _βγ  = 0.

The model was implemented in COMSOL Multiphysics 5.2a using user-defined weak form equations and solved numerical with first order Lagrange elements for both the chemical potential and the displacement from the surface.

Contact angle

The contact angle of an inclusion on a flat surface according to the sharp interface theory is given by Young’s equation,

$$ \cos \left(2\theta \right)=\frac{\sigma_{\alpha \gamma}}{2{\sigma}_{\alpha \beta}} $$

(50)

A parameter sweep was conducted in which bubbles with an initial projected radius of 5 μm were initialized on a 10 μm domain and permitted to evolve to their equilibrium shapes for surface energy ratios of σ _αγ /σ _αβ  = 0.174 to 1.9696, corresponding to theoretical angles of 10° to 85°, and transition sizes of a = 0.05 to 0.3 μm, respectively. As discussed in the “Stationary point” section, one would expect to recover the sharp interface limit in the limit of zero transition size.

The 1D model allows an extremely fine finite-element mesh size of 1 nm to be created to eliminate potential effects of discretization error. A mesh sensitivity study was conducted separately in the “Mesh sensitivity study” section to investigate this issue. The simulated equilibrium geometries are shown in Fig. 6 for the 0.3 and 0.05 μm transition sizes. As expected, the impact of the transition size is visible at the edge of the bubble (i.e. r ≈ 0.5 μm) in which the 0.3 μm transition size produces a much smoother corner compared to the 0.05 μm transition size. As a result of the smoother transition, the interface width is visibly larger for the 0.3 μm case. According to Eq. 49, the approximate interface widths for 85° should be d _0.3 μm = 0.0529 μm and d _0.05 μm = 0.0088 μm, which is roughly consistent with the observed smoothed region.

The simulated contact angle, ϕ, was calculated using two different techniques for comparison purposes. The first technique performs this calculation directly by finding the gradient of the bubble shape and converting it to an angle relative to the horizontal via inverse tangent function. The post-processing of this method is simple since the minimum gradient in the model occurs at the edge of the bubble and can therefore be found as the minimum on the finite-element domain, ϕ = atan(min(h _tau)).

The angle calculated by this method is presented in Fig. 7. The results indicate a clear trend converging towards the theoretical sharp interface angle θ as the transition size decreases for the whole range of angles, as expected. The difference between the angle observed in the simulations and the sharp interface prediction generally increases with the angle for large transition sizes; however, this trend diminishes as the transition size is decreased such that it becomes negligible for the 0.1 and 0.05 μm transition sizes.

As seen in Fig. 6, the finite interface thickness smooths the corner of the bubble, thereby reducing the angle calculated by this method. Although this provides a good measure of the sharpness of the interface, it does not indicate if the curvature is behaving as expected in the bulk material. In order to avoid this difficulty, the second method calculates the expected contact angle based on the values away from the transition region assuming the geometry in Fig. 8. Due to symmetry, the centre of the sphere defining the boundary of the bubble is located along the axis of symmetry (i.e. r = 0) displaced below the origin a distance Z ₀. The contact angle can be found done using the bubble height at the origin, h ₀, and the height at any other location on the sphere, h(r). Using this methodology, the intersection angle of this sphere with the h = 0 axis can be calculated according to

$$ {Z}_0=\frac{r^2+{\left( h(r)\right)}^2+{\left({h}_0\right)}^2}{2\left( h(r)-{h}_0\right)}, $$

(51)

$$ \theta =\mathrm{acos}\left(\frac{Z_0}{h_0+{Z}_0}\right). $$

(52)

Since this is valid for all points on the β phase in the model, the average value is utilised to reduce numerical noise and obtain a single value for comparison. Note that this method is complimentary to the gradient-based method, since it is not calculating the contact angle actually observed in the model, but rather, it calculates the equivalent contact angle for a sphere of the same radius of curvature and location.

The contact angles calculated via this method is shown in Fig. 9 (left) for the same set of simulation results from Fig. 6. This second method shows better agreement with the theoretical values since it is less sensitive to smoothing of the edges in the transition zone. The error relative to the sharp interface limit is shown in Fig. 9 (right). These results also show a clear trend of converging towards the sharp interface limit as the transition size is decreased, which is in agreement with Eq. 48.

The simulated contact angle underpredicts the theoretical angle for all of the simulations in the sweep as a function of the transition size as expected by Eq. 47. The magnitude of the error observed is relatively small, roughly one third of the magnitude of the error calculated at the edge of the bubble, and can be managed via transition size and triple junction energy.

Mesh sensitivity study

It is necessary to develop an understanding of the meshing requirements in order to make efficient use of the model. To this end, a mesh sensitivity study was conducted for a fixed transition size of a = 0.1 μm and mesh density ranging from 0.1 to 100 mesh elements per transition size (i.e. a unitless dimension) for contact angles of 10° to 85°. Following the same procedure from the “Contact angle” section, the effective contact angle from the simulation was calculated using both the minimum gradient at the αβγ-triple junction method and the radius of curvature from the bulk αβ-interface method. These results are presented in Figs. 10 and 11, respectively. For all angles, it is observed that significant mesh dependence (i.e. non-converged values) results from excessively coarse meshes. For a sufficiently small mesh size, mesh sensitivity becomes small and the results converge a value close to the sharp interface limit.

Equation 49 has been superimposed on the simulation results assuming that the interface width (d) is equal to the mesh size (equivalent to having one mesh element per interface width). As expected, the threshold mesh size required for a converged solution appears to be roughly proportional to this prediction, plus some additional mesh density for the larger contact angles to compensate for the numerical difficulty associated with large gradients.

Similar to the results from the transition size study, the mesh sensitivity calculated from the radius of curvature method demonstrates comparatively smaller error measurements and smoother convergence. Once again, more mesh sensitivity is observed at higher contact angles.

Overall, the results suggest that efficient meshes should contain approximately 1–10 linear mesh elements per interface width, with higher contact angles requiring denser meshes. These results are consistent with typical mesh densities used in phase-field or other simulations with varying topologies (Welland et al. 2014).

Comparison against the Gibbs-Thompson effect

In order to demonstrate that the model captures mesoscale effects such as the Gibbs-Thompson effect correctly, simulations were performed varying the initial bubble radius over several orders of magnitude and determining the equilibrium bubble radius and corresponding chemical potential. Simulations were run using interfacial energies of σ _αβ /σ _αγ  = cos(30°)/2 and σ _βγ  = 0 which correspond to a contact angle θ = 30° on a flat surface according to Eq. 50. The classical Gibbs-Thompson theory predicts (Lupis 1983)

$$ {\mu}_r-{\mu}_{\infty }=\frac{2{\sigma}_{\alpha \beta}}{r_{eq}}, $$

(53)

where μ _r and μ _∞ are the chemical potentials for a bubble of radius r and a flat interface, respectively. Figure 12 shows the calculated μ _r as a function of the equilibrium bubble radius, along with Eq. 53. As the spatial extent of the simulation is increased by several orders of magnitude, the transition size may be increased accordingly to reduce computational cost; thus, several overlapping datasets are depicted.

It is noted that as r → ∞, μ _r  → 0, implying that the flat interface chemical potential μ _∞ = 0. It should be emphasised here that this does not preclude a standard reference potential but, rather, that this shifts the results uniformly.

The agreement between the simulated μ _r and the corresponding value computed with the analytical solution (i.e. Eq. 53) with μ _∞ = 0 is excellent, implying that the effect of precipitates with high surface curvature modifying the solubility of c in the surrounding material is correctly captured. This effect leads to various coarsening phenomena, along with bubble collapse if energetically unfavourable (Welland et al. 2015b).

As a demonstration of the applicability of the model in 3D, a simulation of the evolution of intergranular porosity in UO₂ nuclear fuel was performed. The initial porosity is controlled as part of the manufacturing processes in which UO₂ powdered compacts are sintered at high temperatures to form solid pellets. The sintering process also continues during fuel irradiation due to a combination of high operational temperatures, radiation fields, and hydrostatic pressure. During operation in-reactor, the grain boundaries accumulate insoluble fission products—such as xenon and krypton—which pressurise and grow intergranular porosity. The behaviour of these intergranular bubbles is important to fuel performance as they contribute to design limiting phenomena including fuel swelling, degradation of thermal conductivity, and potential releases of fission product inventories. The bubbles grow, coalesce, and percolate along grain faces and grain edges, forming a complex network of tunnels, as shown in Fig. 13 (Ahmed et al. 2016; Rokkam et al. 2009; White and Tucker 1983; Turnbull and Cornell 1971; Jackson and Catlow 1985; White 2001; Pastore 2012). Once a continuous tunnel reaches a crack or the outer surface of a fuel pellet, all the gas in the interconnected tunnel may be vented and the tunnels may collapse.

The intergranular porosity is modelled as phase confined to the grain boundaries of a polycrystalline material. The grain geometry is represented as a truncated octahedron, which is a tetrakaidecahedron with eight regular hexagonal faces and six square faces, that can be tiled infinitely to represent a polycrystalline lattice (White and Tucker 1983). This geometry was constructed from 2D planer surfaces arranged in 3D space to produce the truncated octahedron shape with the focal point, f, at the 3D centre. A grain approximately 15 μm across was meshed with a maximum element size of 0.69 μm, which from Eq. 49 corresponds to five mesh elements per interface width. A total of 31,974 quadrilateral elements in a mapped/structured grid on square faces and 268,546 triangular elements were used on the hexagonal faces. The resulting model contains 332,498 degrees of freedom (DOF). The mesh nodes along the edges are shared between the adjoining faces which enforces continuity of the h field and therefore surface S between the faces.

For comparison, meshing this volume with tetrahedral elements of uniform size with the same maximum size would require approximately 36 million 3D elements and approximately 120 times the DOF. Additionally, the computational expense of the 2D model is lower compared to the 3D model since the 2D finite elements also produce greater matrix sparsity associated with the reduction in dimensionality.

Results of a sample calculation on this grain are provided in Fig. 14, which was initialized with a random initial h field uniformly between 22.6 and 223 nm, equivalent to 7.7% initial intergranular porosity (Additional file 1). These results are based off surface energies corresponding to a contact of 30° with respect to the computational domain X and a transition size τ of 0.1 μm. The porosity was assumed to be reflected above and below each face, which produced an open bubble section on the grain edges and at the vertices, as seen in Fig. 14.

The timescale in this simulation is arbitrary and depends on the material properties such as orientation-dependent interfacial energies and the surface mobility.

The initial randomly fluctuating porosity rapidly smooths to reduce the surface area and areas of high curvature. The smoothed surfaces then begin to decompose into bubbles with the set contact angle (beading). The phase decomposition begins at the vertices and edges and propagates inwards since these are the lowest energy locations. The bubbles then coarsen and coalesce with the other bubbles nearby.

It is interesting to note that the porosity slowly moves towards the grain boundary edges and vertices, which is energetically favourable due to the increase in interfaces. Edge and vertex adherence is therefore implicitly captured within the model. This phenomenon is observed experimentally in post-irradiation examination of irradiated fuel samples, where rounded prismatic tunnels form along the grain edges (White and Tucker 1983). With sufficient grain edge porosity, these tunnels can form an interconnected network connecting to macroscopic cracks venting the fission gas accumulated on the grain boundary faces to the free volume.

This simulation result demonstrates preliminarily the applicability of this model to the problem of grain boundary porosity, including prediction of key features, such as bubble coalescence and edge and vertex adherence, in a computationally efficient manner. The use of this work in analysing intergranular porosity in nuclear fuel requires additional physics such as orientation-dependent grain boundary energies and source terms for fission gas. The kinetics of the model could be determined by calibration against experimental data or by including information from lower length-scale calculations.