_{1}

^{*}

Phase-field modeling for three-dimensional foam structures is presented. The foam structure, which is generally applicable for porous material design, is geometrically approximated with a space-filling structure, and hence, the analysis of the space-filling structure was performed using the phase field model. An additional term was introduced to the conventional multi-phase field model to satisfy the volume constraint condition. Then, the equations were numerically solved using the finite difference method, and simulations were carried out for several nuclei settings. First, the nuclei were set on complete lattice points for a bcc or fcc arrangement, with a truncated hexagonal structure, which is known as a Kelvin cell, or a rhombic dodecahedron being obtained, respectively. Then, an irregularity was introduced in the initial nuclei arrangement. The results revealed that the truncated hexagonal structure was stable against a slight irregularity, whereas the rhombic polyhedral was destroyed by the instability. Finally, the nuclei were placed randomly, and the relaxation process of a certain cell was traced with the result that every cell leads to a convex polyhedron shape.

Porous materials are some of the most prospective advanced materials and have begun to be widely used in various engineering fields, utilizing their advantageous properties such as the light weight, heat adiabaticity, and sound and vibration insulation properties. In particular, metal foam materials are expected to make a great contribution to reducing the weight of parts and conserve energy to run machines [

Porous materials are divided into two types: open and closed cells. The latter, which is focused in this study, consists of many small cells separated by a wall made of the base material, and the cells are usually polyhedral with rounded corners. These structures are approximated by foam structures, typical examples of which are found in soap and beer, even though the wall of the foam is usually liquid. Neglecting the fluidity of the wall, the geometrical characteristics have many similarities, and hence, the analysis of such foam structures is expected to make a great contribution to the design of porous materials.

The analysis of foam structures dates back to the historical works of Plateau in the 1870s, and his insight on the cell boundary remains used with his name, the Plateau boundary. Lord Kelvin also leaves his name in the Kelvin cell, which is known as a preferable candidate of the foam model and was long considered the best solution for the so-called Kelvin problem, “which shape of cells have the least surface area when the space is divided into cells having the same volume?,” before Weaire and Phelan found a better solution in 1994 [

In our previous study [

where,

and

Here, φ_{i} (i = 1 − N) represents the multi-phase field variable, where N is the number of phases considered; m_{ij}, a_{ij}, w_{ij}, and Δf_{ij} are the parameters depending on the combination of i and j; and n is the number of phases for which φ_{i} is non-zero. The last term _{i} is the volume of the i-th cell and

The phase field Equation (1) is numerically solved using the finite difference method (FDM). A cubic domain is divided into 100 × 100 × 100 isotropic lattices, and periodic boundary conditions are imposed in all directions. The differential terms in the right-hand side are discretized on the FDM grids by central difference scheme, and the time derivative is explicitly solved. Independent phase field variables φ_{i} (i = 1 − N) are assigned to every cell such that φ_{i} = 1 represents the inside region of the i-th cell. This value shifts to 0 in a thin domain surrounding the cell; two different phase-field variables have non-zero values in the cell wall region, and three or more variables have a certain value at edge domains. Note that in any case, the summation of the phase field variables is always maintained to be 1

ture is formed by growing the cells from the nuclei, which is disposed as the initial condition. In this study, the nuclei are set on the regular body-centered cubic (bcc) or face-centered cubic (fcc) lattice points in Section 4.1 and Section 4.2, and random assignment is performed in Section 4.3. The computational parameters Δx and Δt are also listed in

_{i}^{2}; the cell face and edge are represented mostly by green and blue, respectively.

m_{ij} | a_{ij} | w_{ij} | Δf_{ij} | Δx | Δt | |||
---|---|---|---|---|---|---|---|---|

for i ≠ j | 1.0 | 1.0 | 4.93 | for i or j = 0 | 4.31 | 0 | 0.15 | 0.0045 |

for i = j | 0 | 0 | 0 | for others | 0 | 0.04 |

Because all the nuclei are set on a completely regular interval, the final cell distribution is also completely regular; these structures actually correspond to the Voronoi tessellations of bcc or fcc lattices. For the bcc arrangement, as shown in

In the previous section, the nuclei were set on the perfect lattice points of a bcc or fcc arrangement, resulting in a completely regular structure. In this section, the initial nuclei are displaced from the perfect lattice points to demonstrate the stability of the structure. The deviation of each nucleus is determined using a random number within a sphere of radius Δr_{d}.

_{max} and G_{min} for Δr_{d} = 0.12 and 0.25 for the bcc arrangement, where the cell size is measured by the number of grids in each cell. In the early stage, every cell grows freely without any interruption, and all the cells have an identical volume. Some of the cells then collide with each other. In this duration, the growth rate is affected; some cells are accelerated, and others are delayed. Therefore, a deviation in the cell size is generated, and the difference between G_{max} and G_{min} becomes apparent. The difference in the cell size becomes larger as Δr_{d} increases because the irregularity in the initial nuclei enhances the instability of the colliding cells. This type of instability is intensified in the usual phase-field model; small cells become smaller and absorbed by larger cells. In the present model, however, the volumetric balancing term f ^{V} in Equation (3) works efficiently, and all the cells converge to an average size.

Simultaneously, in this stage, the shape of each cell is relaxed such that every face is flattened, and sharp corners are blunted. _{d} = 0.25, and two of the cells are selected and depicted in

_{d} = 0.12 and 0.25 for the fcc arrangement. Similar to the case for bcc, a difference in the cell size is generated during the growth from the nuclei. The deviation is relatively larger, and the maximum cell size exceeds the average size given later. However, the overshoot is observed only temporarily, and all the cells settle shortly into an average size.

hexagonal, as observed in

Finally, nuclei are set on random sites to simulate cell growth under a more natural condition. The number of nuclei N is varied, and the cases for N = 16, 24, and 40 are presented in

is more apparent in this case. Overshoots in the variation of cell size is more apparent for N = 40 because the spatial dispersion of the initial nuclei are more evident. A shoulder in G_{max} curve for N = 24 is due to the largest cell is shifted.

A phase field model for investigating the three-dimensional space-filling structures that were typically observed in metal foam materials was presented. An additional term to balance the volume of every cell was introduced to the conventional multi-phase field model, and the stability of the obtained structures was investigated. The truncated hexagonal structure, which is obtained by cell growth from nuclei arrangement on a bcc lattice, is more stable than the rhombic dodecahedron structure obtained from an fcc arrangement. In addition, when the nuclei are set on random sites, distorted cells are first generated; however, they settle into certain convex regular shapes of some type of polyhedron.

Further modeling is required for the detailed evaluation of the stability of the structures, such as physical interpretation of the volume constraint term introduced in the model. Presumably, this term can be explained by the consideration of the gaseous pressure in the cells. Fluidity during the cell growth process should strongly affect the morphology of the formed structure. These points will be addressed in our future work; however, it can be concluded that the availability of the model presented in this study is fairly validated.

TakuyaUehara, (2015) Phase-Field Modeling for the Three-Dimensional Space-Filling Structure of Metal Foam Materials. Open Journal of Modelling and Simulation,03,120-125. doi: 10.4236/ojmsi.2015.33013