A computational method for the computer simulation of the liquid phase sintering of metallic systems

The growth of solid particles during liquid phase sintering was modeled by the Cellular Automata method. The binary phase diagram and Fickian approach for the diffusion process were applied to simulate the chemical composition variation in liquid and solid phases during sintering. The OswaldRipening effect was considered during the dissolution of the solid phase in the liquid phase. It is used to define the probability of solid-phase dissolution by the liquid phase and develop the model to simulate the alloy with solid solubility. So, the microstructure could be modeled in the liquid phase sintering process.

metals 3D printing technology. Liquid phase sintering of two metals or alloys could provide some useful manufacturing methods for the metals 3D printing and the 3D printing of metal matrix composite with or without postprocessing.
The presence of the liquid phase in the LPS method is the key parameter that has the role of activation in the diffusion process. In this method, the temperature during sintering is selected between the two elemental melting points of mixed powders. So, the element with a low melting point is melted. It is the first stage of the liquid phase sintering. In the second stage, solid particles are partially dissolved by the produced liquid phase. Solid particles start to dissolve from their higher energy points on their surface.
Consequently, the low size particles (higher surface curvature) are thoroughly dissolved in the liquid phase, and it helps the growth of the larger solid particles. It is the so-called Oswald-Ripening effect. The two important parameters of the liquid phase sintering process are the wetting conditions of the solid particles by the liquid phase and the solubility of solid particles into the liquid phase. At a specified temperature, solidification of the liquid phase is the function of the chemical composition of the liquid phase [14].
The growth mechanism and the shape of the solid particles depend on surface energy. Solid particles with anisotropic surface energy (i.e., WC particles in Co liquid) have the preferential growth on the special crystal planes. These kinds of solid particles have spatial geometry with flat surfaces. If the surface energy of the solid particles is isotropic, no preferential crystal planes exist for the growth. The achieved topology of these particles is the non-symmetry sphere [15].
So, the physics of the liquid phase sintering process consists of solidification, dissolution, and diffusion phenomena in the solid or liquid phases. Metallography of the liquid phase sintered samples reveals the channel between the two nearest particles or completely attached surfaces of the two particles. The presence of this channel is related to the penetration of the liquid phase across the two compacted solid particles. The penetration of the liquid phase is a function of the wetting and dihedral angle [16]. Some stochastic models have been developed to study the changing of physical parameters during liquid phase sintering. J. Aldazabal et al. introduced the model using the Monte Carlo method [15]. This model has only considered the precipitation of the solid-phase using the predefined chemical composition for the liquid phase as the initial condition. So, this model cannot be able to simulate the Oswald-Ripening effect and the formation of the channel between the adjacent solid particles under the use of the wetting and dihedral angles concepts. This model relates the presence of the channel between the two solid particles to the restriction of their growth by the chemical composition of the liquid phase. Another The present computer model has considered the solution of the solid particles in the liquid phase as a stochastic process to simulate the Oswald-Ripening effect, but this model can not able to simulate the penetration of the liquid phase across the two adjacent solid particles. In addition, the pure liquid phase of the low melting point element, which has completely enclosed the solid particles, was assumed as an initial condition. The diffusion in the solid particle was modeled because of its significance to determine the chemical composition at the surface of the solid particles to define the probability of solid-phase dissolution by the liquid phase. It also develops the model to simulate the alloy with solid solubility during sintering. This model uses the same physical dimension and time scale for both the stochastic method and the finite difference method, which are used in solidification/dissolution and diffusion processes, respectively.

Model description
The differential equation of the diffusion process was solved by the finite difference method. The solidification and dissolution were also modeled using a stochastic method. Both methods need a discretized environment. Each discrete part of the model is called an element. At the center of each element, a node was considered, and the calculation results on each node were extended to its related element as a mean value.
A weighted coordination number was defined for each element. In two dimensional, each element has two kinds of relationships with their neighbor elements. In the first case, two neighbor's elements only contact by their corners and their faces (Fig. 1). The weighted coordination number of the two contacted elements by their corners defines 0.25, and it also defines 1 when the two neighbor elements contacts by their corners. This weighted coordination number can be used as a representation of the surface energy. If the coordination number of the solid element at the surface of the solid particles increases, the surface energy at this point decreases. The same relation can be used between the coordination number and curvature of the surface. So, the higher surface energy and higher curvature are equal to lower coordination numbers and vise versa. These weighted coordination numbers were used to define the Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 12 September 2020 doi:10.20944/preprints202009.0262.v1 probability numbers for dissolution and solidification. Figure 2 shows some possible weighted coordination numbers of a liquid element relate to its adjacent solid elements.  The red color is related to the solid elements , and the gray one is the liquid element.
Two numbers were defined between 0 and 1 to set up the stochastic concept in the discretized model.
The first number is related to the dissolution probability of an element at the surface of the solid particles, and the second number is related to the solidification probability for the liquid elements adjacent to solid elements.
For the diffusion process, the parabolic differential equation can be solved using a finite difference method. The differential equation and its finite difference equation of the diffusion process have been shown in equations 1 and 2, respectively.
has the value lower than the 0.5. Figure 3 shows the initial physical environment of the model. Each solid particle shown in figure 3 has its own color as its own characteristic. Each new element added to the surface of the solid particle gains this color. So, the boundary between the two neighbor's solid particles can be distinguishable. randomly generated then compared with that defined probability number. If the value of a randomly generated number is the lower value of the defined probability number, that pixel changes its state.
Each element of the model has four properties. The first is for the distinction of the phase, solid, or liquid. The second is for the chemical composition of an element. The third is for the weighted coordination number, and finally, the fourth is the position of that element.

Result and discussion
As an example, the Fe -Cu system was considered. At the 1300 o C temperature, it was assumed that the molten Cu has completely enclosed solid Fe particles. The model has 300 pixels in height and the 300 in its width (Fig. 3). Each pixel, which is square, has the 0.5 µm length in width and 0.5 µm in height ( Fig. 4).  First condition: Fe weight percent of a liquid element has a value higher than the 87.5 wt%. Therefore, this element has a chance of solidification.
Second condition: Fe weight percent of a liquid element has a value lower than 87.5 wt% and higher than 8.5 wt%. In this case, that pixel also has a chance of solidification. But if the pixel solidifies, then the new solidified pixel has 87.5 weight percent of Fe, and the additional new liquid phase is also produced with 8.5 weight percent of Fe according to the phase diagram. This new liquid phase changes the chemical composition of its adjacent liquid elements.
Third condition: Fe weight percent of a liquid element has a value lower than 8.5 wt%. In this case, there is no chance of solidification. So, to solidify this liquid element, some solid elements must be dissolved by the liquid phase to change the weight percent of Fe in liquid to the permissible range.
Fe Cu If there is no restriction based on the chemical composition of the liquid phase for solidification, the probability number or chance of the solidification must be defined for each liquid element. First, the solid adjacent pixels of each liquid element must be specified. Then based on table 1, a number between 0 and 0.5 related to the weighted coordination number will be assigned. The (-1) number was defined for the case of no chance of solidification. The numbers in table 1 are defined based on the best microstructure results by trials and errors. Table 1 -Relation between the coordination number and the probability number of a liquid element.    with solubility both in the liquid phase and solid one, due to its ability to consider the diffusion process both in solid and liquid phases.