**3.1. Monte Carlo simulation of grain growth**

The MC grain growth simulation originates from Ising and Potts models for magnetic do‐ main evolution [46]. The Ising model consists of two spin states, namely up and down, and the Potts model allows multiple states (*Q* states) for each particle in the system. The Potts model has been widely used for modeling material behaviors, such as grain growth and tex‐ ture evolution, e.g. [47-49].

During the MC grain growth simulation, a continuum microstructure is mapped onto a 2D MC lattice, which can be either triangular or rectangular lattice [50]. In order to initialize the lattice, an integer number *Si* (between 1 and *Q*) is assigned to each lattice site, where Q rep‐ resents the total number of orientations in the system. Two adjacent sites with different grain orientation numbers are regarded as being separated by a grain boundary and each pair of unlike neighboring sites contributes a unit of grain boundary energy, *J*, to the system. A group of sites having the same orientation number and surrounded by grain boundaries are considered as a grain. The total energy of the system, *E*, is calculated by the grain boun‐ dary energy contributions throughout all the sites.

$$E = \int \sum\_{} \left( 1 - \delta\_{S\_i S\_j} \right) \tag{1}$$

where the sum of *i* is over all *N*MC sites in the system, the sum of *j* is over all the nearestneighbor sites of the site *i*, and *δij* is the Kronecker delta.

The Monte Carlo method iteratively simulates the grain growth process by the following key steps [51-53].


#### **3.2. Monte Carlo simulation of recrystallization**

The major differences between the simulation of recrystallization and grain growth are the bulk stored energy and the nucleation process. A fraction of the energy associated with the deformation of material is stored in the metal, mainly in the form of dislocations. The distri‐ bution of stored energy is heterogeneous, and therefore each site contributes an amount of stored energy, *H(Si )*, to the system. The total energy of the system, *E*, is calculated by sum‐ ming the volume stored energy and the grain boundary energy contributions throughout all the sites.

$$E = I \sum\_{} (1 - \delta\_{S\_i S\_j}) + \sum\_i H(S\_i) \tag{2}$$

**4. Multiscale simulation algorithms**

nections during thermal cycling [38, 39].

**4.1. Thermal cycling test and model assumptions**

ed in the simulation.

an 18 minute dwell time.

The treatment of heterogeneous nucleation and inhomogeneously deformed material re‐ mains a challenge for the MC method. Hybrid methods are needed to perform the task. As steps in this direction, Rollett et al. developed a hybrid model for mesoscopic simula‐ tion of recrystallization by combining the MC and Cellular Automaton methods [43]. Song et al. presented a hybrid MC model for studying recovery and recrystallization of titanium at various annealing temperatures after inhomogeneous deformation [44]. Yu et al. combined the MC method with the finite element (FE) method in order to simulate the microstructure of structural materials under forging and rolling [45]. However, most of the early studies are about the simulations of static recrystallization. Recently, Li et al. presented a hybrid algorithm for simulating dynamic recrystallization of solder intercon‐

Simulation of Dynamic Recrystallization in Solder Interconnections during Thermal Cycling

http://dx.doi.org/10.5772/53820

97

The details of the multiscale simulation of dynamic recrystallization of solder interconnec‐ tions are presented as follows. The finite element method is utilized to model macroscale in‐ homogeneous deformation, and the Monte Carlo Potts model is utilized to model the mesoscale microstructural evolution. Compared to the in situ experimental observations, a correlation between real time and MC simulation time is established. In addition, the effects of intermetallic particles (Cu6Sn5 and Ag3Sn) on recrystallization in solder matrix are includ‐

Thermal cycling (TC) tests are accelerated fatigue tests, which subject the components and solder interconnects to alternating high and low temperature extremes [35, 36]. The tests are conducted to determine the ability of the parts to resist a specified number of temperature cycles from a specified high temperature to a specified low temperature with a certain ramp

**Figure 4.** A typical temperature profile with temperature range from -40 ºC to 125 ºC, with a 6 minute ramp time and

Each thermal cycle can be regarded as 'deformation + annealing' and TC tests normally last several thousand cycles before failures of the solder interconnections are detected. The algo‐

rate and dwell time. A typical temperature profile for a TC test is shown in Fig. 4.

In static recrystallization simulations, the stored energy of each site is positive for unrecrys‐ tallized sites and zero for recrystallized sites. However, in the case of dynamic recrystalliza‐ tion the stored energy is a function of both time and position as new energy is added to the lattice continuously.

The nucleation process is modeled by introducing nuclei (small embryos with zero stored energy) into the lattice at random positions. Embryos have orientations that differ from all the other grains of the original microstructure. Embryos can be added to the lattice at the beginning of the simulation or at a regular interval during the simulation. In the reorienta‐ tion process, if the randomly selected site is unrecrystallized, it will be recrystallized under the condition that the total energy of the system is reduced. If the selected site is recrystal‐ lized, the reorientation process is a simulation of the nucleus growth process or the grain growth process.

#### **3.3. Monte Carlo simulation of recrystallization with the presence of particles**

The particles are normally introduced to the MC sites at the beginning of the simulation, and those sites have an orientation different from any other grains. The particles do not re‐ act, dissolve or grow during the simulation, and thereby are named as inert particles [50]. These assumptions have been proved to be effective, especially for small particles. However, the influence on nucleation stimulation should be considered for large particles. Large parti‐ cles exert localized stress and strain concentrations and cause the increase of dislocation density in the particle-affected deformation regions, which provide favorable sites for nucle‐ ation of recrystallization.
