**4. Multiscale simulation algorithms**

**d.** Calculate the energy change ∆*E*associated the site orientation change

tion of the site is recovered

96 Recent Developments in the Study of Recrystallization

stored energy, *H(Si*

lattice continuously.

growth process.

ation of recrystallization.

the sites.

**g.** Go to 'step a' until the end of the simulation

**3.2. Monte Carlo simulation of recrystallization**

**e.** If ∆*E*is non-positive, the attempted reorientation is accepted, otherwise, the old orienta‐

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

ming the volume stored energy and the grain boundary energy contributions throughout all

(1 ) ( ) *i j S S <sup>i</sup>*

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

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

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‐

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

*H S*

*ij i*

d

*E J*

*)*, to the system. The total energy of the system, *E*, is calculated by sum‐


**f.** Increment time regardless of whether the reorientation attempt is accepted or not

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‐ nections during thermal cycling [38, 39].

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‐ ed in the simulation.

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

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

**Figure 4.** A typical temperature profile with temperature range from -40 ºC to 125 ºC, with a 6 minute ramp time and an 18 minute dwell time.

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‐ rithm of dynamic recrystallization is based on the principle that the stored energy of solder is gradually increased during each thermal cycle. Even though recovery consumes a certain amount of the energy, the net change of the energy is always assumed to be positive due to the fact that newly recrystallized grains appear after a certain number of thermal cycles. When a critical value of the energy is reached, recrystallization is initiated. The stored ener‐ gy is released through the nucleation and growth of new grains, which gradually consume the strain-hardened matrix of high dislocation density.

#### **4.2. Multiscale simulation process**

In order to schematically describe the simulation process, a flow chart is shown in Fig. 5. There are three major steps and all the key inputs for the simulation are listed in the boxes, which are on the left side of each step. In Step I, the finite element method is employed to calculate the inelastic strain energy density of the solder interconnections under thermal cy‐ cling loads. As discussed above, it is assumed that the net increase of the stored energy takes place after every thermal cycle. In Step II (scaling processes), the stored energy, as the driv‐ ing force for recrystallization, is mapped onto the lattice of the MC model, and moreover, a correlation is established to convert real time to MC simulation time with the help of the in situ test results. In Step III, the grain boundary energy and the volume stored energy are taken into consideration in the energy minimization calculations to simulate the recrystalli‐ zation and grain growth processes. Furthermore, intermetallic particles are treated as inert particles and their influence on the distribution of stored energy is included.

**Figure 6.** Schematic show of the BGA component under study [39].

**Figure 7.** Finite element model for the thermomechanical calculation [39].

The 3-D finite element analysis (FEA) was performed with the help of the commercial finite element software ANSYS v.12.0. The model was built up according to the experimental set‐ up where the ball grid array (BGA) components were cut along the diagonal line before the

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in situ test. A schematic drawing of the BGA component is shown in Fig. 6.

**4.3. Finite element model**

**Figure 5.** Flow chart for the simulation of microstructural changes in solder interconnections [39].

Simulation of Dynamic Recrystallization in Solder Interconnections during Thermal Cycling http://dx.doi.org/10.5772/53820 99

**Figure 6.** Schematic show of the BGA component under study [39].

#### **4.3. Finite element model**

rithm of dynamic recrystallization is based on the principle that the stored energy of solder is gradually increased during each thermal cycle. Even though recovery consumes a certain amount of the energy, the net change of the energy is always assumed to be positive due to the fact that newly recrystallized grains appear after a certain number of thermal cycles. When a critical value of the energy is reached, recrystallization is initiated. The stored ener‐ gy is released through the nucleation and growth of new grains, which gradually consume

In order to schematically describe the simulation process, a flow chart is shown in Fig. 5. There are three major steps and all the key inputs for the simulation are listed in the boxes, which are on the left side of each step. In Step I, the finite element method is employed to calculate the inelastic strain energy density of the solder interconnections under thermal cy‐ cling loads. As discussed above, it is assumed that the net increase of the stored energy takes place after every thermal cycle. In Step II (scaling processes), the stored energy, as the driv‐ ing force for recrystallization, is mapped onto the lattice of the MC model, and moreover, a correlation is established to convert real time to MC simulation time with the help of the in situ test results. In Step III, the grain boundary energy and the volume stored energy are taken into consideration in the energy minimization calculations to simulate the recrystalli‐ zation and grain growth processes. Furthermore, intermetallic particles are treated as inert

particles and their influence on the distribution of stored energy is included.

**Figure 5.** Flow chart for the simulation of microstructural changes in solder interconnections [39].

the strain-hardened matrix of high dislocation density.

**4.2. Multiscale simulation process**

98 Recent Developments in the Study of Recrystallization

The 3-D finite element analysis (FEA) was performed with the help of the commercial finite element software ANSYS v.12.0. The model was built up according to the experimental set‐ up where the ball grid array (BGA) components were cut along the diagonal line before the in situ test. A schematic drawing of the BGA component is shown in Fig. 6.

**Figure 7.** Finite element model for the thermomechanical calculation [39].

Symmetrical design of the component board enabled the employment of the one-fourth model of the package during FE calculation (see Fig. 7). The symmetry boundary conditions were applied to the symmetric surfaces as mechanical constraints, and the central node of the bottom of the PWB was fixed to prevent rigid body motion. Each solder interconnection was meshed with 540 SOLID185 elements as SOLID185 has plasticity, viscoplasticity, and large strain capabilities. The rest of the model was meshed with SOLID45 elements. The to‐ tal number of nodes and elements of the model was 158424 and 134379, respectively. The SnAgCu solder was modeled by Anand's constitutive model with the parameters provided by Reinikainen et al. [54]. The Anand model is often used for modeling metals' behaviors under elevated temperature when the behaviors become very sensitive to strain rate, tem‐ perature, history of strain rate and temperature. The model is composed of a flow equation and three evolution equations that describes strain hardening or softening during the pri‐ mary stage of creep and the secondary creep stage [55]. The inelastic strain energy is calcu‐ lated by the integral of the stress with respect to the plastic strain increment.

the size of the MC domain in this case is a 200×200 square lattice, which covers the 185×185

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**Figure 8.** Microstructure of the outermost solder interconnection observed with polarized light (*a*) after solidification, (*b*) after 1000 thermal cycles observed with polarized light, (*c*) close-up view of the top right corner of the cross sec‐

One Monte Carlo time step (MCS) is defined as *N*MC reorientation attempts, where *N*MC is the total number of sites in the MC lattice. This means that each site is given an opportunity to change its orientation. A correlation between the simulation time *t*MC [MCS] and real time *t* [s] is usually expressed in the following form including an apparent activation energy factor

m

However, this time scaling process is not employed in the current study due to three main concerns. Firstly, the value of activation energy factor (*Q*m) for tin is seldom reported in the literatures. Secondly, the time scaling is extremely sensitive to the value of *Q*m. As shown in

*RT* - (3)

MC exp( ) *<sup>Q</sup> tv t*

where *R* is the universal gas constant, *T* is the temperature, and *t* is the time.

(*Q*m) as well as an atomic vibration frequency (around 1013 Hz ).

region. Thus, the unit boundary length of the MC model, *s*, equals 0.925µm.

µm2

tion [39].

*4.5.2. Time scaling*

#### **4.4. In situ experiments**

The in situ thermal cycling tests were carried out for the verifications. During the tests, the components were taken out of the test vehicle after every 500 cycles. The solder interconnec‐ tions were repolished, examined, and then put back to the test vehicle to continue the test. The microstructures of the solder cross sections were examined by optical microscopy with polarized light, which shows the areas of different orientations with different colors (see Fig. 8). The boundaries between the areas of different contrast are the high angle grain bounda‐ ries. Fig. 8 (a) and Fig. 8 (b) present the typical microstructures of an outermost solder inter‐ connection after solidification and after 1000 in situ thermal cycling test respectively, and Fig. 8 (c) is the close-up view of the recrystallized region. The dashed rectangle in Fig. 8 (b) shows the domain for the microstructural simulations.

#### **4.5. Scaling processes**

Generally speaking, there is no physically meaningful time and length scale in MC simu‐ lation, and therefore, it is difficult to compare the simulation results to experimental ob‐ servations. Some calibration procedures are necessary in order to establish a relationship between real time and MC simulation time. Furthermore, the calculated inelastic strain energy needs to be converted to stored energy via a scaling process before being map‐ ped onto the MC lattice. In the following, length scaling, time scaling and energy scaling are addressed respectively.

### *4.5.1. Length scaling*

MC simulation does not model the behavior of single atoms and accordingly is performed at the mesoscale level. An MC lattice site represents a large cluster of atoms with the typical size being in the order of micrometers. The domain of the 2-D MC simulation covers the chosen region, which belongs to the cross section of the solder interconnection. For instance, the size of the MC domain in this case is a 200×200 square lattice, which covers the 185×185 µm2 region. Thus, the unit boundary length of the MC model, *s*, equals 0.925µm.

**Figure 8.** Microstructure of the outermost solder interconnection observed with polarized light (*a*) after solidification, (*b*) after 1000 thermal cycles observed with polarized light, (*c*) close-up view of the top right corner of the cross sec‐ tion [39].

#### *4.5.2. Time scaling*

Symmetrical design of the component board enabled the employment of the one-fourth model of the package during FE calculation (see Fig. 7). The symmetry boundary conditions were applied to the symmetric surfaces as mechanical constraints, and the central node of the bottom of the PWB was fixed to prevent rigid body motion. Each solder interconnection was meshed with 540 SOLID185 elements as SOLID185 has plasticity, viscoplasticity, and large strain capabilities. The rest of the model was meshed with SOLID45 elements. The to‐ tal number of nodes and elements of the model was 158424 and 134379, respectively. The SnAgCu solder was modeled by Anand's constitutive model with the parameters provided by Reinikainen et al. [54]. The Anand model is often used for modeling metals' behaviors under elevated temperature when the behaviors become very sensitive to strain rate, tem‐ perature, history of strain rate and temperature. The model is composed of a flow equation and three evolution equations that describes strain hardening or softening during the pri‐ mary stage of creep and the secondary creep stage [55]. The inelastic strain energy is calcu‐

The in situ thermal cycling tests were carried out for the verifications. During the tests, the components were taken out of the test vehicle after every 500 cycles. The solder interconnec‐ tions were repolished, examined, and then put back to the test vehicle to continue the test. The microstructures of the solder cross sections were examined by optical microscopy with polarized light, which shows the areas of different orientations with different colors (see Fig. 8). The boundaries between the areas of different contrast are the high angle grain bounda‐ ries. Fig. 8 (a) and Fig. 8 (b) present the typical microstructures of an outermost solder inter‐ connection after solidification and after 1000 in situ thermal cycling test respectively, and Fig. 8 (c) is the close-up view of the recrystallized region. The dashed rectangle in Fig. 8 (b)

Generally speaking, there is no physically meaningful time and length scale in MC simu‐ lation, and therefore, it is difficult to compare the simulation results to experimental ob‐ servations. Some calibration procedures are necessary in order to establish a relationship between real time and MC simulation time. Furthermore, the calculated inelastic strain energy needs to be converted to stored energy via a scaling process before being map‐ ped onto the MC lattice. In the following, length scaling, time scaling and energy scaling

MC simulation does not model the behavior of single atoms and accordingly is performed at the mesoscale level. An MC lattice site represents a large cluster of atoms with the typical size being in the order of micrometers. The domain of the 2-D MC simulation covers the chosen region, which belongs to the cross section of the solder interconnection. For instance,

lated by the integral of the stress with respect to the plastic strain increment.

shows the domain for the microstructural simulations.

**4.4. In situ experiments**

100 Recent Developments in the Study of Recrystallization

**4.5. Scaling processes**

are addressed respectively.

*4.5.1. Length scaling*

One Monte Carlo time step (MCS) is defined as *N*MC reorientation attempts, where *N*MC is the total number of sites in the MC lattice. This means that each site is given an opportunity to change its orientation. A correlation between the simulation time *t*MC [MCS] and real time *t* [s] is usually expressed in the following form including an apparent activation energy factor (*Q*m) as well as an atomic vibration frequency (around 1013 Hz ).

$$t\_{\rm MC} = v \exp(-\frac{Q\_{\rm m}}{RT})t\tag{3}$$

where *R* is the universal gas constant, *T* is the temperature, and *t* is the time.

However, this time scaling process is not employed in the current study due to three main concerns. Firstly, the value of activation energy factor (*Q*m) for tin is seldom reported in the literatures. Secondly, the time scaling is extremely sensitive to the value of *Q*m. As shown in Table 1, a possible error of *Q*m (within the range from 20 to 107 kJ/mole) leads to a significant difference in time scale, which will finally result in unreliable simulation results. Thirdly, during each thermal cycle, the temperature alternates between a low temperature and a high temperature (e.g. from -40 ºC to 125 ºC, see Fig. 1), which makes it difficult to use Eq. (3).


**Table 1.** Time scales with different activation energy factors

Besides the scaling approach in Eq. (3), other real time scaling approaches have also been developed and successfully applied to various applications. For instance, Safran et al. [56] set the time scale by multiplying the transition probability with a basic attempt frequency, and Raabe [57] scaled the real time step by a rate theory of grain boundary motion.

In order to improve both the accuracy and the efficiency, a new correlation between the sim‐ ulation time *t*MC [MCS] and real time *t* [TC] is established as follows.

$$t\_{\rm MCC} = \frac{c\_1}{c\_2}t\tag{4}$$

large average grain size as well as long and narrow grain shapes. Fig 9 (c) is regarded as a good representative of the studied microstructure in terms of the similar average grain size and more or less equiaxed grain shapes. Hence, '*c*1 = 50' was used for the rest of the simula‐

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In contrast to the situation in a static recrystallization process, the stored energy of the MC sites accumulates during dynamic loading. The correlation between the external work and the stored energy is discussed as follows. It is well known that a fraction of the external work, typically varying between 1% and 15%, is stored in the metallic material during defor‐ mation [58]. According to the parameter study carried out in [38] with considering the effect of recovery, the most suitable retained fraction, 5%, is used for the unrecrystallized region in the current MC simulation. Zero is assumed to be the retained fraction for the recrystallized region. This assumption is valid due to the fact that only primary recrystallization has been found in the experimental observation of solder interconnections. It should be mentioned that if non-zero retained fraction is used for the recrystallized region, the present model is capable of predicting secondary recrystallization, which may be applicable in other cases.

The energy scaling is based on the principle that the ratios of the volume stored energy to the grain boundary energy should be equated in the MC model and the physical system [51]. The driving force due to the grain boundary energy, *P*grgr, equals *γ/<r>*, where *γ* is the grain boundary energy per unit area and *<r>* is the mean grain radius. Since high angle

**Figure 9.** Simulated microstructures with four typical *c*1 values [39].

*4.5.3. Energy scaling*

tions.

where *c*1 and *c*2 are model parameters with the units [MCS] and [TC], respectively.

One thermal cycle is defined as the unit of time instead of using seconds. In this way, the relatively complicated temperature change within a thermal cycle is simplified and included only in the FE simulation and not in the MC simulation. The physical meaning of *c*1 is the number of Monte Carlo time steps required for the growth of the newly recrystallized grains during each simulation time interval (STI). A certain amount of external energy is added to the MC lattice at the beginning of each STI and the amount of energy is calculated according to a certain number of thermal cycles, i.e. *c*2. Therefore, the parameter *c*2 can be considered as a time compression factor and the number of simulation time intervals is equal to *t*/*c*2.

A series of numerical experiments were carried out and the simulated microstructures were compared to Fig. 8 (c) in order to decide the suitable model parameters, *c*1 and *c*2, in Eq. 4 for the time scaling process. In theory, the value of the time compression factor, *c*2, can range from 1 to *N*TC, where *N*TC is the total number of thermal cycles. A larger value of *c*<sup>2</sup> leads to a more efficient calculation with less accuracy. One extreme case is when *c*2 equals *N*TC, and then, there will be only one simulation time interval and the process will be similar to a stat‐ ic recrystallization simulation. Considering efficiency and accuracy, *c*<sup>2</sup> was assumed to be 100 thermal cycles, making 10 simulation time intervals for the case *N*TC =1000. Besides *c*2, a number of different *c*1 were studied. The microstructures of several typical values of *c*<sup>1</sup> are shown in Fig. 9. A small value of *c*1, e.g. *c*1 = 10 (see Fig. 9 (a)) results in a small average grain size and immature microstructures, where newly introduced embryos do not have enough time to grow up. A large value of *c*1, for instance *c*<sup>1</sup> = 100 (see Fig. 9 (d)), leads to a relatively large average grain size as well as long and narrow grain shapes. Fig 9 (c) is regarded as a good representative of the studied microstructure in terms of the similar average grain size and more or less equiaxed grain shapes. Hence, '*c*1 = 50' was used for the rest of the simula‐ tions.

**Figure 9.** Simulated microstructures with four typical *c*1 values [39].

#### *4.5.3. Energy scaling*

Table 1, a possible error of *Q*m (within the range from 20 to 107 kJ/mole) leads to a significant difference in time scale, which will finally result in unreliable simulation results. Thirdly, during each thermal cycle, the temperature alternates between a low temperature and a high temperature (e.g. from -40 ºC to 125 ºC, see Fig. 1), which makes it difficult to use Eq. (3).

*Qm (kJ/mole)* 20 40 64 70 90 107 *tMC /t (MCS/s)* 1.9e+10 4.7e+7 3.3e+4 5.4e+3 1.3e+1 7.5e-2

Besides the scaling approach in Eq. (3), other real time scaling approaches have also been developed and successfully applied to various applications. For instance, Safran et al. [56] set the time scale by multiplying the transition probability with a basic attempt frequency,

In order to improve both the accuracy and the efficiency, a new correlation between the sim‐

One thermal cycle is defined as the unit of time instead of using seconds. In this way, the relatively complicated temperature change within a thermal cycle is simplified and included only in the FE simulation and not in the MC simulation. The physical meaning of *c*1 is the number of Monte Carlo time steps required for the growth of the newly recrystallized grains during each simulation time interval (STI). A certain amount of external energy is added to the MC lattice at the beginning of each STI and the amount of energy is calculated according to a certain number of thermal cycles, i.e. *c*2. Therefore, the parameter *c*2 can be considered as a time compression factor and the number of simulation time intervals is equal to *t*/*c*2.

A series of numerical experiments were carried out and the simulated microstructures were compared to Fig. 8 (c) in order to decide the suitable model parameters, *c*1 and *c*2, in Eq. 4 for the time scaling process. In theory, the value of the time compression factor, *c*2, can range from 1 to *N*TC, where *N*TC is the total number of thermal cycles. A larger value of *c*<sup>2</sup> leads to a more efficient calculation with less accuracy. One extreme case is when *c*2 equals *N*TC, and then, there will be only one simulation time interval and the process will be similar to a stat‐ ic recrystallization simulation. Considering efficiency and accuracy, *c*<sup>2</sup> was assumed to be 100 thermal cycles, making 10 simulation time intervals for the case *N*TC =1000. Besides *c*2, a number of different *c*1 were studied. The microstructures of several typical values of *c*<sup>1</sup> are shown in Fig. 9. A small value of *c*1, e.g. *c*1 = 10 (see Fig. 9 (a)) results in a small average grain size and immature microstructures, where newly introduced embryos do not have enough time to grow up. A large value of *c*1, for instance *c*<sup>1</sup> = 100 (see Fig. 9 (d)), leads to a relatively

*t* (4)

and Raabe [57] scaled the real time step by a rate theory of grain boundary motion.

*<sup>t</sup>*MC <sup>=</sup> *<sup>c</sup>*<sup>1</sup> *c*2

where *c*1 and *c*2 are model parameters with the units [MCS] and [TC], respectively.

ulation time *t*MC [MCS] and real time *t* [TC] is established as follows.

**Table 1.** Time scales with different activation energy factors

102 Recent Developments in the Study of Recrystallization

In contrast to the situation in a static recrystallization process, the stored energy of the MC sites accumulates during dynamic loading. The correlation between the external work and the stored energy is discussed as follows. It is well known that a fraction of the external work, typically varying between 1% and 15%, is stored in the metallic material during defor‐ mation [58]. According to the parameter study carried out in [38] with considering the effect of recovery, the most suitable retained fraction, 5%, is used for the unrecrystallized region in the current MC simulation. Zero is assumed to be the retained fraction for the recrystallized region. This assumption is valid due to the fact that only primary recrystallization has been found in the experimental observation of solder interconnections. It should be mentioned that if non-zero retained fraction is used for the recrystallized region, the present model is capable of predicting secondary recrystallization, which may be applicable in other cases.

The energy scaling is based on the principle that the ratios of the volume stored energy to the grain boundary energy should be equated in the MC model and the physical system [51]. The driving force due to the grain boundary energy, *P*grgr, equals *γ/<r>*, where *γ* is the grain boundary energy per unit area and *<r>* is the mean grain radius. Since high angle grain boundaries are of interest to the present investigation, the value *γ* =0.164 J/m2 is used for the high angle grain boundary energy of Sn [22].

In the MC model, the driving force due to the stored energy density is given by*P*MC vol=*H* / *A*, where *H* is the volume stored energy of a site and *A* is the area of that site (*A* = *s2* in the 2-D square lattice). The driving force due to the grain boundary energy is given by

$$P\_{\rm MC}^{\rm greg} = \frac{\chi\_{\rm MC}}{\_{\rm MC}} = \frac{I}{s\_{\rm MC}} = \frac{I}{s} \tag{5}$$

**4.6. Nucleation**

(25×25 µm2

were presented in the reference [51].

**4.7. Treatment of large intermetallic particles**

ticles have a significant influence on stimulating nucleation.

introduced in order to consider the effects of IMPs.

Nucleation stage is very crucial for the simulation of recrystallization. The recrystallization process is modeled by introducing nuclei (small embryos with zero stored energy) into the lattice at a constant rate, i.e. continuous nucleation. According to the locations of the valid nuclei, the nucleation is non-uniform. Although the locations of nuclei are randomly chosen, the volume stored energy of the chosen sites has to be larger than the critical stored energy, *H*cr, before the nuclei are placed in their sites. In this way, the sites with high stored energy (e.g. near interfaces, grain boundaries, and IMPs) will have a high probability of nucleation. The critical stored energy, *H*cr, is set to be 2*J* so that a single-site isolated embryo can grow as a new grain. Detailed discussion about the critical stored energy and critical embryo size

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The intermetallic particles in SnAgCu solder are mainly Cu6Sn5 and Ag3Sn. The size of IMPs varies a lot and only coarse IMPs (particle size of 1 µm or above) are studied in the MC sim‐ ulation. Zener-type particle pinning is not considered in the current model for the sake of simplicity. It is believed that fine particles do not remarkably affect the distribution of stored energy within the grains; however, coarse particles exert localized stress concentrations due to the mismatch of mechanical properties and thermal expansion coefficients. The large par‐

The IMPs are introduced into the MC simulation as inert particles. They are assigned an orientation different from any of the surrounding grains and are not allowed to be reor‐ iented during the simulation. Thus, the inert particles do not grow or move. To include the IMPs in the FE model is not realistic and too computationally expensive in view of the fact that the size and shape of IMPs vary a lot and the locations of IMPs are ran‐ domly distributed in the bulk solder. Instead, the size of the particle-affected deforma‐ tion region is estimated and an energy amplification factor (EAF) distribution is

A 2-D FE simulation was carried out to study the particle-affected deformation region and the energy amplification factor distribution. The model was composed of a solder matrix

solder matrix and the hard IMP were from the reference [59]. The loading was defined by applying the displacement on the top and right edges, i.e. bi-axial tension. The von Mises stress at the edges was not influenced by the IMP, and thereby, it was defined as one unit for the sake of normalization. The calculated stress contour and the 'EAF vs. distance' curve are shown in Fig. 10. According to the FE simulation results, the following assumptions are made with the purpose of treating IMPs in the MC simulation. Within a distance of approxi‐ mately one particle radius, the calculated stored energy is amplified by a certain energy am‐ plification factor before mapped onto the Monte Carlo lattice. The EAF for the site close to the IMP is about '1.12' and the EAF decreases linearly to '1'for the site more than one radius distance away from the IMP. In order to realize this, a new matrix storing the EAF distribu‐ tion is introduced to the Monte Carlo algorithm. Every time the stored energy of a site is

) and a round IMP (radius = 5 µm). The material properties of the relatively soft

where *γ*MC and *<r>*MC are the grain boundary energy per unit and the mean grain radius in the MC model, respectively. Each unlike pair of nearest neighboring sites contributes a unit of grain boundary energy *J* to the system. Because of the existence of the length scale factor, *s*, an ideal prediction should satisfy the requirement that the mean grain radiuses in the model and in the physical system are equal, *<r>*MC =*<r>*.

In the physical system, the ratio of the volume stored energy to the grain boundary energy per unit volume is

$$\frac{P^{\text{vol}}}{P^{\text{grgr}}} = \frac{P^{\text{vol}} < r >}{\gamma} \tag{6}$$

In the MC model, the ratio is given by

$$\frac{P\_{\rm MC}^{\rm vol}}{P\_{\rm MC}^{\rm ggrr}} = \frac{H < r >}{J \; s} \tag{7}$$

Equating the ratios of the model and the physical system, and rearranging, gives

$$H = \left(\frac{P^{\text{vol}} \, s}{\gamma}\right) J \tag{8}$$

The increment of *H* can be easily calculated by Eq. (8), where *s* and *γ* are known parameters. *P*vol distribution is obtained by scaling and mapping the energy density distribution calculat‐ ed by FEM onto the MC lattice. *J* is a unit of grain boundary energy in the MC model. It is noteworthy that the absolute value of *J* is not essential and knowing the ratio, *H*/*J*, is suffi‐ cient for the MC simulation.

## **4.6. Nucleation**

is used

vol=*H* / *A*,

(5)

(8)

grain boundaries are of interest to the present investigation, the value *γ* =0.164 J/m2

In the MC model, the driving force due to the stored energy density is given by*P*MC

MC MC *J J <sup>P</sup>*

*r sr s r*

where *γ*MC and *<r>*MC are the grain boundary energy per unit and the mean grain radius in the MC model, respectively. Each unlike pair of nearest neighboring sites contributes a unit of grain boundary energy *J* to the system. Because of the existence of the length scale factor, *s*, an ideal prediction should satisfy the requirement that the mean grain radiuses in the

In the physical system, the ratio of the volume stored energy to the grain boundary energy

(6)

(7)

vol vol

*P P r*

grgr

vol MC grgr MC

Equating the ratios of the model and the physical system, and rearranging, gives

vol *P s H J* 

æ ö ç ÷ ç ÷ è ø

The increment of *H* can be easily calculated by Eq. (8), where *s* and *γ* are known parameters. *P*vol distribution is obtained by scaling and mapping the energy density distribution calculat‐ ed by FEM onto the MC lattice. *J* is a unit of grain boundary energy in the MC model. It is noteworthy that the absolute value of *J* is not essential and knowing the ratio, *H*/*J*, is suffi‐

*P H r P J s*

*P*

square lattice). The driving force due to the grain boundary energy is given by

grgr MC

MC

model and in the physical system are equal, *<r>*MC =*<r>*.

per unit volume is

In the MC model, the ratio is given by

cient for the MC simulation.

where *H* is the volume stored energy of a site and *A* is the area of that site (*A* = *s2* in the 2-D

for the high angle grain boundary energy of Sn [22].

104 Recent Developments in the Study of Recrystallization

Nucleation stage is very crucial for the simulation of recrystallization. The recrystallization process is modeled by introducing nuclei (small embryos with zero stored energy) into the lattice at a constant rate, i.e. continuous nucleation. According to the locations of the valid nuclei, the nucleation is non-uniform. Although the locations of nuclei are randomly chosen, the volume stored energy of the chosen sites has to be larger than the critical stored energy, *H*cr, before the nuclei are placed in their sites. In this way, the sites with high stored energy (e.g. near interfaces, grain boundaries, and IMPs) will have a high probability of nucleation. The critical stored energy, *H*cr, is set to be 2*J* so that a single-site isolated embryo can grow as a new grain. Detailed discussion about the critical stored energy and critical embryo size were presented in the reference [51].

## **4.7. Treatment of large intermetallic particles**

The intermetallic particles in SnAgCu solder are mainly Cu6Sn5 and Ag3Sn. The size of IMPs varies a lot and only coarse IMPs (particle size of 1 µm or above) are studied in the MC sim‐ ulation. Zener-type particle pinning is not considered in the current model for the sake of simplicity. It is believed that fine particles do not remarkably affect the distribution of stored energy within the grains; however, coarse particles exert localized stress concentrations due to the mismatch of mechanical properties and thermal expansion coefficients. The large par‐ ticles have a significant influence on stimulating nucleation.

The IMPs are introduced into the MC simulation as inert particles. They are assigned an orientation different from any of the surrounding grains and are not allowed to be reor‐ iented during the simulation. Thus, the inert particles do not grow or move. To include the IMPs in the FE model is not realistic and too computationally expensive in view of the fact that the size and shape of IMPs vary a lot and the locations of IMPs are ran‐ domly distributed in the bulk solder. Instead, the size of the particle-affected deforma‐ tion region is estimated and an energy amplification factor (EAF) distribution is introduced in order to consider the effects of IMPs.

A 2-D FE simulation was carried out to study the particle-affected deformation region and the energy amplification factor distribution. The model was composed of a solder matrix (25×25 µm2 ) and a round IMP (radius = 5 µm). The material properties of the relatively soft solder matrix and the hard IMP were from the reference [59]. The loading was defined by applying the displacement on the top and right edges, i.e. bi-axial tension. The von Mises stress at the edges was not influenced by the IMP, and thereby, it was defined as one unit for the sake of normalization. The calculated stress contour and the 'EAF vs. distance' curve are shown in Fig. 10. According to the FE simulation results, the following assumptions are made with the purpose of treating IMPs in the MC simulation. Within a distance of approxi‐ mately one particle radius, the calculated stored energy is amplified by a certain energy am‐ plification factor before mapped onto the Monte Carlo lattice. The EAF for the site close to the IMP is about '1.12' and the EAF decreases linearly to '1'for the site more than one radius distance away from the IMP. In order to realize this, a new matrix storing the EAF distribu‐ tion is introduced to the Monte Carlo algorithm. Every time the stored energy of a site is updated during the recrystallization simulation, the energy increment is multiplied by the associated EAF before being added to the site. By introducing the EAF, stored energy densi‐ ty close to IMPs is higher than usual, leading to a higher driving force for nucleation and growth of recrystallized grains. Thus, the particle stimulated nucleation is well taken into consideration in the MC simulation.

the simulated microstructures, the related micrographs are presented. According to the sim‐ ulation results, the incubation time for the recrystallization is about 400 TCs. During the in‐ cubation period, the stored energy is accumulated, but the magnitude remains below the critical value. As a result, no new grains are formed before 400 TCs. The upper right corner of the solder interconnection is the location where the highest inelastic strain energy is con‐ centrated (see Fig. 11 (b)). It is this very same location where the magnitude of the stored energy first exceeds the critical value and recrystallization is initiated (see Fig. 12 (a) and Fig. 12 (d)). Then, as shown in Fig. 12 (e), the recrystallized region expands towards the lower left of the interconnection, which is in good agreement with the experimental finding (see Fig. 12 (b)). By comparing Fig. 12 (e) and 12 (*f*), it is found that the migration rate of the re‐ crystallization front slows down during the period from 1000 TCs to 1500 TCs due to the decreasing driving force in the lattice. In the micrograph, Fig. 12 (c), cracks and voids are obvious, meaning that the continuity assumption of the finite element model is no longer valid. Therefore, the difference between the experimental finding and the simulated micro‐ structure, Fig. 12 (f), is understandable. A possible solution is to simulate the behaviors of cracks and voids in the Monte Carlo model, output the microstructures to the finite element model, and then, use the calculated results as the inputs for the next round Monte Carlo

Simulation of Dynamic Recrystallization in Solder Interconnections during Thermal Cycling

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

107

**Figure 11.** *a*) Plastic deformation of the solder interconnection after 1000 thermal cycles, (*b*) FEM-calculated inelastic strain energy density distribution, dashed rectangle shows the domain of the microstructural simulation [39].

There was no obvious IMP-affected recrystallization in any of the in situ samples. Most of the observed recrystallized microstructures were located close to the interface region where the stored energy density was the highest. In order to focus on the influence of the IMPs and exclude the effects of the heterogeneous energy distribution, a uniform stored energy densi‐ ty distribution was assumed during the simulation. The assumption is valid when the calcu‐ lation domain is located in the center part of the solder interconnection, where the energy magnitude is relatively low and energy distribution is quite uniform. Furthermore, the ener‐

simulation.

**5.2. Simulation with presence of IMPs**

**Figure 10.** von Mises stress contour and 'EAF vs. distance' curve [39].
