## Numerical Analysis of Coalescence Characteristics of Low Melting Point

## Alloy Fillers Using a Non-Equilibrium Phase Field Model

### Jae Hyung Kim

1### , Jung Hee Lee

2### , Jong Min Kim

3### and Seong Hyuk Lee

3;* 1Korea Institute of Machinery & Materials, 104 Sinseongno, Yuseong-gu, Daejon 305-343, Korea
2_{Function Bay Inc., Gyeonggi-do 463-824, Korea}

3_{School of Mechanical Engineering, Chung-Ang University, Seoul 156-756, Korea}

A non-equilibrium phase ﬁeld model (NPFM) is proposed to examine the three-dimensional coalescence characteristics of low melting point alloy (LMPA) ﬁllers, widely used for self-organized interconnection applications. This model suﬃciently considers the non-equilibrium process during phase transitions, resulting in accurate prediction of interfaces between two diﬀerent phases with a high density ratio. The preliminary simulation is carried out for validation of the present model and numerical predictions are in good agreement with analytical solutions for a one-dimensional solidiﬁcation problem. The proposed NPFM successfully predicts the phase transition, the heat transfer from solid to liquid, and the coalescence behaviors of LMPA ﬁllers, whereas the conventional enthalpy method fails to describe the non-equilibrium phase transition. Moreover, the results show that there exists grid dependency in the mush zone, suggesting that special care should be taken of the mush zone for better prediction of interfaces between diﬀerent phases. [doi:10.2320/matertrans.MF200915]

(Received January 7, 2009; Accepted April 27, 2009; Published June 17, 2009)

Keywords: non-equilibrium phase ﬁeld model (NPFM), self-organized interconnection, low melting point alloy (LMPA), coalescence, Phase transition

1. Introduction

In recent years, much attention has been paid to
eco-friendly alternative bonding material to conventional lead-tin
solders being of toxicity and environmental hazard for the
self-organized interconnection using anisotropic conductive
adhesive (ACA) with low melting point alloy (LMPA)
ﬁllers.1–3) _{In a self-organized interconnection procedure, as}
shown in Fig. 1, conductive ﬁllers are initially in solid phase
but during the reﬂow (thermal and compression) process,
they become molten and coalesced with adjacent ﬁllers, and
ﬁnally some of larger ﬁllers make conduction paths, resulting
in forming interconnection. For designing the optimal ACA
process, we should understand detailed physics behind phase
change and coalescence characteristics. In fact, because of
limitations in time and space for current measurement,
computational modeling oﬀers a promising way to get the
detailed information on transient behavior of phase change
and the corresponding coalescence among ﬁllers.
Unfortu-nately, few studies have been reported for simulating the
coalescence of LMPA ﬁllers for interconnection process.

For a simulation of conduction path formation in the self-organized interconnection of ACA, the two phase ﬂow including molten ﬁllers and resin should be ﬁrst studied. Figure 2 illustrates the formation of diﬀerent interfaces during phase transition. For two-phase ﬂows, a time evolution of distinct interfaces between two diﬀerent materials seems diﬃcult to be described because of complex-ity of related phenomena. In this case, the surface tension becomes more dominant than the gravity force. The analysis of the mush zone at the phase boundary aﬀects the mass conservation. Lee et al.4) performed a two-dimensional simulation on the self-organized interconnection using ACA with LMPA ﬁllers and analyzed wetting and coales-cence characteristics during the process as seen in Fig. 3.

However, their results were obtained for non-thermal single liquid phase with a high density ratio.

Generally, the surface tension makes pressure and velocity
discontinuous and thus treating the phase boundary of the
ﬂows in the case of high density changes is important. The
present study adopts the cubic interpolated propagation (CIP)
method5–10) _{using a semi-Lagrangian method for a }
multi-phase ﬂow problem. This method consists of the advection
term and the non-advection term. Not that the former is
related to especially solving the wave equation for a
convection part. Welland et al.11) _{applied the phase ﬁeld}
function to a two phase conduction problem considering the
phase change with the treatment of the numerical mush zone.
Shyy et al.12) used the phase ﬁeld value as a function of
temperature using the enthalpy method. In addition, Takada
et al.14) investigated the coalescence of single liquid drop
into a liquid ﬁlm in a gas under gravity with a high density
ratio in three dimensions.

Fig. 1 The schematic of the self-organized interconnection of ACA with the low melting point alloy ﬁller bonding process.

Fig. 2 Formation of diﬀerent interfaces during the phase transition from solid to liquid (solid: solid tin, liquid 1: molten tin, liquid 2: resin).

*_{Corresponding author, E-mail: [email protected]}

As a matter of fact, the present study can be regarded as the ﬁrst step towards developing a computational model for the self-organized interconnection process. The non-equilibrium phase ﬁeld model (NPFM) is proposed in the present study for analyzing the three-dimensional coalescence character-istics including phase transition. Unlike the previous works,4) the thermal-induced phase change is simulated with the use of the CIP method and the implicit phase ﬁeld method. The numerical solutions using the proposed method are validated by being compared with the analytic solutions for the one dimensional solidiﬁcation problem. In particular, the present study compares the NPFM with the conventional enthalpy method which has been widely utilized in many commercial codes, and discusses the diﬀerence between two models. The numerical results for the phase change from solid to liquid and subsequent coalescence of two diﬀerent LMPA ﬁllers are explained in detail.

2. Theoretical Approach and Mathematical

Represen-tation

2.1 Non-equilibrium phase ﬁeld method (NPFM)

Typically, the molten process of solid ﬁllers undergoes clearly non-equilibrium phase change for a very short time. However, the conventional enthalpy method assumes that phase change takes place under the thermodynamic equi-librium. The present study utilizes the non-equilibrium phase ﬁeld method (NPFM) associated with the Gibbs free energy for phase transition. This study solves the mass, momentum (including pressure), and energy conservation equations by using the C-CUP (CIP-Combined Uniﬁed Procedure) method.8)

@

@t þu rþr u¼0 ð1Þ

@u

@t þu ru¼ rp

þQuþgþ

rf

ð2Þ

@p

@t þu rp¼ c

2

sr u ð3Þ

@T

@t þu rT ¼ Tr uþ 1

r

k

Cp

rT

@qðÞ

@

H

Cp

þKuðÞ

Cp

_

ð4Þ

whererepresents the density,uis the velocity, andpis the
pressure. g, , , and cs indicate the gravitational
accel-eration, the surface tension coeﬃcient, the curvature, and
the speed of sound, respectively. The symbol is deﬁned
the phase-ﬁeld parameter which indicates the diﬀusive
interface proﬁle in the phase-ﬁeld method and its range is
0< <1. If the value of is set to unity, the cell is
composed of phase. In the present study, the smoothed
phase-ﬁeld parameter f is also used to avoid the singularity
appearing at the interface.7) _{k} _{and} _{C}

p are the thermal conductivity and the speciﬁc heat, respectively. In the mush zone, the value of the sound of speed cs inside the mush zone is interpolated linearly between constants of solid ﬁllers and liquid ﬁllers. The same interpolation rule is applied to the viscosity,, the thermal conductivity,k, and the speciﬁc heat, Cp. The force term Qu represents the

viscous force. The present model calculates the surface tension by using the method of continuum surface tension by Brackbill.13) In the right hand side of eq. (4), the third term represents the heat source for general-phase change and the rate of change in -phase can be expressed as follows:

@

@t þu r

¼ r uM

_{1}

T
_{@}_{q}_{ð}_{}_{Þ}

@ Gþ

@KuðÞ

@

"2_{}r2

ð5Þ

whereG is the diﬀerence of Gibbs free energygg, and the smoothing function qðÞ and Ku are obtained as follows:11)

qðÞ ¼3ð6215þ10Þ ð6Þ

KuðÞ ¼W2ð1Þ2 ð7Þ

The function Ku describes the excess surface energy of the materials on the boundary between solid and liquid. The constants,W,", andM, are determined from11)

W¼6= ð8Þ

"2_{}¼12=T ð9Þ

M¼k 5

T

H

2

ð10Þ

where is the mush zone thickness, T is the saturation temperature.H is the latent heat, representing enthalpy diﬀerence betweenand phases.

2.2 Numerical algorithm

The governing equations are divided into the convection term and the non-convection term. The equations presenting the convective term are solved by using the CIP method as follows:8)

**Width, ****x****/**µ**m**

**Height, **

**y**

**/**

µ

**m**

-150 -100 -50 0 150

0 100

(a) Initial state

**Width, ****x****/**µ

50

**m**

**Height, **

**y**

**/**

µ

**m**

-150 -100 -50 0 150

0 100

(b)*t* = 10 ms at µresin= 0.38 kg/m•s

50 100

100

[image:2.595.64.276.70.283.2]_{}_{}n

t þu

n_{ r}_{}n_{¼}_{0} _{ð}_{11}_{Þ}

u_{}_{u}n

t þu

n_{ r}_{u}n _{¼}_{0} _{ð}_{12}_{Þ}

p_{}_{p}n

t þu

n_{ r}_{p}n_{¼}_{0} _{ð}_{13}_{Þ}

T_{}_{T}n

t þu

n

rTn¼0 ð14Þ

_{}_{}n

t þu

n

rn¼0 ð15Þ

where _{,}_{u}_{,} _{p}_{,}_{T}_{, and} _{} _{indicate the predicted values}

from the CIP method, whereasn_{,}_{u}n_{,} _{p}n_{,}_{T}n_{, and}_{}n _{mean}

the old values obtained at the previous time. The time stept

istnþ1_{}_{t}n_{. On the other hand, the non-advection term can}

be described as

nþ1

t ¼

r unþ1 ð16Þ

unþ1_{}_{u}

t ¼

rpnþ1

þQ

uþgþ

rf

ð17Þ

pnþ1_{}_{p}

t ¼

_{c}2

sr u

nþ1 _{ð}_{18}_{Þ}

Tnþ1_{}_{T}

t ¼ T

_{r }_{u}nþ1_{þ} 1

r k

Cp

rTnþ1

@qð
nþ1_{Þ}

@

H

_{C}

p

þKuð
nþ1_{Þ}

_{C}

p

_

nþ1 ð19Þ

nþ1_{}_{}

t ¼

_{r }_{u}nþ1_{}_{M} 1

Tnþ1

@qð_{Þ}

@ Gþ

@KuðÞ

@

"2_{}r2nþ1

ð20Þ

By taking divergence of eq. (17) and substituting it into eq. (18), we obtain consequently the pressure equation as follows:

r 1

rp

¼ 1

_{c}2
st2

pþ 1
tr *u*

ð21Þ

u_{}_{u}

t ¼

rp

þQ

uþgþ

rf

ð22Þ

p¼pnþ1p ð23Þ

3. Results and Discussion

3.1 Analysis for one-dimensional solidiﬁcation

In order to validate the proposed model, the present study solves the well-known one-dimensional solidiﬁcation prob-lem for water as shown in Fig. 4. In this probprob-lem, water is changed to ice and their interface moves along the direction of solidiﬁcation with the phase boundary.13) The mesh is a structured uniform grid. The grid system consists of55 50and the minimum mesh size is 0.02 m. The time step of the simulation is 0.01 s. Because the exact solution can be obtained from the equilibrium assumption, the

one-dimen-sional equilibrium simulation is performed for consistent comparison by using the equilibrium phase ﬁeld (EPF) method. The analytical solution can be easily determined as

x¼2y ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃkt=Cp

p

whereyis determined from

ﬃﬃﬃ

p

yey2erfðyÞ ¼ ðTfT0ÞCp=Lsf ð24Þ

where Lsf is the latent heat, k is thermal conductivity, is density, andCP is speciﬁc heat.Tf is the temperature of water and T0 is the temperature of boundary. The Stephan number, Ste, is deﬁned as (Tf T0) CP=Lsf, and its value is 0.0622 in the present case. It is assumed that the phase boundary is determined when the phase ﬁeld value becomes 0.5. Figure 5 shows analytically and numerically predicted positions of the phase boundary during solidiﬁcation. The numerical predictions are in good agreement with the analytic solutions, indicating that the present model would be suitable for simulation of phase transition problem. Fig. 4 The schematic of the one-dimensional solidiﬁcation problem.

**Time, ****t ****/ s**

**Position (m)**

10 20 30 40 50

0 0.02 0.04 0.06 0.08 0.1

**Theoretical results**
**PFM in equilibrium state**
**x = 2y k/( C )t**√ ρ **p**

**position**

3.2 Phase transition and coalescence characteristics of alloy ﬁllers

In the real circumstance, solid ﬁllers undergoing phase
changes are in the non-equilibrium state and when the
melting process is completed, larger liquid drops are yielded
from the coalescence process. Due to the complicated phase
change process, most of researchers5)_{did not directly include}
the thermal-induced phase change process in their
simula-tion, and instead assumed that solid ﬁllers have been already
molten. Consequently, the liquid phase has been only treated
in their simulation. Unlike the previous works, the present
study concentrates mainly on the coalescence process and
the phase change of LMPA ﬁllers.

Figure 6 illustrates the schematic of the coalescence
simulation and shows that two solid ﬁllers are initially
located in the liquid resin. Using the structured uniform grid,
the computational grid system consists of202020with
the minimum mesh size of 5mm. The time step of simulation
is taken as 1109_{s. Symmetric boundary conditions are}
applied for the momentum conservation equation, Constant
temperature conditions are used at two boundaries in the x

direction, and their temperatures equally are 535 K, which is higher than the melting temperature of 506.2 K. In addition, the adiabatic condition is used for other boundaries. The physical properties are listed in Tables 1 through 3.

Figure 7 depicts the normalized mass predicted by the enthalpy method with respect to time during the phase change. The normalized mass can be determined by the normalization of the mass of each phase with the total mass.

When the mass conservation is satisﬁed, the normalized mass of sum of both phases should be equal to unity. The solid phase does not completely change to the liquid phase and the mass fractions of two phases are nearly equal, which indicates such a compulsive equilibrium state. Note that the phase equilibrium assumption in the enthalpy method violates the real existence of the non-equilibrium state during the phase change.

Figure 8 shows the transient behaviors of two diﬀerent ﬁllers undergoing the phase transition. An initial solid state begins to change toward mixed phase state, indicating that liquid and solid phases coexist as time goes on. The melting process is completed after 1.2ms, when two diﬀerent liquid phases of molten ﬁllers and resin simultaneously exist with the high density ratio.

A mush zone is deﬁned to be an artiﬁcial region having
mixed diﬀerent phases. Note that the thickness of the mush
zone may be an important factor to obtain precise solutions
of multiphase problems of interest. Figures 911 show the
variation of the predicted normalized mass during the phase
change for diﬀerent mush zone thicknesses, i.e., 0.5, 1.0,
**X**

**Y**

**Z**

X

Y Z

**Resin**

**Filler**

**Uniform**
**temperature**
**T=535 K**
**Adiabatic**

**100 m**µ
**100 m**µ

**100 m**µ **Uniform _{temperature}**

**T=535 K**

Fig. 6 The schematic of the coalescence simulation for two diﬀerent solid ﬁllers during the phase transition.

Table 1 Physical properties of resin.15Þ

Property Value

Density,resin/kgm3 1150

Viscosity,resin/Pas 0.38

Speciﬁc heat,Cp,resin/Jkg1_{}_{K}1 _{213}
Thermal conductivity,kresin/Wm1_{}_{K}1 _{43.2}
Surface tension coeﬃcient,resin/Nm1 0.043

Sound speed,cs,resin/ms1 _{932.5}

Table 3 Physical properties for phase change.

Property Value

Melting temperature,Tmelt/K 506.168

Evaporation temperature,Tevap/K 2875.15
Melting heat,Lsolid,liquid/Jm3 _{595}_{}_{10}3

Latent heat,Lliquid,gas/Jm3 _{60}_{:}_{5}_{}_{10}5

**Time, ****t ****/ **µ**s**

**Normalized mass (%)**

0 0.05 0.1 0.15 0.2 0.25 0.3

0 50 100 150 200

**Solid Tin**
**Melted tin**
**Solid + Melted tin**

**Enthalpy Method**

Fig. 7 The normalized mass variation predicted by the enthalpy method. Table 2 Physical properties of solid and molten ﬁller.15Þ

Property Solid ﬁller Molten ﬁller Density,tin/kgm3 7290 6986

Viscosity,tin/Pas 1010 0.00181

Speciﬁc heat,Cp,tin/Jkg1_{}_{K}1 _{213} _{213}
Thermal conductivity,ktin/Wm1K1 63.2 63.2

Surface tension coeﬃcient,tin/Nm1 0.574 0.574

[image:4.595.305.549.221.291.2] [image:4.595.313.533.226.500.2] [image:4.595.47.292.316.416.2]and 1.5 times of unit mesh size, respectively. Compared to
the enthalpy method, the NPFM can predict the complete
phase change from solid to liquid. The enthalpy method
with the equilibrium assumption determines the phase ﬁeld
value from only the temperature ﬁeld, which excludes the
diﬀusion term M"2_{}, whereas the NPFM includes the
diﬀusion eﬀect as well as the surface tension eﬀect at the
phase boundary by estimating the rate of change in by

using eqs. (5) and (20). Thus, the enthalpy method would be inappropriate for the simulation of the self-organized interconnection process where the transient phase change becomes substantially important, in which the non-equi-librium state should be considered for these kinds of calculations. From the results, the times to reach the phase equilibrium are estimated 0.16, 0.42, and 0.84ms for the mush zone thicknesses of 0.5, 1.0 and 1.5 times the unit mesh size, respectively. For the mush zone thicknessequal to 1.0 and 1.5 times the unit mesh size, the ratios of increase in ﬁnal ﬁller mass to initial ﬁller mass are estimated 37.8% and 33.7%, respectively, compared to 2.75% for the case of 0.5 times the mesh size. These results indicate the normalized mass of molten ﬁllers greater than unity. These unreasonable results obtained show that the predictions of the phase change highly depends on the numerical mush zone thickness.

From the sensitivity analysis with respect to the mush zone thickness, the mass conservation is violated, while the normalized mass must be unity in the ideal case. However, it turns out that the results from the NPFM simulation are not unity in cases of the mush zone thicknesses of 1.0 and 1.5 times the unit mesh size. This is due to the basic assumption related to the derivation ofW,", andM. The constants, W,", andM, related to the mush zone thickness in eq. (5),

were derived by the assumption of sharp interface.11) By using the sharp interface assumption, the left part of eq. (25) is ignored and the constants are derived.

Fig. 8 Evolution of the estimated phase transition from solid to liquid and the corresponding coalescence of two ﬁllers.

**Time, ****t ****/ **µ**s**

**Normalized mass, kg/kg**

0 0.05 0.15 0.2 0

0.5 1 1.5 2

**Solid Tin**
**Melted tin**
**Solid + Melted tin**

**Mush zone thickness**
**= 0.5 times of mesh size**

**t****_peq _{=0.16}**µ

_{s}0.1 0.25 0.3

Fig. 9 The normalized mass variation during the phase change when the mush zone thickness is equal to 0.5 times of mesh size.

**Time, ****t ****/ **µ**s**

**Normalized mass, kg/kg**

0 0.4 0.8 1.2 1.6 0

0.5 1 1.5 2

**Solid Tin**
**Melted tin**
**Solid + Melted tin**

**Mush zone thickness**
**= 1.5 times of mesh size**

**t****_peq=0.82**µ**s**

Fig. 11 The normalized mass variation during the phase change when the mush zone thickness is equal to 1.5 times of mesh size.

**Time, ****t ****/ **µ**s**

**Normalized mass, kg/kg**

0 0.2 0.4 0.6 0.8 0

0.5 1 1.5 2

**Solid Tin**
**Melted tin**
**Solid + Melted tin**

**Mush zone thickness**
**= 1.0 times of mesh size**

**t****_peq _{=0.42}**µ

_{s}1 M

@

@t þu rþr u

¼ 1

T

@qðÞ

@ Gþ

@KuðÞ

@

"2_{}r2

ð25Þ

However, if the 1

M

2_{is big, the left part of eq. (25) cannot}
be neglected and this may result in dissatisfaction of the
conservation. To solve this problem, as the future work, the
material derivative term of the phase ﬁeld parameter in
eq. (25) should be included and treated carefully for ﬁnding
the optimal solution for mush zone thickness.

4. Conclusions

This article describes the simulation of coalescence and phase transition characteristics of low melting point alloy (LMPA) ﬁllers by using non-equilibrium phase ﬁeld method with the use of the Navier-Stokes equation. This work has great meaning as the ﬁrst step towards simulating three-dimensional self-organized interconnection process includ-ing the ACA process. The conclusions are summarized as follows: First, the proposed non-equilibrium phase ﬁeld model (NPFM) successfully predicted the thermal-induced phase change from solid to liquid, whereas the enthalpy method failed to describe the non-equilibrium phase change. The proposed model was validated for the one-dimensional solidiﬁcation problem, and the corresponding predictions were in quite good agreement with the analytic solutions. It turns out that the enthalpy method would be inappropriate for the simulation of the self-organized interconnection process where transient phase change becomes substantially impor-tant. Note that the non-equilibrium state during phase changes should be necessarily considered for calculations. Finally, the presence of the grid dependency on the mush zone indicates that the mush zone treatments in capturing

interfaces between two diﬀerent phases should be carefully made for better predictions.

Acknowledgements

This work is the outcome of a Manpower Development Program for Energy & Resources supported by the Ministry of Knowledge and Economy (MKE), and it was additionally supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by Korea government (MOST) (No. R01-2006-000-10702-0).

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