the effective permeability of the underfill flow domain in flip-chip packaging

Applied Mathematical Modelling 37 (2013) 1177–1186

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

The effective permeability of the underfill flow domain inflip-chip packaging

Connie Yang a, Wen-Bin Young b,⇑a Department of Systems and Naval Mechatronic Engineering, National Cheng Kung University, Tainan, Taiwan, ROCb Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan, ROC

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 October 2010Received in revised form 16 March 2012Accepted 20 March 2012Available online 29 March 2012

Keywords:UnderfillFlip-chipCapillary flowPower-law fluid

0307-904X/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.apm.2012.03.036

⇑ Corresponding author.E-mail address: [email protected] (W.

In order to reach the goals of high electrical performance and dense packaging within thelimited space, the flip-chip technology becomes popular in electronics packaging. In theflip-chip assembly, difference between thermal expansion coefficients of the chip and sub-strate may cause thermal fatigue at solder joints. To avoid this thermal fatigue, epoxyencapsulant is filled into the gap between the substrate and chip by the capillary force.Because of the small space in the flow domain, the underfilling flow can be assumed asa flow in porous medium. Permeability is used to characterize the flow field of the spaceamong the substrate, chip, and solder bumps. In this study, a numerical method is usedto determine the effective permeability for the underfilling flow domain. Analysis of thethree dimensional flow in a unit cell of the underfill flow domain is performed. The result-ing average velocity and pressure gradient are used to calculate the apparent permeability.Comparison with the analytical approximation for the permeability in literature is also per-formed. The effective permeability calculated using the proposed numerical method givesreasonable prediction of the underfill flow as compared to the experimental result.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Flip-chip packaging is an integrated circuit (IC) packaging technique that uses solder bumps to connect chip with sub-strate. The mismatch of thermal expansion coefficients tends to cause fatigue at solder junctions. Epoxy encapsulant is usedto solve this problem and improve the reliability of flip-chip packaging. The encapsulant is filled into the gap between thechip and substrate by the capillary force so that the thermal stresses may disperse into the underfill materials to avoid crackgeneration.

Plenty of studies related to the underfill process can be found in literatures. The Washburn model for flow in betweenparallel plates is the most usual assumption to describe the filling flow in underfill encapsulation [1–9]. Han and Wang[5] used a Hele–Shaw model to perform both theoretical and experimental studies. Since they underestimated the effectof solder bumps, the results were deviated from the actual flow. Nguyen et al. [10] used quartz dies to perform underfillexperiments with different bump arrangements. They found that the velocity of the flow front at boundary was larger thanthose at the center due to the edge effect. Fine et al. [11] used different bump densities at center and outside regions. Theyreported that lowering the bump density at center region can get a more uniform flow front, and lower the boundary effects.Young and Yang [2], Young [4] treated the underfill flow domain as a porous medium, and described the flow field with amodified Hele–Shaw model. The fluid was treated as a highly viscous fluid, and the flow was driven by the capillary force

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1178 C. Yang, W.-B. Young / Applied Mathematical Modelling 37 (2013) 1177–1186

among the chip, substrate, and bumps. Together with Darcy’s law, they predicted the flow front evolution in the underfill.Young and Yang [12] studied the effect of contact angle and bump arrangement. They found smaller contact angles have lar-ger capillary pressures. They also reported both experimentally and theoretically that increased the bump pitch did not alterthe filling time much until it reached a critical value. Wan et al. [13] proposed a numerical model for the prediction of flip-chip underfill flow. In this model, the power-law constitutive equation was used to describe the non-Newtonian behavior ofencapsulant fluids and a time-dependent velocity boundary condition was applied.

Epoxy molding compound with silica fillers is used as the underfill material, and it usually exhibits a non-Newtonianbehavior during the filling process. Thus, a non-Newtonian flow effect must be considered in modeling the underfill flow.Because the underfill flow domain contains many solder bumps, it is quite complex in geometry for the flow simulation.A simpler way to avoid complex geometry is to treat the flow domain as a porous medium. Under this assumption, the cor-responding permeability is the major characteristics of the flow domain. In this study, we analyze the flow field of a three-dimensional unit cell in the flow domain. The underfill material is treated as a non-Newtonian fluid. The effective permeabil-ity can be determined by the corresponding average velocity and pressure gradient derived from the flow field of the unitcell. Comparison with the analytical approximation for the permeability in literature is also performed. The effects of thebump stand height, pressure gradient, and bump size on the effective permeability are discussed.

2. Permeability of the non-Newtonian flow

In order to simplify the analysis of the underfill flow, the flow field is assumed as a flow through a space between twoparallel plates with a cylinder bank, as shown in Fig. 1. The cylinders represent the bumps while the parallel plates representthe chip and substrate. To further simplify the analysis, the flow field is treated as a flow through porous medium. For a por-ous medium the Darcy’s law reads:

u*¼ �

��kl� rP; ð1Þ

where u*

is the velocity vector, ��k is the permeability tensor l is the viscosity, and p is the pressure. The permeability is a valueused to characterize the fluidity of the porous medium and is usually has a unit of m2. For a non-Newtonian fluid, the powerlaw can be used to model the shear rate dependent viscosity, and is written as:

l ¼ m _cn�1; ð2Þ

where m is the consistency index, _c s the shear rate, and n is the power-law index. For n < 1 the fluid is called a shear thinningfluid, or pseudo plastic fluid. For n > 1 the fluid is called a shear thickening, or dilatant fluid. The power-law viscosity modelhas good accuracy at high shear rate, but it tends to overestimate the viscosity at low shear rate. To solve this problem, alimit value for the viscosity is given, and the power-law is modified as:

l ¼ m _cn�1 for _c > _cc;

l ¼ mo ¼ m _cn�1c for _c 6 _cc;

ð3Þ

where _cc is the critical shear rate for the underfill encapsulant. The power law will result in an unreal large value of viscosity atthe low shear rate. In reality, the viscosity of a non-Newtonian fluid usually reaches a constant value called the zero-shear-rateviscosity as the shear rate is getting lower. Therefore, it can be defined a critical shear rate to avoid the failure of the power-lawmodel at the low shear rate. A constant viscosity is used when the shear rate is below this value. For flow between two parallelplates, the average viscosity can be derived as:

Fig. 1. The schematic diagram of the flow domain in a flip-chip underfilling process.

C. Yang, W.-B. Young / Applied Mathematical Modelling 37 (2013) 1177–1186 1179

�l ¼ 2h

Z h2

0m

1nz

n�1n

dPdx

��

n�1n

!dz ¼ d

2n�1n

n� 12n� 1

þ n2n� 1

� �m

1n

dPdx

��

n�1n h

2

� �n�1n

; ð4Þ

where h is the distance between the chip and substrate or the bump height, and the average velocity is:

�u ¼ 2h

Z h2

0udz ¼ 1

m

� �1n n

2nþ 1

� �dPdx

��

1n h

2

� �nþ1n

: ð5Þ

Using the above model of average viscosity and velocity, we can derive the analytical permeability between two parallelinfinite flat plats based on Darcy’s law. The permeability is given as:

k ¼ n4ð2nþ 1Þ d

2n�1n

n� 12n� 1

þ n2n� 1

� �h2 ¼ kh2

; ð6Þ

where

k ¼ n4ð2nþ 1Þ d

2n�1n

n� 12n� 1

þ n2n� 1

� �; ð7Þ

d ¼ 2m _cnc

hjdP=dxj : ð8Þ

Consider the permeability for two parallel plates with a cylinder bank, the permeability model should be modified as follow[14]:

�k ¼ 1

kh2 þR d

20

dxkc3

R d20 cdx

d2

� �2

0@

1A�1

¼ k1

h2 þ2Sd� p

2

� �Z d2

0c�3dx

!�1

; ð9Þ

where

c ¼ S�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2 � 4x2

q; ð10Þ

S is the bump pitch, and d is the bump diameter. In Eq. (9), we use a simple model to calculate the permeability of the under-fill flow domain by averaging the permeability between two parallel plates and solder bumps. This analytical model neglectsthe effects of three-dimensional flow.

In order to understand the three-dimensional effects on the permeability, a three-dimensional unit cell flow model isused to simulate the underfill flow, as shown in Fig. 2. The unit cell is defined between two solder bump and representsthe basic space for the flow domain in the underfill. By analyzing the steady-state flow through the unit cell, the fluidityof the unit cell can be characterized by the effective permeability defined in the following. ANSYS FLUENT is used to simulatethe flow in the unit cell model. No-slip boundary conditions are applied at the bottom boundary and cylinder walls whichrepresent the substrate and bumps, respectively. The top boundary where we apply the symmetry boundary condition is setat the mid-plane between the substrate and the chip. Pressure difference is applied across the front and rear boundaries.

Fig. 2. A three-dimensional unit cell model for the underfill flow.

Fig. 3. The finite element mesh of the three-dimensional unit cell model.

Fig. 4. The viscosity distribution on the unit cell model.


From the average velocity and pressure gradient obtained by the simulation, one can find the effective three-dimensionalpermeability. In the simulations, the fluid density is set at 1600 kg/m3, the consistency is 0.924 N-s/m2, the power-law indexis 0.76, the critical shear rate is 0.016 s�1, the bump diameter is 100 lm, the bump pitch is 200 lm, and the height is 50 lm.The effective permeability is defined as:

ke ¼ ��unm0

jdP=dxj ; ð11Þ

Fig. 6. Analytical and numerical effective permeability for different power law index.

Fig. 5. A typical velocity field at the mid-plane between the chip and substrate.


where �un is the average velocity from the numerical simulation. The analysis is based on the steady-state pressure drivenflow through the unit cell to determine the permeability of the geometric structure. The effective permeability also dependson the pressure gradient nonlinearly. The approximation for the effective permeability in the underfill simulation will


depend on the pressure gradient at different locations. Based on the argument, the effective permeability with respect topressure gradient must be determined for a given underfill geometric structure. After that, the simulation can be performedwith assumed porous medium for the underfill flow domain. Based on the geometric data and the pressure difference at eachlocation, the corresponding permeability can be determined for the calculation. In actual underfill flow, the pressure differ-ence is low and its effect on the permeability is low and can be neglected for simplicity. Certainly, some error in associatedwith this assumption may occur in the simulation. The effective permeability can be related to the permeability as:

Fig. 7.the hei

ke ¼�k�l

m0: ð12Þ

The effective permeability as a function of pressure gradient for a given geometry, and the bump diameter is 100 lm, the bump pitch is 200 lm, andght is 50 lm.

Fig. 8. The effective permeability as a function of the bump pitch, and the bump diameter is 100 lm, and the height is 50 lm.

3. Results and discussions


In the numerical simulation, the meshing number should be large enough to reach a converging flow rate. The numericalerror may reduce to an acceptable range as the flow rate converges. In our studies, the flow rate converges as the meshingnumber reaches 500,000. The finite element model of the three-dimensional unit cell after meshing is shown in Fig. 3.

For a non-Newtonian fluid, the viscosity in the flow domain varies with locations, as shown in Fig. 4. Since the underfillencapsulant in this study is considered as a shear thinning material, larger viscosity value is obtained at locations with smal-ler shear rate, and vice versa. The viscosity is quite low at the wall of the bump due to the high shear rate in this region. Atypical velocity distribution at the mid-plane between the chip and substrate is show in Fig. 5. Higher velocity is observed atthe region between the bumps because of the narrow width.

The effective permeability as a function of power-law index is shown in Fig. 6 both analytically and numerically. Theanalytical data are obtained by Eqs. (9) and (12), and the numerical data are obtained by applying the numerical resultsto Eq. (11). For the Newtonian flow as n = 1, the analytical and numerical effective permabilities are also quite different.The numerical result is smaller, and the difference comes from the three-dimensional effect that is neglected in the analyticalmodel. The effect of the three dimensional flow on the effective permeability is significant and must be considered in order tohave correct analysis of the underfill flow. It is obvious that the effective permeability determined by this numerical methodwill give the better approximation for the underfill flow. It must be noticed that definition of effective permeability in Eq.

Fig. 9. The effective permeability as a function of the bump height, and the bump diameter is 100 lm, and the bump pitch is 200 lm.

Fig. 10. The effective permeability as a function of the bump diameter, and the bump pitch is 200 lm, and the height is 50 lm.


(11) using the zero shear rate viscosity instead of the viscosity. During the simulation of an underfill flow using the Darcy’slaw, the average velocity is calculated instead of the actual velocity. Therefore, the true shear rate cannot be available todetermine the viscosity in the simulation. With the effective permeability, a constant zero shear rate viscosity is used inthe simulation and the dependent of the flow on the shear rate is accounted for by the effective permeability.

The effective permeability defined in this study is not only a function of the geometry but also the velocity. Since thevelocity is related to the pressure gradient, the effective permeability may change with the pressure gradient for a givengeometry. Fig. 7 shows the dependance of effective permeability on the pressure gradient, it increases with the pressuregradient.

Bump pitch is the distance between two nearby bump centers. The effective permeability increases as bump pitch in-creases, and approaches a constant value, as shown in Fig. 8. For large length of bump pitch, the viscous force caused bythe bumps is much less that by the chip and substrate. Thus, the effective permeability will cease to vary with the bump

Fig. 11. Numerical effective permeability for different power law index and bump pitch.

Fig. 12. Numerical effective permeability for different power law index and bump diameter.


pitch and be dominated by the bump height. Figs. 9 and 10 show the effects of bump height and diameter on the effectivepermeability, respectively. Under the same pressure gradient, as the bump height increases, the fluid has more space to flow,and the effective permeability increases. As the bump diameter increases, the permeability continuously decreases andreaches zero until the bump diameter is equal to bump pitch. Nearly linear relation is obtained for the bump diameter.

Figs. 11–13 shows the ke � n diagrams with different bump pitch, diameter, and height, respectively. The effective per-meability decreases as power-law index increases for all cases. The effective permeability slightly increases as bump pitchincreases, diameter decreases, or height increases. The results also show that the effective permeability is highly depends onthe power-law index. Different bump size or pitch has little effects on the effective permeability. Compared to the bumppitch and size, the bump height has more effect on the effective permeability.

To compare with the available experimental result of the underfill flow, a chip size of 25 � 25 mm2 with a full array ofbump pattern is selected from in the literature [15]. The underfill process is performed under a constant temperature,90 �C. The contact angle between the encapsulant and glass substrate and chip is 34.73�. The contact angle between theencapsulant and solder bump is 65.26�. The surface tension is 0.0241 N/m. A constant average viscosity, 0.48 Pa s, is used

Fig. 13. Numerical effective permeability for different power law index and bump height.

Fig. 14. Calculated and experimental results of underfill filling rate with respect to time for a flip-chip on board.


in the simulations for the encapsulant assuming that the degree of reaction is low during the underfill process. The bumpdiameter, bump pitch, and bump height are 150, 300, and 100 lm, respectively. From the three-dimensional unit cell flowsimulation, the effective permeability for this flip-chip design is 4.07 � 10�10 m2. The pressure dependence is neglected forsimplicity due to its minor effect at the low pressure difference in actual underfill flow. The corresponding capillary pressurefor the flip-chip design is 471.91 N/m2 based on the formula in the literature [15]. The underfill flow of the flip-chip can bederived from Eq. (1) for a Newtonian fluid as:

l2 ¼ 2Poketmo

; ð13Þ

where l is the underfill flow distance and Po is the capillary pressure. With Eq. (13), the comparison of calculated and exper-imental filling percentage of the flip-chip is shown in Fig. 14. The prediction of the filling flow gives a reasonable result basedon the effective permeability.

4. Conclusions

In the flip-chip underfilling flow field, the flow domain is simplified as two parallel plates with cylinder bumps, and non-Newtonian power-law is used to model the viscosity of the encapsulant. A three-dimensional unit cell model is proposed tosimulate the underfill flow. The simulated average velocity and pressure gradient are used to determine the effective per-meability. Using the non-Newtonian viscosity model to predict the permeability gives a result closer to the actual flow field,especially for small scale systems. The effect of the three dimensional flow on the effective permeability is significant andmust be considered in order to have correct analysis of the underfill flow. To obtain a higher effective permeability, a smallerpower-law index is desired. For underfill encapsulant that has the power-law index around 0.7–0.8, the corresponding non-Newtonian effects are not significant. The most direct way to increase the permeability is to change the bump pitch and size.Increasing the bump pitch, increasing the bump height, or decreasing the bump diameter can all increase the permeability.

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the effective permeability of the underfill flow domain in flip-chip packaging

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