Mathematical modeling of the wall effect on drag forcesin molding ¯ow using optical ®ber sensing data

Yu-Lung Lo*, Hsin-Yi Lai, Ming-Hong Tsai

Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan

Received 15 August 1998

Abstract

This paper studies the wall effect on drag forces in molding ¯ow using optical ®ber sensing data. The results indicate that the effect of the

wall cavity is signi®cant in the calculation of the drag forces and is required to be considered in molding. The study also shows the

effectiveness of the approach in characterizing the behavior of the wire sweep in the encapsulation of semiconductor chips. An in-line ®ber

etalon (ILFE) sensor is designed and fused in the middle of the optical ®ber for accurate strain measurement. This is done by ®rst laying an

optical ®ber in the mid-plane of a simple rectangular mold cavity, and then measuring the strain at the mid-span of the optical ®ber that is

subjected to the ¯ow of homogeneous ¯uid. For a given ¯ow ®eld, several drag force models have been used to calculate the drag forces

over the optical ®ber. The resulting strain of the optical ®ber is estimated analytically ®rst by FEM and then compared with experimental

results. To study the wall effect, different sizes of mold cavities are employed for different b/a ratios and used in Takaisi's model.

Subsequently, Takaisi's model is then modi®ed to comply with the experimental data obtained by using various b/a ratios for comparison

with the numerical drag force model computed by Han and Wang. The result indicates that a modi®ed Takaisi model in compliance with the

requirement of b/a within the range 20±200 is in good agreement with the numerical model proposed by Han and Wang who had taken

effect of the wall cavity into consideration for drag-force calculation. The modi®ed model based on a smaller b/a value is somewhat closer

to the numerical model in a higher average cavity velocity. The wall effect of molding ¯ow is concluded to be an important factor for

various tests conducted at a greater average cavity velocity or a smaller value of b/a. # 2000 Elsevier Science S.A. All rights reserved.

Keywords: Wall effect; Molding ¯ow; Drag force; Optical ®ber sensors

1. Introduction

Mathematical modeling of wire-sweep problems has

emerged as a research topic of great importance in recent

years, since the technique can be used to go beyond tradi-

tional means in the quality control of IC packaging pro-

cesses. The wire-sweep problem has been studied by many

researchers and several key papers have been published

recently. Reusch [1] used an electrical analog to analyze

the wire-sweep problem. However, a quantitative model to

relate wire sweep to ¯ow parameters was not offered.

Nguyen [2] proposed an experimental setup that was done

by encapsulating the chips with an epoxy-molding com-

pound. X-ray scanning was used to reveal the wire shape

inside the package. Han and Wang [3] studied drag forces of

the gold wire sweep by using a video system. The recorded

deformation data was measured on the screen from the

image.

Lo et al. [4] proposed a new experimental setup using an

optical ®ber to measure the deformation induced by the drag

force for molding ¯uid ¯owing over the ®ber object. Since

the optical ®ber is a perfectly circular cylinder, and the

properties of the optical ®ber are perfectly elastic, it can be

used to verify the existing drag force models precisely.

Following this methodology, in the present work the wall

effect is investigated by changing the size of the mold cavity.

Subsequently, the axial strain is computed using the FEM

method. Various drag force models, including Sherman's

model [5], Lamb's model [6], and Takaisi's model [7], are

used to compare with experimental results for different ¯ow

velocities. Also, the Takaisi drag force model for different

mold cavity sizes is modi®ed based upon experimental data.

Finally, the results are compared to the predictions of the

numerical models proposed by Han and Wang.

Journal of Materials Processing Technology 97 (2000) 174±179

*Corresponding author. Tel.: �886-275-7575; fax: �886-235-2973

E-mail addresses: loyl@mail.ncku.edu.tw (Y.-L. Lo),

hylai@mail.ncku.edu.tw (H.-Y. Lai)

0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved.

PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 3 4 3 - X

The major objectives here are to mathematically present

the drag force model and to check the effects of a nearby the

wall and the channel size on both the drag force and lateral

deformation of the wire in mold ¯ow. A comprehensive

modeling example is to be presented at the end of the paper

to verify the effectiveness of the present approach.

2. Drag force modeling in molding flow by Takaisi

Various drag force models have been proposed for study-

ing the behavior of the wire sweep in the encapsulation of

semiconductor chips [2] These models include Sherman's

model [5], Lamb's model [6] and Takaisi's model [7].

According to Han and Wang [3] and Lo et al. [4], the

Takaisi drag force model is close to the expected analytical

model; hence, this paper introduces Takaisi's model for

modi®cation. The ®rst step in wire-sweep analysis is to

obtain the velocity distribution inside the mold cavity

neglecting the details of the optical ®bers or wires. For a

fully developed ¯ow of an incompressible Newtonian ¯uid

in a rectangular mold cavity, the velocity distribution can be

described in terms of the coordinate system given in Fig. 1

[8] as:

u � ÿ6�uy

Hm

� �2

ÿ y

Hm

� �" #; (1)

where �u is the average velocity in the mold cavity section

and Hm is the height of the mold cavity. To consider the

effects of the cavity wall on the drag force, Takaisi [7]

proposed an equation to represent the drag force on the

circular cylinder. Takaisi's drag force equation is given as

D � 4��b0; (2)

where D is the drag force per unit length of the circular

cylinder, � is the fluid viscosity, and b0 is a parameter that

can be expressed as

b0 � u

ln�b=a�ÿ0:9156� 1:7243�a=b�2 ; (3)

where a is the radius of the circular cylinder and b is the

distance between the circular cylinder and the cavity wall as

shown in Fig. 1. Therefore, by substituting Eq. (3) into Eq.

(2), the drag force on the optical fiber can be computed. It

should be noted that the limited value of b/a in 20±200 is

needed, since Takaisi took the cavity wall effects into

consideration in this drag force model.

3. Design of an in-line fiber etalon for sensing

This section describes the optical arrangement and the

implementation of a demodulation scheme speci®cally

designed for in-line ®ber etalon (ILFE) sensors. These

sensors, which are a variation of the EFPI sensor [9], are

composed of a hollow core ®ber fused between two standard

single mode optical ®bers (see Fig. 2) [10,11]. The two

re¯ected waves interfere to produce an optical signal, the

optical phase of which is proportional to the cavity length of

the sensor. The change in this optical phase is given by

D�� 4�

�0

"zzLs; (4)

where �0 is the wavelength of the launching light, "zz is the

axial strain in ILFE, and Ls is the cavity length of the sensor.

Also, the ILFE sensor has the low thermal apparent strain.

Therefore, an ILFE sensor can be applied in this study

properly. The demodulation scheme used in the present

experiments to determine the optical phase change in ILFE

sensors is based on path matched differential interferometry

(PMDI). Fabry±Perot fiber sensors using PMDI techniques

were first proposed by Cielo [12] and there have been many

subsequent variations [10,13]. Fig. 3 shows a PMDI con-

figuration ion that uses a low finesse Fabry±Perot cavity as

the read-out interferometer. The broad-band light is passed

through a fiber-optic coupler to an ILFE strain sensor. The

Fig. 1. Configuration of a mold cavity.

Fig. 2. An ILFE sensor.

Fig. 3. Demodulation system for ILFE sensors.

Y.-L. Lo et al. / Journal of Materials Processing Technology 97 (2000) 174±179 175

reflected light from the ILFE is again passed through a fiber-

optic coupler that formed a low finesse Fabry±Perot cavity

with a mirror. The mirror in the Fabry±Perot cavity is

bonded to a PZT stack that is placed on a linear transition

stage. This stack is used to provide a phase generated carrier

for ILFE demodulation and the translation on stage is used to

adjust the optical path of the read-out interferometer to

match that of the ILFE. This configuration is used to test

the strain measurement capability of the ILFE sensor by

driving he PZT stack with ramp function and using single

channel phase tracker [14] for demodulation. This scheme is

based on an electronic feedback phase nulling technique

using a AD639 trigonometric simulator chip. The concept of

electronic phase nulling is similar to the concept of the

active homodyne technique [15].

4. Analytical model by FEM

Since an ILFE strain sensor is made of a hollow core

implemented in the middle of a single straight optical ®ber

that is attached to the mid-plane of a rectangular cavity, the

FEM model for the geometry of an ILFE is developed by

including the existing drag force models to compute the

axial strain along the ®ber in the middle. In this study, a 3D

solid mold (SOLID 45 can be declared for a large deforma-

tion) is constructed by a well-known structure analysis code

(ANSYS). The calculated strain in the ILFE sensor in the

FEM model illustrated in Fig. 4 needs to be de®ned. Since

the coordinate A is (�x1, �y1, �z1) and the coordinate B is

(�x2 � Ls, �y2, �z2), the change in gage length Ls can be

expressed as

D ������������������������������������������������������������������������������������������������Dx2 � LSÿDx1�2 � �Dy2ÿDy1�2 � �Dz2ÿDz1�2

q:

(5)

Therefore, the axial strain in an ILFE sensor is obtained

by

"x � DLs

: (6)

The analytical results are compared with the experimental

data measured by a novel optical ®ber sensor. Since the

optical ®ber has a perfect circular cylinder shape and

possesses perfect elastic properties, it is suitable to analyze

the molding ¯ow in drag force models in the elastic range. In

this case of a single straight optical ®ber attached to the mid-

plane of a rectangular cavity, simple clamping on both ends

of the optical ®ber can be perfectly analyzed by a FEM

model with Takaisi's drag force models in molding ¯uid

¯ow. Table 1 shows the analytical parameters for the FEM

model. It should be noted that the deformation of an optical

®ber that is ®xed on both ends is very small whilst the ¯uid

passes through it. The drag force, therefore, can be modeled

simply as an original single straight optical ®ber without

considering the deformed shape. Furthermore, due to the

conservative system in a perfect elastic optical ®ber, it is

analyzed by FEM path-independently. Strictly speaking, a

gold wire of elastic±plastic properties needs to be used for

analysis in FEM with the assumption of path-dependent

deformation in molding ¯ow conditions.

5. Experimental setup

The broad-band light (FWHM � 50 nm) from a pigtailed

super luminescent diode source with a nominal wavelength

1.3 mm, 150 mW optical power, was passed through a ®ber±

optic coupler to an ILFE strain sensor. The PMDI con®g-

uration is used to test the strain measurement capability of

the ILFE sensor by driving the PZT stack with a 6 kHz

sinusoidal carrier frequency and 800 Hz cut-off frequency of

a low-pass ®lter in a single channel phase tracker selected for

low frequency response in a molding ¯ow test. Lo et al. [4]

have demonstrated that this optical ®ber system is suitable

for molding ¯ow tests. A schematic diagram of this experi-

mental setup is illustrated in Fig. 5(a). A high pressure gas

tank is used to force the ¯uid to ¯ow through the mold cavity

system. A single straight optical ®ber with an ILFE is

attached to the mid-plane of a rectangular cavity (see Fig.

5(b)) where the ¯ow of a clear and homogeneous ¯uid is

passing through it. An ILFE is implemented in the middle of

the optical ®ber with cavity length of 100 mm. Silicone oil is

chosen, and is injected at a temperature of 228C into the

mold cavity using a high pressure gas tank. In these experi-

ments, the ¯uid velocities are chosen in the range 1±12 mm/

s, for which the calculated shear rate is much smaller than

100/s [4]. Therefore, the viscosity of the silicon oil can be

treated as constant at 300 poise. To investigate the wall

effects on the molding ¯ow to the mold cavity, the different

sizes of the mold cavities are designed such that b/a are 40,

80, and 320. It can be seen that a value of b/a 40 and 80 are inFig. 4. Constructed model of an ILFE.

Table 1

Analytical parameters

Diameter of optica1 fibers (D) 0.125 mm

Length of optical fiber (L) 12 mm

Young's modulus of optical fibers (E) 69 GPa

Silicone oil density (�) 970 kg/m3

Silicone oil viscosity (�) 300 poise (1 poise � 0.1 N s/m2)

176 Y.-L. Lo et al. / Journal of Materials Processing Technology 97 (2000) 174±179

compliance with the requirement of b/a to be within the

range 20±200 in the Takaisi drag force model: the other b/a

value of 320 is beyond the requirements.

The next section introduces three modi®ed Takaisi models

that are developed for a comparison with the numerical

model proposed by Han and Wang, who considered the

effects of the cavity wall in the drag-force calculation.

6. Numerical verification

The calculated axial strains by a FEM method with the

Takaisi model for the different values of b/a � 40, 80, and

320 are compared with experimental results at different ¯ow

velocities as illustrated in Fig. 6(a)±(c). As can be seen that

the calculated results from the Takaisi model for the values

of b/a of 40, 80, and 320 have a little deviation from the

experimental data. Also, it should be noted that the experi-

mental data values for values of b/a of 40 and 80 are below

the calculated results, and that for the value of b/a of 320 the

experimental data is beyond the calculated result. Therefore,

to make a comparison with the drag force model developed

numerically by Han and Wang for investigating the wall

effects, the three modi®ed Takaisi drag force models are

developed according to Fig. 6 by changing some of para-

meters in the models as

b0 � u

ln�b=a�ÿ0:455� 1:7243�a=b�2 : (7)

b0 � u

ln�b=a�ÿ0:473� 1:7243�a=b�2 : (8)

and

b0 � u

ln�b=a�ÿ1:466� 1:7243�a=b�2 : (9)

After a Takaisi model has been modi®ed by the three

different experimental data, the three modi®ed models are

used to predict the center deformation on the optical ®ber in

the cases proposed by Han and Wang. Table 2 lists some of

Fig. 5. Showing: (a) a schematic diagram of the experimental setup; (b) an

ILFE sensor in the mold cavity.

Fig. 6. Axial strain versus average fluid velocities.

Y.-L. Lo et al. / Journal of Materials Processing Technology 97 (2000) 174±179 177

the parameters in gold wire molding designed by Han and

Wang. The size of mold cavity provides a b/a value of 154,

which is in the range of b/a � 20±200 in the Takaisi model.

In Fig. 7, the calculated results from the three modi®ed

Takaisi models are compared to the numerical results cal-

culated by Han and Wang based on the center deformation in

an optical ®ber. It can be seen that the modi®ed Takaisi

model for values of b/a of 40 and 80 is better than the other

modi®ed Takaisi model for a value of b/a of 320. The

modi®ed Takaisi models are in good agreement for a lower

average cavity velocity which can be explained the wall

effects playing an important role at higher average cavity

velocity, to enable the three modi®ed Takaisi models to be

distinguished. It can be seen that the modi®ed model based

on the smaller value of b/a is closer to the numerical model

for higher average cavity velocity. Accordingly, the wall

effect is a key factor in molding ¯ow when the b/a value is

small.

7. Conclusions

A modi®ed Takaisi model that is in compliance with the

requirement of b/a being within the range 20±200 provides

good agreement with the numerical model proposed by Han

and Wang. The modi®ed model based on the smaller value

of b/a is closer to the numerical model for higher average

cavity velocity. The wall effect in molding ¯ow is an

important factor when tests are conducted at a higher

average cavity velocity or a smaller value of b/a. Further-

more, a modi®ed Takaisi model, a close form in a mathe-

matical model, is different from the numerical model, which

later needs to be analyzed case by case; hence, a modi®ed

Takaisi model can be used in more easily in molding ¯ow

tests.

Acknowledgements

The authors would like to acknowledge Dr. Huang,

Sheng-Jye, for his support with the molding ¯ow apparatus

in the Department of Mechanical Engineering at the

National Cheng Kung University, Taiwan, ROC. This work

has been supported by National Science Council Grant

(NSC87-2212-E-006-047) to the National Cheng Kung

University.

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Table 2

Parameters in gold wire molding

Diameter of gold wires (D) 0.025 mm

Length of gold wires (L) 13.3 mm

Young's modulus of gold wires (E) E � 40 GPa

Silicone oil density 980 kg/m3

Silicone oil viscosity 320 poise

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178 Y.-L. Lo et al. / Journal of Materials Processing Technology 97 (2000) 174±179

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