"A Model of the Underfill Flow Process: Particle Distribution Effects," Y. Guo, G.L. Lehmann, T. Driscoll, E.J. Cotts,
to be published in the Proceedings for the 1999 IEEE/EIA CPMT Electronic Components and Technology Conference.
A Model of the Underfill Flow Process: Particle Distribution Effects
Y. Guo, G.L. Lehmann
T. Driscoll, E.J. Cotts
Department of Mechanical Engineering
Physics Department
State University of New York at Binghamton
PO Box 6000; Binghamton, NY 13902
[email protected]
Abstract – Key features of the underfill flow process are
simulated by investigating the capillary flow of a dense
suspension into a plane channel. A flow model is posed which
includes sub-models for wetting and rheology.
The
infiltration rate is successfully predicted if the velocity and
particle concentration fields are modeled by coupled transport
equations.
I.
INTRODUCTION
A key step in the manufacture of direct–chipattachment underfill packaging is the capillary flow of an
underfill material into the chip-to-substrate standoff created by
an array of solder bumps. Underfill materials are a mixture of
solid particles suspended in a liquid. It is a challenge to
model these materials [1-3]. This paper is a continuation of
the effort at Binghamton [2,3] to develop a flow model that
will: 1) predict the infiltration rate of underfill materials and 2)
predict the evolution of the particle distribution during the
underfill flow process.
II.
APPROACH
In this study the approach taken is to model the solidliquid mixture by replacing it with an effective single phase
fluid. In such a "mixture-fluid" model, the solid volume
fraction will be treated as a point-wise vary scalar field, φ. φ
will be termed the particle concentration field. This approach
has been widely adopted in the suspension literature [4].
The motion of a slowly flowing single-phase fluid
must, at each instant in time, satisfy a local force balance.
r
∇p − ρg = ∇ ⋅ τ
(1).
r
Here p is the pressure, ρg is the body force due to gravity and
τ is the viscous stress tensor. It is necessary to adopt a
rheology model to describe the viscous stress and in addition a
wetting model for capillary-driven processes. The material
properties used to characterize the flowing system are defined
by these models and are as follows. The mixture density (ρ)
and the mixture viscosity (η) vary with the solid volume
fraction (φ) in the suspension. The mixture-air interface will
be characterized by an effective surface tension (σ). The
contact angle (θ) provides a boundary condition for the
prediction of the interface shape. This angle is a function of
the three materials (solid, mixture and air) present at the
contact line. In the following all process are taken to be
isothermal. For brevity the effective properties ρ, η and σ will
be simply termed the density, the viscosity and the surface
tension.
In the following measurements of material properties
are presented for 1) a Newtonian liquid, 2) a commercial
underfill and 3) a model-mixture. The model-mixture consists
of polystyrene spheres suspended in propylene glycol. The
spheres have a narrow, monomodal size distribution with a
mean diameter of 5µm. The model-mixture allows for a
controlled variation of the solid volume fraction. The outline
of the paper is as follows. First the development of a wetting
model is presented. Measurements of θ and σ are listed along
with the measurement techniques. Second the development of
a mixture viscosity model is discussed.
Viscosity
measurements for the commercial underfill and the modelmixture are presented. The mixture viscosity will exhibit a
strong dependence on the solid volume fraction.
Implementation of the viscosity model requires knowledge of
the particle concentration field. To accommodate this need a
transport model to predict the evolution of the particle
concentration field is presented.
Finally capillary-driven
infiltration of the model-mixture into a plane channel will
serve as a simulation of the underfill process. The couple flow
and particle concentration fields are solved to predict the
infiltration rate. A comparison with the measured infiltration
rate is used to evaluate the flow model.
III.
WETTING MODEL
We seek simple methods to characterize the wetting
parameters that are used in the modeling of capillary flow.
The static contact angle, θa, provides a measure of the ability
of a liquid to wet a solid surface. Upon placing a small drop
of liquid on a homogenous surface, the contact line will adjust
until a static shape, essentially a spherical cap, is obtained.
The contact line will form a circle of radius, R. The static
contact angle is related to R and the volume (∀) by the
following formula:
1
sin θ a
3
[2 − 3 cosθ
a
]
+ cos 3 θ a =
3∀
πR 3
(2).
Measurements are conducted on clean glass slides. The slide
is massed before and after the drop is placed. The volume is
calculated from the change in mass ( ∀ = ∆mass / ρ ). The
drop image is captured from a line of sight perpendicular to
the slide. The contact line radius is then determined using
1
image analysis. Representative values of θa are listed in
Table 1.
A vertical capillary rise apparatus is used to estimate
the wetting parameter σcosθa. It is constructed using two
parallel plates of spacing s to form a plane channel. Let the
channel extend into a reservoir of fluid such that the channel
and reservoir are both exposed to the ambient pressure (pamb).
The liquid will rise into the channel and asymptotically
approach an equilibrium height, hc.
The formula for the
equilibrium height hc = 2σ cosθ a sρg is deduced from
Eq (1). hc is measured for a range of s values and the wetting
parameter deduced from the slope:
σ cosθ a =
ρg dhc
2 d (1 / s )
(3).
Figure 1: Infiltration of a liquid into a plane channel.
Static angle, θa [ deg ]
Mineral oil
Underfill A
Model-mixture
20
19
23
Table 1: Measured contact angle values.
∂  ∂u 
dp
− ρg sin ( β ) =  µ 
∂z  ∂z 
dx
Representative values of the wetting parameter are listed in
Table 2.
Using the model-mixture measurements were
obtained for various values of the solid volume fraction (φ).
The data indicate that σcosθa is independent of the fill ratio
and essentially equal to that obtained with the particles absent
(φ=0).
A
B
C
0
36.6
0.1
36.9
0.2
37.1
0.28
36.5
0.4
35.5
0.45
36.9
Table 2. Measured values of the wetting parameter.
(6).
σ cos(θ )
s
(7).
As the liquid front rises up the channel the rate of change of
the front position (xf) will equal the mean speed of the liquid,
i.e. dxf/dt = um and the pressure gradient in the liquid will
essentially be –dp/dx = ∆pc/xf. Combining Eq. (6) and (7) one
can obtain the infiltration rate of a Newtonian liquid as,
Capillary Rise of a Newtonian Liquid
Let us next consider the capillary infiltration of a
Newtonian liquid, displacing air in a plane channel. As
sketched in Figure 1, let (x,z) be the rectangular coordinates
used to describe a plane in which a uni-directional flow exists
for which the velocity components are (u(z),w=0). τzx and γzx
= ∂u / ∂z are the nonzero components of the stress and
strain-rate tensors respectively. For a Newtonian liquid the
viscous stress law provides:
τ zx
Z
2
1 W
 dp
 s
u m = ∫ u ( z )dz − 
− ρg sin ( β )
s − ZW
 dx
 12 µ
∆p c = 2
15.0
9.9
7.6
∂u
=µ
∂z
Solution of Eq. (5) provides the velocity profile, u(z).
Integration of this profile provides the Poiseuille flow result,
In a plane channel the fluid-air interface will approach a
circular arc. The curvature of the interface will produce a
capillary pressure drop across the interface that obeys
Wetting parameter, σcos(θa) [ mN/m ]
Mineral
Underfill
Model-mixture
oil
φ
29.4
(5).
(4).
In a slender channel the equation of motion simplifies to:
 2σ cos(θ )
 s2
=
− ρg sin (β )
 s x (t )
 12 µ
dt
f


dx f
(8).
The contact angle, as defined in Eq. (7), will vary with the
speed of the contact line [8]. This is commonly correlated by
the Capillary number, Ca = umσ/µ. We have found that
Hoffman’s mobility relation (see [4])
cos(θ a ) − cos(θ )
= tanh( 4.96Ca 0.702 )
cos(θ a ) + 1
(9).
provides a satisfactory model for a number of liquids wetting
glass. At large time dxf/dt à 0, θ à θa and xf -> hc. In this
limit Eq. (8) reduces to the standard capillary rise formula.
2
At sufficiently large shear-rates, hydrodynamic
forces will dominate the particle-particle interactions. Many
suspensions exhibit a “high shear-rate regime” in which η is
independent of γ and is well correlated in the form, η/ηο =
f(φ/φm). Here ηo is the unfilled liquid carrier viscosity and φm
is the maximum packing or fluidity limit. The Kreiger
formula is a common correlation:
2.5
2.0
1.5

φ
η H = η o 1 −
 φm
1.0
0.5
Capillary rise of Mineral oil
s = 256 µm
4 seperate trials
0.0
0
20
40
60
80
time [ s ]
Figure 2: Capillary rise of a Newtonian liquid in a
vertical plane channel.
The Capillary Flow Cell
Capillary flow experiments are conducted in a
channel formed by two parallel glass plates. The plate
separation (s) is maintained using precision stainless steel
spacers and separations covering a range of 38 mm to 300 mm
have been utilized. The flow cell measurements are conducted
in a controlled temperature/humidity chamber. Experiments
have been conducted using Newtonian liquids, model
suspensions and commercial underfill mixtures. Figure 2
shows the variation of the measured front position with time in
typical runs. Mineral oil serves as a Newtonian liquid (µ =
0.19 Pa s) and the flow orientation is vertical (β = π/2). Data
are shown for four distinct runs, all at a spacing of s = 256 µm.
The front rises with time and asymptotically reaches the
equilibrium rise height (hc). The product σcos(θa) is first
determined from hc. We than used Eq. (8) with the Hoffman
mobility relation and simulated the motion of the front. This
calculation is shown as the solid line on Figure 2. Good
agreement is observed between the model calculation and the
data. This provides some qualification of the wetting model
without the complications added by solid particles in
suspension.
IV.
MIXTURES
The rheology of a dense suspension is commonly
described by adopting the generalized Newtonian fluid
τ zx = η (φ , γ )
∂u
∂z
(10).
Here η denotes the mixture viscosity and γ = | du/dz| is the
shear-rate magnitude. For an isothermal suspension η may
vary with both the solid volume fraction and the shear-rate. In
a general flow both φ and γ will exhibit spatial variation. The
γ field is determined from the equation of motion (Eq. (1)).
The φ field will require an additional model.



p
(11)
where φm and p are empirical constants. The subscript H is
used here to identify the value of η in the high shear rate
regime.
Figures 3 and 4 contain viscosity data obtained for a
commercial underfill and a model suspension. The model
suspension serves as a mixture in which the bulk solid volume
fraction, φb, may be varied in a systematic way.
Measurements of η as a function of γ are obtained in a shearrate range relevant to the underfill flow process. Both
materials exhibit shear thinning while approaching an
asymptotic limit at the higher γ values.
Measurements of viscosity are conducted using a
CSL2 500 rheometer manufactured by TA Instruments. The
data shown in Figures 3 and 4 have been obtained using a
rotating parallel disk configuration, with a disk radius of 2 cm.
Measurements are conducted for a range of gap spacings. No
correlation with gap spacing is observed. See Li, et.al. [3] for
further discussion of the measurement technique.
The model-mixture data suggested a correlation of
the form η = ηH(φ/φm) G(γ) where G will approach 1 at
sufficiently large γ and the high shear-rate limit of Eq. (11)
will be recovered. The data has been correlated in the form
12
10
8
η [ Poise ]
xf [ cm ]
Model prediction
6
o
Commercial underfill at T = 80 C.
4
2
0
0
2
4
6
8
10
12
14
16
18
20
γ [1/s]
Figure 3: The effect of shear-rate on the
measured viscosity of commercial underfill C.
3
particle interactions, commonly termed particle migration. Nz
is the particle migration flux in the z-direction.
The
development of models for the particle migration flux is a
relatively recent topic of research. Zhang and Acrivos [4]
report a model for monodisperse spherical particles suspended
in a Newtonian liquid. The model has been developed and
tested for application in the high shear-rate regime. In a
horizontal plane channel flow Nz is composed of three terms:
3
polystyrene particles suspended in propylene glycol
η [ Poise ]
φ = 0.45
2
φ = 0.2
1.0
φ = 0.28
N z = − Dc
∂φ
∂γ
− Ds
− N g (14).
∂z
∂z
0.5
φ = 0.1
0
200
400
600
800
1000
1200
γ [ 1/s ]
Figure 4: The variation of viscosity with shear-rate for
the model-mixture of polystyrene particles suspended in
propylene glycol. The correlation fit and the measured
data (symbols) are shown.

φ
η = η o 1 −
 φm
Here Dc and Ds are the diffusion coefficients of the particle
flux due to gradients in the φ and γ profiles. The last term, Ng,
describes the settling flux that arises due to a difference in the
particle and liquid density. For the present model suspension
the density difference is sufficiently small so that Ng may be
neglected. This may not be the case for an actual underfill
mixture. The empirical correlations cited by Zhang et.al. [4]
for Dc and Ds are as follows:

1 dη 

Dc = a 2γ  0.43φ + 0.65φ 2
η dφ 

p

 G (γ )

D s = 0.43a 2 φ 2 (16)
G = 0.959 + 0.102e ( −γ / 31.84 ) +
0.736e ( −γ / 12.3) + 0.167e (−γ / 1823 )
(12)
with p = -2.5φm and φm = 0.68. Figure 4 shows the variation
of the measured η with γ at φ = 0.1, 0.2, 0.28 and 0.45. The
correlation (solid lines) provides satisfactory agreement with
the measured data (symbols). Therefore Eq. (12) will serve as
our rheology model.
The Concentration Profile
For dense suspensions η is a strong function of the
solid volume fraction, as demonstrated by Eq. (11). Thus it is
necessary to pose a model for the spatial distribution of the
particles. Let φb denote the particle concentration that would
exist if the particles are uniformly distributed throughout the
mixture and term this a homogenous mixture. In fact several
studies have documented that gradients in φ can exist and are
strongly influenced by the fluid flow motion. To predict the
point-wise variation of φ, a transport equation based on
conservation of solid volume may be introduced. In the
present case of uni-directional channel flow the local
conservation statement may be written as follows:
∂φ ∂
∂
+ (uφ ) = − N z
∂t ∂x
∂z
(15)
(13).
The solid particles are transported along the flow streamlines
by the bulk motion and across the streamlines by irreversible
a is the particle radius.
The Infiltration Rate of a Mixture-Fluid
Let us now consider the infiltration of a mixture-fluid
into a plane channel under capillary action. Invoking the
slender channel approximations and using Eq.(10), the
equation of motion reduces to:
dp
∂  ∂u 
− ρg sin ( β ) = η  (17).
dx
∂z  ∂z 
For a homogenous mixture η = ηH(φb)G(γ). In this cases η
and u are independent of x so that, at an instant in time, –
dp/dx = ∆pc/xf(t). The integration of Eq.(17) is accomplished
with the rheology model used to evaluate η[(γ(z)].
 2σ cos(θ )
 zw z r
=
− ρg sin (β ) ∫ ∫ drdz (18)
dt
 sx f
 − zw − zw η
dx f
Alternately one can use the particle migration transport
equation to determine the evolution of the concentration field.
We assume that the entrance concentration remains fixed as:
φ ( x = 0, z , t ) = φ b
(19)
and the boundary conditions are zero flux at each wall, i.e.
4
N z ( x ,− z w , t ) = N z ( x , z w , t ) = 0
(20)
0.5
( )
p( x = 0 ) = p amb
p x (f−) = p amb − ∆pc
φb=0.45, s= 254 µm
0.4
0.3
0.2
φ
Now one must solve the coupled u and φ fields, using Eq. (17)
and (13) with the rheology and particle flux models as
described above. An additional complication arises in that
dp/dx now varies with x as the axial variation of φ results in a
notable change in η with axial position. Thus one is forced to
solve an elliptic system that satisfies the pressure boundary
conditions
t = 10 s
t = 100 s
1
3
t = 300 s
t = 500 s
0.1
0.0
(21).
0
2
The calculation of the u and φ fields may be obtained by
computational methods, as discussed by Guo [6].
4
5
6
7
8
x [ cm ]
Particle Migration Model Results
The assembled model has been used to predict the
evolution of the concentration field and infiltration rate of the
model suspension. Key results are presented here for the case
of φb = 0.45 at a channel spacing of s = 254 µm. Flow enters
channel with a uniform profile of φ = φb. As the flow
continues particles migrate from the channel walls towards the
centerline. Figure 5 shows the concentration field 300 s after
the flow has entered the channel and the front has reach an
axial position of about 38 mm. The redistribution of the solid
volume fraction is significant as illustrated by the profile
presented in Figure 5.
0.7
Figure 6: The particle concentration axial profile
φ(x,zw,t) at the channel wall; φb=0.45 and s = 254 µm.
result the net viscous drag at the two walls is significantly less
than would be experienced if the particle distribution had
remained homogenous.
Figure 7 shows the variation of the square of the front
position with time. Measured data obtained using a horizontal
capillary flow cell is shown as solid squares. The prediction
made using the particle migration model so that both the flow
field and concentration fields are evolving with time is shown
as a solid line. The model and data are in excellent agreement.
In contrast the calculation obtained using a homogenous φ
model is a poor prediction. This is not surprising given the
overwhelming evidence that particle migration is an essential
feature of dense suspensions in pressure-driven channel flow.
Additional calculations have been made in which the channel
0.6
0.5
0.4
φ
45
40
0.3
0.2
20
X[
0.0
100
150
30
mm
200
40
]
50
250
z [ µm ]
t =300s, s =254µm
s =254 µm, φ=0.45, T = 27 c
o
2
[cm ]
50
10
2
0
30
Xf
0.1
Particle migration model
35
25
20
15
Uniform profile, φ=φb
10
5
Figure 5: The predicted evolution of the field for the
model-mixture, 300 s after the entering the channel:
for φb = 0.45 and s = 254 µm.
In Figure 6 the axial concentration at the solid surface
(z = + zw) is plotted at t = 10s, 100s, 300s and 500 s after first
entering the channel. The profiles start at φ = φb at the inlet
(x=0) and decrease with distance traveled into the channel.
The position of the front (xf(t)) is located at the step drop to
zero. The value of φ at the wall decreases by over 20%. As a
0
0
100
200
300
400
500
time [s]
Figure 7: The infiltration of the model-mixture, under
the conditions of Figure 6, is presented as the variation
of the square of the front position with time. The
measured data (symbols) are compared with the uniform
φ profile and variable φ profile models.
5
spacing or φb has been varied. Good agreement between the
measured data and the particle migration based model was
observed in all cases.
V.
CONCLUSIONS
A model for the capillary motion of dense suspension
mixtures has been posed, including sub-models for wetting
and rheology. Simple techniques to measure σ and θa were
presented. Viscosity models must account for variation in
both the solid volume fraction and shear rate. Also the
temporal and spatial evolution of the particle concentration
appears to play a significant role in the capillary-driven flows
encountered in the underfill flow process. A flow model
based on a homogenous distribution of the particles performed
poorly.
VI.
ACKNOWLEDGEMENTS
Mr. P.C. Li and Mr. J. Casio contributed to the
laboratory measurements discussed in this paper. This
research was funded by DARPA contract #N00164-96-C-0074
and The Integrated Electronics Engineering Research Center
(IEEC) located in the Watson School at Binghamton
University. The IEEC receives funding from the New York
State Science and Technology Foundation, the National
Science Foundation and a consortium of industrial members.
References:
[1] Steidel, C.A., et.al., “Material Science and the Electronic
Packaging Roadmap”, Spring 95 MRS Symp. On Electronic
Packaging Material Science.
[2]Lehmann, G.L., et.al., “Underflow Process for Direct-ChipAttachment Packaging”, IEEC Trans on Comp. Pack, and
Manf. Tech, Part A, Vol 21, No 2.
[3] Li, P.C. et. al., “Viscosity Measurements and Models of
Underfill Mixtures”, 3rd Int. Conf. Adhesive Joining and
Coating Technology in Electronics Manufacturing,
Binghamton, NY Sept. 1998.
[4]Zhang, K. and A. Acrivos, “Viscous Resuspension in Fully
Developed Laminar Pipe Flow”, Int. J. Multiphase Flow, 20,
No.3: 579-591, 1994.
[5] Dussan V, E.B., “On The Spreading of Liquids on Solid
Surfaces: Static and Dynamic Contact Lines”, Ann. Rev. Fluid
Mechanics, 1979,11:371-400.
[6] Guo, Y. M.S. Thesis, Mechanical Engineering,
Binghamton University, Binghamton, NY, May 1999.
6