Horizontal capillary flow of a Newtonian liquid in a narrow gap between a plane wall and a sinusoidal wall

Fluid Dynamics Research 40 (2008) 779 – 802

Horizontal capillary flow of a Newtonian liquid in a narrow gap between a plane wall and a sinusoidal wall D. Davidsona , G.L. Lehmanna,1 , E.J. Cottsb,∗ a Mechanical Engineering Department, State University of New York at Binghamton, Binghamton, NY 13902-6000, USA b Physics Department, State University of New York at Binghamton, Binghamton, NY 13902-6000, USA

Received 27 November 2007; accepted 25 April 2008 Available online 5 June 2008 Communicated by Z. Warhaft

Abstract An experimental and theoretical study of the capillary flow of a Newtonian liquid (mineral oil) in a Hele-Shaw cell in which the gap varies sinusoidally in one coordinate direction, and flow takes place in the direction of constant channel cross-sectional area is reported. The geometric non-uniformity of the gap is observed to produce interface fingering. Finger length is observed to increase with decreasing spacing between plates of fixed shape, and with increasing gross penetration distance. In the regime of interest, finger length is observed to increase slowly with increasing interface advancement, motivating a quasi-steady model in which gross interface advancement is predicted by a Lucas–Washburn model and interface fingering is predicted by a Hele-Shaw model of steady flow. The steady interface velocity in the Hele-Shaw model is set equal to the instantaneous interface velocity predicted by the Lucas–Washburn model. Fingering predicted by the quasi-steady model matches the experimentally observed trends with regards to plate spacing and gross penetration distance. © 2008 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. PACS: 47.15.gp; 47.55.Ca; 47.55.nb; 47.63.mf Keywords: Capillary flow; Corrugated channel; Hydrostatic equilibrium; Lucas–Washburn model; Hele-Shaw model

∗ Corresponding author. Tel.: +1 607 777 2217/4371; fax: +1 607 777 6841.

E-mail address: [email protected] (E.J. Cotts). 1 Deceased 6/05.

0169-5983/$32.00 © 2008 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. doi:10.1016/j.fluiddyn.2008.04.003

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1. Introduction 1.1. Background and motivation In the underfill process step in flip-chip electronics manufacturing, liquid epoxy fills (encapsulates) the narrow (∼ 100 m) gap between the chip and the printed circuit board by capillary action (Machuga et al., 1992). The epoxy contains a high volume fraction of silica particles, and is cured at elevated temperature once the gap is filled. The side length of the chip is typically ∼ 10 mm. The chip and board are oriented horizontally with respect to gravity during the filling process. Epoxy is usually introduced along one or two edges of the chip. The gap between the chip and the board typically contains an array of solder ball electrical connections. Electrical trace lines may be found on the surface of the board. Geometrically, the solder balls resemble outward-bulging cylindrical posts and the trace lines resemble earthen dams. Two aspects of the underfill process which many researchers seek to understand are the time rate of gap filling, and the occurrence of air bubbles in the filled gap. In some cases, distortion of the advancing air–liquid interface causes air bubble entrapment. Thus, interface shape in capillary flow is studied. Typically, a plane channel populated by various arrangements of solderballs is the flow cell chosen for studying the underfill process. Fine et al. (2000) report an experimental study using commercialtype underfill fluid. Bogoyavlenskiy et al. (2004) carry out Hele-Shaw modeling of a Newtonian liquid advancing through an array of solder balls. So far, little attention has been given to the trace lines. To the best of the author’s knowledge, Davidson et al. (2002) is the only publication associated with the underfill application which seeks to understand the effect of trace lines on the board. Davidson et al. (2002) report experimental observations of interface fingering in a Hele-Shaw cell composed of a flat glass plate and a printed circuit board with parallel trace lines; the advancing liquid is mineral oil. In this work, we continue to study the experimental system introduced by Davidson et al. (2002). To review, we simplify the underfill flow process by replacing the epoxy–silica mixture with a Newtonian oil (mineral oil) of similar surface tension and viscosity, and simpler rheological behavior. We remove the solder balls from the gap, and populate the board with parallel arrays of trace lines. The trace lines impart a sinusoidal shape to the board. The silicon chip is represented by a flat glass plate. The resulting flow cell is shown in Fig. 1. The gravity vector is orientated in the negative z-direction. Flow is initiated in

Fig. 1. Section view of the flow cell. The gap between the plates S(x) is a sinusoidal function of coordinate x. Smin and Smax  are the minimum and maximum of S(x) with respect to x, and Savg = (1/) 0 S(x) dx. HT = Smax − Smin . The static wetting angles of the advancing liquid on the plates are eq,1 and eq,2 .

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the y-direction from a reservoir of liquid in the half-space y < 0. We previously reported (Davidson et al., 2002) experimental observations of the interface fingering illustrated in Fig. 2. This interface fingering is undesirable because of the possibility of air entrapment; improved understanding could assist in reducing it. Our study is continued by first presenting new experimental observations in sufficient detail that the overall trends in interface fingering are apparent. We then use the experimental observations to motivate detailed modeling of the interface fingering. 1.2. Theoretical studies reported in the literature concerning similar corrugated channels Borhan et al. (1991) and Borhan and Rungta (1993) study the capillary rise of a Newtonian liquid between a pair of vertically oriented, sinusoidal plates. The channel cross-section is invariant in the vertical direction (y-direction). Unlike our flow cell (Fig. 1), their plates are mirror-imaged about the gap mid-plane, with identical surface chemistry. A numerical model of hydrostatic equilibrium is obtained (Borhan et al., 1991) by minimizing the system energy (gravitational and surface) computed from the discretized interface shape. The equilibrium interface shape is predicted for three system configurations 2 / = .4, (described here using the terminology from our Fig. 1): HT /Savg = .2, .6, and 1, with  ∗ g ∗ Savg eq = 60◦ , and /Savg = 1. The deviation of their predicted interface shape from the plane y = constant is significant only near the wall for these parameters. Their predicted variation in elevation (y) of the wall contact line, which we call finger length L F , is a small percentage of the mean gap Savg . Interface fingertips are predicted to be aligned with the gap maxima. The variation in elevation (y) of their predicted interface shape at the center of the gap (z = Smax − S(x)/2, c.f. Fig. 1) is negligible in comparison with Savg . The rise height at the center of the gap is therefore a convenient representation of the gross rise height. For the three system configurations, the predicted centerline rise height increases by .6%, 17.8%, and 44.6% in comparison with the rise height in a plane channel of equal cross-sectional area ( A x z ). The predicted centerline rise height is computed for an additional set of system configurations {.5 < /Savg < 2.5, 2 / = .4, 15◦ <  < 60◦ }. Centerline rise height is predicted to increase 0 < HT /Savg < 1.4,  ∗ g ∗ Savg eq with HT /Savg and decrease with /Savg . For the channel with the shortest wavelength and the largest amplitude, the predicted centerline rise height is approximately three times the rise height in the plane channel of equal cross-sectional area ( A x z ). For all channels with the longest wavelength {/Savg = 2.5}, the predicted centerline rise height increases by less than 10% over the rise height in the plane channel of equal cross-sectional area ( A x z ). In a later publication, Borhan and Rungta (1993) developed a Lucas–Washburn-type model (Lucas, 1918; Washburn, 1921) for the time rate of capillary rise of a Newtonian liquid in their vertically oriented sinusoidal channel. They report a numerical solution for the dimensionless flow conductance K  in Poiseuille flow  · V0 K = (1) 2 j P Savg · jy 4 in the parameter range {.5 < /Savg < 2.5, 0 < HT /Savg < 1}. The liquid viscosity is , the pressure is P, and V0 is the mean velocity in the vertical (y) direction. For a plane channel, K  = .33. K  of the sinusoidal channel is predicted to decrease with decreasing /Savg and increasing HT /Savg . The predicted K  for the channel with the largest amplitude and the shortest wavelength is .08. The predicted K  for the

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Fig. 2. Finger-like shape of the air–liquid interface (pictorial and three views) in the flow cell depicted in Fig. 1. A liquid displaces air in the positive y-direction. The interface shown is predicted by a Hele-Shaw model of steady flow (Sections 3.5 and 4.7), with contact angles fixed at the static values, Smax = 40 m, Smin = 29 m, HT = 11 m,  = 251 m, Ca =  ∗ V0 / = .0024, Re=  ∗V0 ∗ Smin / =.0026, eq,1 =26◦ , and eq,2 =12◦ . The predicted value of interface finger length L F (L F =Ynarrow −Ywide )  is 67 m. The chosen value of YF (YF = (1/) 0 Y (x) dx) for the plot is arbitrary.

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channel with the largest amplitude and the longest wavelength is .29. The Poiseuille flow solution for flow conductance is used in combination with the hydrostatic equilibrium solution for interface pressure from the previous study (Borhan et al., 1991) to obtain a Lucas–Washburn model for the centerline rise height as a function of time. The time required to attain 99% of the equilibrium rise height (te ) is reported 2 / = .4,  = 60◦ }. As H /S in the parameter range {.5 < /Savg < 2.5, 0 < HT /Savg < 1,  ∗ g ∗ Savg eq T avg increases and /Savg decreases, the predicted rise time te increases because of reduced flow conductance. For the channel with the largest amplitude and the shortest wavelength, predicted rise time increases nearly 12-fold over that of the plane channel of equal cross-sectional area ( A x z ). For the channel with the largest amplitude and the longest wavelength, predicted rise time is less than 20% greater than that of the plane channel of equal cross-sectional area. While the work reported by Borhan et al. (1991) and Borhan and Rungta (1993) provides useful insights, a direct prediction of the shape of the interface in horizontal capillary flow is not given. A numerical solution for a Hele-Shaw model of capillary-driven, creeping flow of a Newtonian liquid between horizontal, sinusoidal plates of identical surface chemistry was obtained by Bogoyavlenskiy (unpublished work). The plates are mirror-imaged about the gap mid-plane, and flow occurs in the direction of constant flow cell cross-sectional area (the y-direction in Fig. 1). The spatial gradient of the gap between the plates is assumed small (dS/dx >1), and the pressure in the liquid at the interface is approximated by P=

−2 ·  · cos(eq ) S(x)

(2)

The wetting angle is fixed at the static value ( = eq ). Eq. (2) is nearly exact for a straight interface in a plane channel at small Bond number, gap Reynolds number, and capillary number. The shape of the air–liquid interface is represented by the plane curve Y (x, t), with Y (x, t = 0) taken to be a small positive constant. Bogoyavlenskiy predicts the following quantities (see Fig. 1): Ywide = Y (x = 0, t) and Ynarrow = Y (x = /2, t), the finger length L F = Ynarrow − Ywide , and the ratio of the time derivative of Ynarrow to the time derivative of Ywide , vnarrow /vwide . Gross penetration distance is measured by  1  YF = Y (x, t) dx.  0

Solutions are obtained for .2 < HT /Savg < 1.8 and 0 < YF < . The chosen upper limit for YF is several orders of magnitude smaller than the typical side length of a silicon chip in the industrial application. In the initial phase of the filling process, Bogoyavlenskiy predicts that the fingertip grows at the gap maxima (0 < vnarrow /vwide < 1). At a certain point, while YF remains less than /4, the ratio vnarrow /vwide becomes greater than 1, and the fingertip shifts from the gap maxima to the gap minima. For .2 < HT /Savg < 1.8, the interface shape changes continuously over the entire filling process (0 < YF < ). The effect of increasing the value of HT /Savg is to increase (Ynarrow − Ywide )/ in absolute value, and to increase the value of YF at which Ynarrow − Ywide changes from negative to positive. The largest predicted value of L F in the solution set is less than . While the approach of Bogoyavlenskiy involves only the usual Hele-Shaw approximations and could offer considerable physical realism, we do not choose to follow it because the computational cost of his numerical method (the same numerical method employed in Bogoyavlenskiy et al., 2004) restricts its practical application to infiltration distances which are much shorter than typically encountered in the underfill application. In our work discussed here, we predict finger length at arbitrary gross penetration distance.

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2. Experimental apparatus and procedure Capillary-driven flows are carried out in a Hele-Shaw cell composed of a flat glass plate and a soldermask-coated printed circuit board with parallel trace lines (traces) of 250 m pitch. Referring to Fig. 1, the width of the flow cell in the x-direction is 25 mm, the length of the flow cell in the y-direction is 15 mm, and the gap Smax is one of 19, 25, or 39 m, according to the thickness of selected polymer shims. Mineral oil is the experimental liquid. In order to produce capillary filling of the flow cell, the mineral oil is introduced via syringe along one edge of the flow cell (the plane y = 0 in Fig. 1). A CCD camera transmits a sequence of images of the flow cell as viewed through the glass plate to a VHS recorder (resolution 320×240 and 30 frames per second). Lenses provide a field of view ranging from .8 to 25 mm.

3. Theory 3.1. Overview: a quasi-steady model for predicting filling kinetics and finger growth A quasi-steady model is posed for horizontal, capillary-driven creeping flow of a Newtonian liquid as depicted in Fig. 2. The time rate of advancement of the mean position   1 YF = · Y (x) · dx 

0

of the air–liquid interface (dYF /dt), or interface velocity, is predicted using a Lucas–Washburn-type model (Lucas, 1918; Washburn, 1921). The finger-like-shape of the contact line on the flat plate (Y (x, t)) is predicted using a Hele-Shaw model for steady flow, with the steady interface velocity (dYF /dt) set equal to the interface velocity predicted by the Lucas–Washburn model (LWM). This quasi-steady model is based on the assumption that the finger shape Y (x, t) in horizontal capillary-driven flow changes very gradually with gross penetration distance YF , so that the solution for finger shape Y (x, t) in steady flow is a good approximation for finger shape Y (x, t) in capillary flow. 3.2. Predicting finger length in hydrostatic equilibrium This section describes a model of hydrostatic equilibrium of the system in Fig. 2. The finger-like-shape of the contact line on the flat plate (Y (x)) predicted by this hydrostatic equilibrium model provides a reference point for Y (x, t) in capillary flow, and the predicted interface pressure is a building block of the LWM to be discussed in Section 3.4. The hydrostatic equilibrium model is obtained by making a simplifying assumption regarding the geometry of the interface. The Young–Laplace equation is the relationship between the pressure P in the liquid at a point on the air–liquid interface, the surface tension , and the local mean curvature m of the interface: P = −2 ·  · m

(3)

Conservation of linear momentum for a liquid in hydrostatic equilibrium is  P +  · g = 0 −

(4)

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Using primes to denote scaled variables: 

 P + −

2  · g · Smax



· gˆ = 0

(5)

In Eq. (5), the scale of length is Smax and the scale of pressure is /Smax . The second term in Eq. (5) is the product of the Bond number and the unit vector in the direction of the body force vector. Assuming that the Bond number is very small compared to 1, the spatial gradient of pressure in the sessile liquid is very small, so that the pressure in the liquid is approximately uniform. Therefore, by Eq. (3), the interface shape in hydrostatic equilibrium is one of uniform mean curvature m . The interface shape has symmetry properties corresponding to the spatial periodicity of the sinusoidal channel in the x-direction, and exhibits the required static wetting angles on the plates (eq,1 and eq,2 ). The contact line on the flat plate is labeled with the subscript 1: x1 = x · xˆ + Y (x) · yˆ + Smax · zˆ with Y assumed to be a one-to-one function of x The mean curvature m of a surface at point P on the surface is equal to one-half the sum of the normal curvatures n of two curves in the surface intersecting at point P at right angles. The normal curvature of a curve is the product of its principal curvature K, and the vector inner product of the principal unit normal Nˆ to the curve with the unit normal nˆ to the surface. Thus, the mean curvature of a surface is m = 21 · (n,1 + n,2 ) = 21 · (K 1 · Nˆ 1 · nˆ + K 2 · Nˆ 2 · n) ˆ

(6)

The subscripts 1 and 2 in Eq. (6) refer to two curves in the surface intersecting at point P. Now, let the subscript 1 refer to the flat plate, and the subscript 2 refer to the sinusoidal plate in Fig. 2. To obtain an approximate expression for the mean curvature of the air–liquid interface at a point on the flat-plate-contact-line x1 = x · xˆ + Y (x) · yˆ + Smax · zˆ , we assume that the intersection of the interface with the plane perpendicular to the curve x1 is the circular arc , of radius R = S(x)/(cos(1 ) + cos(2 )), where  is the wetting angle. The existence of the circular arc  is a simplifying assumption which permits the approximate analytical solution for hydrostatic equilibrium. The reader will note that the contact angle boundary condition on the sinusoidal plate is not rigorously satisfied by this geometric approximation. Choosing x1 and  as the two perpendicular curves in Eq. (6), an approximate expression for the mean curvature of the air–liquid interface at the point x1 is   1 cos(1 ) + cos(2 ) (7) m = x y · sin(1 ) + 2 S(x) where x y is the curvature of the plane curve x1 = x · xˆ + Y (x) · yˆ + Smax · zˆ , and is expressed as x y =

d2 /dx 2 (Y (x)) (1 + (d/dx(Y (x)))2 )3/2

(8)

Given x1 = x · xˆ + Y (x) · yˆ + Smax · zˆ , the assumption of the circular arc  permits the generation of the rest of the interface. For each point on x1 , the circular arc  of radius R = S(x)/(cos(1 ) + cos(2 )) may be used to obtain an approximation for the position x2 of the contact line on the sinusoidal plate. Let Nˆ 1 be the principal unit normal to x1 = x · xˆ + Y (x) · yˆ + Smax · zˆ : x2 = x · xˆ + Y (x) · yˆ + Smax · zˆ + R · (sin(1 ) − sin(2 )) · Nˆ 1 − S(x) · zˆ

(9)

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Fig. 3. Top view of the flow cell from Fig. 1, showing the contact line on the flat plate x1 = {x, y = Y (x), z = Smax }. Boundary conditions on pressure P pertain to a Hele-Shaw model of steady flow (Section 3.5).

The position of the back curve of the interface xb is determined in a similar manner: xb = x · xˆ + Y (x) · yˆ + Smax · zˆ − R · (1 − sin(1 )) · Nˆ 1 − R · cos(1 ) · zˆ Substituting Eq. (8) into Eq. (3):   cos(1 ) + cos(2 ) P = −2 ·  · m = − · x y · sin(1 ) + S(x)

(10)

(11)

Bogoyavlenskiy omits the first term in parentheses in Eq. (11). Because the pressure in the liquid is approximated as uniform, and the gap thickness is a sinusoidal function of the x-coordinate (see Fig. 1),   HT 2· ·x S(x) = Savg + (12) · cos 2  the balance of the two terms in Eq. (11) results in a finger-like interface shape. The following relationships are obtained (see Fig. 3). Arc length measured along the flat-plate contact-line x1 is denoted by s, and

is the angle between the unit tangent to x1 and the x-axis. x y =

j

js

(13) 

sin( (x)) =

x

x y · dx

(14)

0

Solving Eq. (11) for x y and substituting in Eq. (14):   x P (cos(1 ) + cos(2 )) − sin( ) = − · dx  · sin(1 ) S(x) · sin(1 ) 0  x dx P·x (cos(1 ) + cos(2 )) = − −  · sin(1 ) sin(1 ) 0 S(x)

(15)

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By the spatial periodicity of the flow cell, the angle is zero at x = /2, which permits the hydrostatic equilibrium pressure P0 to be determined by the rearrangement of Eq. (15):   · (cos(1 ) + cos(2 )) /2 dx (16) P0 = − /2 S(x) 0 Substituting Eq. (12) in Eq. (16), a simple closed-form expression for pressure in hydrostatic equilibrium in the system of Fig. 2 is obtained as  · (cos(1 ) + cos(2 ))  · (cos(1 ) + cos(2 ))  P0 = − √ =− √ 2 Smax · Smax − HT 2 − HT Savg 4

The finger shape y = Y (x) is obtained using  x y = Y (x) = tan( ) · dx

(17)

(18)

0

The tangent of in Eq. (18) is found using Eq. (15), with P set equal to P0 as given by Eq. (17). The finger shape Y (x) predicted by the hydrostatic equilibrium model discussed in this section includes an inflection point xi , where x y equals zero and changes sign. The relationship between xi and Smax is obtained by combining Eqs. (17) and (11), setting x y to zero, and solving for xi :      Smax 2 · Smax Smax −1 · cos · −1 − +1 (19) xi = 2· 2· HT HT HT Eq. (19) shows that as Smax decreases from +∞, the inflection point xi moves from the gap maximum x = 0 (AD in Fig. 3) to the gap minimum x = /2 (BC in Fig. 3). Let Scrit denote the value of Smax at which the angle at the inflection point xi equals /2, the interface shape Y (x) becomes tangent to the vertical line x = xi , and the finger length L F becomes infinite. The value of Scrit is obtained by setting

= /2 in Eq. (15), combining with Eq. (19), and numerically finding the root. 3.3. Modeling fully-developed internal flow The fully developed internal flow (Poiseuille flow) solution is a necessary component of the LWM for filling kinetics, to be described in Section 3.4. Conservation of linear momentum in the y-direction for the fully developed, internal creeping flow is   2 j v j2 v dP (20) + 2 +· 0=− dy jx 2 jz Eq. (20) is a Poisson problem for v(x, z) on the cross-hatched domain ABIH in Fig. 2. The boundary conditions for the Poisson problem are no-slip on the plates (v=0 on AB and HI) and symmetry (jv/jx =0 on AH and BI). We obtain two solutions for Poiseuille flow. In the first, the Poisson problem for v(x, z) is solved numerically using the finite-element (F.E.) method. In the second, the lubrication approximation allows an analytical solution. We call the second solution a Hele-Shaw solution for Poiseuille flow. Because the

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gap width (Smax ) is much smaller than the lateral dimensions of the flow cell (), the momentum balance is approximated in the Hele-Shaw solution by 0=−

dP j2 v +· 2 dy jz

The flow conductance K is defined as      Smax K dP 1 · v · dx · dz = · − v¯ = Savg ·  HT /2·(1−cos(2· ·x/)) 0  dy

(21)

(22)

After integrating Eq. (21), applying Eq. (12), and substituting the result in Eq. (22), an analytical expression for the flow conductance of the sinusoidal channel is obtained via the Hele-Shaw approximation as   2 Savg H2 1 K H.S. = · S(x)3 · dx = + T (23) 12 · Savg ·  x=0 12 32 3.4. A LWM for predicting filling kinetics As before (see Section 3.2), let YF denote the average y-coordinate of the flat plate contact line    1 Y (x) · dx . YF = · 

0

In this study, we consider the situation in which the interface finger length L F (see Fig. 3) is much smaller than YF (|L F |/YF >1). In this case, we assume that the bulk of the advancing liquid (at sufficient distance from the flow inlet and the interface) obeys the Poiseuille flow model (Section 3.3) at every instant in time. The evolution in time of YF is predicted by a Lucas–Washburn-type model (Lucas, 1918; Washburn, 1921). The advancing liquid is approximated by an advancing slug of length YF in the y-direction. The pressure in the liquid at y = YF is given by the solution for liquid pressure in hydrostatic equilibrium P0 (see Section 3.2 and Eq. (17)), while the pressure at the inlet y = 0 is atmospheric (P = 0). The internal flow is predicted using Eq. (23) (see Section 3.3). Identifying the mean velocity V0 in Poiseuille flow as the time derivative of YF , neglecting variation in wetting angle with contact line speed, and neglecting inertial effects:     K K dYF dP 0 − P0 V0 = = = (24) · − · dt  dy  YF  2   Savg YF2 HT2  · (cos(1 ) + cos(2 )) 2 K (25) · P0 = − · = −2 · + · −√ √ t   12 32 Smax · Smax − HT For fixed wetting angles, YF is a quadratic function of time. The analogous prediction for YF (t) in a plane channel of gap spacing Hflat is: 2 YF,flat

t

=

Hflat ·  · (cos(1 ) + cos(2 )) 6·

(26)

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Comparing the sinusoidal plate system and the plane channel system (with contact angles fixed at the static values), the ratio of YF at time t is a function of the flow cell geometry only  √ YF (1+1.5 · (HT /(2 · Smax −HT ))2 · (Smax −(HT /2))2 · 1/(Smax · 1−(HT /Smax )) = (27) YF,flat Hflat To include the effect of wetting angle variation with contact line speed, the correlation of Hoffman’s data (1975) due to Jiang et al. (1979) is used in Eq. (24): ⎛ ⎡ dY ⎤.702 ⎞ F · ⎜ cos(eq ) − cos() ⎢ dt ⎥ ⎟ (28) 4.96 · ⎣ = tanh ⎜ ⎦ ⎟ ⎠ ⎝ cos(eq ) + 1  When the dynamic angle effect is included, YF (t) is obtained numerically. 3.5. Predicting finger length in steady flow using a Hele-Shaw model To complete the quasi-steady model of horizontal capillary flow introduced in Section 3.1, we obtain a Hele-Shaw model for steady flow in the vicinity of the interface (yFE < y < Y (x); c.f. Fig. 3). In the region (0 < y < yFE ) upstream from the interface, the flow is assumed to be Poiseuille flow (see Section 3.3). The shape of the interface (y = Y (x)), the velocity field, and the pressure field are to be determined in the vicinity of the interface (yFE < y < Y (x)). Using the Hele-Shaw approximations in the region yFE < y < Y (x): 1 jP · · (S(x) · z − z 2 ), i = 1, 2 2 ·  jxi  jP 1 1 · u i · dz = − ui = · S(x)2 · , i = 1, 2 S(x) z 12 ·  jxi

ui = −

j jx

(S(x) · u) +

j jy

(S(x) · v) = 0

1 · S(x)3 12 ·      j jP j jP M(x) · + M (x) · =0 jx jx jy jy

M(x) =

(29) (30) (31) (32) (33)

Referring to Fig. 3, Eq. (33) applies in the region FECD. On FD and EC, the boundary condition is symmetry (j P/jx = 0). The flow in the region upstream of FE (0 < y < yFE ) is assumed to be the Hele-Shaw solution for Poiseuille flow discussed above. Because P = P(y only) in Poiseuille flow, the boundary condition on FE is constant pressure. The pressure value PFE is determined as a part of the iterative solution. The distance L in Fig. 3 is increased until the predicted interface shape no longer changes significantly. Both the kinematic condition and the dynamic condition must be satisfied on the

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interface CD. In a coordinate system moving in the y-direction at the mean flow speed V0 , the Hele-Shaw kinematic condition is given by Eq. (35): jP jP U  = S(x) · (u¯  · xˆ + v¯  · yˆ ) = −M(x) · · xˆ − M(x) · · yˆ − S(x) · V0 · yˆ jx jy

(34)

U  · Nˆ G E = 0

(35)

The dynamic condition on CD is approximated by   cos(1 ) + cos(2 ) P = −2 ·  · m = − · x y · sin(1 ) + S(x)

(36)

The approximate expression we use for x y was given in Eq. (8). By solving Eq. (33), only one boundary condition can be applied on each part of the boundary. A second boundary condition is enforced through iteration. The dynamic update method for enforcing both the kinematic and dynamic conditions on CD is used in the manner of Lowndes (1980). Eq. (33) is solved numerically by mapping FECD to a rectangular domain and using finite differences. The input parameter set for the Hele-Shaw model of steady flow in the vicinity of the interface is eq,1 ;

eq,2 ;

HT 

;

Smax 

; Ca =

 · V0 

(37)

3.6. Summary: a quasi-steady model for predicting filling kinetics and finger growth A quasi-steady model is posed for horizontal, capillary-driven creeping flow of the Newtonian liquid in Fig. 2. The time rate of advancement of the mean position   1 YF = · Y (x) · dx 

0

of the air–liquid interface (dYF /dt), or interface velocity, is predicted using a Lucas–Washburn-type model (Lucas, 1918; Washburn, 1921) as detailed in Section 3.4. The shape of the contact line on the flat plate (Y (x, t)) is predicted by a Hele-Shaw model for steady flow, as detailed in Section 3.5. The steady interface velocity in the Hele-Shaw model is taken to be the instantaneous interface velocity predicted by the LWM. This quasi-steady model assumes that Y (x, t) in horizontal capillary-driven flow changes very gradually with YF , so that the solution for Y (x, t) in steady flow is a good approximation of Y (x, t) in capillary flow. 4. Results and discussion 4.1. Overview of results In Sections 4.2 and 4.3, experimental observations of interface finger length and gross penetration kinetics in capillary flow in the system described in Section 2 are used to illustrate the important role of flow cell plate spacing. In Sections 4.4–4.8, the theoretical models discussed in Section 3 are used to predict the behavior of the experimental system described in Section 2. Sections 4.4–4.8 are placed

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in order to increase modeling complexity (i.e. the results of the submodels are given first), building up to a prediction of interface finger length in capillary flow in Section 4.8. To understand the effects of the modeling approximations on the results in Section 4.8, the interested reader is encouraged to follow through Sections 4.4–4.7. As discussed in Section 2, a real printed circuit board is used to construct the flow cell. Such boards are not particularly flat. Therefore, the flow cell gap may deviate from the thickness of the spacer between the board and the glass plate. The results to follow may be considered as an assessment of the effectiveness of highly simplified modeling in describing the behavior of a realistic underfill flow test vehicle. 4.2. Experimental observations of fingering in capillary flow The geometry of the traces on the printed circuit board is measured using a stylus profilometer and is closely approximated by a sine wave of amplitude 5.5 m and wavelength 251 m {HT = 11 m;  = 251 m}. The characteristic form error of the float glass plate is 4 m per 25 mm. The measured maximumadvancing static wetting angle of mineral oil at 22 ◦ C is 12◦ on soldermask, and 26◦ on glass. Viscosity and surface tension of mineral oil at 22 ◦ C are .19 Pa s and .0313 N/m (Guo et al., 1999). In all experimental images, the camera focuses down at the horizontal flow cell, through the flat glass plate (the negative z-direction in Fig. 1). Fig. 4 shows sequential images in time of horizontal, capillarydriven flow at Smax = 19 m; the field of view is the entire 25 mm wide by 15 mm long cell. The traces are vertically oriented in the images, with roughly 100 traces from left to right. Fingers in tests at larger gaps (Smax = 25 m and above) are too small to resolve at this magnification. In Figs. 5 and 6, the camera is focused approximately halfway down the channel (x = 13 mm, y = 8 mm), and the image width spans three trace wavelengths (.8 mm ∼ 3 ∗ ). Fingertips are observed at the gap minima (S(x) = Smin ). Finger length L F is observed to decrease with increasing Smax , from L F ∼ 1 mm at Smax =19 m, to L F ∼ 30 m at Smax = 25 m, to L F >10 microns at Smax = 39 m. Values of Smax of 19, 25, and 38 m correspond to values of Smin of 8, 14, and 27 m. Experimental measurements of L F versus gross penetration distance  YF YF = 1/ · 0 Y (x) · dxare shown in Fig. 7. Finger length L F in horizontal, capillary-driven flow is observed to increase with decreasing Smax and increasing YF . 4.3. Experimental measurements of capillary filling rate Fig. 8 shows the square of the gross penetration distance   1 Y (x) · dx YF = · 

0

as a function of time for experiments at Smax = 19, 25, and 39 m. The relationship between YF2 and t is approximately linear; flow speed increases with Smax . The experimental values of capillary number (based on mean flow speed V0 ; Ca =  ∗ V0 /) and Reynolds number (ReSmin =  ∗ Smin ∗ V0 /) are small compared to 1. 4.4. Predicted finger length in hydrostatic equilibrium For illustrative purposes, the hydrostatic equilibrium model set forth in Section 3.2 is applied to the experimental system described in Sections 2 and 4.1.

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Fig. 4. Experimental observations of horizontal capillary flow of 22 ◦ C mineral oil ( = .019 Pa s and  = .0313 N/m according to Guo et al., 1999) in the flow cell illustrated in Fig. 1 (width = 25 mm, Smax = 19 m, Smin = 8 m, HT = 11 m,  = 251 m, eq,1 = 26◦ , eq,2 = 12◦ ), which is composed of a flat glass plate and a sinusoidal printed circuit board (see Section 2). Interface fingertips are observed at the locations where S(x) = Smin . In these images, the interface fingers are thin, fine lines (.13 mm in width, 1 mm in length, 100 fingers from left to right in each image). The advancing liquid is the darker region, the dry printed circuit board is the lighter region, and the fingers appear as a gray band between the two. Image resolution is .013 pixels per m at 30 frames per second.

Eq. (18) generates finger solutions for Smax greater than Scrit = 36.4 m; Scrit is computed using Eqs. (15) and (19) as explained at the end of Section 3.2. For Smax < Scrit , Eq. (18) generates closed curves (“bubbles”) instead of fingers. The pressure in the liquid P0 predicted using Eq. (17) is slightly more negative than that corresponding to the plane channel system (i.e. two flat, parallel plates) of gap spacing S = Savg .

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Fig. 5. Experimental observations of horizontal capillary flow of 22 ◦ C mineral oil ( = .019 Pa s and  = .0313 N/m according to Guo et al., 1999) in the flow cell illustrated in Fig. 1 (width = 25 mm, Smax = 19 m, Smin = 8 m, HT = 11 m,  = 251 m, eq,1 = 26◦ , eq,2 = 12◦ ), which is composed of a flat glass plate and a sinusoidal printed circuit board (see Section 2). Interface fingertips are observed at the locations where S(x) = Smin . The advancing liquid is the darker region. YF ∼ 8 mm  (YF = (1/) 0 Y (x) · dx) and L F ∼ 1 mm (L F = Ynarrow − Ywide ). Image resolution is .4 pixels per m at 30 frames per second.

Fig. 9 shows interface shape predictions (0 < x < /2) generated using Eq. (18) at four values of Smax (100, 80, 60, and 40 m). The three-dimensional interface shape is represented pictorially by plotting three characteristic curves in the air–liquid interface: the contact line x2 = x · xˆ + Y (x) · yˆ + Smax · zˆ + R · (sin(1 ) − sin(2 )) · Nˆ 1 − S(x) · zˆ on the sinusoidal plate, the contact line x1 = x · xˆ + Y (x) · yˆ + Smax · zˆ on the flat plate, and the back curve xb = x · xˆ +Y (x)· yˆ + Smax · zˆ − R ·(1−sin(1 ))· Nˆ 1 − R ·cos(1 )· zˆ . The liquid is predicted to advance farthest in the gap minima, which is simply a consequence of the balance of terms in Eq. (11). Predicted finger length L F (L F = Y (x = /2) − Y (x = 0) = Ynarrow − Ywide ) is shown in Fig. 10; L F is predicted to increase with decreasing Savg . Using a full numerical model free of geometric assumptions concerning interface shape, Borhan et al. (1991) predict that sessile liquid advances farthest in the gap maxima in a chosen parameter space (see Section 1.2). Their parameter space and model geometry are different than ours, and direct comparison of predicted finger shape is inappropriate. 4.5. Fully developed internal flow prediction Continuing an illustrative series of predictions for the experimental system discussed in Sections 2 and 4.1, the Poiseuille flow model set forth in Section 3.3 is applied. Table 1 gives the flow conductance K in the sinusoidal channel (HT = 11 m,  = 251 m) predicted by both the F.E. numerical solution and the Hele-Shaw solution (Eq. (23)); the two solutions agree in the parameter space of interest. Let K denote the flow conductance of a sinusoidal channel (Fig. 1), and K flat the flow conductance of a plane channel

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Fig. 6. Experimental observations of horizontal capillary flow of 22 ◦ C mineral oil ( = .019 Pa s and  = .0313 N/m according to Guo et al., 1999) in the flow cell illustrated in Fig. 1 (width = 25 mm, Smax = 25 m, Smin = 14 m, HT = 11 m,  = 251 m, eq,1 = 26◦ , eq,2 = 12◦ ), which is composed of a flat glass plate and a sinusoidal printed circuit board (see Section 2). Interface fingertips are observed at the locations where S(x) = Smin . The advancing liquid is the darker region. YF ∼ 8 mm  (YF = (1/) 0 Y (x) · dx) and L F ∼ 30 m (L F = Ynarrow − Ywide ). Image resolution is .4 pixels per m at 30 frames per second.

Fig. 7. Experimental measurements (see Sections 2 and 4.2) and model predictions (quasi-steady model of capillary flow with wetting angles given by Eq. (28); see Sections 3.6 and 3.8) of interface finger length L F in horizontal capillary flow in the channel shown in Fig. 1. Parameter values used in the model are HT = 11 m,  = 251 m,  = .019 Pa s,  = .0313 N/m,  eq,1 = 26◦ , eq,2 = 12◦ . Note: YF = (1/) 0 Y (x) · dx and L F = Ynarrow − Ywide . The numerical solution of the quasi-steady model of capillary flow fails to converge below Smax = 37 m for the chosen parameters, while fingers of significant length are experimentally observed at Smax = 19 and 25 m.

(i.e. parallel flat plates) having the same cross-sectional area as the sinusoidal channel (i.e. S = Savg ). The value of K flat /(Savg /2)2 is 1/3. As Savg decreases from +∞, the flow conductance K becomes larger than K flat . When Smin = 0, K is 52% larger than K flat . For Savg = (Scrit − HT /2) = 30.9 m (see beginning of Section 4.3), K is less than 6% greater than K flat .

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Fig. 8. Experimental measurements (see Sections 2 and 4.3) and Lucas–Washburn model (LWM) predictions (see Sections 3.6  and 4.6) of the history of the square of the gross penetration distance YF (YF = (1/) · 0 Y (x) · dx) in horizontal capillary flow in the channel shown in Fig. 1. The sketches at upper left refer to (A) the sinusoidal plate system shown in Fig. 1, and ((B)–(D)) three plane channel systems. Open symbols are experimental observations in system (A). Lines are LWM predictions for systems (A)–(D), as labeled. Dashed lines are the version of the LWM with wetting angles given by Eq. (28); solid lines are the version of the LWM with wetting angles fixed at the static values. Experimental flow cell width is 25 mm. Parameter values are HT = 11 m,  = 251 m,  = .019 Pa s,  = .0313 N/m, eq,1 = 26◦ , eq,2 = 12◦ . The LWM is inapplicable to system (A) for Smax less than Scrit = 37 m for the chosen parameters (see Sections 3.2 and 3.4).

4.6. LWM predictions of capillary filling history Continuing an illustrative series of predictions for the experimental system discussed in Sections 2, 4.1, and 4.2, Fig. 8 shows the square of the gross penetration distance YF =

1 

 · 0



Y (x) · dx

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Fig. 9. Top view of the predicted air–liquid interface shape in hydrostatic equilibrium in the channel shown in Fig. 1 (see Section 3.2). The contact line x1 on the flat plate, the contact line x2 on the sinusoidal plate, and the back curve xb of the interface are shown. Parameter values are HT = 11 m,  = 251 m,  = .0313 N/m, eq,1 = 26◦ , eq,2 = 12◦ . A finger-like shape of the interface is predicted for Smax > Scrit = 36.4 m for the chosen parameters (see Section 3.2).

in horizontal capillary flow predicted using several versions of the LWM set forth in Section 3.4. Fig. 8 also includes the capillary flow experimental measurements discussed in Section 4.2. At Smax = 39 m, six LWM versions are shown (sinusoidal channel, plane channel at the maximum and minimum gaps Smax and Smin , with and without the dynamic wetting angle correlation of Eq. (28)). At Smax = 19 and 25 m, Smax is smaller than Scrit = 36.4 m (see Section 4.3), so the LWM for the sinusoidal channel cannot be evaluated because an interface pressure prediction is lacking (see Sections 3.4 and 3.2). At Smax = 19 and 25 m, four versions of the plane-channel LWM are shown (plane channel at the maximum and minimum gaps Smax and Smin , with and without the dynamic wetting angle correlation of Eq. (28)). The LWM predictions for the plane channel and the sinusoidal channel are further compared in Fig. 11 using Eq. (27) with the parameters corresponding to the experimental system discussed in Sections 2

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Fig. 10. Model predictions for interface finger length L F = Ynarrow − Ywide in both hydrostatic equilibrium (see Sections 3.2 and 4.4) and capillary flow (see Sections 3.6 and 4.8) in the sinusoidal plate flow cell shown in Fig. 1. Parameter values are HT = 11 m,  = 251 m,  = .019 Pa s,  = .0313 N/m, eq,1 = 26◦ , eq,2 = 12◦ . For capillary flow, solutions are shown for  YF = (1/) 0 Y (x) · dx = 1, 2, 4, 13, and 75 mm.

Table 1 Finite-element (F.E.) and lubrication (L) model predictions of the flow conductance (K) for Poiseuille flow of a Newtonian liquid in the sinusoidal channel in Fig. 1 Savg (m)

HT /2/Savg

/(Savg /2)

K F.E. (m2 )

K L (m2 )

K F.E. /(Savg /2)2

94.5 54.5 29.5 19.5 14.5 10.5 8.5 6.5 5.5

0.06 0.10 0.19 0.28 0.38 0.52 0.65 0.85 1.00

5.31 9.21 17.02 25.74 34.62 47.81 59.06 77.23 91.27

7.42E−10 2.49E−10 7.56E−11 3.52E−11 2.11E−11 1.29E−11 9.73E−12 7.25E−12 6.26E−12

7.48E−10 2.51E−10 7.63E−11 3.55E−11 2.13E−11 1.30E−11 9.80E−12 7.30E−12 6.30E−12

0.33 0.34 0.35 0.37 0.40 0.47 0.54 0.69 0.83

The values of HT and  are 11 and 251 m.

and 4.1. The gross penetration distance YF (t) in horizontal capillary flow is predicted to be only a few percent greater at a given time in the sinusoidal channel than in the plane channel of equal cross-sectional area for {Scrit = 36.4 m < Smax < 100 m}. Borhan and Rungta (1993) also predict a small difference between YF (t) for capillary rise in their sinusoidal channel and YF (t) for capillary rise in the plane channel of equal cross-sectional area, for the parameter range {/Savg = 2.5, HT /Savg < 1}. Specifically, their predicted rise time te in their sinusoidal channel for {/Savg = 2.5, HT /Savg < 1} is less than 20% greater than te in the plane channel of equal cross-sectional area.

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Fig. 11. Time history of gross penetration distance in flow cells A–D predicted by a Lucas–Washburn model, with contact angles fixed at static values  (see Section 3.4). The curves in the upper-left plot represent the ratio at time t of gross penetration distance YF (YF = (1/) · 0 Y (x) · dx) in flow cell J (where J is one of A–D) to YF in flow cell C. Parameter values are HT = 11 m,  = 251 m,  = .0313 N/m, eq,1 = 26◦ , and eq,2 = 12◦ , and  = .019 Pa s.

As seen in Fig. 8, the LWM predictions of filling rate fall significantly short of the experimental measurements at all gaps. The experimental flow channel gap spacing may differ from the polymer shim thickness due to printed circuit board form error. “Side racing” in the experiment could alter the capillary flow kinetics. Side racing is an artifact of the finite width of the flow cell, and of insufficient sealing of the flow cell sides, which can increase the driving pressure gradient in the liquid by increasing the pressure in the liquid at the flow cell sides. 4.7. Predicted finger length in steady flow (Hele-Shaw model) Continuing an illustrative series of predictions for the experimental system discussed in Sections 2 and 4.1, the Hele-Shaw model of steady flow set forth in Section 3.5 is applied. Convergence of the iterative solution for interface shape was obtained for Smax 37 m and larger. A typical solution is shown in Fig. 12. The point labels F, E, G, C, and D are consistent with those in Fig. 3. The curved boundary CD is the predicted position of the contact line on the flat plate x1 = {x, y = Y (x), z = Smax }. The predicted gap-integrated velocity on FE is nearly identical to that predicted by the Poiseuille flow model (Sections 3.3 and 4.4). Fig. 12 shows a component of flow tangent to the interface, directed from the gap maximum

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Fig. 12. Velocity vectors (arrows), pressure contours, and interface shape Y (x) in the sinusoidal channel (Fig. 1), predicted by a Hele-Shaw model of steady flow (see Section 3.5). The velocity is referenced to a coordinate system translating in the positive y-direction at the mean speed of the flow V0 . The lettering (C, D, E, F, and G) is consistent with Fig. 3. Parameter values are Smax = 40 m, Smin = 29 m, Ca =  ∗ V0 / = .0024, Re =  ∗ V0 ∗ Smin / = .0026, HT = 11 m,  = 251 m, eq,1 = 26◦ , eq,2 = 12◦ . The predicted L F = Ynarrow − Ywide = 67 m.

Smax to the gap minimum Smin . This component of flow is associated with the non-uniform upstream (Poiseuille) velocity distribution on yFE ; the faster-moving fluid at the gap maximum Smax must turn at the interface. This type of flow is often called a “fountain flow.” A set of predictions of interface finger length L F = Y (x = /2) − Y (x = 0) = Ynarrow − Ywide is shown in Fig. 13. Increasing the mean flow velocity V0 at fixed Savg is predicted to decrease L F from the hydrostatic equilibrium value at V0 = 0 (Section 3.2 and 4.3). At fixed V0 , L F is predicted to decrease with increasing Savg . When contact angles vary with V0 according to Eq. (28) instead of remaining fixed at the static values, L F is predicted to decrease at fixed V0 . Fig. 2 is a three-dimensional plot of a typical predicted interface shape. 4.8. Quasi-steady model-predicted finger length in capillary flow Completing an illustrative series of predictions for the experimental system discussed in Sections 2 and 4.1, the quasi-steady model set forth in Section 3.6 is applied. Convergence of the iterative solution for interface shape was obtained for Smax 37 m and larger. Predicted interface finger length L F (contact angles fixed at the static values) is shown in Fig. 10 as a function of flow cell gap Savg for several values of gross penetration distance YF . As YF increases at fixed Savg , L F is predicted to approach the hydrostatic equilibrium model prediction (Section 4.3). Fig. 14 compares the predicted effect on L F of fixed contact angles versus dynamic contact angles (Eq. (28)).

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Fig. 13. Predictions of interface finger length (L F = Y (x = /2) − Y (x = 0) = Ynarrow − Ywide ) in steady flow in a sinusoidal channel (Fig. 1) given by a Hele-Shaw model (see Section 3.5). Finger length L F is predicted to decrease with increasing mean flow speed V0 at fixed gap spacing Smax . For the model predictions connected by solid lines, the contact angles are fixed at the static values. For the model predictions connected by dotted lines, the contact angles vary with V0 according to Eq. (28). The parameter set is HT = 11 m,  = 251 m, eq,1 = 26◦ , eq,2 = 12◦ ,  = .0313 N/m, and  = .019 Pa s.

Model predictions with dynamic contact angles (Eq. (28)) are plotted alongside experimental measurements in Fig. 7 (see Section 4.1). Model predictions at the experimental gaps of Smax = 19 and 25 m could not be obtained due to lack of convergence of the iterative solution below Smax = 37 m. The model and the experiment exhibit the same qualitative trends.

5. Conclusions A novel approximate model (“quasi-steady model”) is introduced for predicting interface finger length and penetration kinetics for a Newtonian liquid displacing a gas by capillary action in a long, narrow sinusoidal channel. The quasi-steady model predicts capillary flow penetration kinetics using Lucas–Washburn model, and predicts capillary flow interface finger length using Hele-Shaw model of steady infiltration. The Lucas–Washburn model employs a novel hydrostatic equilibrium model which gives a closed-form solution for interface pressure, and a lubrication approach for Poiseuille flow which gives a closed-form solution for flow conductance; the Lucas–Washburn model thus gives a closed-form expression for penetration distance history in capillary flow. The hydrostatic equilibrium model prediction of interface finger shape requires only a single numerical integration. The solution of the quasi-steady model requires far

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Fig. 14. Interface finger length (L F = Y (x = /2) − Y (x = 0) = Ynarrow − Ywide ) at fixed flow cell gap Smax in capillary flow in a sinusoidal channel (Fig. 1), predicted by a quasi-steady model (see Section 3.6). The dotted lines correspond to a version of the model with contact angles given by Eq. (28); the solid lines correspond to a version of the model with contact angles fixed at the static values. Parameter values are HT = 11 m,  = 251 m,  = .019 Pa s,  = .0313 N/m, eq,1 = 26◦ , eq,2 = 12◦ . Note:  (YF = (1/) · 0 Y (x) · dx).

less computing time than the numerical solution of the Hele-Shaw model undertaken by Bogoyavlenskiy, allowing predictions of fingering in capillary flow at arbitrary gross penetration distance. The modeling approach is illustrated by application to an experimental system consisting of mineral oil between a flat glass plate and a sinusoidal printed circuit board (a simplified system for the study of underfill encapsulation). Interface finger length is experimentally observed to be highly sensitive to flow cell gap spacing. Interface finger length is observed to increase with decreasing mean flow cell gap, and with increasing gross penetration distance. In every case, finger length is observed to increase with decreasing mean flow speed. Interface fingertips are observed at the minima of the sinusoidal gap in each flow cell. The quasi-steady model predicts the qualitative trends in finger length observed experimentally. Finger length given by the hydrostatic equilibrium model is an upper bound for the finger length given by the quasi-steady model, and requires only a single numerical integration. Acknowledgements This research was funded by NSF award #DMI 9908332 and the Integrated Electronics Engineering Research Center (IEEC) located in the Watson School at Binghamton University. The IEEC receives

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funding from the New York State Science and Technology Foundation, the National Science Foundation and a consortium of industrial members. Drew Davidson wishes to thank Dorel Homentcovschi (M.E. Department Binghamton University) for helpful discussions. References Bogoyavlenskiy, V.A., unpublished work. Bogoyavlenskiy, V.A., Giamis, A.C., Cotts, E.J., 2004. Mean-field dynamics of free surface flows through obstacle arrays in a narrow passage: ammendments of the washburn model. Fluid Dyn. Res. 35, 23–35. Borhan, A., Rungta, K.K., 1993. Lucas–Washburn kinetics for capillary penetration between periodically corrugated plates. J. Colloid Interface Sci. 155, 438–443. Borhan, A., Rungta, K.K., Marmur, A., 1991. Capillary penetration of liquids between periodically corrugated plates. J. Colloid Interface Sci. 146, 425–433. Davidson, D.A., Lehmann, G.L., Cotts, E.J., 2002. Effect of geometric surface features on void formation: application to the underfill process. In: Proceedings of 52nd Electronic Components and Technology Conference, pp. 1411–1418. Fine, P., Cobb, B., Nguyen, L., 2000. Flip chip underfill flow characteristics and prediction. IEEE Trans. Components Packag. Technol. 23, 420–427. Guo, Y., Lehmann, G.L., Driscoll, T., Cotts, E.J., 1999. A model of the underfill flow process: particle distribution effects. In: Proceedings of 1999 IEEE/EIA CPMT Electronic Components and Technology Conference. Hoffman, R., 1975. A Study of the advancing interface. I. Interface shape in liquid–gas systems. J. Colloid Interface Sci. 50, 228–241. Jiang, T.S., Oh, S.G., Slatterly, J.C., 1979. Correlation for dynamic contact angle. J. Colloid Interface Sci. 69, 74–77. Lowndes, J., 1980. The numerical simulation of the steady movement of a fluid meniscus in a capillary tube. J. Fluid Mech. 101, 631–646. Lucas, R., 1918. Ueber das Zeitgesetz des Kapillaren Aufstiegs von Flussigkeiten. Kolloid Z. 23, 15–22. Machuga, S.C., Lindsey, S.E., Moore, K.D., Skipor, A.F., 1992. Encapsulation of flip chip structures. In: Proceedings of IEEE/CHMT International Electronics Manufacturing Technology Symposium, pp. 53–58. Washburn, E.W., 1921. The dynamics of capillary flow. Phys. Rev. 17, 273–283.