3222
Langmuir 1991, 7, 3222-3228
Evaluation of Two-Dimensional and Three-Dimensional Axisymmetric Fluid Interface Shapes with Boundary Conditions S. Rooks,? L. M. Racz,l J. Szekely,l B. Benhabib,t and A. W. Neumann*pt Department of Mechanical Engineering, University of Toronto, Toronto, Ontario, Canada M5S lA4, and Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received January 28,1991. In Final Form: October 21, 1991 This paper evaluates the equilibrium shapes of two-dimensional and three-dimensionalaxisymmetric fluid interfaces which result when liquid menisci are in contact with horizontal and vertical intersecting surfaces, subject to boundary conditions on the contact angles and the initial value of the first principal radius of curvature. The motivation for the use of these particular boundary conditions is explained, and an extensive literature review is presented. The two-dimensionalcase is evaluated based on existing work done on menisci between two vertical plates, and an original technique is also presented. For the threedimensional case, a numerical technique is introduced to transform the Laplace equation of capillarity from a two-point boundary value problem to an iterative initial value problem. The solution of the threedimensional case is further simplified by an approximation using elliptic integrals to calculate the height of the capillary surface, while a fourth-order Runge-Kutta technique is used to calculate its volume. Finally, the linear relationship between the fluid volume and the height of the surface is investigated. 1. Introduction
A liquid meniscus in contact with intersecting horizontal and vertical surfaces will rise up the vertical surface and extend out along the horizontal surface until the system reaches equilibrium. Equilibrium is reached when the free energy change due to the exchange of solid/vapor interfacial area with solid/liquid interfacial area is balanced by the free energy change due to the increase in liquid/vapor interfacial area and the work done against gravity. With sufficient liquid volume and horizontal wettable surface area, the liquid meniscus will extend outward until the first principal radius of curvature is infinite and the contact angle equals the phenomenological contact angle for the particular surface material and roughness at the three-phase line of contact with the horizontal surface. On high-energy horizontal surfaces, the equilibrium shape of the fluid interface can be considered level with the horizontal surface at some distance from the three-phase line of contact with the vertical surface, and, therefore, the fluid interface can be considered unbounded, that is, as extending to infinity. In this case, the unbounded Laplace equation of capillarity can be solved analytically to determine the equilibrium shape of the fluid interface under hydrostatic and surface tension forces and, hence, can be used to determine the volume of the liquid meniscus. 1.1. Literature Review. Extensive previous research has been done on determining equilibrium shapes of fluid interfaces. Much of this work dealt with the unbounded cases of particles, plane surfaces, or rotationally symmetric bodies in contact with fluid of infinite extent. White and Tallmadge,l for example, presented solutions of the Laplace equation for static menisci on the outside of cylinders. P r i n ~ e ndeveloped ~.~ a solution for the shape of a liquid drop at a liquid-liquid interface. Huh and Scriven4 + University of Toronto.
Massachusetts Institute of Technology. (1) White, D. A.; Tallmadge, J. A. Static Menisci on the outside of cylinders. J. Fluid Mech. 1965, 23, 325. (2) Princen, H. M. Shape of a Fluid Drop at a Liquid-Liquid Interface. J. Colloid Sci. 1963, 18, 178. (3)Princen, H. M. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley-Interscience: New York, 1969; Vol. 11, Chapter 1. 4
0743-7463/91/2407-3222$02.50/0
described an improved method for calculating the shape of an unbounded axisymmetric fluid interface, and Rapachietta and Neumann5g6presented a modified RungeKutta technique for calculating the meniscus for small spherical and cylindrical particles floating on the surface of a fluid. Other authors have, in more recent publications, addressed the solution of bounded menisci. Princen? for example, addressed the problem of a meniscus between two vertical plates, and Hornung and Mittelmann7 determine numerically the shapes of capillary surfaces of prescribed volume which are constrained by arbitrarily chosen planes in space. There exists, however, a strong motivation to solve the Laplace equation of capillarity with the particular boundary conditions presented in this paper. Aside from its theoretical interest, this formulation is applicable to the determination of optimal liquid solder volumes to produce concave menisci with specific heights between the copper pads of printed circuit boards and the leads of electronic components. Some literature aimed at this particular industry does exist; Wassink: for instance, presented an unbounded solution to the Laplace equation to estimate volumes of tin/lead solder menisci. Similar work has been done by Jopek? who determined empirically an acceptable solder volume range, which he finds to be 17-45% higher than the theoretical prediction. The disparity between the experimental results and the theoretical prediction is presumably due to the use of the unbounded analytical solution and assumptions made (4) Huh, C.; Scriven, L. E. Shapes of Axisymmetric Fluid Interfaces of Unbounded Extent. J . Colloid Interface Sci. 1969, 30, 323. (5) Rapachietta,A. V.; Neumann, A. W. ForceandFreeEnergy Analyses of Small Particles at Fluid Interfaces: I. Cylinders. J. Colloid and Interface Sci. 1977, 59,555. (6) Rapachietta,A.V.; Neumann, A. W. Force and Free Energy Analyses of Small Particles at Fluid Interfaces: 11. Spheres. J. Colloid Interface Sci. 1977, 59, 555. (7) Hornung, U.; Mittelmann, H. D. A Finite Element Method for Capillary Surfaces with Volume Constraints. J.Comput. Phys. 1990,87,
---. 1%
(8) Klein Wassink, R. J. Soldering in Electronics; Electrochemical Publications, Ltd.: Ayr, Scotland, 1989. (9) Jopek, S. Solder Volume Determination for Fine Pitch Packages. Printed Circuit Assembly 1989, 3, 15.
0 1991 American Chemical Society
Langmuir, Vol. 7, No. 12, 1991 3223
Equilibrium Shapes of Axisymmetric Fluid Interfaces
2-D Case: Meniscus Wetting t o Intersecting Plane Surfaces
a
3-D Axisymmetric Case: Meniscus Wetting t o a Cylinder and a Plane Surface which h a s Limited Wettable Surface Area
Figure 1. Examples of liquid menisci with finite volumes in contact with intersecting vertical and horizontal surfaces. regarding solder volume and geometry. ChulO developed a bounded solution for printed circuit board components with "through-hole" geometries in which he determines the shapes of lowest-energy configurations but does not perform analyses on surface height or solder volume. Heinrich et al." presented an analytical model of solder joint formation. The model expresses the two-dimensional Laplace equation in an explicit integral form which they solve for the two limiting cases of infinitesimal and infinite wettable areas. The purpose of this paper, then, is to explore solutions of the Laplace equation that are motivated by the microelectronics industry: to determine the equilibrium shapes and volumes of two-dimensional (2-D) and threedimensional (3-D) axisymmetric liquid menisci, both of which possess many of the generic attributes of circuit components, which result when these menisci, subject to appropriate boundary conditions, are in contact with intersectinghorizontal and vertical surfaces. In particular, the Laplace equation of capillarity will be applied to determine the volumes of liquid menisci in contact with (Figure 1) (a) intersecting horizontal and vertical plane surfacesand (b) a vertical cylinder intersecting a horizontal plane surface having limited wettable surface area. 2. The Laplace Equation of Capillarity The well-known Laplace equation of capillarity combined with the equation for hydrostatic pressure relates the pressure difference AP across a fluid interface within a gravitational field to the curvature of the interface and the interfacial tension
where, referring to Figure 2, PI and P2 are the pressures on either side of the interface between phases 1and 2, Ap is the difference between the densities of phases 1and 2, g is the acceleration due to gravity, y is the height of the meniscus above the horizontal plane surface, R1 and R2 are the principal radii of curvature of the interface, and 7 1 2 is the interfacial tension. If PI is greater than P2, the interface will be curved toward phase 1,and R1 and R2 are positive or negative depending on whether the centers of curvature are located inside phase 1or 2, respectively. In principle, therefore, the Laplace equation permits the calculation of the equilibrium shape of any curved interface (10) Chu, T. Y. A Hydrostatic Model of Solder Fillets. Western Electric, The Engineer Condensation Soldering Sequence; April, 1985; p 13. (11) Heinrich, S. M.; et al. Solder Joint Formation in Surface Mount Technology; ASME J. Electron. Packaging 1990, 112, 210.
Yt Meniscus Profile
I
z
X
0'
Figure 2. Laplace equation of capillarity: liquid meniscus profile. between two fluid phases, in this case,the interface between a liquid meniscus (phase 1) and its vapor (phase 2). Knowing the shape of the liquid meniscus subject to the particular boundary conditions addressed in this study permits the determination of the liquid volume. As mentioned previously, the principal radii of curvature, R1 and R2, are positive when they lie inside the liquid meniscus and negative when they lie outside the liquid meniscus. Referring to Figure 2, R1 always rotates in the plane of the paper, and R2 rotates in a plane perpendicular to the paper. If the liquid meniscus has rotational symmetry about an axis, R2 sweeps out a cone coaxial with this axis (x = 0), otherwise, it is infinite. If the liquid is to the left of the meniscus profile in Figure 2, the slope and the principle radii of curvature of the meniscus can be expressed as -= d~ dx
-tan4 -1 &c sin 4 d+
R, =-
(3)
X R2 = sin 4
(4)
where x is the distance from the rotational axis of symmetry and 4 is the angle that the tangent to the curve at (x,y) makes with the horizontal.8 2.1. Boundary Conditionson the Laplace Equation. Equation 1 implies that the pressure difference at the datum, y = 0, is zero and, therefore, that the principal radii of curvature at the datum, R1° and R2O, are infinite resulting in an unbounded interface. with 4 0 as x This unbound.ed case of the Laplace equation can be used to model the shape of a liquid meniscus in contact with a plate or a wire in a liquid reservoir or the various other cases mentioned in section 1.1. However, as shown in Figure 3, when the volume of liquid is limited, a pressure difference exists at the datum, and this formulation is no longer valid. The first principal radius of curvature now has a finite value, RlO,at the datum, and the Laplace equation is altered in the following manner
- -
00,
where
therefore APgY =-Y++---
1
R2
'I
R,O
(7)
In addition to the boundary condition on the first principal
Rooks et al.
3224 Langmuir,Vol. 7,No. 12, 1991
sectional area and shape. Combining eqs 3 and 7 with R2 equal to infinity the Laplace equation becomes
Rearranging eq 9 yields
Integrating eq 10 analytically yields an expression for determining meniscus height Figure 3. Boundary conditionson the Laplace equation for 2-D and 3-D axisymmetric cases.
radius of curvature at the datum, boundary conditions also exist on 4 at either end of the liquid meniscus. A three-phase line of contact exists between the liquid, its vapor, and the solid surface with which the liquid is in contact at either end of the liquid meniscus. At these lines of contact, 4 should equal the phenomenological contact angle between the liquid and the solid surface. Therefore, as shown in Figure 3, should equal the phenomenological contact angle, CXH between the liquid and the material of the horizontal plane surface. Likewise, 4' should depend on the phenomenological contact angle, a,, between the liquid and the material of the vertical surface such that
41 = 900 - a,
Also taking the differential dy of eq 11 and substituting into eq 2 results in
and from the differential expression for area dA, = ydx
(8)
(13)
However, in many applications involving high-energy surfaces, the extent of the liquid meniscus is limited by a nonwettable mask surrounding the wettable surface area. If the liquid meniscus extends to the contact line between the wettable surface area and the nonwettable mask, the boundary condition on 4O no longer holds, and at this juncture 4O will take on some value between CYHand 180°, which is the phenomenological contact angle between any liquid and a nonwettable surface.
In general, the differential equations (12 and 13) do not have analytical solutions. However, in the regime of interest for practical applications, that of concave menisci, the solution for eqs 9, 12, and 13 corresponds to that for a meniscus between two vertical plate^.^ Moreover, the effect of gravity is generally negligible in the regime of interest. This is determined by the fact that the Bond number
3. Solutions of the Bounded Laplace Equation In general, the combination of eqs 2,3,4, and 7 leads to a second-order, nonlinear differential equation for which an analytical solution exists for only a few cases. In particular, an analytical solution is only obtainable for y when the second radius of curvature, R2, is infinite, which occurs when a liquid is in contact with intersecting planes of infinite width. When R2 is finite (for example, when a liquid is in contact with an axisymmetric solid that intersects a plane perpendicular to its axis), generally the Laplace equation can only be solved numerically. However, as will become evident in subsequent discussion, an approximation by elliptic integrals is possible for concave menisci, and an analytical approximation is valid when gravity is negligible.3 3.1. 2-D Case: Menisci in Contact with an Intersecting Plane Surface. When a liquid meniscus is in contact with an intersecting plane surface, the surface cannot be considered infinite in width since the second radius of curvature of the meniscus, R2, does have a finite value near the edges of the plane surface. However, the shape of the central portion of the liquid meniscus where the three-phase line of contact is straight should not be significantly affected by the edge effects. The 2-D analytical solution of the Laplace equation can be used to model this region, which in turn is a good approximation of the whole meniscus; thus, this technique can be used to obtain an accurate estimate of the meniscus' cross-
Y (14) a dimensionless group relating surface tension forces to gravitational forces, L being the characteristic dimension of the surface, is sufficiently small.12 In this regime, the arcs formed by the menisci are essentially circular" and are thus given by
A, =
cos
y = -R;(cos
a. - sin al)
(15)
x = -R:(cos
a1- sin ao)
(16)
a.
1 cos a,- -sin 4 2ao -
which vastly simplifiesthe calculations necessary to obtain the meniscus shape. 3.1.1. Application of the Bounded 2-DLaplace Equation. To demonstrate the application of the bounded 2-D solution of the Laplace equation, eqs 11-13 were applied to determine the cross-sectional areas of liquid tin/lead solder menisci of various heights in contact with intersecting vertical and horizontal copper plane surfaces (Figure 1). Two different sets of boundary conditions were applied to demonstrate the different effects on the shape and cross-sectional area of the liquid meniscus: (1) a ' and # l = 90" boundary condition only on R1° with = 0 (12) Szekely, J.FluidFlou, Phenomena inMetals Processing; Academic Press: New York, 1979.
Equilibrium Shapes of Axisymmetric Fluid Interfaces Y
Langmuir, Vol. 7, No. 12,1991 3225 Y
0.16
("1
("1
("1
0.080 0.120 0.160 0.200 0.241 0.282 0.323
0.080
0.080 0.0014 0.120 0.0031
0.120 0.160 0.200 0.240 0.280 0.320
0160 0 200 0.241 0.281 0.322
yU
("1
("2)
4ro)
-Rt
b" ("1
@
a , v ("2)
("3)
0.130 0.332 0.506 10.01' 0.0071 0.0088 0.350 0.477 6.46 0.0073 O . W g 2 0.140 0.350 0.606 10 lT 0.0084 0.0107
0.0055
0.0086 0.0124 0.0169 0.0221
1////A
0 1W
I 0.350 I 10971 I 23.50 /0.0127 10,01671
0.08 a 0 I
I
I
0.08
0.16
I
= 0' I
0.24 0.32 0.40
I
X
0.48 (mm)
Figure 4. Profiles of liquid menisci contacting intersectingplane surfaces with boundary conditions on RlO. Y (")
Figure 6. Profiles of liquid menisci contacting a cylinder and a horizontal plane surface having limited wettable surface area. Table I. Comparison of General and Low-Gravity Approximation Calculations of the Meniscus Shape for the 2-DCase; Ideal State
-
0.24
-Ria 0.16-
0.08
a, = I
I
0.08 0.16
I
I
0.24
0.32
o.ko
io'
X o,k8 (mm)
Figure 5. Profiles of liquid menisci contacting intersectingplane surfaces with boundary conditions on RI0and 6. (Figure 4); (2) boundary conditions on both R1° and 4, with 40 = loo and c#+ = 75O, and reduced liquid surface tension (Figure 5). The boundary condition on RlO, which has the greatest effect on meniscus shape, was determined from the desired meniscus height ymaxby the following expression derived from eq 11
The following values were used for the material properties of eutectic solder (63 Sn-37 Pb) for the evaluations of eqs 11-14? 1. liquid surface tension at 250 OC in an inert atmosphere (Figure 4) mJ m2
yLv = 458 -
.using flux (0.2 % C1 activation) (Figures 5 and 6): mJ m2
YLV = 375 -
2. density Ap = 8420
m3
Figures 4 and 5 were generated for the desired meniscus heights under the different conditions using eq 11 by first calculating the necessary R1° from eq 14. The crosssectional area of the meniscus, A,, and the meniscus length, xm, (y = 0), were then determined numerically from the
0.080 0.120 0.160 0.241 0.282 0.323
y(x=O)
= 0) x ( y = 0) 0.080 0.080 0.120 0.120 0.160 0.160 0.240 0.240 0.280 0.281 0.320 0.322
y(x
A, (zerog) 0.0014 0.080 0.0031 0.120 0.0055 0.160 0.0124 0.241 0.0169 0.282 0.0221 0.323
x(y=O)
(zerog) 0.080 0.120 0.160 0.241 0.282 0.323
A, (zerog) 0.0014 0.0031 0.0055 0.0125 0.0171 0.0224
Table 11. Comparison of General and Zero-Gravity Approximation Calculations of the Meniscus Shape for the 2-DCase; Reflow with Flux -R~O 0.125 0.187 0.250 0.313 0.377 0.442 0.507
y(x = 0)
xb = 0)
0.080 0.120 0.160 0.200 0.240 0.280 0.320
0.095 0.143 0.191 0.239 0.287 0.335 0.384
= 01 (zerogj 0.080 0.120 0.161 0.201 0.242 0.284 0.326
v(x "~
A, 0.0024 0.0054 0.0097 0.0151 0.0218 0.0297 0.0388
x(v =O)
A,
(ierogj 0.096 0.143 0.191 0.240 0.289 0.339 0.388
(zerig) 0.0024 0.0054 0.0097 0.0152 0.0221 0.0304 0.0400
~~
differential system of eqs 12 and 13, using a fourth-order Runge-Kutta numerical technique with an adaptive step size control based on the truncation error estimate.13This was implemented in a C language program which is available from the authors. Figure 4 represents an ideal state with boundary conditions placed only on the first principal radius of curvature, RlO,and the contact angles equal to zero. Figure 5 represents the conditions when solder is reflowed using flux to remove surface oxides, which is reflected by the surface tension of the liquid solder and the nonzero contact angles. Setting boundary conditions on a0 and a1 results in a greater cross-sectional area for an equivalent meniscus height over the ideal state in Figure 4. The above calculations were then reproduced, excluding the last cases in Figures 4 and 5, in which gravity has a significant effect, using the zero-gravity analytical approximation (eqs 15-17). The results obtained are tabulated in Tables I and I1 alongside the generalcalculations, where it is evident that the results of Figures 4 and 5 are reproduced to within 3% accuracy. It is thus clear that for the practical considerations of the microelectronics industry, the use of the analytical approximation is much preferable. Furthermore, the values for the maximum meniscus height ymaxand the cross-sectional area A, were calculated (13)Press, W. H.; et al. NumericaERecipes in C: The Art of Scientific Computing; Cambridge University Press: Cambridge, 1989.
Rooks et al.
3226 Langmuir, Vol. 7, No. 12, 1991 using the unbounded solution of the Laplace equation, that is, with R1O = w for comparison with the results of the bounded solution. The value for the meniscus length in Figure 5 is bracketed because, though it is definitely finite, its accuracy is questionable. As R1° a, the calculated value for the meniscus length becomes very sensitive to the accuracy of the numerical solution. To obtain another estimate, the reader is here referred to Princen’s solution for a meniscus between flat platesa3 Clearly, from Tables I and I1 and Figures 4 and 5, applying boundary conditions to the Laplace equation greatly reduces the calculated values for the height and cross-sectional area of the liquid solder meniscus as compared to the unbounded Laplace equation. Finally, to compare this approach of estimating volumes of liquid solder menisci to Jopeks approach? the optimum volume of a liquid solder meniscus in contact with a tinned copper lead and a copper pad was calculated for the same geometry that Jopek considered. Using the unbounded 2-D Laplace equation for the particular geometry under consideration, Jopek calculated that the theoretical optimum liquid solder volume for a meniscus height of 0.25 mm was 0.0097 mm3 but determined experimentally that the acceptable range was 0.0114-0.0147 mm3. However, using the bounded 2-D Laplace equation for the same geometry,the optimum liquid solder volume was calculated to be 0.0140 mm3, which compares well with the experimental results reported by Jopek. Thus, it is clear that the use of the bounded 2-D Laplace equation more accurately models the shape and volume of a liquid meniscus in contact with intersecting plane surfaces. 3.2. 3-D Axisymmetric Case: Menisci in Contact with a Cylinder Intersecting a Plane Surface. In Figure 1, if the cylinder’s axis is perpendicular to the normal of the circular horizontal plane surface and is aligned with the circular surface’scenter, a liquid meniscus in contact with the cylinder will have rotational symmetry about the cylinder’s axis. Though perfect centering of the cylinder on the circular surface is unachievable in actual practice, the strong surface tension forces acting to center the cylinder ensure that the results based on the assumption of rotational symmetry are reasonably accurate. When a liquid meniscus has rotational symmetry about an axis, the second radius of curvature, Rz, has a finite value, which is expressed by eq 4. This results in the combination of eqs 2,3,4, and 7 leading to a second-order, nonlinear, differential equation which, in general, requires a numerical solution. With R2 having a finite value, eq 7 becomes
equation may be parametrized with 4 as the independent variable and using eq 2 to derive an expression for d X
-
-x cos
dX
c#J
(23)
The above system of differential equations (23) constitutes a two-point boundary-value problem which must satisfy the following boundary conditions at the desired starting and ending points (Figure 3): 1. (XO, YO): (a) liquid meniscus extends to the edge of the wettable area xO
= Rwettable
area
(%Y2
Yo = 0 -w
