Flip-Chip Underfill

What is modelled

The gap under a flip-chip die is a narrow slot between two plates. A wetting resin is pulled into it by capillary pressure and slowed by viscous friction, so the front slows down as it goes: the distance grows with the square root of time. Heating lowers the viscosity and speeds the fill, but also speeds the cure, so there is a window between fast enough and gelled.

Equations

• Capillary pressure Δp = 2 γ cosθ / h (Young-Laplace for two plates).
• Plate flow, mean speed u = h² Δp / (12 μ L), so L dL/dt = h γ cosθ f / (6 μ) and L² = L0² + K t, K = h γ cosθ f / (3 μ).
• Bump blockage f = (1 − φ)² with area fraction φ = π d² / 4p².
• Viscosity μ(T) = μ25 exp[Ea/R (1/T − 1/298)]; gel time tgel = tref exp[Ec/R (1/T − 1/373)] with 600 s at 100 C.

Checks run in the page

• Runge-Kutta stepping of the front against the closed form.
• The parabolic velocity profile integrated across the gap against h²Δp/12μL.
• The capillary pressure from the slope of the surface energy.
• The fill time of a die twice as long against the exact scaling law.

Honest simplifications

The bump blockage is a single empirical factor, not a computed flow through the bump array, the contact angle and surface tension are constants, the viscosity does not rise as the resin cures (only the gel time marks the end), the front is flat across the gap and no voids form. It shows why large dies, low temperatures and tight gaps are slow, not a process window.

• Press Large die and watch the fill time grow with the square of the distance.
• Press Cold substrate (slow) to see the fill time soar as the viscosity rises, then press Too hot: gels first for the opposite failure.
• Narrow the gap and the bump pitch to see capillary pressure rise but flow resistance rise faster.