Modeling Slip Velocity and Pressure Gradients across Smooth Copper Foils
Modelling slip velocity and pressure gradients across smooth copper foils prevents resin starvation and locks dielectric thickness tolerances during lamination.
Reynolds Lubrication Equation
Mathematical Model
A partial differential equation relates the pressure and thickness of a thin fluid film to the relative velocity of the surrounding surfaces. In the field of electronics manufacturing, the Reynolds lubrication equation predicts the flow of underfill resins into the narrow space between a silicon die and a substrate. This formula accounts for the effects of surface tension and viscous forces in high-aspect-ratio gaps.
It allows process engineers to estimate the time required for a liquid to completely encapsulate the solder bumps.
Gap Management
Narrow gaps increase the resistance to fluid movement and require a precise balance of properties. Calculations using the Reynolds lubrication equation show that the flow velocity is highly sensitive to the height of the standoff. Small changes in the distance between the chip and the board have a large impact on the production throughput.
Underfill Application
Capillary action drives the flow in these applications. Using the Reynolds lubrication equation, designers can select the optimal viscosity for a given component size and bump pitch. This ensures that the resin reaches the center of the chip without leaving any trapped air.
If the flow is not uniform, the resulting voids can lead to solder bridge formation or moisture ingress. Careful control of the substrate temperature during the dispensing process maintains the fluid properties required by the model.