A model to predict the critical velocity for liquid loss from a receding meniscus

Contact angles and liquid loss behavior of high index fluids

“…This research group has developed a model (discussed in greater detail in prior works [1]) for predicting the critical liquid loss velocity. This model combines the effect of both observed of liquid loss mechanisms: film pulling and inertial instability.…”

Advances in Patterning Materials for 193 nm Immersion Lithography

“…Fluid loss has been found to occur via two mechanisms: film pulling and inertial instability (as shown in Figure ). ,− Film pulling occurs when the dynamic receding contact angle approaches zero and a thin layer of fluid is pulled out behind the receding meniscus . The film pulling velocity (υ fp ) can be modeled by υ fp = C 1 υ ̅ ca θ s , r 3 wherein C 1 is an empirical constant, θ s,r is the static, receding contact angle, and the capillary velocity υ̅ ca is defined as υ ̅ ca = γ μ wherein γ is the surface tension and μ is the dynamic viscosity of the fluid. ,, Film pulling of water is frequently observed at low velocities on hydrophilic surfaces (i.e., those with low SRCA values).…”

“…Failure modes shown are (a) film pulling on a surface with a low SRCA and (b) inertial instability on a surface with higher SRCA. Reprinted with permission from ref . Copyright 2006 SPIE.…”

“…Fluid loss by inertial instability occurs when the fluid surface tension is insufficient to counteract the inertial force imposed by the fluid at the receding meniscus. ,,, In this case, bulk fluid loss occurs when droplets of fluid break off from the extended receding meniscus. Failure by inertial instability is frequently observed at high wafer accelerations and velocities on more hydrophobic surfaces (i.e., those with higher SRCA values).…”

“…Failure by inertial instability is frequently observed at high wafer accelerations and velocities on more hydrophobic surfaces (i.e., those with higher SRCA values). The inertial instability velocity (υ in ) can be modeled by υ in = C 2 υ ̅ in 3 / 4 υ ̅ ca 1 / 4 sin 3 / 4 true( θ s , r 2 true) wherein C 2 is an empirical constant and υ̅ in is given by υ ̅ in = γ ρ κ − 1 wherein ρ is the density of the fluid and κ −1 is κ − 1 = γ ρ g wherein g is the acceleration due to gravity. , …”