Direct bonding interfaces exhibit, before annealing, a narrow density gap due to the contact interactions of facing surface asperities and their incomplete compression. This gap makes a narrow channel (width<1nm) along which water is able to move. This is evidenced clearly from the defect distribution in Si/Si hydrophilic bonding showing clearly a rim of defects whose extent is related to the kinetics of water penetration [1]. A simple model of capillary flow has been used in a first attempt to describe the kinetics of water penetration [2] under such very high confinement but this model fails to reproduce some of the experimental observations as e.g. the increase of wetting front width with penetration length, or the water distribution in case of reversed flow. We shall describe these experimental results, recording kinetics of water flow under varying amplitude and direction of chemical potential gradients (e.g. relative humidity) and different temperatures. These experiments allow a direct test of the dependence of flow kinetics with various parameters such as viscosity and surface energies.

Finally we shall establish from basic principles the equation of propagation, which is shown to be a porous medium equation type [3]:

∂ρ/∂t= D/2ρ₀ ∂²ρ²/∂x²

where ρ is the interface areal density of water and D a constant “diffusion” coefficient depending on the wetting characteristics of the surface , the geometry of the gap and the water viscosity. Solutions of this equation will be given with boundary conditions corresponding to the wafer bonding interface situation. They exhibit features intermediate between sharp water front propagation (as described by Lucas-Washburn type of models) and diffusion equation, in agreement with experiments.

REFERENCES

[1] M. Tedjini et al. Appl. Phys. Lett. 109, 111603 (2016)

[2] F. Rieutord,et al ECS Trans. 2016 volume 75( 9) , 163-167

[3] J. L. Vazquez, The Porous Medium Equation: Mathematical Theory, Clarendon Press Oxford (2007)