DLVO Theory, Acoustic Cavitation Dynamics & Supercritical CO2 Drying Physics

01Classical DLVO Theory of Colloidal Stability

The balance between particle adhesion and removal in liquid media is quantitatively governed by DLVO Theory (Derjaguin, Landau, Verwey, and Overbeek). Total interaction potential energy $V_{total}$ between a particle and a flat surface is the sum of attractive van der Waals potential ( $V_{VDW}$ ) and electrostatic double-layer repulsive potential ( $V_{EDL}$ ):

V_{total} (z) = V_{VDW} (z) + V_{EDL} (z)

Potential Energy VSchematic · source diagram

Mathematical Formulations

van der Waals Potential (Sphere-Plate Geometry):

V_{VDW} (z) = - \frac{A _{H} \cdot R _{p}}{6 z}

Where $A_{H}$ is the composite Hamaker constant, $R_{p}$ is particle radius, and z is separation distance.

Electrostatic Double-Layer Potential (Linearized PB Approximation):

V_{EDL} (z) = 16 π ϵ_{r} ϵ_{0} R_{p} (\frac{k _{B} T}{e})^{2} tanh (\frac{e ζ _{p}}{4 k _{B} T}) tanh (\frac{e ζ _{w}}{4 k _{B} T}) exp (- κ z)

Where $ζ_{p}$ and $ζ_{w}$ are particle and wafer Zeta potentials, and $κ^{- 1}$ is the Debye screening length:

κ = \frac{2 e ^{2} I}{ϵ _{r} ϵ _{0} k _{B} T}

To prevent particle re-deposition, the Primary Energy Barrier $V_{max}$ must exceed $15 - 20 k_{B} T$ . This is achieved by increasing Zeta potentials ( $ζ < - 30 mV$ ) and reducing ionic strength I (dilute chemistries).

02Megasonic Acoustic Cavitation & Micro-Streaming Physics

Megasonic energy induces two distinct physical phenomena in liquid media:

A. Acoustic Micro-Streaming (Schlichting Boundary Layer Streaming)

Attenuated sound waves drive momentum into the liquid, generating localized boundary layer micro-vortices. The acoustic streaming velocity $u_{stream}$ near the wall is given by:

u_{stream} = \frac{3}{8} \frac{α E _{ac}}{ρ _{0} c ω} sin (2 k x)

Where $α$ is sound absorption coefficient, $E_{ac}$ is acoustic energy density, and $ω$ is angular frequency. This fluid drag exerts a net mechanical moment that rolls particles out of the primary energy well.

B. Stable Cavitation vs. Transient Bubble Collapse

The threshold pressure for acoustic cavitation is governed by the Blake Threshold $P_{B}$ :

P_{B} = P_{0} + \frac{0.854 σ}{R _{0}}

At megasonic frequencies ( $> 1 MHz$ ), the acoustic period ( $τ \sim 1 μ s$ ) is shorter than the time required for transient bubble growth. Bubbles oscillate stably around their equilibrium radius $R_{0}$ , generating gentle pressure waves ( $Δ P \sim 1 - 5 MPa$ ) that detach particles without exceeding the yield strength of silicon nanosheets.

03Supercritical CO2 ( $scCO_{2}$ ) Drying Physics

As transistor features scale to sub-2nm Gate-All-Around (GAA) nanosheets and high-aspect-ratio (> 100:1) 3D NAND memory structures, even Marangoni drying causes Pattern Collapse due to capillary forces during liquid withdrawal:

F_{capillary} = \frac{2 γ cos θ}{d} \cdot A

Where $γ$ is liquid surface tension, $θ$ is contact angle, and d is trench spacing.

Process schematicSchematic · source diagram

The Supercritical Phase

Supercritical carbon dioxide ( $scCO_{2}$ ) operates beyond its thermodynamic critical point ( $T_{c} = 31. 1^{\circ} C, P_{c} = 7.38 MPa$ ):

Zero Surface Tension ( $γ = 0$ ): Capillary force drops precisely to zero ( $F_{capillary} = 0$ ).
Gas-Like Diffusivity & Liquid-Like Density: $scCO_{2}$ penetrates $1 nm$ features effortlessly, dissolving organic solvents (IPA) and completely evacuating high-aspect-ratio trenches with zero structural damage.