Numerical investigation of terminal residual thickness of the water film and surface flow in the entrainment region in Marangoni drying

Abstract

Marangoni drying is widely employed in integrated circuit manufacturing to remove water from the wafer surface. The organic vapor blown at the meniscus induces a strong Marangoni effect, which strips the entrained water film induced by the wafer’s withdrawal from a deionization water bath, and realizes ultra-clean drying of the wafer. The present work implements a numerical investigation of the terminal residual film and free surface flow in the water entrainment region during Marangoni drying based on a trans-scale model containing the thin water film, meniscus, and bulk region. The results show that the terminal thickness of the water film is 30 nm, which approximately agrees with the previous experimental and numerical results. In addition, the stagnation point moves away from the thin film region compared with that of wafer withdrawn in the absence of the Marangoni effect. The magnitude of free surface velocity in the thin film region after drying depends only on the withdrawal velocity, and is irrelevant to the Marangoni number. The analyses of the entrained water film thinning and surface flow provide useful guidance about controlling the drying process.

Keywords:

• Wafer drying
• Marangoni effect
• terminal thickness
• surface velocity
• entrainment region

Nomenclature

Letters

u	=	velocity, m/s
uc	=	convection velocity of nodes, m/s
Vs	=	surface velocity, m/s
V0	=	withdrawal velocity, mm/s
p	=	pressure, Pa
e	=	unit vector
n	=	normal vector at specified boundary
T	=	total stress tensor
I	=	identity tensor
g	=	gravitational acceleration, m/s2
x, y	=	rectangular coordinates, mm
s	=	air–water interface, mm
t	=	time, s
Ma	=	Marangoni number, 1
h	=	film thickness, μm

Greek symbols

$\mu$	=	dynamic viscosity, Pa s
$\rho$	=	density, kg/m3
$\sigma$	=	surface tension, N/m
${\tau }_0$	=	surface tension gradient, N/m2

Subscripts

0	=	initial value
R	=	residual value
$\mathrm{\infty }$	=	thin film region
st	=	stagnation point
e	=	terminal value

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors greatly appreciate the financial support provided by the National Natural Science Foundation of China (Grants No. 51305227 and 91323302).