deformation characteristics of ultra-thin liquid film
TRANSCRIPT
TECHNICAL PAPER
Deformation characteristics of ultra-thin liquid film consideringtemperature and film thickness dependence of surface tension
Three-dimensional analyses by the unsteady and linearized long wave equation
Hiroshige Matsuoka • Koji Oka • Yusuke Yamashita •
Fumihiro Saeki • Shigehisa Fukui
Received: 31 August 2010 / Accepted: 28 December 2010 / Published online: 18 January 2011
� Springer-Verlag 2011
Abstract Thermocapillary deformations of an ultra-thin
liquid film caused by temperature distribution were three-
dimensionally analyzed using the unsteady and linearized
long wave equation considering the temperature and film
thickness dependence of surface tension. The temperature
and film thickness dependence equation for the surface
tension of a liquid was firstly established. The temperature
dependence of the surface tension was obtained experi-
mentally using a surface tensiometer and the film thickness
dependence was obtained theoretically from the corrected
van der Waals pressure equation for a symmetric multi-
layer system. Time evolutions of depression and groove of
the ultra-thin liquid film caused by local heating were
obtained quantitatively.
1 Introduction
In current magnetic storage systems, the spacing between
the flying head and the disk has been dramatically
decreased to \10 nm in order to realize ultra-high density
recording. When the flying height of the head is of the
same order as the lubricant film thickness, lubricant
deformation affects the static and dynamic flying charac-
teristics of the slider. Therefore, it is very important to
investigate the deformation and flow characteristics of the
lubricant on the recording disk. In particular, in heat-
assisted magnetic recording (HAMR), we need to consider
heat conduction on the nanometer scale, the evaporation of
the lubricant, the distribution of surface tension, and the
distribution of viscosity by local laser heating, which may
cause deformation of the lubricant film (Oka et al. 2009;
Wu 2007).
In the present paper, we focus on lubricant film defor-
mation due to thermocapillary effects. We first establish
the temperature and film thickness dependence equation for
the surface tension. The temperature dependence was
obtained by measuring the relationship between surface
tension and temperature by means of a surface tensiometer,
while the film thickness dependence was obtained based on
the theoretical considerations of the van der Waals pressure
equation for a symmetric multilayer system. Using the
unsteady and linearized long wave equation considering
the temperature and film thickness dependence of the sur-
face tension, we analyzed the liquid film deformation
caused by the temperature distribution three-dimensionally,
and the basic characteristics of the liquid film deformation
due to the thermocapillary effects are described.
2 Long wave equation for lubricant film deformation
We assume that a thin liquid film is placed on a solid
surface and that the liquid surface is exposed to a gas, as
shown in Fig. 1. The film thickness is denoted by
hL(x, y, t), where the x and y coordinates show the in-plane
directions, and t denotes the time. Assuming that the liquid
film satisfies the continuum hypothesis and that the char-
acteristic length in the in-plane directions is much larger
than the film thickness, the surface deformations caused by
stresses acting on the liquid surface are described by the
long wave equation (Oron et al. 1997; Fukui et al. 2007;
Saeki et al. 2009). The equation for the unsteady state is
written as follows:
H. Matsuoka (&) � K. Oka � Y. Yamashita � F. Saeki � S. Fukui
Department of Mechanical and Aerospace Engineering,
Graduate School of Engineering, Tottori University,
4-101 Minami, Koyama, Tottori 680-8552, Japan
e-mail: [email protected]
123
Microsyst Technol (2011) 17:983–990
DOI 10.1007/s00542-011-1223-0
ohL
ot� 1
3lL
o
oxh3
L
opL
ox
� �þ 1
2lL
o
oxh2
L sLxjz¼hL
� �n o
þ uDohL
ox� 1
3lL
o
oyh3
L
opL
oy
� �þ 1
2lL
o
oyh2
L sLy
��z¼hL
� �n o
þ vDohL
oy¼ 0; ð1Þ
and
pL ¼ pG � cGL
o2hL
ox2þ o2hL
oy2
� �þ A1232
6ph3L
þ qghL; ð2Þ
sLxjz¼hL¼ sGx þ
ocGL
oxand sLy
��z¼hL¼ sGy þ
ocGL
oy; ð3Þ
where pL is the liquid pressure, pG is the gas pressure, lL is
the liquid viscosity, uD and vD are the speeds of the solid
surface in the x and y directions, respectively, cGL is the
surface tension of the liquid, sLx and sLy are the liquid
shear stresses, and sGx and sGy are the gas shear stresses. In
addition, A1232 is the Hamaker constant, which is a function
of the refractive indices of the materials (A1232 =
-4.68 9 10-20 J in the case of a perfluoropolyether (PFPE,
Fomblin Z03) film on a diamond-like carbon (DLC) sur-
face), g is the gravitational acceleration and q is the liquid
density.
3 Temperature dependence of surface tension
The surface tension of a liquid depends on the temperature,
and the surface tension distribution caused by the temper-
ature distribution can deform the liquid surface (thermo-
capillary effect). The surface tension of Fomblin Z03 was
measured by a surface tensiometer. The temperature of Z03
was controlled in the range of 10–180�C. Figure 2 shows
the measurement results for the temperature dependence of
the surface tension of Z03. The surface tension decreased
linearly with increasing temperature. The following linear
relation between the surface tension cGL (mN/m) and
temperature h (�C) was obtained from Fig. 2 by least
squares fitting:
cGL ¼ cGL0 1þ c
cGL0
h� h0ð Þ� �
; ð4Þ
where h0 (=20�C) is the room temperature, cGL0 (=18.8
mN/m) is the surface tension at h0, and c (=-0.0635 mN/m �C)
is the inclination of the fitted line.
4 Film thickness dependence of surface tension
The surface tension of a bulk liquid is a material constant,
but it depends on the film thickness due to the effect of the
substrate when the film is ultra-thin. The film thickness
dependence of the surface tension can be obtained by the
integration of the corrected van der Waals pressure equa-
tion (Matsuoka et al. 2005) for a symmetric five-layer
system shown in Fig. 3 and is given by (Matsuoka et al.
2010a, b)
cGL ¼ cGL0 1þ 2A2312
A2323
D20
D0 þ hLð Þ2þ A1212
A2323
D20
D0 þ 2hLð Þ2
( );
ð5Þ
and
cGL0 ¼A2323
24pD20
; ð6Þ
where D0 is the cut-off distance [D0 = 0.165 nm
(Israelachvili 1992)] and Aijkl is the Hamaker constant,
which is given by
Fig. 1 Head-disk interface and balance of the gas–liquid interface 0 100 2000
10
20
30
Temperature, θ, °C
Surf
ace
tens
ion,
γG
L, m
N/m
measured (Z03)
fitted (Z03)
Fig. 2 Temperature dependence of surface tension of Fomblin Z03
984 Microsyst Technol (2011) 17:983–990
123
Aijkl ¼3�hx0
8ffiffiffi2p
p
n2i � n2
j
� �n2
k � n2l
�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2
i þ n2j
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2
k þ n2l
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2
i þ n2j
qþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2
k þ n2l
pn o :ð7Þ
Here, �h is Planck’s constant (=1.05 9 10-34 Js), x0 is the
principle absorption frequency (=1.88 9 106 rad/s), and ni
is the refractive index of material i. In the present study, the
Hamaker constants are A2323 = 2.84 9 10-20 J, A2312 =
4.68 9 10-20 J, and A1212 = 7.94 9 10-20 J for a Z03
film on DLC. When hL ? ?, the surface tension cGL
approaches cGL0, which is the surface tension of the bulk
liquid at a certain temperature.
5 Temperature and film thickness dependence
of surface tension
In the present study, the effects of the temperature and film
thickness on the surface tension are assumed to be inde-
pendent. Coupling Eqs. 4 and 5, the equation for temper-
ature and film thickness dependence of the surface tension
is obtained as follows:
cGL ¼ cGL0 1þ c
cGL0
ðh� h0Þ� �
1þ 2A2312
A2323
D20
ðD0 þ hLÞ2
(
þA1212
A2323
D20
ðD0 þ 2hLÞ2
): ð8Þ
The surface tension is a function of x and y because the tem-
perature h and the film thickness hL are functions of x and
y. This equation is used in the long wave equation (1).
6 Linearization of long wave equation
The long wave equation (1) is linearized assuming a small
deformation of the liquid film thickness, i.e.,
hL
hL0
¼ HL ¼ 1þ HL; HL ¼ DhL=hL0; jHLj � 1; ð9Þ
where hL0 is the average film thickness, HL is the nondi-
mensional film thickness, and DhL is the film thickness
fluctuation.
The linearized long wave equation in a nondimensional
form is
rLoHL
oT�4
o
oXð1þ3HLÞ
oPG
oX
� �
þ4eLo
oX
o
oXCA
o2HL
oX2þ 1
B2
o2HL
oY2
� �� ��
�4o
oXG�A
2
� �oHL
oX
� �
þ 6
eL
o
oXTGxþ
oCA
oXþ o
oXðCBHLÞ
� �þ2 TGxþ
oCA
oX
� �HL
� �
þ2KLxoHL
oX� 4
B2
o
oY1þ3HL
�oPG
oY
� �
þ4eL
B2
o
oY
o
oYCA
o2HL
oX2þ 1
B2
o2HL
oY2
� �� ��
� 4
B2
o
oYG�A
2
� �oHL
oY
� �þ 6
eLB
o
oY
� TGyþ1
B
oCA
oYþ 1
B
o
oYCBHL
�� ��
þ2 TGyþ1
B
oCA
oY
� �HL
�þ2KLy
oHL
oY¼0; ð10Þ
hL
D 3
21
hL 21
Liquid (Z03, n2=1.3)
Gas (Air, n3=1.0)
Solid (DLC, n1=1.9)
Solid (DLC, n1=1.9)
Liquid (Z03, n2=1.3)
hL
D 3
21
hL 21
Liquid (Z03, n2=1.3)
Gas (Air, n3=1.0)
Solid (DLC, n1=1.9)
Solid (DLC, n1=1.9)
Liquid (Z03, n2=1.3)
Fig. 3 Symmetric five-layer system
Local heating
LiquidDisk
l = 50 μm
b = 50 μm
hL0 = 2nm
uD
hL0 = 2 nm (average film thickness)l = 50 μm (characteristic length) b = 50 μm (characteristic length) pa = 101325 Pa (atmosphere pressure) μL = 54.6×10-3 Pa·s (viscosity) ρ0 = 1.82×103 kg/m3 (density) vD = 0 m/s (disk speed in y direction) X = 0.5 (nondim. heating position)
Y = 0.5 (nondim. heating position) σx = 0.02 (nondim. standard deviation) σy = 0.02 (nondim. standard deviation) pG = 0 Pa (gas pressure) τGx = τGy = 0 Pa (gas shear stresses)
Fig. 4 Calculation conditions
Microsyst Technol (2011) 17:983–990 985
123
where X (=x/l) and Y (=y/b) are the nondimensional coor-
dinates, l and b are the characteristic lengths in x and
y directions, respectively, T (=xct) is the nondimensional
time, xc is the characteristic frequency, PG (=pG/pa) is
the nondimensional gas pressure, SGx (=sGx/pa) and SGy
(=sGy/pa) are the nondimensional gas shear stresses, pa is
the ambient pressure, A (=A1232/ppahL03 ) is the nondi-
mensional Hamaker constant, G (=qghL0/pa) is the non-
dimensional gravitational acceleration, B = b/l and
eL = hL0/l (�1).
The linearized surface tension CGL (=cGL/pal) is given by
CGL ¼ CA þ CBHL; ð11Þ
where
CA ¼cGL0
pal1þ a
cGL0
h� 20ð Þ� �
1þ 2A2312
A2323
D0
D0 þ hL0
� �2(
þA1212
A2323
D0
D0 þ 2hL0
� �2)
and ð12Þ
Fig. 5 Calculation results for a stationary disk (ah = 5�C, uD = 0 m/s,
t = 0.08 s)
0 1 2 30.8
0.9
1
1.1
Time, t , s
HL
max
and
HL
min
HL max
HL min
Fig. 6 Maximum liquid height HLmax and minimum liquid height
HLmin as functions of time (ah = 5�C, uD = 0 m/s)
0 1 2 3 4 5 6 7 8 90.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Time, t , s
Hm
ax a
nd H
min
HNL max
HNL min
HL max
HL min
1.05
Fig. 7 Comparison of maximum liquid heights HLmax and minimum
liquid heights HLmin of linear and nonlinear calculation results (suffix
NL shows nonlinear; ah = 5�C, uD = 0 m/s)
986 Microsyst Technol (2011) 17:983–990
123
Fig. 8 Calculation results for a running disk (uD = 0.01 m/s)
Microsyst Technol (2011) 17:983–990 987
123
CB ¼cGL0
pal1þ a
cGL0
ðh� 20Þ� �
2A2312
A2323
D0
D0 þ hL0
� �2(
� 2
D0=hL0ð Þ þ 1
� �þ A1212
A2323
D0
D0 þ 2hL0
� �2
� 2
D0=2hL0ð Þ þ 1
� ��: ð13Þ
We employed a density of q = 1.82 9 103 kg/m3 and
viscosity lL = 5.46 9 10-2 Pa s at 20�C.
7 Results and discussions
We performed three-dimensional liquid film deformation
analyses for a temperature distribution. The calculation
conditions are shown in Fig. 4. The temperature distribu-
tion is given by the Gaussian distribution:
hðxÞ ¼ h0 þ ah exp �ðX ��XÞ2
2r2x
� ðY ��YÞ2
2r2y
( ); ð14Þ
where ah is the maximum temperature rise, �X and Y are the
nondimensional center position of heating, and rx and ry
are the nondimensional standard deviations of the tem-
perature distribution. A periodic boundary condition was
adopted in the present study. The gas pressure pG and the
gas shear stresses sGx and sGy are not considered here
because we intend to clarify the effects of the temperature
distribution on the liquid surface deformation.
The calculation results of the surface tension and the
liquid film deformation for ah = 5�C and uD = 0 m/s (a
stationary disk) at t = 0.08 s are shown in Fig. 5. The
surface tension (Fig. 5a) decreased as the temperature
increased due to the temperature dependence shown in
Fig. 2. The liquid film thickness (Fig. 5b) was also found
to decrease as the temperature increased and upheaval can
be observed around the depression. The liquid is pulled to
the low-temperature area because the surface tension in the
low-temperature area is larger than that in the high-tem-
perature area (around X = Y = 0.5). It was also found that
a small temperature change can cause a relatively large
deformation of the liquid surface in this case (uD = 0 m/s).
Figure 6 shows the maximum liquid height HLmax and the
minimum liquid height HLmin as functions of time. Both
approach constant values as time passes, that is, the liquid
film shape approaches a steady shape.
In order to verify the validity of the linear calculation,
the results obtained by the linear calculation are compared
with the results by the nonlinear calculation. The calcula-
tion conditions are the same as the conditions in Figs. 5
and 6. Figure 7 shows the maximum liquid heights HLmax
and the minimum liquid heights HLmin of linear and
nonlinear calculation results. The errors of HLmax and
HLmin between the linear and the nonlinear calculation
results are \0.01% and 0.8%, respectively. Therefore, the
linear analysis presented in this study can give sufficiently
good calculation results when a deformation is smaller than
the deformation shown in Figs. 5 and 6. Furthermore, the
calculation time of the linear analysis is about 2.7 h,
whereas the nonlinear calculation takes 7.4 h. The linear
calculation takes much less calculation time.
Figure 8 shows the calculation results when the disk
runs at uD = 0.01 m/s when ah = 50�C (a–c) and 100�C
(d–f). In this case, a groove is formed. The groove is
deepened stepwise at the heating point (X = Y = 0.5) as
shown in Fig. 9, and the liquid film shape approaches a
steady shape as time passes. Naturally, the higher tem-
perature rise gives the larger liquid film deformation.
Figure 10 shows the effect of the disk speed uD on the
time evolution of the groove at ah = 50�C and Fig. 11
shows the effect of the disk speed uD on the maximum
liquid height HLmax and the minimum liquid height HLmin.
The liquid surface deformation decreases as the disk speed
increases, but the liquid film deformation is less sensitive
to the disk speed than to the temperature change. The liquid
surface deformations for uD = 1 m/s are found to be close
to those for uD = 0.1 m/s. The deformation is not so dif-
ferent for much faster disk speed.
8 Conclusions
A temperature and film thickness dependence equation for
surface tension was established. The temperature depen-
dence term was a linear function and the coefficients of the
function are obtained using the experimental data. The film
0 0.02 0.04 0.06 0.080.8
0.9
1
1.1
Time, t, s
HL
max
and
HL
min
Max of Temperature
aθ = 50 aθ = 100 HL max
HL min
Fig. 9 Effect of ah on maximum liquid height HLmax and minimum
liquid height HLmin (uD = 0.01 m/s)
988 Microsyst Technol (2011) 17:983–990
123
Fig. 10 Effect of disk speed uD (ah = 50�C)
Microsyst Technol (2011) 17:983–990 989
123
thickness dependence term was derived theoretically by
integrating the corrected van der Waals pressure equation
for a symmetric multilayer system.
Using the surface tension equation and the unsteady and
linearized long wave equation, the liquid film deformation
due to the temperature distribution was calculated numer-
ically. Time evolutions of depression and groove of the
ultra-thin liquid film caused by local heating were obtained
quantitatively.
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0 0.02 0.04 0.06 0.080.94
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
Time, t, s
HL
max
and
HL
min Speed of Disk, uD
0.01 m/s 0.1 m/s HL max
HL min
1.0 m/s
Fig. 11 Effect of uD on maximum liquid height HLmax and minimum
liquid height HLmin (ah = 50�C)
990 Microsyst Technol (2011) 17:983–990
123