deformation characteristics of ultra-thin liquid ﬁlm

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)


hL

D 3

21

hL 21


Gas (Air, n3=1.0)

Solid (DLC, n1=1.9)

Solid (DLC, n1=1.9)


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)


123

Fig. 8 Calculation results for a running disk (uD = 0.01 m/s)


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)


123

Fig. 10 Effect of disk speed uD (ah = 50�C)


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.

References

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(2009) Deformation characteristics of ultra-thin liquid film

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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)


123

deformation characteristics of ultra-thin liquid ﬁlm

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