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R EPORT
LIQUID DROPLET IN CONTACT WITHA SOLID SURFACE
S. Maruyam a, T. Kurash ige, S. Matsum oto, Y. Yam aguch i,
and T. Kim uraDepartm ent o f Mechan ical En gin eerin g, The Un iv ersity o f Tokyo , Tokyo, Japan
A liqu id droplet in contact with a solid su rface was sim u la ted by the m olecu lar dyn am ics
m ethod, in order to study the m icroscopic aspects of th e liqu id ± solid con tact ph enom ena an d
ph ase-chan ge h eat tran sfer. Measu red ``contact angle’’ was well correla ted by the depth of
the in tegrated poten tia l of th e su rface. Th e layered liqu id stru cture n ear the su rface was also
explain ed with the in tegrated poten tia l field. Fu rth erm ore, ev aporation an d conden sation
throu gh the droplet were sim u la ted by preparin g two solid su rfaces with temperatu re
differences on the top and bottom of the calcu la tion dom ain . Ov erall h eat flu x an d
temperature distribu tions of droplets were m easu red.
The microscopic mechanism of solid ] liquid contact is fundamental to under-
standing phase-change phenomena such as dropwise condensation and the collapse
of a liquid film on a heated surface. We have applied the molecular dynamics
w xmethod to understand inte rphase phenomena such as surface tension 1, 2 and
w x w xliquid ] solid contact 3 . O ur previous study 3 demonstrated that the drople t could
be regarded as a semisphe rical shape except for a few layers of liquid structure on
the surface , and that the cosine of the `̀ contact angle ’ ’ was a linear function of the
energy scale of the inte raction potential. In this report, by changing more parame-
ters such as the length scale of the inte raction potential, system size, and tempera-
ture, we give a more general de scription of the phenomenon. As surface molecules
we re expressed by a single layer of fixed molecules in the previous report, in this
report we show two different levels of approximation for the surface : an even
simpler one-dimensional function, and more complex three layers of harmonic
molecules. Finally, through the simulation of actual phase change , the microscopic
characte ristics of heat transfer are discussed.
Received 12 August 1997; accepted 1 Octobe r 1997.
S. Matsumoto ’s pre sent addre ss is Mechanical Enginee ring Laboratory, Agency of Industrial
Science and Technology r MITI, Ibaraki, Japan.
Addre ss corre sponde nce to Prof. Shige o Maruyama, Departme nt of Mechanical Engineering,
The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan. E -mail: maruyama@ photon.t.
u-tokyo.ac.jp
M icroscale Th erm op h ys ica l En gin eer in g, 2 :49 ± 62 , 1998
Cop yr igh t Q 1998 Taylor & F ran cis
1089-3954 rrrrr 98 $12.00 H .00 49
S. M ARUYAM A ET AL.50
EQUILIBRIUM DROPLET
w xSimilar simulations as in the previous report 3 we re pe rformed with the
modification of the surface condition. A molecular dynamics simulation was
de signed to achieve an equilibrium state of the liquid drople t and its vapor on a
solid surface under the NVE ensemble . Figure 1 illustrate s the initial configuration
and liquid ] vapor equilibrium condition after 1,000 ps. Here , the liquid and vapor
molecules were distinguished by the potential fie ld felt by each molecule . The
liquid drople t and the vapor consisted of Lennard-Jones molecule s with the
interaction potential expressed as
12 6s ss . s .f r s 4 e y 1t / t /r r
For physical insights, we regarded the molecule as argon with the potential
Ê y 21parameters set as: s s 3.4 A, e s 1.67 = 10 J. The calculation region hadAR AR
speriodic boundaries for four side surfaces and a mirror boundary or a hard wall
.boundary for the top surface . The bottom surface was expressed by e ither a
one-dimensional potential function, a fixed single layer of solid molecule s, or three
s .layers of harmonic molecule s. We mode led the fcc 1, 1, 1 surface with the
Ênearest-neighbor distance s s 2.77 A, the spring constant k s 46.8 N r m, andSy 26 s .mass m s 32.39 = 10 kg the platinum crystal was mode led . The inte ractionS
potential between solid and argon molecules was also expressed by the Lennard-
Jones potential with the parameters s and e . For the one-dimensionalINT INT
Figu r e 1. Snapshots of a liquid drople t on the solid surface . Gray, dark gray, and white
circles corre spond to solid, liquid, and vapor mole cules, re spectively.
LIQUID DROPLE T IN CONTACT WITH A SOLID SURFACE 51
s .potential function we have integrated one layer of the fcc 1, 1, 1 surface as
2 10 4X4 3 p e s s sINT INT INT INTs . s .F z s 2 y 5 2
2 w 5t / t /15 z zs S
where z is the coordinate normal to the surface . Another one-dimensional func-
w xtion is available in a textbook 4 : the full integration of uniform solid molecules.
Howe ver, we be lieve that our form of one -layer integration must give a better
representation of the potential field very near the surface , and that it is suitable for
a direct comparison with our othe r representations of the surface. The second and
third solid layers can be taken into account by adding the same function with a
shift of z of the lattice constant. Here, it should be noticed that the minimum
s .e of this function in Eq. 2 appears when z s s asSU RF INT
2X4 3 p s INTs .e s e 3SU RF INT2t / t /5 s S
The equation of motion was integrated numerically utilizing the V erle t algorithm
w x4 with a time step of 0.01 ps. Equilibration and data acquisition we re the same as
w xin our previous report 3 . Note that the effect of the initial condition was
negligible for the equilibrium results, and the same equilibrium state could be
obtained even from a dense gas-phase initial condition.
w xFigure 2 shows the two-dimensional density profiles 3 for various sizes of
droplets. With the increase of the number of liquid molecules, the relative
pe rcentage of bulk liquid increases. The contact angle u was measured by fitting a
w xcircle to the density contour, ignoring the layered part as before 3 . It is rather
surprising that the contact angle was nearly the same even with only a few hundred
molecules constituting the droplet. The smaller drople t was made mostly of the
w xlayered structure 5 , and the measurement of the contact angle was difficult.
The e ffect of the temperature on the contact angle was too small to be
de scribed within our uncertainty, though the two-dimensional density distributions
seemed quite different. The liquid ] vapor interface was more diffuse, and the first
liquid layer was more prominent for higher temperature.
It was found that the contact angle was correlated with the integrated depth
w s .xof the surface potential e see Eq. 3 , as demonstrated in Figure 3. Parame-SU RF
ters of the inte raction potential s and e were changed from comple teINT INT
s . s .we tting u s 0 8 through comple te drying u s 180 8 , as shown in Table 1. The
representation of the surface molecule s did not affect the result as long as the
« was the same . Furthermore, the two-dimensional density and poten-SU RF
tial distributions we re almost nondistinguishable among three different surface
conditions.
Here, the we ll-known Young’s equation for the macroscopic contact angle is
expressed as
s . s .cos u s g y g r g 4s g s l l g
S. M ARUYAM A ET AL.52
Figu re 2. The effe ct of liquid size on the two-dim ensional de nsity profile .
LIQUID DROPLE T IN CONTACT WITH A SOLID SURFACE 53
s .Figu re 2. The effect of liquid size on the two-dim ensional de nsity profile Con tin u ed .
S. M ARUYAM A ET AL.54
Figu re 3. D epe nde nce of the contact
angle on the de pth of the integrated
potentia l. V ariab le « and s corre -IN T IN T
spond to cond itions E x and S x in Table
1, re spe ctive ly.
where g , g , and g are the surface energie s of the solid ] vapor, solid ] liquid, ands g s l l g
liquid ] vapor interfaces, respectively. Even though the early molecular dynamics
study concluded that the Young’s equation was not applicable to the microscopic
w x w xsystem 6 , later studie s suggested opposite results 7, 8 . If we assume that the
difference of the solid-related surface energy g y g is proportional to thes g s l
w xpotential parameter as expected from 7, 8 , our re sult supports Young’s equation
for the main body of the droplet.
The layered structure is unique to the microscopic droplets. As the drople t
size becomes smalle r, characteristics of this structure dominate the whole shape as
shown in Figure 2. Figure 4 compares the profile s of liquid density for various
energy scale s e . The first peak appeared at s from the solid molecule , andINT INT
ssuccessive peaks appeared at intervals of s a similar picture with varying sAR INT
.supported this relationship . These re lations can be clearly understood by conside r-
ing Figure 5, which compares the density profile with the integrated potential
function. The similarity of the density profile and the integrated potential suggests
that the density profile is simply dete rmined by the effective potential fie ld and the
temperature. As the first liquid layer is built up by the surface molecules, it works
as the imperfect surface for the second liquid layer and so on.
(INEQUILIBRIUM DROPLET PHASE-CHANGE)HEAT TRANSFER
Characte ristics of phase-change heat transfer we re studied with the system as
shown in Figure 6. Solid surface s on the top and bottom of the calculation domain
we re represented by three layers of harmonic molecule s with additional `̀ phantom’ ’
w xmolecules 9 . The phantom molecules, which mimicked the continuous bulk solid,
we re introduced to give constant-temperature boundary conditions. Solid molecules
sin the third layer we re connected to phantom molecules through springs the
.vertical and horizontal spring constants we re 0.5k and 2 k , re spective ly . Then,
sphantom molecules were connected to the fixed frame by springs the vertical and
.horizontal constants were 3.5k and 2 k , re spectively and dampers with the damping
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55
S. M ARUYAM A ET AL.56
Figu re 4. De nsity distributions of liquid
ne ar the solid su rface . Se e Table 1 for E x
labe ls.
y 12 w xconstant a s 5.184 = 10 kg r s . The phantom molecule s we re furthe r excited
by artificial random force with standard deviation s s 2 a k T r D t , where k isXF B B
the Boltzmann constant and D t is the time step.
Figure 7 shows the time sequence of the calculation. We prepared a solid
surface and a drople t separate ly in equilibrium at 100 K, and merged them to the
configuration of Figure 6. Then, after calculating 200 ps for the relaxation, top and
bottom phantom molecule s were suddenly set to 85 and 115 K, re spectively. The
phase-change process can be clearly seen in the change of the number of moleculescond s . evap s . s .N droplet in condensing side , N evaporating side , and N vapor in theL L V
bottom panel of Figure 7. Since N was almost constant, the condensation rateV
and the evaporation rate we re almost the same. The temperature of the solidevap s . cond s .surfaces T bottom and T top quickly jumped to the phantom tempera-S S
ture, but the temperature difference of the two liquid drople ts was very small. This
Figu r e 5. Re lationsh ip of the density
profile and the inte grate d poten tial func-
s .tion E 2 conditio n in Table 1 .
LIQUID DROPLE T IN CONTACT WITH A SOLID SURFACE 57
Figu re 6. A snapshot of inequilibrium drople ts on solid
surface s.
re sult showed that the liquid ] solid contact thermal re sistance was dominant
compared with heat conduction and phase-change thermal re sistances. We as-
ssumed that the heat transfer was quasi-steady the rate of phase change was
.constant after 500 ps, and measured the heat flux, ve locity, and temperature
distributions. The average heat flux measured at the middle of the calculation
domain was 24 MW r m 2 , which is the order of the maximum heat flux possible in
phase change .
S. M ARUYAM A ET AL.58
Figu re 7. Variations of temperature and num be r of molecule s.
T and N are tempe rature and num be r of mole cules, respe c-
tive ly. Subscripts S, L , and V re pre sent solid, liquid, and vapor,
respe ctively, and supe rscripts evap and cond repre se nt evapora-
s . s .tion side bottom and condensation side top , re spective ly.
Figure 8 shows the ve locity and temperature distributions on the evaporation
sside . The velocity distribution compared with the density distribution top panel of
. sFigure 8 shows that the evaporation at the three -phase contact line or the first
.layer of liquid was most dominant, and some supply flow of liquid from the middle
of the drople t to the first layer was observed. The highe r drople t temperature at
the three -phase contact line also supports this observation. The temperature of the
first layer of the solid was lowe r under this area and higher under the center of the
droplet due to the the rmal insulation provided by the liquid.
The same representations on the condensing side are shown in Figure 9.
Here, the specialness at the three -phase contact line was le ss pronounced, and
rathe r uniform condensation was observed over the whole liquid surface. The
temperature distribution of the liquid exhibited a simple linear increase along the z
direction, and the decrease of solid surface temperature at the center is simply
explained by the thermal insulating e ffect of the liquid droplet.
The measured contact angle was almost the same for both condensing and
evaporating drople ts, as recognized in Figures 8 and 9. It seems that the contact
angle was principally determined by the surface ene rgy balance and was not very
sensitive to the lower-energy temperature distributions.
We noticed that three -layer solid molecules had been too thin to consider
such a problem with the temperature distribution in the radial direction. Even
though we we re confident of the use of the phantom technique through equilib-
LIQUID DROPLE T IN CONTACT WITH A SOLID SURFACE 59
Figu re 8. Characte ristics of evaporating drople t. To p , density profile
and ve locity vectors; m id dle, tempe rature distributio n in the drople t;
bottom , temperature distributio n of the first laye r of solid surface .
S. M ARUYAM A ET AL.60
Figu re 9. Characte ristics of conde nsing drople t. To p , de nsity profile
and ve locity vectors; m id dle, tempe rature distributio n in the drople t;
bottom , temperature distributio n of the first laye r of solid surface .
LIQUID DROPLE T IN CONTACT WITH A SOLID SURFACE 61
rium calculations of othe r systems, imposing the uniform temperature just three
layers be low the first layer with temperature distribution as seen in Figures 8 and 9
was actually unacceptable. If we could use reasonably thicker solid layers, probably
the solid-layer temperature distribution would be more pronounced. Furthe rmore,
s .the slight upper shift about 3 K of the solid temperature from the phantom set
value observed in Figure 7 seems to be due to this problem. Even though computer
time and memory crucially limit the number of solid molecule s, calculations with a
thicker solid layer are desired in the future.
CONCLUSIONS
A liquid droplet in contact with a solid surface was simulated by the
molecular dynamics method. The liquid drople t and surrounding vapor we re
realized by Lennard-Jones molecules, and the solid surface was represented by
three types of mode ls: three layers of harmonic molecule s, one layer of fixed
molecules, or a simple one-dimensional function. The interaction potential be -
tween solid molecules and liquid r vapor molecule s was represented by the
Lennard-Jones potential with various length and energy parameters.
It was found that the cosine of the contact angle was well expressed by a
linear function of the depth of the integrated interaction potential, regardle ss of
surface representations. Assuming that the surface ene rgy was proportional to the
potential depth, the macroscopic Young’s equation was still valid even for such a
small drople t. O n the other hand, the layered structure that appeared in the
two-dimensional density profile of drople t was well explained through the shape of
the integrated interaction potential.
By preparing two solid surfaces with different temperatures, evaporation in
one drople t and condensation on the other were simulated simultaneously. The
contact angle was almost the same as the equilibrium condition. V e locity and
temperature distributions of drople ts evaporating and condensing as we ll as the
heat flux we re measured for the quasi-steady condition. The importance of
the three -phase contact line in the evaporation process, in clear contrast to the
condensation process, was observed.
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