R EPO RT
LIQUID DROPLET IN CONTACT WITH A SOLID SURFACE
S. Maru ya m a, T. Ku rash ige, S. Matsu m oto, Y. Yam agu ch i, an d T. Kim u ra
D epartm en t o f Mec h an ic al E n gin eerin g, Th e Un iv ersity o f Tokyo , To kyo, Japan
A liqu id droplet in contact with a solid su rface was sim u la ted by th e m olecu lar dyn a m ics m eth od , in ord er to stu d y the m icroscop ic a sp ects of th e liqu id± solid con tact ph en om en a an d p h ase-ch an ge h eat tra n sfer. Measu red ``con tact a n gle’’ wa s well correla ted by th e d epth of the in tegrated poten tia l of th e su rfa ce. Th e la yered liqu id stru ctu re n ear th e su rfa ce wa s a lso explain ed with th e in tegrated p oten tia l field. F u rth erm ore, ev a poration an d con den sa tion throu gh th e droplet were sim u la ted by prepa rin g two solid su rfa ces with tem p era tu re d ifferences on the top a n d bottom of th e calcu la tion d om a in . Ov era ll h eat flu x an d tem perature d istribu tion s of droplets were m easu red .
The microscopic me chanism of solid] liquid contact is fundame ntal to unde r- standing phase -change phenome na such as dropwise condensation and the collapse of a liquid film on a heate d surface. W e have applie d the molecular dynamics
w x
me thod to unde rstand inte rphase phe nome na such as surface te nsion 1, 2 and
w x w x
liquid] solid contact 3 . O ur pre vious study 3 demonstrate d that the drople t could be regarde d as a se misphe rical shape e xcept for a fe w layers of liquid structure on the surface , and that the cosine of the ``contact angle ’’ was a line ar function of the e nergy scale of the inte raction pote ntial. In this re port, by changing more parame - ters such as the le ngth scale of the inte raction pote ntial, system size, and te mpera- ture, we give a more ge ne ral de scription of the phe nomenon. As surface m olecules we re e xpresse d by a single laye r of fixe d mole cules in the pre vious report, in this re port we show two different le vels of approximation for the surface : an eve n simpler one-dim ensional function, and more com plex thre e laye rs of harmonic mole cules. Finally, through the simulation of actual phase change , the microscopic characte ristics of he at transfer are discussed.
Receive d 12 August 1997; acce pte d 1 O ctobe r 1997.
S. Matsumoto ’s pre se nt addre ss is Me chanical E ngine e ring Laboratory, Age ncy of Industrial Scie nce and TechnologyrMITI, Ibaraki, Japan.
Addre ss corre sponde nce to Prof. Shige o Maruyama, De partme nt of Me chanical E ngine ering, The Unive rsity of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan. E -mail: m aruyama@ photon.t.
u-tokyo.ac.jp
M icr o s ca le Th er m op h ys ica l E n gin eer in g , 2 :49 ± 62 , 1 9 98 C o p yr igh tQ1 99 8 Ta ylo r & F r a n cis
1 08 9 -39 54rrrrr98 $ 12 .0 0H .0 0 4 9
inte raction potential expressed as
12 6
s s
s . s .
f r s 4e y 1
t / t /
r rFor physical insights, we regarde d the mole cule as argon with the potential
Ê y2 1
parame te rs set as: s AR s 3.4 A,e A Rs 1.67= 10 J. The calculation region had spe riodic 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-dim ensional potential function, a fixed single layer of solid mole cule s, or thre e
s .
layers of harmonic mole cule s. W e m ode le d the fcc 1, 1, 1 surface with the Êne are st-neighbor distance s s 2.77 A, the spring constant ks 46.8 Nrm , andS
y2 6 s .
mass mSs 32.39= 10 kg the platinum crystal was mode le d . The inte raction pote ntial betwe en solid and argon m olecules was also e xpre ssed by the Le nnard- Jone s potential with the parame te rs s INT and e INT. For the one -dime nsional
F ig u 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 m ole cules, re spe ctively.
L IQ UID D R O PLE T IN C O NTAC T W ITH A SO L ID SU R F ACE 5 1
s .
pote ntial function we have inte grated one layer of the fcc 1, 1, 1 surface as
2 10 4
4 3X p e INTs INT s INT s INT
s . s .
F z s 2 y 5 22
w 5 t / t /
15 z zs Swhe re z is the coordinate norm al to the surface . Another one-dime nsional func- tion is available in a textbook 4 : the full integration of uniform solid molecules.w x Howe ver, we be lieve that our form of one -layer inte gration must give a be tter re prese ntation of the potential field ve ry near the surface , and that it is suitable for a dire ct comparison with our othe r re prese ntations of the surface. The se cond and third solid layers can be take n into account by adding the sam e function with a shift of z of the lattice constant. He re, it should be notice d that the minimum e SU R F of this function in E q. 2 appe ars whe ns . zs s INT as
X 2
4 3p s INT
e SU R Fs
t
5/ t /s S2 e INT s .3
The equation of motion was integrated nume rically utilizing the V e rle t algorithm w x4 with a time ste p of 0.01 ps. E quilibration and data acquisition we re the same as in our pre vious report 3 . Note that the effect of the initial condition wasw x ne gligible for the e quilibrium re sults, and the same e quilibrium state could be obtained e ven from a de nse gas-phase initial condition.
Figure 2 shows the two-dime nsional density profiles 3 for various sizes ofw x droplets. W ith the incre ase of the numbe r of liquid m olecules, the relative pe rcentage of bulk liquid incre ase s. The contact angle u was me asure d by fitting a circle to the de nsity contour, ignoring the layered part as before 3 . It is ratherw x surprising that the contact angle was ne arly the same eve n with only a fe w hundre d mole cules constituting the droplet. The smaller drople t was made mostly of the layere d structure 5 , and the measureme nt of the contact angle was difficult.w x
The e ffe ct of the tem pe rature on the contact angle was too sm all to be de scribed within our uncertainty, though the two-dime nsional density distributions se em ed quite differe nt. The liquid] vapor interface was more diffuse, and the first liquid layer was more promine nt for higher tem pe rature.
It was found that the contact angle was correlate d with the integrated de pth
w s .x
of the surface pote ntial e SU R F se e E q. 3 , as dem onstrated in Figure 3. Param e- ters of the inte raction pote ntial s INT and e INT we re change d from comple te
s . s .
we tting u s 08 through comple te drying u s 1808 , as shown in Table 1. The re prese ntation of the surface mole cule s did not affe ct the result as long as the
« was the same . Furthe rmore, the two-dime nsional de nsity and poten-SU R F tial distributions we re almost nondistinguishable am ong three different surface conditions.
He re, the we ll-known Young’s equation for the m acroscopic contact angle is e xpre sse d as
s . s .
cosu s g s gy g s l rg l g 4
F igu r e 2 . The effe ct of liquid size on the two-dim ensional de nsity profile .
L IQ UID D R O PLE T IN C O NTAC T W ITH A SO L ID SU R F ACE 5 3
s .
F igu r e 2 . The effect of liquid size on the two-dim ensional de nsity profile C o n tin u ed.
F igu r e 3 . D e pe nde nce of th e contact a ngle on th e de pth o f the in te g ra te d po te ntia l. V a riab le « IN T a nd s I N T corre -
spon d to cond itio ns Exa nd Sxin T a ble 1 , re spe ctive ly.
whe reg s g,g s l, andg l g are the surface e nergie s of the solid] vapor, solid] liquid, and liquid] vapor interfaces, respe ctively. E ve n though the early mole cular dynamics study conclude d that the Young’ s equation was not applicable to the microscopic
w x w x
system 6 , late r studie s suggeste d opposite results 7, 8 . If we assume that the differe nce of the solid-relate d surface e nergy g s gy g s l is proportional to the
w x
pote ntial parame te r as e xpe cted from 7, 8 , our re sult supports Young’s e quation for the m ain body of the droplet.
The layere d structure is unique to the microscopic droplets. As the drople t size become s smalle r, characteristics of this structure dominate the whole shape as shown in Figure 2. Figure 4 compare s the profile s of liquid density for various e nergy scale s e INT. The first pe ak appe ared at s INT from the solid molecule , and ssuccessive pe aks appe ared at intervals of s a similar picture with varying sAR INT
supported this relationship . These re lations can be cle arly understood by conside r-. ing Figure 5, which compares the density profile with the integrated potential function. The sim ilarity of the de nsity profile and the integrated pote ntial sugge sts that the density profile is sim ply dete rmine d by the effective pote ntial fie ld and the tem pe rature. As the first liquid layer is built up by the surface mole cules, it works as the im pe rfect surface for the se cond liquid layer and so on.
INEQUILIBRIUM DROPLET PHASE-CHANGE ( HEAT TRANSFER )
Characte ristics of phase-change heat transfe r we re studied with the system as shown in Figure 6. Solid surface s on the top and bottom of the calculation dom ain we re re pre se nte d by thre e layers of harm onic mole cule s with additional ``phantom’’
mole cules 9 . The phantom mole cules, which mimicke d the continuous bulk solid,w x we re introduce d to give constant-tem pe rature boundary conditions. Solid m olecules sin the third laye r we re conne cte d to phantom molecules through springs the ve rtical and horizontal spring constants we re 0.5k and 2k, re spective ly . The n,. sphantom m olecules we re connecte d to the fixe d frame by springs the ve rtical and horizontal constants we re 3.5kand 2k, re spe ctively and dampe rs with the damping.
Table1.Calculationconditionsandcontactangle One-dimensionalfunctionFixedone-layer seINTINT Uy21ÊÊÊwxwxwxwxwxwxwxwxwxwxLabel=10JAeTKNRAudegudegTKNRAudegudegSURFvDNSPOTvDNSPOT E00.3093.0851.0089.511314.9149.6153.590.012514.7157.0162.9 E10.4003.0851.2991.911015.2139.1142.695.113415.0135.4134.1 E20.5753.0851.8690.810117.4105.2104.594.611317.0105.8101.7 E30.7503.0852.4294.810520.285.587.994.410119.687.087.0 E40.9253.0852.9994.29333.051.052.293.98730.655.255.6 E51.1003.0853.5694.576;`00 S10.5752.5731.2992.711514.8149.2157.394.212514.8140.4142.3 S30.5753.5232.4291.49620.384.281.693.711421.477.976.6 S40.5753.9132.9990.77553.636.844.995.58654.236.842.9 3-layersofharmonicmolecules V20.4683.0851.8692.212816.7106.0140.6 eU sere;T,temperature;N,numberofvapor;R,radiusoffittingcircle,u,contactanglemeasuredfromdensitySURFSURFARvDNS profile;u,contactanglemeasuredfrompotentialprofile.POT
5 5
F igu r e 4. D e nsity distrib utions of liq uid ne ar the solid su rfa ce . Se e T a ble 1 fo r Ex lab e ls.
y1 2 w x
constant a s 5.184= 10 kgrs . The phantom mole cule s we re furthe r excited by artificial random force with standard deviation s Fs
X
2a k TB rD t, whe re kB is the Boltzmann constant and D t is the time ste p.Figure 7 shows the time se que nce of the calculation. W e pre pare d a solid surface and a drople t se parate ly in e quilibrium at 100 K, and me rge d them to the configuration of Figure 6. Then, after calculating 200 ps for the relaxation, top and bottom phantom mole cule s we re suddenly set to 85 and 115 K, re spectively. The phase -change proce ss can be cle arly see n in the change of the numbe r of mole cules
co nd s . e vap s . s .
NL droplet in conde nsing side , NL e vaporating side , and NV vapor in the bottom panel of Figure 7. Since NV was almost constant, the conde nsation rate and the e vaporation rate we re almost the sam e. The te mperature of the solid
e vap s . co nd s .
surfaces TS bottom and TS top quickly jumped to the phantom te mpera- ture, but the tempe rature diffe re nce of the two liquid drople ts was very sm all. This
F igu r e 5. R e latio n sh ip of th e de n sity profile and the in te grate d pote n tial func-
s .
tion E 2 conditio n in T able 1 .
L IQ UID D R O PLE T IN C O NTAC T W ITH A SO L ID SU R F ACE 5 7
F igu r e 6 . A snapshot of ine quilibrium drople ts on solid surface s.
re sult showed that the liquid] solid contact thermal re sistance was dom inant compared with he at conduction and phase -change thermal re sistance s. W e as- ssume d that the he at transfe r was quasi-ste ady the rate of phase change was constant. after 500 ps, and me asured the heat flux, ve locity, and te mperature distributions. The average heat flux me asure d at the middle of the calculation domain was 24 MWrm2, which is the order of the maximum he at flux possible in phase change .
F igu r e 7 . V ariations of tem pe rature and num be r of mole cule s.
T and N are te mpe rature and num be r of mole cules, re spe c-
tive ly. Subscripts S,L, andV re pre se nt solid, liquid, and vapor, re spe ctively, and supe rscripts e vap and cond re pre se nt evapora-
s . s .
tion side bottom and conde nsation side top , re spe ctive ly.
Figure 8 shows the ve locity and te mpe rature distributions on the e vaporation sside . The velocity distribution compare d with the density distribution top panel of
. s
Figure 8 shows that the e vaporation at the thre e -phase contact line or the first layer of liquid was m ost dominant, and som e supply flow of liquid from the m iddle. of the drople t to the first layer was obse rve d. The highe r drople t te mperature at the three -phase contact line also supports this observation. The tempe rature of the first laye r of the solid was lowe r unde r this area and highe r unde r the center of the droplet due to the the rmal insulation provide d by the liquid.
The same represe ntations on the conde nsing side are shown in Figure 9.
He re, the specialness at the thre e -phase contact line was le ss pronounce d, and rathe r uniform conde nsation was obse rve d ove r the whole liquid surface. The tem pe rature distribution of the liquid exhibited a sim ple line ar incre ase along the z dire ction, and the de cre ase of solid surface tem pe rature at the center is simply e xplaine d by the therm al insulating e ffe ct of the liquid droplet.
The me asure d contact angle was alm ost the same for both conde nsing and e vaporating drople ts, as re cognized in Figures 8 and 9. It se ems that the contact angle was principally de termined by the surface ene rgy balance and was not very se nsitive to the lowe r-e nergy tempe rature distributions.
W e noticed that thre e -layer solid molecules had bee n too thin to consider such a proble m with the tem pe rature distribution in the radial dire ction. E ve n though we we re confide nt of the use of the phantom te chnique through equilib-
L IQ UID D R O PLE T IN C O NTAC T W ITH A SO L ID SU R F ACE 5 9
F igu r e 8 . Characte ristics of evaporating drople t.To p, de nsity profile and ve locity vectors; m id d le, te mpe rature distributio n in the drople t;
b o ttom, te mpe rature distributio n of the first laye r of solid surface .
F igu r e 9 . Characte ristics of conde nsing drople t.To p, de nsity profile and ve locity vectors; m id d le, te mpe rature distributio n in the drople t;
b o ttom, te mpe rature distributio n of the first laye r of solid surface .
L IQ UID D R O PLE T IN C O NTAC T W ITH A SO L ID SU R F ACE 6 1
rium calculations of othe r system s, imposing the uniform te mperature just thre e layers be low the first layer with te mperature distribution as see n in Figures 8 and 9 was actually unacceptable. If we could use re asonably thicker solid layers, probably the solid-laye r te mperature distribution would be more pronounce d. Furthe rmore,
s .
the slight uppe r shift about 3 K of the solid tem pe rature from the phantom set value observe d in Figure 7 se em s to be due to this problem . E ve n though compute r time and mem ory crucially limit the numbe r of solid molecule s, calculations with a thicke r solid layer are desired in the future.
CONCLUSIONS
A liquid droplet in contact with a solid surface was sim ulate d by the mole cular dynam ics m ethod. The liquid drople t and surrounding vapor we re re alize d by Le nnard-Jones m olecules, and the solid surface was repre sented by three types of mode ls: three laye rs of harmonic mole cule s, one laye r of fixe d mole cules, or a simple one-dim ensional function. The interaction potential be - twe en solid molecules and liquidrvapor mole cule s was re pre sented by the Le nnard-Jone s pote ntial with various le ngth and e nergy param eters.
It was found that the cosine of the contact angle was we ll expresse d by a linear function of the depth of the integrated interaction pote ntial, regardle ss of surface represe ntations. Assuming that the surface ene rgy was proportional to the pote ntial depth, the m acroscopic Young’s equation was still valid e ven for such a small drople t. O n the other hand, the layere d structure that appe are d in the two-dimensional density profile of drople t was we ll e xplaine d through the shape of the integrated inte raction potential.
By preparing two solid surfaces with diffe re nt tem pe ratures, evaporation in one drople t and conde nsation on the other we re sim ulated simultaneously. The contact angle was almost the sam e as the equilibrium condition. V e locity and tem pe rature distributions of drople ts evaporating and condensing as we ll as the he at flux we re m easured for the quasi-steady condition. The importance of the three -phase contact line in the e vaporation process, in cle ar contrast to the condensation process, was observe d.
REFERENCES
1. S. Maruyam a, S. Matsumoto, and A. O gita, Surface Phe nom ena of Mole cular Cluste rs by Mole cular Dynamics Method,Th erm al Sci. E n g., vol. 2, no. 1, pp. 77] 84, 1994.
2. S. Maruyam a, S. Matsumoto, M. Shoji, and A. O gita, A Mole cular Dynamics Study of Interface Phe nom ena of a Liquid Droplet, Proc.10th In t. Heat Tran sfer Co n f., vol. 3, pp.
409] 414, Brighton, U.K., 1994.
3. S. Matsumoto, S. Maruyam a, and H. Saruwatari, A Mole cular Dynamics Simulation of a Liquid Droplet on a Solid Surface , Pro c. ASMErJSME Th erm al E n gin eerin g Jo in t C on f., vol. 2, pp. 557] 562, Maui, HI, 1995.
4. M. P. Allen and D. J. Tilde sle y,C om pu ter Sim u la tio n of L iq u id s, O xford University Pre ss, O xford, 1987.
5. J. N. Israe lachvili, In term o lec u la r an d Su rface Fo rces, Acade mic Pre ss, London, 1985.
6. G. Saville , Compute r Simulation of the Liquid-Solid-V apour Contact Angle, J. Ch em. So c. F arad ay Tran s.2, vol. 73, pp. 1122] 1132, 1977.
