Revista Mexicana de Física (Suplemento) 32 No. SI (1986) S49.S99

SOME RECENTADVANCES IN COLLOID

AND INTERFACESCIENCE:

FLUID CONFINED BETWEEN SaLIDSURFACESTHE DYNAMICS OF WETTING

H. Ted OavisDepartment of Chemical Engineering and Materials Science

University of MinnesotaMinneapol is. Minnesota 55455

S 49

Estas notas constan de dos partes: i) Fluidos Confinados entre Superficies Sólidas y ii) la Dinámica de Mojado. En la sección (i), examinamos simulaciones de dinámica molecular de la estructura, presión, te;sión y difusividad de un fluido simple confinado entre paredes sólidas -lisas. Mostramos que la teoría generalizada de van der Waals propuesta porNordholm y sus colaboradores, reproduce la estructura del fluido y expli-ca las fuerzas de superficie que se observan. Los resultados sobre la difusividad están relacionados con la viscosidad aparente de las capas de -fluido confinadas entre las paredes. En la sección (ii) presentamos desarrollos recientes de la teoría de flujos dispersos y de su uso en la ex--plicación del comportamiento de gotas de agua colocadas sobre láminasde vidrio. Las pelícuias delgadas y las gotas se consideran de un tamañosuficiente como para justificar la descripción del transporte a lo largodel substrato solvente como un flujo convectivo.

550

ABSTRACT

These lectures notes consist of two independent parts: i) FluidsConfined between Solid Surfaces and ii) The Dynamics oE Wetting. In part(i), we examine molecular dynamical simulations oE the structure, pres-sure, tension and diffusivity oE a simple fluid confined between flat,structureless solid walls. We show that the generalized van der Waalstheory oE Nordholm and coworhers accounts fer the fluid structure andexplains the observed surface forces. The diffusivity results have im-plications fer the apparent viscosity oE the fluid layers confined be-tween smooth walls. In section (ii) we present sorne recent developmentsoE the theory oE spreading flows and its use in explaining the behavioroE water droplets placed on glass slides. The drops and thin films areconsidered thick enough to justify description of the transport across asolvent substrate as convective flow.

FLUIDS CONFINED BETWEEN SOLID SURFACES

Introduction

Perhaps the most 81gnlftcant advancement In collold a.nd Interface sclence toappear In the last decade Is the development of the surface force apparatus.whlch can measure the forces between solld surfaces as a functlon of separatIondown to molecular dlstances. The measurements can be made In vacuo or In thepresence of a fluld.1.2 Operated In a dynamlc mode. the surface force apparatuscan be used to deduce the vlseoslty oC thln layers oC eonf1ned tluld.3 The surCaceCorees are adequately predleted by DLVO theoI1-"~ or the L1Cshltz-??? theoryS lCthe layer oC fluld 15sutneIently thIck.2 However. at solld separatlons oC the arderoC flve or ten tlmes the dlameter of the conftned fluid molecules. the observedCorees (called -dlsJolnlng" forces by DeI1-'agln6 and "so)vatlon Corees" by the Aus-trallan sehool7) are qulte dlfferent from those predIcted by these theorles. Theobserved surCace forces are osclllatoI1-' as a CunctIon of separatlon. The oscllla-tlons arlse CroID molecular layerlng Induced by the conftnlng sollds and thus atheoretlcal understandlng of the surface forces for molecular-seale separatIonsreQuIres a more precise mlcroscoplc descrlptlon than 15 provlded by the DLVO orLP theory.

In thIs lecture we examIne molecular dynamIcal slmlllatlons oC the structure.pressure. tenslon and dltfuslvlty of a sImple fluId conflned between flato structure-less solld walls. For thls system. we shall 8ee that the generallzed van der \Vaals(GVD\V) theory of J\'ordholm and co\",'orkers8.9 Quantltatlvely accounts Cor thefluid structure. whIch In turo Qualltatlvel)-' explalns the observed behavIor oC sur-Cace Corees. The molecular dynamIcs results for dlffuslvlty have Impllcatlons forthe apparent vlscoslty of thln layers of a fluid conflned between smooth walls.

551

Thermodynamics of Fluid between Flat Solid Surfaces

I! two tlat salid surCaces are brought elose to ene another In the presence oC afluid (FIgure 1) the normal pressure PN exerted by the fluid 00 the solld surCacedltfers from the bulk pressure Pb, The dl5jolnlng force 15deftned as the force ITAwhlch must be applled to the upper surCace to malntaln a separatlon h oC the flatsurCaces. From hydrostatlc eQull1brlum lt Collows that the dlsJolnlng pressure 15

IlA. disjoining or

1solvalion forceIl" Pw Pb

BULK FLUIDAT T, Pb ,¡1i

h

x

Area, A

~(Z) is lhe transverse pressure PNis lhe normal pressure

Figure 1. lIIuslration 01 Ihe disjoining pressure n ,

552(1 )

Although ooL drawn to scale In Figure 1, we are Imaglnlng that the distan ce h oC

separatlon oC the opposed ftat taces oC sollds ls very small compared te thelengths oC the surCaces In the yz-plane. In thls case the normal component oCpressure PN 15 constanL throughout the fluid fUm between the sollds. except tor a

smalJ reglan oC edge effects near the perlphery oC the flat surCace oC the suppersolld. The transverse component oC pressure PT depends. however, 00 the dls-

tance N from, say. the botlOm surCace. The film tenslon '"YF15 deflned as the

work requlred to lncrease the area A oC each ftat sol1dsurCaceby amount dA.

namely.

h

1FdA = J ( Ph - PT(z) ) dzdAo

or

h

1F = J (Pb - PT(z) ) dz.o

(2)

In a reversIble process the change In thermodynamlc energy oC the fluid sys-tero encloslng sol1ds A and B 1s

edU = TdS + ~1¡;dN, - PbdV

1=1

e= TdS + ~I¡,d:-¡, - PhdV - ITAdh + 1FdA.

1=1

(3)

553

In thls energy balance appears not only the usual work or expanslon or bulk nuld.-PbdV. but also the dlsJolnlng work -ITAdh or changlng the connned nIm thlck-ness by amount dh and the work I'rdA stretchlng the area oC the conflnlng spaceby amount dA. T, S. ¡Jil and Ni are the temperature and entropy oC the systemand ¡Ji andN¡ are the chemlcal patentlal and number oC molecules oC specles 1 oCthe C-component fluId.

The dlsjolnlng pressure TI and film tenslon Ir can In principie be computedfroIDelther the mechanlcal expresslon, Eq. (1) and Eq. (2), or the thermodynamlcrelatlons

1 [au)11= - A ah S,V,N.A

and

( au)"YF= --aA S,V,N,h

derlved from Eq. (3).

Qne can define a Glbus free energy as

G "" U - TS + PbV

and substltute It Into Eq. (3) to obtaln

dG = - SdT + I;1l,dN, - I1Adh + "YdA+ VdPb

Thls expresslon can be Integrated at constant T. Pb• J-ll' and h to ftnd

(4)

(5)

(6)

(7)

a result whlch when dltTerentlated and subtracted froIn Eq. (7) glves the Glbbs-

554

Duhem equatton

0= - SdT - EN,dl', - Ad'j - llAdh + Vd?b (8)

In surCace sclence lt 15 customary to subtract troro Eq. (8) the Glbbs-Duhemequatlon ror a bulk fluid In volume V aL temperature T. pressurePb• and cheml-cal potentlals IJ¡O The result 15 the sllt pare adsorptlon lsotherrn

where

o = 25'dT + E2r,d¡J, + d'j + ndh.,

5' == (S - Sb)/2A and r,==(N, - N'b)/2A.

(Q)

Sb and N1b are the entropy and nurober oC molecules oC specles 1 In a volume oCthe bulk tluld. Equatlon (g) ylelds the Maxwell relatlon between dlsJotnlng pres-sure and pare wldth (separatlon h):

(10)

and the relatlon

(11 )

Statistical Mechanics: Equilibrium Properties

In the remalnder oC thls lecture we shall assume that Sol1ds A and B areIdentlcal materlals. and are rlgld and lmpervlous to the fluid. and that the fluId 15pureo The normal pressure exerted on Sol1d A by the conflned fluId fllm and by

Solld B 15

555

(12)

where p~s ls the contrlbutlon fram ftuld-soltd lnteractlons and p~s that fram

solld-solld lnteractlons. For a rlgld. lmpervlous solld these two contrlbutlons areIndependent. p~s can be computed fram the expresslon

h duP~s = J p(z)_w_(z)dz + p,(dA,)kT

o dz(13)

where p(z) ls the fluId member denslty at postttan z In the film and uw(z) 15 the

potentlal oC lnteractlon between a fluId molecule aud the sollds p,(dAF) 15 the den-

sIty oC the fluId at the dlstance dAf of closest approach to the wal!. Ir the fluld-

wall repulslon goes contlnuously to Inftnlty P(dAf) = O, but Ir the wall repulslon 15

approxlmated as a hard sphere - hard wall lnteractlon tlIeo P(dAf)rfO.

Ir the fluId-fluid molecule Interactlons are central and palr addltlve, theo thenormal component oC the Irvlng-Klrkwood pressure tensor glves another expres-slon Cor PN:121

NkT 1 z,1 d,,(r,¡)PN = -- - -<I;---

hA hA ij r¡[ dr¡j

+ I;, [z, _d_u_:_f(_z,_)+ (h _ z,) _d_uw_A'_(h_-_Z,_)] >.dz] dz¡

(14)

where the polnted brackets, <0.0>, denote an ensemble average rij Is the lnter-

partlcle separatlon and Zij lts z-component. tP(rij) Is the palr potentlal betweenmolecules 1 and jo

There are two avallable theorles tor estlmatlng PNo One theory ls the DLP

theory In whIch the soltds are treated as dlelectrlc continua and P: ls computed

from the measured dlelectrlc relaxatlon propertles oC the solldso An alternatlveapproxlmatlon Is to assume that the sollds are composed of molecules InteractIngvla palr addltlve central forces. In thls case,

556

(17)

where 4JSt5'(lR.A-R¡Bh. 15 the torce between a molecule ot Sol1d A aL slte RIA and a

molecule ot Sol1d D aL slte RIB. Ir we assume that the molecules In Sollds A and

B are unltormly dlstrlbuted. then we oblato

Pi! =

o 0000

= - 2rrp~p~ J J J <1>,,' ( ,¡(z - Z')2 + p2) dzdz' pdp,-ooh o

(18)

where nl and n~ are the densltles at Sollds A and B. For an lnverse power lawpotentlal, <I>,,(r)= -Emr-v, Equatlon (18) ylelds

211"VPÁp~EABp~ = _ ---------h-I.' + 3

(v - l)(v - 2)(V - 3)(1Q)

For nonpolar dlsperslon rOfces v = 6. and so EQ. (H~) predlcts a sol1d-solld dls-jolnlng pressure talllng off as h-3 wlth lncreaslng separatlon. Thls 15 the same as

the power laVo: predlcted by the DLP theor)' ter separatlons h less than a

mlcron.7j

Ir the ftuld-sol1d Interactlons abey the lnverse power law 4>lf(r) - 00, r < d,land c>.f(r) = -E11r-v• r > dlr• then

21l"p~EAf 2rrp¿EBfIIw(x) = - -----(h - ztV+3 _ ------ Z-V+3

(V-2)(V-3) (v - 2)(V - ;l)(20)

ror dAr<z<h-dA1 wlth the erude approxlmatlon to the film dl'nslty thnt p(z)=O

tor Z<dAf, Z>h-ctAf. and p(z) = Pf tor dAf<Z<h- dA1, the approxlmatlon

21l'PsPfEB1

(V-2)(V-3)

557

(21 )

Is obtalned rrom Eq .. (13). Slmllarly to Eq. (IQ) thls expresslon lmplles that P~'approaches lts asymptatlc value as h-1I+3 w1th lncreaslng flIm thlckness h. As we

shaJl see In what follows. Lhe constant film denslty assumptlon on whlch Eq. (21)

ls based ls a very poor approxlmatlon.

Although In an experImental measurement oC dlsjolnlng forees the contrlbu-tIan p~ 15 always present, In theoretlcal work or In computer slmulatlons we can

Ignore tl115etfect and concentrated onl)' on Lhe Ould-sol1dpart oC Lhe dlsJolnlngforees. In lhe remalnder oC Lile leClure we shall Indeed do just thl5. Incldentally,to Lile extent that the sollds are rlgld and lmpervlous to Lile ftuld Lile contrlbutlon

oC Pi! can always be ellrnlnated fraID an experImental result by subtractlon of

tlle experlmentally deterrntned dlsJolnlng pressure w1th the fluId replaced by avacuum.

There are also two ways to compute the fllrn tenslon. The ftrst Is to lntr~duce the Helmholtz free energy F == U - TS. Substltutton of thts Into Eq. (3)ylelds

and so

edI" = -SdT + EI',dN, - PbdV - l1Adh + 'lFdA,

1=1

[al")I'F = --oA T,N.\'.h

(22)

(23)

The other method uses tlle Irvlng.Klrkwood formula for the transverse com-ponent of the pressure tensor

h

'l J [Pb - PT(z) ]dZo

SS8

1 y,¡ + X,¡ d4>(r'J) NkT- h<Pb + --~------- - -->.

2hA xIJ fij dr1j hA(24)

To compute the dlsjolnlng pressure froro Eq. (12) or the tenslon froro Eq. (15). weneed a theory oC the fluid denslty profiJe and the free energy. For thls purposewe choose the general1zed van der Waals (GVD\V) theory.lO,Il] It 15 the slmplest

avallable theory whlch (as we shall see later) has the capablllty of predlctlng thefluid structure Induced by conftnement between sollds.

If ane expresses the N-body lnteractlon potentlal In the forrn

(25)

then the Helmholtz free energy oC a classlcal fluid can be rlgorously expressed In

the forro

F = FR - kTln < exp - (4)rt + 4>J)/kT>. (26)

where < >R denotes an ensemble average In a reference fluid wlth potentlal4JN=4>/J and FR 15 the free energy oC thls reference fluid. 4>,J 1s the potentlal of1nteraetlon between fluid partleles and external sourees, sueh as solld walls. Wlth

the mean fleld approxlmatlon,

(27)

and for palr-addltlve attraetlve forees and addltlve external rorees, Eq. (25)

becomes

where gR(r,r') 1s the palr correlatlon functton of referenee fluId and Uw(r) Is thepotentlal of external force on a fluId partlele. FR ls based on lntroductlon of anexcluded volume. Suppose the reference fluid were an Ideal gas wlth a local den-slty p(r). Then the referenee free energy would be

559

(2Q)

where ver) 15 the free volume avatlable per molecule. In a homogeneous Ideal gasv = V/N = p-l. In an Ideallnhomogeneous gas v(r)=p(rtl. Accordlng to thevan der Waals theory of homogeneous fluId, v = (V-Nvo)/N=p-l-vo' where Vo lsthe average excluded volume per moJecule. In what 15 known as the van derWaals theory oC lnhomogeneous fluid, the choIce Vo remalns the same as In horno-geneous fluId. Nordholm and coworkers argued that In lnhomogeneous fluId thevolume excluded to a molecule at polnt r In the fluId should reftect the fact thatthe denslty In the nelghborhood oC r dlffers (rom lts average value.

Ir the denslty were p(r) In a volume (4/3)7rd3 about r, where d 15the molecu-lar dlameter. then the number oC molecules In sphere would beÓNav=p(r)(4/3)1rd3• On the other hand. slnce the denslty about r var1es. theactual number of molecules in the sphere ls

6N act = J p( r' )d3r'Ir' - rl<d

Accordlngly, Nordholm and coworkers Introduce a nonlocal excluded volumev(r;{p})=vo6Na.ct/6Nav to correct for the local denslty varIatlon In the viclnltyof r. TheIr local free volume ls then

where

{1 p(r)

v(r) p}) = ---vp(r) p(r) o

p(r) = _3_ J p(r')d3r'471"d

3.' _ ,I<d

(30)

(31)

Comblnlng Eqs. (28). (2Q), and (30) and addlng the slmpllfylng approxlmatlon

that gR(r,r') equals O when Ir-r'l < d and equals 1 when Ir-r'l < d. we obtalnthe generallzed van der Waals (GVOW) free energy

560

F = kT f p(r)ln{ A-3/2p(r¡-'[1 - p(r)vol }d3r

1 -+ 2 f f p(r)p(r')4>A(~ - r'lld3rd3r' + f p(r)uw(r)d3r. (32)

where "j,A(r) = 4>A(r). r>d and "j,A(r) = O, r < d. Ir p(r) 15 replaced by p(r).EQ. (32) becomes the free energy oC van der Waals original theory.

The eQulllbrlum dens\ty dlstrlbutlon p(r) 15 the ane whlch mlnlmlzes thegrand potentlal O( {n}) = F - NI'. where ¡l 15 the chemlcal poten ti al of the ftuld.The mlnlmum can be found from the eondlt1on

[80(n + 'v) ] = O for all v(r).

BE. l=O

Thls condltlon results In the nonl1near Integral equatlon

{ }3 vop(r')

/' - kT In!p(r)A,3/2¡ - 1 - -- f _ d3r41fd3,' _ rl<d 1 - p(r')vo

(33)

(34)

For aplanar system P = pez). and so Eq. (34) can be reduced LO a one-

dlmenslonal nonllnear Integral eQuatlon.

Statistical Mechanics: DitTusional Relaxation

COllslder a fluid conllned between two f1at solld surCaces separated by a dls-

tance h In the z-dlrectlon and oC Intlnlte cross-sectlonal area In the xy-plane. Thedllfuslon coemclent ot a reterence partlcle dlffuslng In the x or y-dlrectlon can becomputed trom the formula

D= 11m <y'>2t

11m <x'>2t

561

(35)

where <--> denotes an ensemble average and x (or ~.) 1'"the dlsplacement In thepartlcle cturlng the time t. Thls dlsplacement can' "l]lllted froID the partlcleveloclty v(t) lIS

t

X = J v,(s)ds.o

Puttlng thls expresslon loto Eq. (35), one can derIve an alternatlve expresslon rorthe dlffuslvlty. nameiy,

00

D = JCw(t)dt. ,,= x or y.o

where Cvv(t) 15 the veloclty autocorrelatlon functlon

(36)

(37)

Slnce the fluid 15 conflned In the z-dlrectlon, the quantlty <Z2>/2t goes to Oas t goes to 00 and so the dlffuslvlty ls zera In the x-dlrectlon. In terms oC theveloclty autocorrelatton functlon, thls property means that the tlme Integral oCC,,(t) 18 zero.

In the dynamlcal theory oC the metloo caused by forclng two sol1ds togetherwlth a conftned fluid film 15 needed. There Is presently no tractable theory oC thtsvtscoslty and molecular dynamlcs or vlscoslty are very costly. Inslght can begalned. however. by st udylng the molecular dynarnlcs oC the self dlffusloncoemclent. whlch at least In bulk fiulds Is an lnverse rneasure oC fluid vlscoslty.In thls lecture we shall dlscuss the lmpllcatlons oC sorne recent molecular dynam-les results.

Molecular Dynamics

1 shall dlscuss a model system of simple non polar molecules for whlch corn-puter slmulatlons are avallable. The ftuld molecules lnteract vla the truncated

5626-12 Lennard-Jones palr potentlal,

<;ó(r)= hJ(r) - <I'LJ(r.) .r < r.

= O ,r> re

where

(38)

(3g)

l and r are the characterlstlc energy and length parameters and re 15 the cut-off

dIstan ce (chosen LObe 3.00 In the work dlscussed hereln).

The salid wall-fluId molecule potentlal to be consldered 15

[( ]

102 Gfs

uw(x) = 211'"fls "5 -Z- h ]3(Z/17,{ + 0.431)3

(40)

Thls potentlal represents the Interactlon oC a fluid molecule wlth ao Inftnlte stackoC layers oC randomly-dlstrlbuted sol1d molecules. the layers havlng a separatlonO'lsand the ftuld-sol1d molecules Interatlng vla a 6-12 Lennard-Jones palr poten-

tlal.

In the molecular dynamlcs studles 1 shall dlscuss, Newton's equatlons oCmotlon were sol ved tor about 50 to 250 ftuld molecules between a palr oC flatsolld waUs separated by a dlstanee h whleh ranged Crom, 2 to 120'. The solld-

fluid parameter ffs and GIs \Vere the same as the fluId-fluid parameters f and 0'.

The chemlcal potentlal and temperature \Vas kept eonstant (at least nearly so) Ina sean over solld separatIon h. Chemleal potentlals oC the model system arekno\Vn from Monte Cario slmulatlons.13] The system was made InflnIte ln the yz-plane by the usual devIee of enCorclng perlodle boundary eondlt1ons. The dIsjoln-lng pressure lutcrCaclal tenslon and dlffuslvlty were computed Crom Equatlons (13)

or (H), (24) and (36). The densIty dlstrlbutlon ln the fluid between the solld

walls was calculated rrom

1 dp(x) = -- < N(O~z»,

A dx

563

(41)

where A ls the cross-sectlon oC the wall and (N(O-z) 15the number oC fluId par tl-eles Iylng between the wall and pasltIoo z. The dlsjolnlng pressure, lnterfaclaltenslon, and dlffuslvlty were computed from Equatlon (13) or (14), (24), and (36),respectlvely. The ensemble average Indlcated by <---> 15 replaced by the long

time average In a molecular d:rnamlcs slmulatlon. Accordlng to the ergodlchypothesls the two are equlvalent. (Ir the hypothesls falled the long time averagewould prevall as the approprlate ane to compare wlth experlment.)

Discussion of Computer Simulations, Mean Field Theory and Experi-mental Results

The molecular dynarnlcs results we shall dlscuss In what follows wereobtalned by Magda el al. for the rnadel descrlbed aboYe. H] Where comparlsonscan be made (equlllbrlum propertles) the results are lndlstlngulshable from the

Monte Cario results of Snook and van Megen.13] Slmllar molecular dynamlcs andMonte Cario studles on hard spheres between hard walls have been reportedrecently by Antonchenko el al.ls. Thelr structural and thermodynamlc are qual1-tatlvely simIlar to those of Magda el al., although the dlffuslvltles of the model ofMagda el al. seem to be more sensltlve to wall separatlon.

In FIgure 2-4 typlcaJ denslty prefiJes pez) are shown for varlous pore wldthsI.e., varlous wall separatlons h. Denslty ls glven In unlts of (1-3 and pore wldthsIn unlts of (1-1. The proftles In FIgures 2 and 3 correspond to a conflned ftlm In

equlllbrlum wlth a IIquld at temperature T = 1.2,/k and bulk densltynb=O,S9a-3, the one In FIgure 4 15 In equlllbrlum wlth a IIquld at T = O,97,/kand nb = 0.67(1-3.

It Is apparent from FIgures 2-4 that the Impenetrabll1ty of the solld to thefluid Induces a great de al of structure In the fluid near the wall. The structure 15essentlally a layerlng oC the fluid near the wal!. That the structure 15 layerlng 15brought out most clearly by Figures 2 and 3. The pare oC wldth h = 3(1 canaccornmodate only two fluid layers and so two peaks appear. As the pore wIdthls Increased a thIrd fluId layer appears: the thlrd layer 15 fully developed In thepare of wlth h = 4(1. As the wldth of the pare contlnues to Increase the densltyproflle at each solld surface belngs to reach the asymptotlc structure oC a bulkfluId agalnst a sIngle wal!' Snook and van Megen and Magda el al. found that

564

Lile structllre aL each waH changes IIttle for pore wldths greater thao about 60'.

Tllus. Lhe denslty proflle shown In FIgure 4 15 representatlve oC a bulk I1quld al a

single wall.

3.6,---,---,----,---"---,

v

3.2 J-

2.8

2.4

2.0p(Z)

1.6

1.2

08

0.4 1-

o 06 12Z

18

\

2.4

FIgure 2. Denslty proftle oC conftned fluId. Unlts oC dIstance and densltyare a and 0'.-3 O, molecular dynamlcs (Ret. 14): £1 Monte Cario

(Rer. 13). h = 3.

The dlsJolnlng pressure oC tlle conflned fUro osclllates. FIgure 5. as a funcHonoC pore wldth In retlectlon oC the layerlng tendency oC Lile OIm. We lnterpret Lhemlnlma aL h = 2.250' and 3.250' as arlslng when Lile spaclng 15optlmal fer form-Ing aue and two fluId layers, respectlvely. The maxlma aL h = 2.750' and 3.75acorrespondlngly arlse at separatlons ¡east favorable to formatlon aL an Integralnumber of layers. Successlve most and least favorable posltlons dlffer by about0/2 as expected wlth thls lnterpretatlon.

The fllm tenslon '1F also oscHlates as a functlan aC pare wldth as shown In

Figure 6. Accardlng to the Maxwell retatlan, Eq. (10), maxlma and mlnlma In '1

should correspond to zeros oC n. That thls ls seen to be so In Figure 6 amounts

565

to a thermodynamlc conslslency test of the molecular dynamlcs results. Inclden-tally. Integratlon of Eq. (10) provldes a means of determlnlng ¡(h) from D(h) rela-

Uve to an Inftnlte pore datum:

3.0

2.5

2.0

p(Z) 15

1.0

0.5

1.0 2.0Z

3.0 4 ..0

Figure 3. Denslty profiJe of conflned fluId. Unlts and symbols are lhe sameas In Figure 2. h = 4.

00

¡(h) = ¡(h = 00) - j[](h')dh'.h

(42)

,\'here lhe Integratlon 15 carrled out al constant T and chernlcal patentla! of aHfluJd components.

The dlffuslvlty versus pore wldth 15 shown In Flg. 7. (The bulk dltfuslvltyDSu1k for lhe Huid equals 0.18 (fo2/rn)I/2 al T = 1.2 f/k and Pb = 0.590-3 andequals 0.11 (w.2/m) al T = 0.97 f./k and Pb = 0.670-3. the latter condltlonsbelng those of lile pore wlth wldth h - 11.57). Also shown In Flg. J' ls lhe averagepore deoslty.

866

2

p (z)1

2 4z

6 8 10 12

FIgure 4. Denstty profile oí conflned fluid. Molecular dynamlcs. Unltssame as In Figure 2. h = 11.57.

4

2P fsN

o

-2

(ASymptote - Bulk Pressure

0-----0------.."...

2 3 4 5h

6 7 8 9

FIgure 5. Normal pressure versus pore wldth. Molecular dynamlcs. UnUsoC dlstance and pressure are (j and (/a.2

567h

PAVE = f p(z)dz/h,o

whlch has local maxtma at pore wldths favorable to layertng and local rnlnlma atpore wldths unfavorable to layerlng. The dlfIuslvlty has local mlnlma and max-lroa correspondlng to the maxlma and mlnlma In the average pore denslty. acorrelatlon whlch Indlcates that dlsruptlon oC fluId layerlng enhances pore

dlffuslvlty.

4 t/t 3.5

3.252

TI O 3.0 -y

-2 2.75\

-4 2.5

2.25 2.75 3.25 3.75h

FIgure 6. DlsJolnlng pressure and tenslon versus wall separat1on.Molecular dynamlcs. Unlts of dlstance, pressure, andtenslon are a, [/0.2

To assess the etrect oC local denslty varlatlons 00 the dlffuslvlty. Magda el alcomputed the autocorrelatlon functlon In ftve dlfferent regloos parallel to the porewalls. The dlvldlng planes of these fluId "sl1ces" are lndlcated by the dashedvertlcal lInes In Flg. 8. The autocorrelatlon functlon and lts Integral wasevaluated for partlcles whlch remalned In thelr lnltlal sllce. The surprlslng resultls that des pIte strong dlfferences In the denslty proftle In dlt'ferent s1lces, thedlt'fuslvltles computed In the d1t'ferent sllces are vlrtually the same. Thus, thepore dlffuslvlty appears to be determlned by the average pore denslty, not by thelocal denslty. Whether thls lnterestlng property would be true when the solldwall Is glven atomlc structure and the ablllty to exchange klnetlc energy wtthfluid partlcles ls an lmportant open questlon. As very llttle rlgorous theory exlstsfor transport In rnlcropores, the prospect that the dlffus1vlty Is constant

568

throughout a rnlcropore of constant wldth 15 an attractlve ane Indeed.

Let liS now evaluate the statlstlcal mechanlcal theorles lntroduced earller Inthe lecture. Clearly. the contlnuum theory, whlch Ignores local denslty

0.0

0.2

0.4

1.0 ~"<Il

0.80

""-O0.6

111097 8h

654

--0_ -- -0- -0_----

Asymplote hVE ~PooLK D/DBulk Á

PAVE / PBu1k O

3

2.0

1.6"'""<Il 1.2

Q..

""-w 0.8;:rQ..

0.4

0,0

Figure 7. Pore wldth dependence of average denslty and dlffuslvlty Dof conftned fluId relatlve to bulk values versus porewldth. Molecular dynarnlcs. DIstance In untts of a.

p (Z)

2

D," .114:l:.00<j

IIIIIIIIII

D;,~.109 03".111 °4~.113 o,~.119

2 4 6 8 10 12Z

FIgure 8. Denslty proflJe and sllce-wlse dlffuslvlty In a pore wldthh = 11.57. Molecular dynamlcs. Untts oC length, denslty.and dlffuslvlty are a, 0'-3 and Jf..(j2/rn

S 69

profiJes. 15 DoL expected to be accurate for mlcropores. Thus, lt 15 no surprlsethat Eq. (21) predlcts that the dlsJolnlng pressure varles monotonlcally wlth

lncreaslng pore wldth. In contradlctlon wlth the behavlor reported In Flg. 5.

The van der Waals (VDW) theory or Inhomogeneous ftuld (Eq. (29) wlthv(r)=p(rfl-vo) predlcts sorne structure In che fluid denslty proflle as 11Iustrated

In Flgs. 9 and 10. IIowever. Che VD\V rnodel compares very poorly wlth COffi-

40 T=1.2

P =0.59B

Q..3.2

>-1-üiZIJJO

24

16

.8

oO 8 1.6 2.4 3.2

DISTANCE FROM PORE WALL, Z

Figure 9. Comparlson of denslty proflles frcm computer slmulatlons(polnts) and generallzed van der \Vaals (GVO\V) theory.(Rer. 16). Unlts same as In Figure 2. h = 4.

puter slm ulatlons. In partlcular. the predlcted denslty proftles have only a singlepeak near each pore wall and the local denstty cannot be larger than V,;I, whlchIs than 3/27ro,3 Ir the trad1tlonal tdentlOcat1on of Vo ls used (d •.......o tor the

Lennard-Jones flutd).

570

3.0 T=1.5flcO.59

2.5 B

el.. 2.0>-....(/) 1.5Zl&JO

1.0

.5

OO 2 4 6 8

DISTANCE FROM PORE WALL, zFigure 10. Comparlson oC denslty profiJes froID computer slmulatlons

(polnts) and GVDW theory (Rer. 16). Unlts same as In

Figure 3. h = 7.5.

The generallzed van der Waals theory, on the other hand. predlcts densltyproftles that compare QuIte well wlth computer slmulatlollS. The predlctedprofiles were taken froro the work oC Bellare el all61 They used vo=a

3 as reCOffi-

mended by Nordholm and coworkers. Thus. the nonlocal excluded volume lotro-

duced In mean fteld theory by Nordholm and coworkers. Eqs. (30) and (31).appears to capture the effects oC repulslve forees responslble for the fluId layerlngIn mlcropores. The normal pressure p~s can be computed (roID the GVDW

theory uslng the thermodynamlc relatlon

pi! = - .2...( 8F )A iJh T,N

(43)

S71

wlth F glven hy Eq. (32). The GVDW predlctlons shown in Flg. 11 are In goodqualltatlve agreement wlth the computer slrnulatlons. The quantltatlve agree-ment ls fiot so good, but thls Is Dot unexpected slnce the VDW equatlon oC stateoC homogeneous fluId predlcts a pbase dlagram ln poor quantltatlve agreementwlth experImento The GVDW predlctlons oC tenslon "fF are slmllarly In goodqualltatlve but poor quantltatlve agreement wlth slmulatlons (Bellare el al.161 ).

Another problem wlth the GVDW theory 15that It predIcts a palr correlatlonfunctlon that allows lnterpenetratlon of hard spheres. The problems arlses (rom

1.00

0.75 fL--3.17T=1.20

0.50 P=0.59

pfs0.25

N 0.00

-0.25

-0.50

- 0.752.0 3.0 4.0 5.0 6.0 7.0 8.0

POREWIDTH. h

Figure 11. Normal pressure versus porewldth predlcted by GVDW theory.(Rer. 16) Unlts same as In Figure 5.

Nordholm and coworkers choIce oC the nonlocal excluded volume. Tarazona haslnvestlgated these problems and has lntrodueed a systematlc procedure forImprovlng the orIgInal GVDW theory.17] Hls reformulatlon takes eare of the palrcorrelatlon Cunctlon Callure and presumably wlll glve more quantltatlve results torthe dIsJoInlng pressure and fUro tenslon oC conftned fluid.

572

Flnally. 1et liS end thIs le('ture by notlng Lhe slml1arlty bet\\'een experImentalresults and Lhecomputer slmulatlons and G\'D\V thcor)". The Australlan schoolhas publ1shed a serIes oC very elegant rneasurements oC Lhe forces between crossed

mIca cyllnders brought together w1th a fluId contlned bet""'een them.1.2] A typlcalexample oC thelr results (rar cyclohexane) 15 shown In Flg. 12 In whlch ratio oC Lheforce te Lhe

2

~E....zE~

Oa:il:

-1

-2

, - '-T,20,

I .:, . :,I r O, ,I, ,, I -20, 1, ,, ,

OI , • 2 4, : ,•

I : ' 1~_.....................I .' .,.. ., ,, l.,,' ~~' I

I.,, , I ,, 1/ , I, , •, I , ••, o ,

, • o •.,:1: I oO/,o. o1: •-l' , ,

2 3 4 5 6SEPARATION h (nm)

Figure 12. SurCace Corce measured a.'5 a runctlon oC separatlon between crosscyllnders oC mica Irnmersed In cyclobexene (Ree. 2). Dashedcurve 15 DLP theory (F/H = -0.9 X 1O-2°J/6h').

radlus oC <,urvalure of the cyllnders 15 plotted versus separatlon D of the

cyllnders. Slnce the radlus of curvature of the cyllndcrs ls large compared to

separatlon D. surfaces conflnlng, the fluid can be thought of as planar surfaces.

The obscrved force osclllates wlth a length scale on the arder of the molecular

dlameter of the rnolecllles leadlng one to Interpret thc osclllatlons In terms of the

573

Jayerlng tendencles observed In the computer slmulallons. AIso shown In FIg. 12are predlctlons oC the contlnuum approxlmatlon, \\'hlch 15 In poor agreement wlthexperImento

The take-home lcsson from thls lecture 15 that the theoretlcal descrlptlon oC

ftulds conftned In mlcrapores and between coalesclng colloldal partlcles !TI ustaccount quantltatlvely for the local molecular structure, In partIcular layel'lngtendencles, oC fluId near lhe salid surCaces. The GVD\V theory 15 a step In therlght dIrectIon, bul a quantltatlve equlllbrlum theory demands a more sophlstl-

cated approach; for example, lmprovement of the excluded vaJume estlmate as Inthe work oC Tarazana. There 15 presently no predlctlve theory oC transpon Inconflned fluid ftlms.

References

1. Horn, R. G. ami Israelachv111, J. N., J. Chem. Phys. 75, 1100 (10SI).

2. Chrlstenson, H. K., Horn, R. G. and Israelaehvlll. J. !':., J. Collold and In-terface Sel. 88, 70 (1OS2).

3. Israelaehvlll,.1. N., J. Collold and Interface Sel. (to appear, 10S6).

4. Chan, D. Y. C. and Horn, R. G., J. Collold and Interface Se\. (to appear.1g86).

5. Sheludko, A., Adv. Collold and Interface Sel, 1, 301 (1067).

6. Dzyaloshlnskll, I. E., L1fshltz, E. M. and Pltaevskll, Soviet Phys. HETP37, 161 (lgOO); Llfshltz, E. M .• Soviet Phys. JETP 2, 73 (lg50).

7. Mahanty, J. and Nlnham, B. \V., "Dlsperslon Forees," Academlc Press,New York, 10713.

8. Deryagln, B.V., Zh. Flz. Khlm. 5, 37g (1g34).

Q. Chrlstenson, H. K. and 110m, R. G., "Chemlca Scrlpta," 25,37 (1085).

574

10. Nordholrn, S. and Hayrnet, Ad. D. J .• Aust. J. Chern. 33, 2013 (1Q80).

11. Johnson, M. and Nordholrn, S.• J. Chern. Phys. 75, 1QS3 (1Q81).

12. Irvlng, J. 1-1.and Klrkwood, J. G., J. Chern. Phys. 18, 817 (1QSO).

13. Snook, I. K. and van Megen, W. J.. J. Chern. Phys. 72, 2Q07 (1Q80); van

Megen, W. J. and Snook, I. K., J. Chern. Phys. 74, 140Q (lQ81).

14. Magda, J. J., Tlrrell, M. V. and Davls, 1-1.T., J. Chern. Phys .• 83, 1888(1Q8S).

15. Antonchenko. V. Ya., II:r1n, V. V., Makoysky, N. N .. Pavlov, A. N. andSokhan. V. P. ,Molecular Physlcs 52, 345 (1984).

16. Bellare. J.• Kerlns, J. E., Scrlven, L. E. and Davls, H. T., J. Chem. Phys.(to be publlshed, 1Q86).

n. Tarazona, P., Phys. Rev. A31, 2672, (1Q8S).

575

THE DYNAMICS OF WETTING

Introduction

Durlng the last half a dozen years a ¡ol of progress has beeo made In the

theory of wettlng transltlons. A fluid drap placed on a solld substrate In thepresence of a second fluId phase w1l1under certaln condltlons relax tnto sorne len-

tIcular contlguratlon w1th lts menlscus lntersectlng the salid surface w1th sornecontact angle e Iylng between o and 180'. By approprlate varlatlon of condl-tlons. e.g., temperature or vapor pressure of a component, a transltlon polnt cangenerally be reached al whtch the drap wl1lspontaneously spread to forro a thln¡ayer or f1lm 00 the solld. The droplet phase 15then sald to be completely or pre-fectly wettlng al the Interface between the solld and the other fluId phase. It hasbeen dlscovered theoretlcally that dependlng on the nature oC the fluld-solldlnteractlons the wettlng transItlon can be a flrst arder thermodynamlc transltlonor a second arder transltlon. Relevant scallng laws have been derlved. Thestatus of theoretlcal work on wettlng transltlons can be assessed frOID a numberof recent revlew artlcles1,2,31. In most cases theorles have been developed forsmooth, lnpervlous, homogeneous sol1ds, although lately InvestIgatlons of flulds atsemlpermeable membranesH] and at dIsordered substrates have begun toappear5,61.

ExperImental results are emerglng to test and gulde theory, although owlng tothe tedlus nature of experlments on Interfaces and thln ftlms experlment has nctbe en rnovlng as fast as theory7-1O).

Much less well developed than the equlllbrlum theory ls the dynarnlcs of thespreadlng flows. These flows can be of practlcal slgnlflcance In lubrlcatloD, wet-tlng, coatlng, prlntlng and the Ilke. At the Unlverslty of Mlnnesota we llave beendeveloplng at a "mlcrofluld mechanIcal" level the dynamlcs of spontaneous wet-tlng (spreadIng)1l-131, dewettlngl4] and convectlon drlven wettlng (wettlng hydr<rdynamlcs)11,15]. The recent revlew of de Gennes on wettlng provldes an excellent

perspectlve of tlle status oC theory In thls are a and of the challenges to futuredevelopment16].

In what follows, I shall present sorne of the recent developments made at theUnlvcrslty of Mlnncsota In the theory of spreadlng ftows and Its use ln explalntng

thls rather surprlslng behavlor of water droplets placed on glass sl1des. We res-trlet our conslderatlons to drops and thln fllms thlck enough to justl1'y

576

descrlptlon of che transport across a salId substratc as convectlve tIow as opposedto dltruslve hopplng. The more general case has also been treated by TeletzkeetaI.11,131.

Thermodynamics of Thick Thin-Films on a Flat Salid

By thlck thIn-ftlms, 'we mean fluId fllms thlck enough that thelr lnterfaclalzones do not overlap. In Flg. 1 a film of fluId B 15shown on a fiat salid In the

BULKFLUID A

THIN FILM ON ASOLIO SURFACE

x

P=P(z)1

'.:.:::::::::::::.-

BULKFLUID B

PRESSUREISOTROPIC

'.: ~::i'h.~'.~'~(~.'~i,:~:l:':':'~':'.. , PRESSURE"';Zon@:::: ..l......••.•;.,...•IS~~R.~~IC ,

h(x) ..••••••••••••::»>.... ...:: ":':', . ::::::::: -:.

P = P(z) I ;....... ..•• . :: .••••••••••••••••• • ••• ';:;: ":':::':':'" 0.' •••

~:~:~:~:.~::i~.f~~¡.~.~.¡~1.:z~.ri~:..•...•.~.•.:... 0,0: ::.>," ..... .;.;.;." ".'

~///////////////////////:SÓ(¡6~//////////////////dFIgure 1. Transltlon of a thlck thIn-film to a thln Lhln-film.

presence of a bulk phase of nuld A. 1'0 Lhe left of Lhe figure the film Is a thlckthln-fllm: the InterfacIal zones beLween saIld and fluId B and between bulk fiuldA and fluId D da nat overlap and In the reglan between the Interfaclal zones thepresure 1s lsatraplc. Although the pressure Is Isotroplc between the Interfaces 1tw1ll depend on dlstance fram the solld mvlng to dlsperslon and electrastatlcInteractlons between fluId and solld materlals. 1'0 the r1ght of the figure thelnterfaclal zones merge and the fllm becomes a th1n thln-film. and as such has noreglon 01' pressure Isotropy. !v101ecular modelllng and computer slmll1atlons have

establ1shed that If the film thlckness h Is two 01' three times the wldth of tllelnterfac1al zones the nIm behaves as a th1ck thln-fllm. h ls deflned by chooslngconvenlent mathematlcal surfaces located In the 1nterfaclal zones (e.g .• a G1bbsdlvldlng surface of zero excess mass). In what follows we shall assume that h lsmuch larger than the thlckness of the Interfac1al zone so that ane does not have

577to be any more precIse about Lhe locatlon oC the dlvldlng surCace. SuIDclently farfrom a crltIcal pOlnt between ftulds A and B the Interfac1al zones are oC the arderoC Leos of angstroms \VIde, and so In thls case nIms In whlch h ls oC the arder oChundreds oC angstroms wlde would be thlck thlo-nlms.

\Ve shall assume Lhat the solld 15a rlgld. Impenetrable body. Thus. lt can be

treated as the souree oC a body force on the fluid. 'fhe fluId pressure tensor Pthen obeys Lhe hydrostatlc equatlon aL equlllbrlum:

(1)

where Pi 15 the denslty oC component 1. Far Lhe purposes oC thls sectlon \Ve

assume Lhat gravltatlonal and centrIfugal forces are negllglble. Thls poses no

speclal IImltatlon 00 fluId statlcs slnce aver dIstan ces typlcal oC Interfaclal wldthsthese forces vary negllgIbly.

For aplanar system. flat In the xy-plane, Uw = uw(z). Pxx = Pyy' and P =

Pxx( 11+ JJ)+Ptzkk. Furthermore. It follows from Eq. (1) that Pxx and Pyy

depend only on the dlstance z perpendIcular to the solld and that P zz obeys theequatlon

dPzz duw i-=-¿;p--' ,dz i l dz (2)

OutsIde the Interfac1al zones P zz = p yy = P zz. Inslde these zones

P xx = p yy .:¡: P zz' The varlatlon of P xx and P yy In tlle Interfaclal zone ls con-

trolled by Intermolecular forces. Varlous approx1mate molecular theorles (usuallybased on the exact theor1es of BufI and Klrkwoodl7] and Irvlng and Klrkwood18J)have been developed durlng the last decade for predIctlng these quantltles.19)

The fllm tenslon and use fui thermodynamlc relatlonshlps can be der1ved byconslderIng a reversIble work experIment as 1I1ustrated In FIg. 2. The reversiblework of dlsplacement of plstons 1 and 2 ls

576

(3)

where dA = wdx. w belng Lile wldth of the system In Lile y~dlrectlon and L ltshelght In Lhe z-dlrectlon. \Ve assume that L Iles ahoye Lhe fluid-fluid Interfaclalzone where In Lhe absence of external forces Lhe pressure 15 lsotroplc and COI1-

stant. Theorel1call.y It su IDees for L LO be eQual LO 11(-)+ lAB' where h(-) 15 Lhe

value of z just below lhe fluId-fluid lnterfaclal zone and lAR 15 Lhe lnterfaclal zone

wldth.

Piston 1

BULK FLUID Aat PRESSURE Pb

Piston 2

:/'/////////

INTERFACIAL ZONES(AREA A)Pxx/Pzz

dV¡

FIgure 2. Schematlc way Lo measure film tenslon.

The total volume change In Lhe experlment ls dVdx •• and so Eq. (3) can be rearranged Lo

dV, + dV,. dV, =Lw

(1)

where the tlllck thln-1Il111 tellslou IF has been denned 8B

L

lF = J!Pb - P,,(z)]dz.o

(5)

579

Frem Eq. (4) lt follow5 that In an arbItrary reversIble process the thermodynamlcenergy change wll1 be

dU = TdS - PbdV + 1FdA + ¿;Jl,dN,., (6 )

\Vlth the usual definltlon of the Helmholtz and Glbbs free energtesand grand

potentlal. F = U - TS. G = F + PbV ancl n = F - ¿;Jl,N,. we obtaln Ihe rela-,tlons

and derive the Glbbs-Duhem equatlon

o = SdT + Ad1F + I;N¡dJl¡

(7)

(8)

Ir we define a hypothetlcal system In whlch the component densltles bave thebulk values of phase A every\'/here and subtract It5 Glbbs-Duhem from Eq. (8) weobtaln far the lsothermal sltuatlon

o = d1F + ¿;r,d¡l¡ •

where surface excess denslty of component 1 ls

L

r¡= Jlp,(z) - p,A]dz.o

(9)

(10)

\Ve can divIde f. tnto the suro of f.SF. the surface excess denslty assoclated

wlth film B, and r¡AB, the surface excess denslty assoclated wlth the fluid-fluIdInterface. In particular. define

580

h(-) L

r,SF = J [p,(z) - p,A¡dz , r,All = J [p,(z) - p,Ajdzo h(_}

(11)

h(-) 15 the value of z Just !JeJow the lnterfaclal zone between flulds A and [3 andas mentloll('d befare lhe uppcr llmlt L could be replaccd by h(+) and /.'\.D-whefe

t.w 15 the wldth of the lnterfaclal zone.

SlmIlarl:-. we define the ~olld-ftuld tenslon and fluId-fluId tenslons as

and

h(-)

ISF = J [1',,(111-1) - I',,(z)]dzo

L

IAB = J [Pb - I',,(z)]dz.hH

( 12)

( 13)

where I\z(h(-) 15 the value of the pressure In Lile lsotroplc reglons Just below and

just aboye lile fluId-fluId Interface. Ir 15 related to these quantltles by the

expresslon

.As h(-) Inereases P zz(h(-) approaches Pb and as expected 110' approaches lhe 311m

oC lhe sol1d-fluld B and fluid A-fluId B lnterCaclal tenslons.

A result that \\.e shall llnd use fui In the next sectloIl can be obtalned b.••.

lntegratlng Eq. (2) to find

, ,

\\'hlch can be uscd to show t..hat

h(-)

du.•••.B J P1(z')--, -(z')dzl

dz(15)

581

and

h(-I hH du

'SF = J [P,,(Z) - ~ J p,(z')~dZ' - P ,,(z)ldxoo ¡ o dz

Theory of Spreading

(le)

(17)

The analysls In tbls secUan wlll be glven tor a Newtonlan, lncompresstble,one-component. ltQuld-lIke fllm B In the presence oC ao Immlsclble gas A. Thetreatment draws frOID the earller work oC Hlgglns and Scrlven.20]

Conslder ftrst the case oC a translatlonally symmetrlc ftow In the x-dlrectloDThe momentum balance equatlon In the ftlm 15

p [ ~: + V°'VV ] + 'V0p - 'V0' + P'V°uw - pf = O (18)

v 15 tbe fluid veloclty (havlng components v and v In the x and z-dlrectlons fortranslatlonally syrnmetrlc tlo"" In the x-dlrectlon). pf 15 ao external body force(gravltatlonal or centrlfugal). Tbe stress tensor has beeo decomposedloto thevlscous stress tensor T arlslng froro Intermolecular lnteractlons. Outslde at tbelntertaclal zanes the pressure tensor P ls 1sotroplc whereas the vlscaus stress ten-sor 1snoto The condltlon at lncampresslb1l1tyat film B Is

au BY'V°v = - + - = 00ax az (IQ)

\Ve slmplll"y the momentum balance equatlon under the tollowlng assump-tlans: (1) translent and nonllnear veloclty terms (av/at and v'\7v) are negllglble.

(2) the shear component at T dom1nates (Txz;»Txx.TuJ. (3) 8u/az > > étv/8x, alld(4) aTu/aZ » aTu/aXo

SB2

FLUID A

FILM Bh(x,t)

FIgure 3. Baste quantltles In translatlonal1y syrnmetrlc thlo-film ftow.

Under these assumptlons Eq. (18) becomes wlth components

\l'P - \l" + P\luw - pf = O, (20)

wlth components

8P xx OTxz.--- ---pr =0ox az x(21 )

f)p u; duw-- + p-- - pr, = O,az dz(22)

We lntegrate EQ. (20) ayer dz froID tlle salId to a posltlon h just under the ftuld-fluId Interface (denoted h(-) In tlle prevlous sectlon) Lo obtaln

a h h

-a Jp,,(x,z)dz - h,P,,(x,h) - ',,(x,h) + ',,(x,O) + J pr,dz = O (23)x o o

where hx = ah/fu. Equatton (21) 15 lntegrated frOID z to h wlth tlle result

h du h

P zz(X,Z) - Pu(x,h) - J p-+ctz' - J pez' dz' = O.:r. dz z

583

(24)

A Jump boundary condltlon across the f1.uld-ftuld Interface can be derlved bylntegratlng Eq. (18) oyer the small volume .ó.n In Figure --1and lettlng the volumego to zero by pll5Slng to the IImlt £>0-00 In the figure r+(O) (rjO») Indlcates posl-tIan on the surCace In the fluid Just outstde (1nslde) the nuld-nuld Inter-

S+óSfr-teldSS

Figure 4. Volume.ó.n used to define a force balance across the fluid A - film BInterface.

facIal zaDe. Lettlng f1 and t denote unlt vectors normal and tangent to the Inter-face. we obtalo

18+"'8

11m { J 'V°IP - TI dV } = 11m J fio¡p(+J - ,l+J¡ r +(O)dO60 ....•0 68 w 6.8_0 e

8+.c:..8 r-+-<8)

- J fio[PC-J - ,l+J]r(O)dO + J ¡(O)o[P - Tldr8 r'¡8)

584

,,(e+t>1!) }

- J ,(O+~O)' IP - TldrL.(8+Q.8)

(25)

As T varles slowly compared LOP across the Interface, we neglect T In the termsIntegrated over dr. Moreover, slnce the Interfaclal curvature ls small P has onlydiagonal components, and so i.p = PTt where PT 15the component oC P In the

plane tangent to the Interface. Thus, the rlght hand slde of EQ. (19) can beexpressed In the forro

f+<,)

8 J .--1 (PN - t'T)drl.ao He)

where It has beeo Doted that the fluid-fluId tenslon 15 glven by

r +f.O)

'AR = J (PN - PT)dr.,-<o

From dlfferentlal geometry we know that

8,(0) = 2Hfi.r(0)80

(26)

(27)

(28)

whefe !lIs the mean curvature oC the translatlonally syrnmetrlc surCace descrlbed

by r(O)dO and

(2Q)

where Vs ls tbe surCace gradlent operatof.

The quantlty f+(O) - rjO) 15equal Lo the wldth oC the fluId-fluid Interface. Incontlnuum mechanlcs thls wldth Is set eQual to zero In whlch case lntegratlon of

EQ. (18) over .6.0, dlvlslon of the result by .6. f} and passage te the UrnIt .6.(} = O

S 85

ylelds the boundary condltlon20].

(30)

From the derlvatlon we have just glven to obtato Eq. (30), lt follows that theboundary condltlon 15ao approxlmatlon valld LOterms of ftrst arder In lnterfaclalwldth. The re asan for glvlng the aboye detalls Is to Indlcate ao area oC potentlalraUure of the current theory as the Interfaclal zone becomes appreclable com-pared LOthe film thlckness.

Phase A 15a gas whlch \Ve assume ls dllute enough to allow neglect oC vlscousstresses and the varlatlon oC the pressure pA In the base. For the translattonallysyrnmetrlc now consldered here

fi=-hxi + k(1+h;)'/'

t =i+b,k

(1 + hx')'/'(31 )

ane! 'V,= i( 1+ h;t'!' %x so that

_ / O'1AB~ ~ = t(l+h ')-' , __v s IAn x 8x (32)

where hx= oh/m ane! hxx= o'h/ox'. The components or Eq. (30) are tben

A hxP,,(x.h) - P - T,,(k.h) + hxT,,(x.h)+ 2/hAB +(1+h

x2)1/1!

=0 (33)

- (l-h;)T,,(k,h) + hxIT,,(x,h)- T,,(x,h)]+ (1+h;)l/' O'1AB = O, (34)OX

where PA 15 the (constant) pressure In the gas phase.

Conslstent wlth the approxlmatlons leadlng to the reductton oC EQ. (18) toEqs. (21) and (22). these boundary condltlons slmpllfy to

586

and

( h 2) ( ) ( 2)1/2 fhlill- 1 - x TXZ x,h + l+hX -- = O,aX

=0 (35)

(36)

Ir We now Integrate Eq. (24) over z trom O to h. dlfTerentlate wlth respect tox, subtract Eq. (23) from the resultlng equatlon trom, note that Pxx(x,h)

P zz(x,h). and ellrnlnate P zz (x,h) uslng Eq. (35). we obtato the equatton

h a ah

T,,(x,h) - T,,(X,O) + f pf,dz + -;¡::- + -a f pf,zdz ,o VA X o

where

h h du'1SF = fIP,,(x,z) - fp-i-dZ' - P,,(x,z)ldz,

o z dz

conslstent wlth the deftnltlon glven tar the sol1d-ftlm tenslon at Eq. (17).

(37)

(38)

Equatlon (38) can be used to ftnd a relatlonshlp between the veloclty fteld andthe film thlckness. We seek such a relatlonshlp tar the case oC small slopes. I.e.,hx« 1. Thus. we can use a lubrlcatloD-type approxlmatlon to the veloclty

profiJe. namely.

(30)

The no slip boundary condltlon. u(x,z = O) = O. ylelds Fo = O. The velocltycomponent v can be determlned Crom u through the lncompresslblllty condltlon,

Eq. (¡O).

587

For the approxlmatlon consldered here TXY = Tla8u/8z, where TIa Is the vIscos-lty of tbe ftlm B (we assume that the vlscostty ts constant everywhere In themm). Thus, T,,(X,O) = T/aF" From Eq. (36) It Collowsthat

T,,(x,h) = (40)

Substltutlon oC TlaF, Cor T,,(X,O) In Eq. (37) leads to

1 {8"YSF 8[F¡ = - -- + h- hXTXy(x,h)+ 21hAB +TIa ex ax

h a h 1+ Tu(x,b) + J pCxdz+ fuJ pC,zdz ,o o

whlch when comblned wlth Eq. (40) determines F¡ and F.

(41)

Te obtato closure of the fluid mechanlcal theory of film spreadlng, we notethat the lmmlsclb1l1ty oC ftulds A and B lropases to the contlnulty equatlon

where

ah a&t + fu Q = o,

h

Q = JUdz.o

(42)

(43)

Q 15 the voJumetrlc ftow rate per unlt film wldth. Wlth the lubrlcatlon approxl-h

matlon JUdz = (h'/2)F¡ + (hJ/3)F •. Solvlng Eqs .. (40) and (41), we obtaln theo

flnal worklng equatlon for the spreadlng oC a IIquld-l1ke ftlm In the presenee oC adllute gas:

S88

h3jl O')SF a I h I a hQ = =-- -h -0- + "-. 2h')AB + - J pr,dz + --;¡::- J pr,zdz

3/ls h v, h o h VA o

a [ h, O')AB ] 3+ - -------- +OX (1+h,')1/2 OX 2h

(l+h,')'/2

(l-h,'1(44)

Conslstentiy wlth the small slope approxlmatlon the two terms In the curlybraCket5 or Eq. ('13) Involvlng O')AB/Ox can be replaced by the single term

(3/2b)o')AJ3/Ox. Moreover. Ir the gas Oow rate were sumclently hlgh to Impart a

constant shear Tx~ stress aL the film surface. then Eq. (43) would be modlfted Intbe small slope approxlmatlon merely by replaclng the terms Involvlng B

'I\B/8x

by (3/2b) IO')AJ3/Ox+ T~I.

Ir tbe external rorce 15gravlty (r = - g ) and Ir tbe Ould-Ould Interraclal ten-

aloo 1AB In constant. then tbe film ftow rate takes the slrnple from obtalned orlgl-nal1y by Teletzke etaI.1l-131:

b3

{ I O')5F a () }Q = -- --- + - 2H').n - pg - pg b31'B h ah &x J"U} x t X(45)

The pbyslCS oC these formulas for tbe film volumetrlc flow rate 15 c1ear. Gra-dlents In tbe film tenslon 1SF curvature h. and fluid-fluid tenslon 'Aa can drlveflow. So cah the external rorces ex and ff.' The resistan ce te the ftow. 3PB/b3,lncreases as the Inverse cube oC film thlckness as the f1Jmthlns. The ftow rate 15equal to the ratio oC the sum oC the drlvlng Corce to the ftow reslstance.

A drop spreadlng on a smooth surface 15more I1kely to have cyl1ndrlcal ftowsymmetry than one-dlmenslonal translatlonal syrnmetry. In th15 case h 15a func-tlOD of radIal postlon r and time t. The contlnulty equatloD Cor film thlckness 15

hah la!-+ -- urdz.8t r 8r o

(46)

where u 15 the radlal component oC velocUy. The components oC the approxlmate

589momentum equatlon analogou5 to Eq5.. (21) and (22) are

8P rr BTrz------pf =08r 8z '

8P zz du.•...-- + p-- - pf,= O.8z dz

(.17)

(48)

Agaln uslng the lubrlcatlon approxlmatlon. Eq. (.H) wlth Fi = F¡(r,h). and retrac-lng the steps In the derlvatlon tor the translatlonally syrnmetrlc case, we ftnd foraxlally syrnmetrlc flow and constant IAB

8h 1 8 ¡rh3 [ 1 8'51" 8- = -- -- - --- - -(2H,,,,)8t r 8r 31'e h 8r 8r .

h 8 h 11- 2...Jpf,dz - 2...-J pf,zdzh o h 8r o

(49)

the result presented earller by Teletzke el al. The mean curvature for thls casecan be expressed as

h"2/l= ----+(1+h,)'/2

h,r(l+h,2)1/2 • (50)

where hr = ahlar and hrr= B2h/8r2. Ir IAJj 15 not constant the small slopeapproxlmatlon 15obtalned by addlng In the square brackets In Eq. ('¡Q) the term

(3/2h)81AB/8r; Ir also the gas Imparts a constant shear Tx~ add LO thls terrn(3/2h)Tx~'

In the next sectlon we apply Eq. (49) Lo the spreadlng oC water drops on ahorIzontal glass surface (fr = O • rz= - g).

590

The Spreading of Water Drops on Glass

In lOSOMarmur and Lelah publ1shed sorne fasclnatlng studles oC smaJl drops(about 0.01 cro3 In volume) spreadlng 00 cleaned glass slldes In a1r partlallysaturated wlth water vapor211.. The results aL the time seemed counter-Intultlve.They found that water drops spread raster on sUdes havlng smaller surCaceareas.

The etfect was pronounced even though the visIble edges oC the drops were farfrom the edges oC the sUde (see Flg. 5). Also when a drop was placed near thecomer oC a glass sUde lt spread raster In the dlrectlon oC the corner thao It dld

away from the cornero

Figure 5. Spreadlng are a versus spreadlng tIme oC a water drop (0.011 mivolurne) placed on a g]ass sl1de. Marmur and Lelah's obser-

vatlons (Rer. 21).

\Vlth the ald or the mlcrofluld mechanlcal theory developed In the prevloussectlons Teletzke etaI.11-13] were able to explaln the observatlons or Marmur and

Lelah. '-Ve report thelr analysls In thls sectlon.

S 91

The lnputs oC the theory are the surface tenslon 1Lv and v\scostty '1w oC water(1AB and 11B oC the theory) and the solld ftlm tenslon IsF' ExperImental valuesare avallable for "'{LVand 1Jw' However. In typlcal experlments 00 thln ftlms Oile

determines the entlre film tenslon IF • not ISF separately. In tact, In the best

experlments avallable for water films 00 glass, the chernlcal poetntlal (deducedfrom the water vapor pressure) and the flIm thlckness h are the measured quantl-tles. Such data were summarlzed by phase In an artlcle by Pashley22] appearlngserendtpltously a few pages ahead oC the Marmur and Lelah arUcle. The connec-tlon between chemlcal potentlal and film tenslon '"YF 15 glven by Eq. (g). Aslde

from the very thln lnterfaclal zones the denslty Pw of the water film can be well

approxlmated by the bulk value Pw of water at the temperature or the experI-

mento The bulk densltles of the water vapor and alr (lr present) In the gas phaseare negl1g1ble In comparlson w1th Pw• Thus. the surface excess denslty. EQ. (10) ls

well approxlmated as r = Pwh and Eq. (g) becomes

Deryagln has lntroduced the concept or a d1sjolnlng pressure TIsemlconfined ftlms (fiulds on a solld substrate) of lnterest here231. Theexpresslon for th1s dIsjolnlng pressure ls

(51)

for thedellnlng

(52)

where J.l ls actual chemlcal potentlal Of the film and Pw Is a reference bulk den-

slty. The general G1bbs lsotherm eQuatlon wrltten In terms of TI Is

rdlF= -dI1 "" hdI1 (53)

Pw

The ratlo r/Pw has the d1menslons of length and Is deflned to be the "film thlck-ness'. h even when the thln-fllm 1s not thlck enough to define separate 50lld-fluldand fluId-fluId Interfaces. In that case h 15 slmply a stand-In for the adsorptlondenslty r. In the present case of thlck thln-films r/Pw 15meanlngfully 1dentlfled

wlth the film thlckness .•It is worth noting as an aside that the disjoining pressure defined here for

semiconfined fluid films is qualitatively diJJeren~ from that defined for confined

S92

fluid films. In the case of conftned tllrns the dlsJolnlng pressure 15lndeed a pres-sure deflned as the dltrerence between the normal pressure exerted LOconfine the

film and the pressure oC a bulk fluid phase In chemlcal equ1l1blrum wlth thecontlned fIlm. The dlsjolnlng pressure oC a semlcanflned film 15 rnerely a chemlcalpotenLlal converted loto unlts oC pressure by multlpllcatlon by a reference den-

slty. Pashley has presented experimental results as f1 versus h, wbere tbe quan-tlty h 15 measured by apLlca1 methods and n ls computed fram n =PwkT1n(P /PSV). where P ls the measured vapor pressure oC the ftlm and pSV 15 the

vapor pressure oC saturated IIquld aL temperature T.

Let liS return Lo the problem aL hand. \Ve need for the theory the Quant1ty'lSFt bUL we obtaln experImentally the quantlty IF' The two are related throughEq. \14). In prIncipIe, we need a molecular theory to proceed further. Teletzkeel al. have Investlgated the van der Waals theory of thlck-thln- films and havefound that tlle fluId-Huid tensIon oC the one component Ilquld-lIke ftlm In equ1l1-brlum wlth Its vapor becomes Insens1tlve to fUm thlckness by the tIme the film 15a few ftuld-nuld Interfaces thlck. Moreover. we note that the dlfferencePe - P tt(h(-») In Eq. (14) represents the normal pressure drop over the wldth oC

an Interface and Is therefore negllglble compared wlth the other term5 on the

rlght hand slde of EQ. (14). Thu3. Lhe approxlmatlon d1r1dh = d1SF/dh 13justlfted, and so from EQ.(53) It follows that

d1sF dO--=h-

dh dh(5-1)

•

Slnce Marmur and Lelah's experlments were under qule5cent, Isothermal con-dltlons, the surCace tenslon 1LV Is expected to be constant. The glass plates werehorizontal and so gr = O. For the symmetrlcally spreadlng drops. then, the

approprlate evolutlon equatIon ls

~ = ~~{ rh3

[_ oTI ~ _ 1LV 0(2H) + pwg~]} (55)ot r or 3ryw oh or or or

Wlth the mean curvature glven by Eq. (51). thls 15 a fourth order partlal

dlfferentlal equatlon.

The boundary condltlons Introduced by Teletzke el al. are

ahor

03h--=O,r=O,or3

593

(56)

comlng from syrnmetry at the drap's pole. and

ahor

03h--=O,r=R,or3 (57)

Imposed to conserve the total drop volume. (Although the glass sUdes are rec-tangular thelr perlmeters \Vere approxlmated as cIrcular In the analysls. Thlsapproxlmatlon would ¡ead to qualltatlve error only when the drap has spreadenough te begln to lose cIrcular syrnrnetry fram edge and corner effects.) Thesharpness oC the edges oC the glass slldes greatly lmpede fiow, a fact whIchJustlftes the no Oow approxlmatlon represented by EQ. (58). Further Justlftcatloncan be derlved froID Marmur and Lelah's observatlon that spreadlng In paramnbarrlers on glass slldes was 51millar to spreadlng 00 glass sil des of the slze deftnedby the paraffin barrlers.

As an lnltlal condltlon the 0.01 cm3 drop was assumed to be a sphere capw1th a base area of 0.5 cm3• At the humldlty of the experlments, the d1sjolnlngpressure data summarlzed by phase justlfy the assumpt10n that a 10 angstromwater fUm exIsts on the glass sUde at the tlme the drop 1s deposlted.

Wlth these Inltlal and boundary condltlons Eq. (55) can he sol ved by ftnltedlfference ar ftnlte element technIques. Teletzke el al. used a ftnlte dlfferencemethad. Befare presentlng thelr quantltatlve results 1t Is lnstructlve to examInethe dlsJalnlng pressure behavIor of varlous flulds on glass or quartz.

Contlnuum approxlmatlons, whlch rnlght be expected to be valld far thlckth1n-ftlms, lead to power law predlctlons of the form

O(h) = ~A,/h' , (58)

where A¡ are so-called Hamaker constants. For nonpolar ftulds the Dzyalosh1n-skll. Llfshltz and Pltaevskll (DLP) theory26] predlcts the asymptotlc result TI(h)= A3/h3 for thlck thln-ftlm less than about half a rnlcron thlck. Thls behavIorhas been confirmed for numerous non polar ftulds on sollds27,28), examples belng

594octane 00 quartz. SF ti 00 glasslO] and hel1um on glass91. The DLP theory agrees

qulte weH wllh thlck thlnk-ftlm results, even for fllms oC the arder oC lOOÁ thlck.For Detane on Quartz the experimental result 15 A3 = O.QX 10-21 Joules. Forpure water, a polar fluid, Deryagln has deduced that Il(h) -/\.2/h2 WILh A2 = 2 X 10-12 Newtons for water ftlms thlcker than 1200 Á onquartz. Ir electrolytes are present In the water, the Deryagln-Landau-Verwey-Overbeek (DLVO) double ¡ayer theory predlcts an electrostatlc contrlbutlon oC

the forro TIe'"'-'exp(-Kh). where K 15 the Debye kappa, whIch 15 proportlonal to thesquare root oC lonte concentratlon. Electrostatlc effects were not lmportant In

Marmur and Leiah's experlments slnce they used trlply dlstllled water.

Ir one uses the expresslon n(h) = ~/h2 In the theory one predlcts from Eq.

(56) that spread1ng ls unaffected by the slze of the glass sUdes and that spreadlngwould take days as opposed to the observed perlod of minutes. It was thusextremely slgnlftcant that phase amassed evldence that the dlsjolnlng pressure ofwater on Quartz was very dlffereot from Deryagln's formula. H1s flndlngs are

plotted 10 FIgure 6. the dlsjolnlng pressure goes as A1/h for h less than 800 Áand Is two orders of magn1tude larger than would be expected from extrapoJatlonof Deryagln's formula, There exlsts no theory whlch expla1ns Jarge magnltude orthe h-I dependence of n. One can appeal 15 merely an Intellectual comfortabJe.

the agreement to be dlscussed below between our spreadlng theory aod Marmurand Lelah's observatlon 15 thus slgnlftcant In that lt furnlshes Indlrect support forthe forro of n phase that Pashley has deduced from equlllbrlum experlments.

Teletzke el al. used the dlsj01nlng pressure shown 10 Flg. 6 wlth a llnear1nterpolatlon between Its values at 80nm and 120001. The surface tens10n ofwater was glven the value 72 dyn/cm and the deoslty the value 1 gm/cm3

•

The1r predlctloos for Marmur and Lelah's drop spread1ng on a 2-cm sUde.

approxlmated as circular rather than sQuare, are shown ln FIg. 7, A broadenlngfront of secoodary thIckenlng (1n5et) of the 1 nm prlmary film up to 100 001 racesoutward from the cap1llary choke at the drop margln to the captllary arrest atthe edge of the plate, reachlng the edge In about a second (whlch corresponds to

a dHJusIvlty of th1ckenlng of around a sQuare centlmeter per second). Thus a prl-mary flIm condult oC oue mlll10n times greater conductancc Cor drop spreadlng 15establlshed all the way to the edge of the sUde wlLhln a second. Tbe spreadlngmargln oC tbe drop follows along behlnd at one-tenth Lhe raleo The two ratesand assoclated tIme scales come Crom the swltch-over 10 dlsjolnlng pressure

dependeoce at film thlcknesses between 80 and 120 nm.

S 95

EXPERIMENTAL:WATER ON QUARTZ

120-150nm60-80nm

\ EXPERIMENTAL:\(NONPOLAR L1QUID

ON QUARTZ'-....__ .¿TI = A

3/h3

wa:::Jen

N/m2enWa::

7 x 105Cl...

<.::>Z 2 x 10-ZOJenO

FILM THICKNESS

FIgure 6. DlsJolnlng pressure versus flIm thlckness tor water and a nonpolar I1Quldon Quartz.

The effect of sUde slze 15shown 10 Flg. 8. Marmur and Lelah observed the

spreadlng drops from below the glass sUde by meaos oC transmltted l1ght andmeasured as a functlon ef time th~ area wlthln a dark rlng whlch they took LObetbe drop's margln. An opUcal analysls (Berry lG76) predlcts that the radlus of

the dark rlng closely corresponds te the radial dlstance trom the crop's axIs aL

whlch the lnterfaclal proftle has maxlmum curvature. Therefore. to compare wlth

Marmur and Lelah's experlment In FIgure 8 the area plotted 15that wlLhln thepolnt ef maxlmum curvature oC the proftle. In summary: Marmur and Lelahobserved that the area wltbln tbe drop margln lncreases more rapldly on a smallsUde tban on a large one and tbe theoretlcal calculatlon based on separate meas-urements oC egu1J1brlllmdlsjolnlng pressure completely accounts Cor tbls and alsocorrectly predlcts the magnltude oC the spreadlng rateo No parameters wereacUusted to achleve a fit to experlment.

1.00.7RADIUS,cm

OAO

596

7

6 SECONDARY FILM ADVANCE

(f)Z 1000 AO 5a::u:2(f) 4(f)wz~ 3 .4 0.7 1.0U RADIUS, cmIf- BULK SPREADING:2 2--'Le O

Figure 7. Predlcted evolutlon oC ftlm prefiJe oC a water drop 00 a 2-crn gla.ss

sllde (Refs. 11 and 12).

The reason for the slze dependence 15 the secondary film thlckenlng that racesoutward to the edge and In effect sends back news oC the barrler there. Marmurand Lelah found that a paraffin escarpment en the sUde has the same etIect asthe edge. Because Lhe raclng 15 dlffuslve. and the conOguratlon radlally syrn-metrlc. lts speed falls off \\11th dIstance. \Vhen the sllde Is S cm or more across,the edge barrler 15too far (rOIDthe drop margln tor Lhe dlffuslng secondary fllm

to transmlt any appreclable thlckness effect back to the drop.

The exIsten ce oC Lhe secondary f1Im 15 ao Important predlctlon oC the theory.Hardy long ago deduced the presence oC such f1.lms. whlch. although InvIsible totlle eye. caused dust partlcles LOmove on a salid surCace ahead oC the vIsIble panoC a spreadlng film. An opt1cal experlment could be deslgned to observe theappearance oC the seconctary film on a vertIcal glass sil de wllen Its lower edge lsdlpped In a pool oC water. tlle theory could be used to predlct tlle rate oC secon-dary film spreadlng as an ald lO tlle deslgn oC tlle experlment.

597

2.0

C\JEo~ 1.0eQ)••<l:

2

1.5 cm Glass Slide

8.0 cm Glass Slide

Ti me. seconds10

Figure 8. Predlcted area oC water drop versus tlme lar 1.5 cm and 8.0 cm glassslldes (Rers. 11 and 12).

It 18 Interestlng to conslder the role played by the dependence oC the dlsJoln-Ing pressure on film Lhlckness In determlnlng the spreadlng bebavlor oC a drop.Thls can be antlclpated by analyzlng the contrlbutlons to the volumetrlc ftow

rate oC the film. For drops oC slze consldered bere gravlty 15oC negl1glble effectand the dlsJolnlng pressure domlnates the other drlvlng Corces tar lnduclng ftow.The resistan ce Lo flow goes as h-3. Thus. Ir TI goes as h-LJ• Its contrlbutlon Lo thevolumetrlc ftaw rate 15proportlonal Lo h-II+2• For Donpolar flulds, v = 3, and so

wlth film thlckenlng the drlvlng Corce tar ftow becomes smaller. Nonpolar nuldwould spread more rapldly on a larger glass sllde (but much more slowly thanwater owlng to the small magnltude ot the dlsjolnlng pressure). It n goes as h-2,the flow rate trom the dlsjolnlng pressure 15 lndependent ot ftlm thlckness andtherefore ot slze oC the sUde. For water n goes as h-1• the contrlbutlon to flowrate as h and so a smaller glass sUde leads to a taster spreadlng rateo

598

References

I. Pandlt. R .• Schlck. M. and Wortls. M •. Phys. Rev. IJ 26. 5112 (1082).

2. Su1l1van, D. E. and Tela da gama, In "FluId Interfaclal Phenomena," edlt-

ed by C. A. Croxton. \Vtley, Kew York. 1085.

3. Davls. H. T. I3enner. H. E.o Teletzke. G. F. and Scrlven. L. E.o In"Proceedlngs of <lhe 5th Internatlonal Symposlum on Surfactants In

Solutlon:' edlted by K. L. ~flttaJ. 1086.

-1. Varea. C., Robledo, A.o and MartIna, E.o "\Vettlng Heglmes al a Seml-permeable Membrane," (lO be publlshed. 19S0).

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