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EPJmanuscriptNo. (willbeinsertedbytheeditor) Evaporation induced flow inside circular wells 9 Yu.Yu.Tarasevich,I.V.Vodolazskaya,O.P.Isakova,andM.S.AbdelLatif 0 0 AstrakhanStateUniversity,20aTatishchevSt.,Astrakhan,414056,Russia,e-mail:[email protected] 2 n a Abstract. Flowfieldandheightaveragedradialvelocityinsideadropletevaporatinginanopencircularwellwere J calculatedfordifferentmodesofliquidevaporation. 7 PACS. 47.55.nb Capillaryandthermocapillaryflows ] n y d 1 Introduction - u l Wideusedtechniqueofpatterningsurfaceswithsolidparticles f . utilizes the evaporationof colloidaldropletsfroma substrate. s c Inparticular,evaporationofliquidsamplesisakeyproblemin i thedevelopmentofmicroarraytechnology(includinglabs-on-a s y chip),especiallyinthecaseofopenreactors[1]. h A plenty of works are devoted to the evaporation of the p sessile droplets[2–5].Oneassumesthata droplethasashape [ like a spherical cap. Nevertheless, investigation of the drops withmorecomplexgeometryisofinterest.Thuswhenahigh 1 v concentratedcolloidaldropletevaporatesfromasubstrate,the 8 solute particles in solution will form a ring like deposit wall 7 nearthe edgeof the drop(Fig.1). One can considerthiscase 7 as evaporationof a liquid drop inside open circular well with 0 verticalwallsformedbygel. . 1 In the paper [1] the flow field inside a liquid sample in 0 verythincircularwellswasmeasured.Theconfidencebetween 9 measuredvelocity[1]andtheestimationsobtainedwiththeory 0 ofringformation[2]isreasonable. : v Inthispaper,therelationshipbetweenhydrodynamicalflow i insidealiquidsampleevaporatinginopencircularwellandthe X modeofevaporationisexamined. r We foundanalyticalexpressionsof heightaveragedradial a velocity for differentmodes of evaporation.Velocity field in- Fig.1.Timeevolutionofthedepositphasegrowth[6] side the circular well was foundfor the particularcase of flat air–liquidinterface. Moreover,wewillsupposethatevaporationisasteadystate 2 Velocity field inside circular well process. This assumption is valid e.g. for evaporation of the drops of aqueous solutions under room temperature and nor- malatmospherepressure,i.e.forthetypicalexperimentalcon- 2.1 Heightaveragedradialvelocity ditions.Apexdynamicsisratherslow[1] Wewillconsideroneparticularbutinterestingcase,becauseof its practical applications, when a liquid inside a circular well may be described as a thin film. So, in paper [1] in circular H(t)=(7.38µm 6.13µm) (0.042µm/s)t. − − wells with a radius of r = 100 µm and a depth of h = f f 6 µm were investigated. Approximately the same ratio depth to radius is typical for the drops of biological fluids used for Inthecasesofmedicaltests,thetypicalvelocityofdropapex medicaltests[7,8]. isapproximately1mm/h. 2 Yu.Yu.Tarasevichetal.:Evaporationinducedflowinsidecircularwells Massconservationgivesthefollowingequationforheight whereL(x,t)= L(0,0)+4j0(1−m−e−m)t 1 x2 . averagedradialvelocity[2] m(1 e−m) − (cid:18) − (cid:19) Anotherevaporativefluxfunctionswasprop(cid:0)osedin[(cid:1)10] v (r,t) = r h i 1 r ∂h 2 ∂h j(x,τ)= j0 , (8) j(r,t) 1+ +ρ rdr, (1) K+L(x,τ) − ρrh  s ∂r ∂t Z (cid:18) (cid:19) 0 wheretheconstantK measuresthedegreeofnon-equilibrium   whereh=h(r,t)isthedropshape,ρisthedensityofsolution, at the evaporating interface and is related to material proper- j(r,t) is the evaporationrate defined as the evaporativemass ties.K 0correspondstoahighlyvolatiledroplet.Thelimit lossperunitsurfaceareaperunittime. K →correspondsto a nonvolatiledroplet. Analytical ex- Consideringthe thindropletsonly(h(r,t) rf)we will pres→sio∞n for height averaged velocity inside circular well can ≪ neglecteverywhereh2(0,t)and ∂h 2 incomparewith1.We beobtainedifvaporfluxisdescribedbyEq.8 ∂r willutilizeapproximativeequationforair–liquidinterface (cid:0) (cid:1) j h(r,t)=h(0,t) 1 r 2 , (2) hvr(x,t)i= 2x(h +L(00,t)(1 x2))× 0 −(cid:18)rf(cid:19) ! 1 L−(0,t)x2 ln 1 whererf isradiusofcircularwell. ×(cid:18)L(0,t) (cid:18) − K+h0+L(0,t)(cid:19)− Sincethedropisthinandthecontactangleissmall,wewill usetheexpression x2 1− x22 (L(0,t)−2(K+h0)) , (9) j (cid:16) (cid:17)(K+h )2  j(r)= 0 (3) 0 2  1 r r −(cid:16)rf(cid:17) whereL(0,t)=2(K+h0)+(L0−2(K+h0))exp (K+j0ht0)2 . fortheevaporationrate,whichhasareciprocalsquare-rootdi- Deeganetal.[2]demonstratedexperimentally,(cid:16)thatifeva(cid:17)p- vergencenearthe contactline[6].Thisexpressioncanbe de- oration is greatest at the center of the droplet, a uniform dis- rivedfromLaplaceequation(see[2,6]fordetails).Thisfunc- tributionofcolloidalparticlesremainedonthesubstrate.Sim- tionalformfor vaporflux is widely used, in particular,it was ulationsby [3] confirmed,that as fluid is lost from the center utilizedinpaper[1]. ofthedroplet,aninwardflowdevelopstoreplenishtheevapo- Some quantities of interest can be expressed analytically ratedfluid.TheevaporativefluxfunctionwasproposedbyFis- exploiting(3).Velocityofdropapexdecreaseis cher[3]tomimicevaporationthatisconcentratedatthecenter dh(0,t) 4j ofthedroplet = 0. (4) j dt − ρ j(x,t)= 0 e Ax2, (10) − L(0,t) Heightaveragedradialvelocityis j 1 √1 x2 2x2+x4 whereAisanadjustableconstant. 0 v (x,t) = − − − , (5) We derived the analytical expression for height averaged h r i − ρx(h +L(0,t)(1 x2)) (cid:0) 0 − (cid:1) velocityforevaporationfunctiongivenbyEq.10 wherex = r ,L = h ,h = hf.h istheheightofvertical rf rf 0 rf f wallofthewell. vr(x,t) = h i ExploitingofEq.3leadstoasingularityforbothvaporflux j 1 e Ax2 + 1 e A x2 x2 2 andgradientoftheheightaveragedvelocityofliquidinsidea 0 − − − − − , (11) dropattheedgeofdroplet.Asmoothingfunctionmaybeused (cid:16) 2Ax(h0+(cid:0)L(x,t))L(cid:1)(0,(cid:0)t) (cid:1)(cid:17) toeliminatephysicallysenselessdivergency[9] where j 1 exp( m√1 x2) j(x,t)= 0 − − − , (6) r √1 x2 1 exp( m) f − − − L(x,t)= wheremisanadjustableconstant.Inthiscase,heightaveraged j (1 e A)t radialvelocitycanbewrittenanalytically 1 x2 (h +L )2 4 0 − − h . 0 0 0 − r − A − ! j0 (cid:0) (cid:1) v (x,t) = h r i x(h0+L(x,t))m(1−e−m)× Fig.2demonstratesourcalculationsforalldescribedabove evaporativefunctions. In all figures, the plots correspondsto: m 1 x2+e−m√1−x2 m e−m initialstage(air–liquidinterfaceisconvex);air–liquidinterface − − − − (cid:18) p x2 isflat; air–liquidinterfaceis concavewith thesame curvature 2 1 m e−m 1 x2 , (7) asat theinitial stage;90% oftime, whenair–liquidinterface − − − 2 touchthebottomofthewell. (cid:18) (cid:19) (cid:19) (cid:0) (cid:1) Yu.Yu.Tarasevichetal.:Evaporationinducedflowinsidecircularwells 3 a) b) c) d) Fig.2.Heightaveragedradialvelocityfordifferentmodesofevaporation:a)(3);b)(6)ifm = 5;c)(8);d)(10).Foradditionalexplanations seethemaintext. 2.2 Velocityfield wherer isradiusofthecircularwell. f Moreover,thereisphysicallyobviousrelation VelocityfieldscalculatedforsessiledropletsfromLaplaceequa- ∂u tion[11]andfromNavier–Stokesequations[3]areverysimi- =0. (13) ∂r lar.Wehopethatidealliquidisreasonableassumptionforour (cid:12)r=0 (cid:12) objectofinterest. Thebottomofthecircula(cid:12)rwellisimpermeable,hence Cylindrical coordinates (r,ϕ,z) will be used, as they are (cid:12) ∂u mostnaturalforthegeometryofinterest.Theoriginischosen =0. (14) inthe centerofthecircularwellonthe substrate.Thecoordi- ∂z (cid:12)z=0 natezisnormaltothesubstrate,andthebottomofthecircular (cid:12) The mass flux inside the dro(cid:12)plet near the air–liquid interface wellisdescribedbyz =0,withzbeingpositiveonthedroplet (cid:12) f(r)isconnectedwithvaporflux side of the space. The coordinates (r,ϕ) are the polar radius and the angle, respectively. Due to the axial symmetry of the ∂u =f(r). (15) problemandourchoiceofthecoordinates,noquantitydepends ∂z (cid:12)freesurface ontheangleϕ,inparticularu=u(r,z). (cid:12) Laplaceequationin(cid:12)cylindricalcoordinatesfor the axially Themassfluxattheverticalwallofcircularwellisabsent, (cid:12) symmetriccaseiswrittenas hence ∂u 1 ∂ ∂u ∂2u =0, (12) r + =0. (16) ∂r(cid:12)r=rf r∂r (cid:18) ∂r(cid:19) ∂z2 (cid:12) (cid:12) (cid:12) 4 Yu.Yu.Tarasevichetal.:Evaporationinducedflowinsidecircularwells Boundaryproblem(12),(14),forequation(16) We computed velocity fields for all evaporation laws de- scribedinSec.2.1.Theresultsshowthatcurrentinsidecircular ∞ µ(n1) µ(n1) wellishorizontalexcludingrathernarrowregionnearthewall u(r,z)= a cosh z J r , (17) n r 0 r and in the center of the well for the case of practical interest f ! f ! nX=1 (h/r 0.1). ∼ whereJ (r) isthe Bessel functionofthe firstkind,orderm, Many authors supposed that evaporationinduced flow in- m µ(1) are the real zeros of Bessel J (r) function: J (r) = 0. sidethe dropletsisindependentofits height.Oursimulations n 1 1 confirmed validity of this very wide used approach for thin Notethatcondition(13)issatisfiedautomatically. droplets. Takingintoaccountrelationbetweenvelocityandpotential v= gradu,wecanfindradialcomponentofvelocity − TheauthorsaregratefultotheRussianFoundationforBasicResearch ∞ µ(n1) µ(n1) µ(n1) forfundingthisworkunderGrantNo.06-02-16027-a. v (r,z) = a cosh z J r . r n 1 h i rf ! rf rf ! n=1 X Height averaged radial velocity, i.e. the same velocity ex- References aminedin[1],canbewrittenas 1. B.Rieger,L.R.vandenDoel,L.J.vanVliet,PhysicalReviewE hf 68,036312(2003). 1 v (r) = v (r,z)dz = 2. R.D.Deegan,O.Bakajin,T.F.Dupont,G.Huber,S.R.Nagel, r r h i hf T.A.Witten,Phys.Rev.E62,756(2000) Z 0 3. B.J.Fischer,Langmuir18,60(2002) 1 ∞ µ(n1) µ(n1) 4. R.Mollaret,K.Sefiane,J.R.E.Christy,D.Veyret,Chem.Eng. ansinh hf J1 r . Res.Design82(A4),471(2004) h r r f f ! f ! n=1 5. H.Hu,R. G. Larson,Langmuir21,3963(2005) X 6. Y.O.Popov,Phys.Rev.E71,036313(2005) Coefficients a can be obtained from (15). For simplicity n 7. L. V. Savina, Crystaloscopical structures of blood serum of wewillassumethatair–liquidinterfaceisflat. healthyandseekhuman(SovyetskayaKuban,Krasnodar,1999) 8. V.N.Shabalin,S.N.Shatokhina,MorphologyofBiologicalFlu- ∂u = ids(Khrizostom,Moscow,2001) ∂z 9. M. Cachile, O. Beńichou, A. M. Cazabat, Langmuir 18, 7985 (cid:12)z=hf (cid:12) (2002) ∞(cid:12)(cid:12) a µ(n1) sinh µ(n1)h J µ(n1)r =f(r). (18) 10. D.M.Anderson,D.M.Davis,PhysicsofFluids7,248(1995) n f 0 rf rf ! rf ! 11. Y.Y.Tarasevich,Phys.Rev.E71,027301(2005) n=1 X Weintroducenotation f(r)rdr =F(r). Z ∞ µ(n1) F(r) c J r = (19) n 1 r r f ! n=1 X isaFourier–BesselseriesofF(r)/r,where µ(1) n c =a sinh h . (20) n n f r f ! Hence, a = n 2 rf µ(1) n J r F(r)dr. (21) rf2sinh µr(nf1)hf J02 µ(n1) Z0 1 rf ! (cid:16) (cid:17) (cid:16) (cid:17) We need to determine f(r) to find the unknown coefficients. Massconservationleadstotherelation ∂u ∂L(r,t) J(r,t) = . (22) ∂z − ∂t − ρ (cid:12)z=hf (cid:12) (cid:12) (cid:12) Yu.Yu.Tarasevichetal.:Evaporationinducedflowinsidecircularwells 5 1,0 1,0 0,8 0,8 0,6 0,6 hz0 hz0 0,4 0,4 0,2 0,2 0 0,0 0 0,2 0,4 0,6 0,8 1,0 0,0 0,2 0,4 0,6 0,8 1,0 r r 1,0 1,0 0,8 0,8 0,6 0,6 z z h0 h0 0,4 0,4 0,2 0,2 0 0 0 0,2 0,4 0,6 0,8 1,0 0 0,2 0,4 0,6 0,8 1,0 r r 1,0 1,0 0,8 0,8 0,6 0,6 hz0 hz0 0,4 0,4 0,2 0,2 0 0,0 0 0,2 0,4 0,6 0,8 1,0 0,0 0,2 0,4 0,6 0,8 1,0 r r 1,0 1,0 0,8 0,8 0,6 0,6 z z h0 h0 0,4 0,4 0,2 0,2 0 0 0 0,2 0,4 0,6 0,8 1,0 0 0,2 0,4 0,6 0,8 1,0 r r Fig.3.Velocityfieldinsidecircularwellfordifferenevaporativemodes.Theair–liquidinterfaceissupposedtobeflat.Fromtoptobottom(3), (6)withm=5,(8),(10).Leftcolumncorrespondstoratioh /r =0.0618,rightcolumncorrespondstoratioh /r =0.99 f f f f