Evaporation induced flow inside circular wells
Yu. Yu. Tarasevich, I. V. Vodolazskaya, O. P. Isakova, M. S. Abdel Latif
aa r X i v : . [ phy s i c s . f l u - dyn ] J a n EPJ manuscript No. (will be inserted by the editor)
Evaporation induced flow inside circular wells
Yu. Yu. Tarasevich, I. V. Vodolazskaya, O. P. Isakova, and M. S. Abdel Latif
Astrakhan State University, 20a Tatishchev St., Astrakhan, 414056, Russia, e-mail: [email protected]
Abstract.
Flow field and height averaged radial velocity inside a droplet evaporating in an open circular well werecalculated for different modes of liquid evaporation.
PACS.
Wide used technique of patterning surfaces with solid particlesutilizes the evaporation of colloidal droplets from a substrate.In particular, evaporation of liquid samples is a key problem inthe development of microarray technology (including labs-on-achip), especially in the case of open reactors [1].A plenty of works are devoted to the evaporation of thesessile droplets [2–5]. One assumes that a droplet has a shapelike a spherical cap. Nevertheless, investigation of the dropswith more complex geometry is of interest. Thus when a highconcentrated colloidal droplet evaporates from a substrate, thesolute particles in solution will form a ring like deposit wallnear the edge of the drop (Fig. 1). One can consider this caseas evaporation of a liquid drop inside open circular well withvertical walls formed by gel.In the paper [1] the flow field inside a liquid sample invery thin circular wells was measured. The confidence betweenmeasured velocity [1] and the estimations obtained with theoryof ring formation [2] is reasonable.In this paper, the relationship between hydrodynamical flowinside a liquid sample evaporating in open circular well and themode of evaporation is examined.We found analytical expressions of height averaged radialvelocity for different modes of evaporation. Velocity field in-side the circular well was found for the particular case of flatair–liquid interface.
We will consider one particular but interesting case, because ofits practical applications, when a liquid inside a circular wellmay be described as a thin film. So, in paper [1] in circularwells with a radius of r f = 100 µ m and a depth of h f =6 µ m were investigated. Approximately the same ratio depthto radius is typical for the drops of biological fluids used formedical tests [7, 8]. Fig. 1.
Time evolution of the deposit phase growth [6]
Moreover, we will suppose that evaporation is a steady stateprocess. This assumption is valid e.g. for evaporation of thedrops of aqueous solutions under room temperature and nor-mal atmosphere pressure, i.e. for the typical experimental con-ditions. Apex dynamics is rather slow [1] H ( t ) = (7 . µ m − . µ m) − (0 . µ m / s) t. In the cases of medical tests, the typical velocity of drop apexis approximately 1 mm/h.
Yu. Yu. Tarasevich et al.: Evaporation induced flow inside circular wells
Mass conservation gives the following equation for heightaveraged radial velocity [2] h v r ( r, t ) i = − ρrh r Z j ( r, t ) s (cid:18) ∂h∂r (cid:19) + ρ ∂h∂t r dr, (1)where h = h ( r, t ) is the drop shape, ρ is the density of solution, j ( r, t ) is the evaporation rate defined as the evaporative massloss per unit surface area per unit time.Considering the thin droplets only ( h ( r, t ) ≪ r f ) we willneglect everywhere h (0 , t ) and (cid:0) ∂h∂r (cid:1) in compare with 1. Wewill utilize approximative equation for air–liquid interface h ( r, t ) = h (0 , t ) − (cid:18) rr f (cid:19) ! , (2)where r f is radius of circular well.Since the drop is thin and the contact angle is small, we willuse the expression j ( r ) = j r − (cid:16) rr f (cid:17) (3)for the evaporation rate, which has a reciprocal square-root di-vergence near the contact line [6]. This expression can be de-rived from Laplace equation (see [2, 6] for details). This func-tional form for vapor flux is widely used, in particular, it wasutilized in paper [1].Some quantities of interest can be expressed analyticallyexploiting (3). Velocity of drop apex decrease is dh (0 , t ) dt = − j ρ . (4)Height averaged radial velocity is h v r ( x, t ) i = − j (cid:0) − √ − x − x + x (cid:1) ρx ( h + L (0 , t )(1 − x )) , (5)where x = rr f , L = hr f , h = h f r f . h f is the height of verticalwall of the well.Exploiting of Eq. 3 leads to a singularity for both vapor fluxand gradient of the height averaged velocity of liquid inside adrop at the edge of droplet. A smoothing function may be usedto eliminate physically senseless divergency [9] j ( x, t ) = j r f √ − x − exp( − m √ − x )1 − exp( − m ) , (6)where m is an adjustable constant. In this case, height averagedradial velocity can be written analytically h v r ( x, t ) i = j x ( h + L ( x, t )) m (1 − e − m ) × (cid:18) m p − x + e − m √ − x − m − e − m − (cid:0) − m − e − m (cid:1) (cid:18) − x (cid:19) x (cid:19) , (7) where L ( x, t ) = (cid:18) L (0 ,
0) + 4 j ( − m − e − m ) tm (1 − e − m ) (cid:19) (cid:0) − x (cid:1) . Another evaporative flux functions was proposed in [10] j ( x, τ ) = j K + L ( x, τ ) , (8)where the constant K measures the degree of non-equilibriumat the evaporating interface and is related to material proper-ties. K → corresponds to a highly volatile droplet. The limit K → ∞ corresponds to a nonvolatile droplet. Analytical ex-pression for height averaged velocity inside circular well canbe obtained if vapor flux is described by Eq. 8 h v r ( x, t ) i = j x ( h + L (0 , t )(1 − x )) ×× (cid:18) L (0 , t ) ln (cid:18) − L (0 , t ) x K + h + L (0 , t ) (cid:19) − x (cid:16) − x (cid:17) ( L (0 , t ) − K + h ))( K + h ) , (9)where L (0 , t ) = 2( K + h )+( L − K + h )) exp (cid:16) j t ( K + h ) (cid:17) . Deegan et al. [2] demonstrated experimentally, that if evap-oration is greatest at the center of the droplet, a uniform dis-tribution of colloidal particles remained on the substrate. Sim-ulations by [3] confirmed, that as fluid is lost from the centerof the droplet, an inward flow develops to replenish the evapo-rated fluid. The evaporative flux function was proposed by Fis-cher [3] to mimic evaporation that is concentrated at the centerof the droplet j ( x, t ) = j L (0 , t ) e − Ax , (10)where A is an adjustable constant.We derived the analytical expression for height averagedvelocity for evaporation function given by Eq. 10 h v r ( x, t ) i = j (cid:16) − e − Ax + (cid:0) − e − A (cid:1) x (cid:0) x − (cid:1)(cid:17) Ax ( h + L ( x, t )) L (0 , t ) , (11)where L ( x, t ) = (cid:0) − x (cid:1) r ( h + L ) − j (1 − e − A ) tA − h ! . Fig. 2 demonstrates our calculations for all described aboveevaporative functions. In all figures, the plots corresponds to:initial stage (air–liquid interface is convex); air–liquid interfaceis flat; air–liquid interface is concave with the same curvatureas at the initial stage; 90 % of time, when air–liquid interfacetouch the bottom of the well. u. Yu. Tarasevich et al.: Evaporation induced flow inside circular wells 3a) b)c) d)
Fig. 2.
Height averaged radial velocity for different modes of evaporation: a) (3); b) (6) if m = 5 ; c) (8); d) (10). For additional explanationssee the main text. Velocity fields calculated for sessile droplets from Laplace equa-tion [11] and from Navier–Stokes equations [3] are very simi-lar. We hope that ideal liquid is reasonable assumption for ourobject of interest.Cylindrical coordinates ( r, ϕ, z ) will be used, as they aremost natural for the geometry of interest. The origin is chosenin the center of the circular well on the substrate. The coordi-nate z is normal to the substrate, and the bottom of the circularwell is described by z = 0 , with z being positive on the dropletside of the space. The coordinates ( r, ϕ ) are the polar radiusand the angle, respectively. Due to the axial symmetry of theproblem and our choice of the coordinates, no quantity dependson the angle ϕ , in particular u = u ( r, z ) .The mass flux at the vertical wall of circular well is absent,hence ∂u∂r (cid:12)(cid:12)(cid:12)(cid:12) r = r f = 0 , (12) where r f is radius of the circular well.Moreover, there is physically obvious relation ∂u∂r (cid:12)(cid:12)(cid:12)(cid:12) r =0 = 0 . (13)The bottom of the circular well is impermeable, hence ∂u∂z (cid:12)(cid:12)(cid:12)(cid:12) z =0 = 0 . (14)The mass flux inside the droplet near the air–liquid interface f ( r ) is connected with vapor flux ∂u∂z (cid:12)(cid:12)(cid:12)(cid:12) free surface = f ( r ) . (15)Laplace equation in cylindrical coordinates for the axiallysymmetric case is written as r ∂∂r (cid:18) r ∂u∂r (cid:19) + ∂ u∂z = 0 . (16) Yu. Yu. Tarasevich et al.: Evaporation induced flow inside circular wells
Boundary problem (12),(14), for equation (16) u ( r, z ) = ∞ X n =1 a n cosh µ (1) n r f z ! J µ (1) n r f r ! , (17)where J m ( r ) is the Bessel function of the first kind, order m , µ (1) n are the real zeros of Bessel J ( r ) function: J ( r ) = 0 .Note that condition (13) is satisfied automatically.Taking into account relation between velocity and potential v = − grad u , we can find radial component of velocity h v r ( r, z ) i = ∞ X n =1 a n cosh µ (1) n r f z ! µ (1) n r f J µ (1) n r f r ! . Height averaged radial velocity, i.e. the same velocity ex-amined in [1], can be written as h v r ( r ) i = 1 h f h f Z v r ( r, z ) dz =1 h f ∞ X n =1 a n sinh µ (1) n r f h f ! J µ (1) n r f r ! . Coefficients a n can be obtained from (15). For simplicitywe will assume that air–liquid interface is flat. ∂u∂z (cid:12)(cid:12)(cid:12)(cid:12) z = h f = ∞ X n =1 a n µ (1) n r f sinh µ (1) n r f h f ! J µ (1) n r f r ! = f ( r ) . (18)We introduce notation Z f ( r ) r dr = F ( r ) . ∞ X n =1 c n J µ (1) n r f r ! = F ( r ) r (19)is a Fourier–Bessel series of F ( r ) /r , where c n = a n sinh µ (1) n r f h f ! . (20)Hence, a n = 2 r f sinh (cid:16) µ (1) n r f h f (cid:17) J (cid:16) µ (1) n (cid:17) r f Z J µ (1) n r f r ! F ( r ) dr. (21)We need to determine f ( r ) to find the unknown coeffi-cients.Mass conservation leads to the relation ∂u∂z (cid:12)(cid:12)(cid:12)(cid:12) z = h f = − ∂L ( r, t ) ∂t − J ( r, t ) ρ . (22) We computed velocity fields for all evaporation laws de-scribed in Sec. 2.1. The results show that current inside circularwell is horizontal excluding rather narrow region near the walland in the center of the well for the case of practical interest( h/r ∼ . ).Many authors supposed that evaporation induced flow in-side the droplets is independent of its height. Our simulationsconfirmed validity of this very wide used approach for thindroplets. The authors are grateful to the Russian Foundation for Basic Researchfor funding this work under Grant No. 06-02-16027-a.
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Morphology of Biological Flu-ids (Khrizostom, Moscow, 2001)9. M. Cachile, O. B´enichou, A. M. Cazabat, Langmuir , 7985(2002)10. D. M. Anderson, D. M. Davis, Physics of Fluids , 248 (1995)11. Y. Y. Tarasevich, Phys. Rev. E , 027301 (2005)u. Yu. Tarasevich et al.: Evaporation induced flow inside circular wells 5 r0 0,2 0,4 0,6 0,8 1,0zh r0 0,2 0,4 0,6 0,8 1,0zh r0 0,2 0,4 0,6 0,8 1,0zh r0 0,2 0,4 0,6 0,8 1,0zh r0,0 0,2 0,4 0,6 0,8 1,0zh r0 0,2 0,4 0,6 0,8 1,0zh Fig. 3.
Velocity field inside circular well for differen evaporative modes. The air–liquid interface is supposed to be flat. From top to bottom (3),(6) with m = 5 , (8), (10). Left column corresponds to ratio h f /r f = 0 . , right column corresponds to ratio h f /r f = 0 ..