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Transport of colloids along corners: visualization of evaporation induced flows beyond the axisymmetric condition Juan Rodrigo Velez-Cordero, Bernardo Yáñez Soto, and Jose Luis Arauz-Lara Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b01998 • Publication Date (Web): 20 Jul 2016 Downloaded from http://pubs.acs.org on July 21, 2016

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Transport of colloids along corners: visualization of evaporation induced flows beyond the axisymmetric condition ∗,† ´ ˜ Soto,† and Jose´ L. Arauz-Lara‡ J. Rodrigo Velez-Cordero, Bernardo Ya´ nez

†CONACYT-Instituto de F´ısica, Universidad Autónoma de San Luis Potos´ı, Alvaro Obregón 64, 78000 San Luis Potos´ı, S.L.P., México ‡Instituto de F´ısica, Universidad Autónoma de San Luis Potos´ı, Alvaro Obregón 64, 78000 San Luis Potos´ı, S.L.P., México E-mail: [email protected]

Abstract Nonhomogeneous evaporation fluxes have shown to promote the formation of internal currents in sessile droplets, explaining the patterns that suspended particles left after the droplet has dried out. While most of the evaporation experiments have been conducted in spherical cap shaped drops, which is essentially an axisymmetric geometry, here we show an example of nonhomogeneous evaporation in asymmetric geometries, which is visualized by following the motion of colloidal particles along liquid fingers forming a meniscus at square corners. It is found that the particle’s velocity increases with the diffusive evaporation factor D(1 − RH)cs for the three tested fluids: water, isopropyl alcohol (IPA) and ethanol (EtOH). Here D is the vapor diffusivity in air, RH the relative amount of vapor in the atmosphere and cs the saturated vapor concentration. We observed that while in water the particle’s trajectories is basically unidirectional, in IPA and EtOH the internal currents promote a 3D spiral motion. By adding 0.25 CMC of SDS surfactant in water, a velocity blast was observed in the whole circulation flow

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pattern, going from O(100)µm/s to nearly O(1000)µm/s in the longitudinal velocity component. In order to assess the effect of breaking the axisymmetric condition on the evaporation flux profile, we numerically solved the diffusive equation in model geometries that preserve the value of the contact angle θ but introduces an additional angle φ that characterize the solid substrate. By testing different combinations of θ and φ we corroborated that the evaporation flux increases when the substrate and the gas-liquid curves meet at corners with increasingly sharpness.

Introduction The liquid-gas phase transition or evaporation processes counts among the most studied problems in classical thermodynamics. Despite this, it is surprising that such daily experience phenomena still offer attractive areas of research, specially when it is coupled to real multi-physical problems or when interfacial fluxes are analyzed in micro-systems or smaller scales. 1–5 Reviewing the literature for the past few decades, one can roughly find two distinct regimes concerning these events. On one side we have systems whose thermodynamic states are close to the coexistence curve. At such temperature-pressure values the evaporation process is driven by the supplied external energy, and a net interfacial flux j = −k∇T · n/hlg is established between the phases. 5 Here k is the thermal conductivity, hlg the latent heat of vaporization, n the normal interfacial vector pointing towards the vapor phase and ∇T the temperature gradient. The most notable advances in this topic have been the study of phase transitions in micro-channels or sessile droplets, where the boiling process is controlled by: a) the close proximity of the interface and the heated wall, 6 b) the effects of curvature, 7 and c) the effects of disjoining pressure, which become relevant for sub-micron liquid layers. 8 On the other side, we have systems whose thermodynamic states are far from the coexistence curve. In such conditions there is no considerable energy input flowing to the system and evaporation only occurs if there is a concentration gradient between the surface of the liquid and the surroundings, i.e., the evaporation is driven by a diffusive process. 2 This is the regime that the 2 ACS Paragon Plus Environment

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present study is focused on.

The simplest example of diffusive evaporation is that of a spherical water droplet suspended in air. If the vapor phase concentration profile adjust rapidly to changes in the droplet shape in comparison to the overall evaporation time process, the time derivatives of the vapor concentration can become negligible and the quasi-steady process can be described by the Laplacian D∇2 c = 0, where c is the vapor phase concentration and D the diffusive coefficient of the vapor in air. 2 Considering that the water vapor concentration at the interface (r = R) corresponds to the saturation concentration cs , and the concentration at r → ∞ is cs RH, RH being the relative humidity, the solution of the Laplacian in spherical coordinates yields the homogeneous evaporation flux: ∂c D(1 − RH)cs . J = −D = ∂r r=R R

(1)

Integrating over the droplet surface yields the net mass transfer rate m ˙ = −4πDRcs (1−RH). One of the striking peculiarities of this system is that when the droplet is not longer spherical but attains other shapes when placed on a surface, on a corner or in a porous media, the evaporation flux becomes nonhomogeneous and differences in the mass flux at the surface generate currents inside the droplet. 9 Such nonhomogeneous evaporation induced by geometrical factors has triggered the publication of a number of scientific papers, either focusing on the internal flows and droplet’s shape or on the deposition patterns of solids produced by the overall drying process. 10–13 The most celebrated example of such evaporation induced patterns is when a drop of coffee dries out: the so-called coffee ring effect. 1 The explanation of these nonhomogeneous evaporation fluxes can be found elsewhere; 2,14 see for instance the analogy of evaporative fluxes and the distribution of electrical charges in Hu & Larson 2 or the “probability of escape” of an evaporative molecule in Deegan et al.

14

The typical example of nonhomogeneous evaporation is a water droplet resting on a surface. The droplet then attains an axisymmetric spherical cap shape with a contact angle θ against the


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surface. In this case the evaporation flux profile is no longer of the form of equation 1 but tend to increase towards the triple contact line, where it achieves the maximum value; the flux profile then has the nonhomogeneous form: 1,2 " r 2 #−λ(θ) D(1 − RH)cs J= 1− f (θ), R R

(2)

where the functions f (θ), λ(θ) are generally found numerically due to the already complex nature of the geometry. By mass conservation, this flux profile at the interface produces currents inside the droplet, and solid tracers suspended in the drop seem to flow radially at the droplet’s base. 1 Hu and Larson also provided us with a formula to compute m ˙ in spherical cap geometries. 2 The combined dependance of m ˙ in R and θ comes from the fact that the evaporation flux is nonhomogeneous as well as that every R-θ combination yields a specific interfacial area available for mass transfer in relation with the droplet’s volume, leading to the result of two droplets of identical volume having different evaporation rates depending on the particular wetting properties. The dependance of the evaporation mass transfer rate on the geometrical parameters is not the only effect brought by nonhomogeneous evaporation. Due to the fact that jhlg = −k∇T · n = q, where q is the heat flux, a nonhomogeneous evaporation induces a nonhomogeneous cooling temperature profile over the droplet surface, leading to thermocapillary effects. 3,15 Other authors have also studied the effect of adding surfactants in the internal flows and patterns generated after the drying process. 12,16 Generally speaking, thermo- and soluto-capillary effects tend to induce interfacial currents against the evaporative induced flows in sessile droplets, i.e., a superficial flow directed towards the droplet’s center. This means that surface tension tend to become higher at that point, either because evaporative cooling is faster there, or because surfactants tend to concentrate near the triple contact line establishing a tension directed towards the droplet’s center.

Almost all the works that we have cited herein have focused on the effects of nonhomogeneous evaporation and Marangoni flows in sessile droplets, where the standard shape is the spherical cap


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model, which is in essence a 2D axisymmetric geometry. A first step towards the study of 3D geometries is, for instance, placing a droplet not on a plane but on a corner having a characteristic separation angle φ between the walls (φ = 180◦ returns to the axisymmetric condition), or pouring a liquid in a container having edges or corners. These kind of general geometries, although not as widely studied as the classical axisymmetric capillary meniscus, have as well solid grounds in the literature, specially concerning the imbibition rate of liquids in open and closed polygonal vessels. 17–20 Perhaps the most famous relation concerning the interfacial shape solution in general geometries is that after Concus and Finn, 21 which states that in any corner having a characteristic angle φ touching a liquid with a contact angle θ, the meniscus will rise indefinitely (ideally) if:

φ < 180◦ − 2θ.

(3)

Hence, for square channels, φ = 90◦ and the critical contact angle becomes θ = 45◦ ; below this value the liquid will rise along the corner of the square channel leaving the bulk meniscus behind. The main statement of the present work is a basic generalization of 2D to 3D geometries: intuitively we expect to have higher evaporation fluxes on curves meeting at points than on curves meeting at lines; this is because in the solution of the Laplacian around corners, the flux lines density increases at sharp corners. Our procedures and results consist in two complementary parts: in the first part we report the velocity of particle tracers moving inside vertical menisci formed along square corners (φ = 90◦ ). Besides using liquids with different values of the evaporation factor D(1 − RH)cs , we report as well the effects of adding an ionic surfactant (SDS) in the formation of countercurrent soluto-Marangoni flows. Perhaps some immediate antecedents of these findings are those by Camassel et al. and Chauvet et al. , where they reported the accumulation of solutes in the corners of square capillary tubes by diffusive evaporation. 22,23 Also worth mentioning in this regard is the work of Eijkel et al. , 24 where the authors reported the importance of sharp corners on the drying process in noncylindrical nanochannels. In the second part of the paper we demonstrated that diffusive evaporation fluxes increase as


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the vertex at which three surface meet becomes more acute (two solid planes and one gas-liquid interface). For this purpose we employed droplet models that ensure that the liquid meets the walls at the prescribed contact angle θ.

Experimental section Experimental Setup. Flow visualizations were conducted in the vertical menisci formed along the corners of rectangular spectrometer cuvettes made by optical glass and quartz (inner volume of 1 × 1 × 4.3cm3 , Buke Scientific). The working liquids were silicon oil (sil-oil) of 7mPa · s (Gelest Inc.), Milli-Q water, ethanol (EtOH) and isopropyl alcohol (IPA). We employed as well two solutions of sodium dodecyl sulfate, SDS, in water to test soluto-Marangoni effects: one below the critical micelle concentration (0.25 CMC or 2.05mM) and another having an excess of SDS (4 CMC). In order to measure the flow velocities inside the menisci we seeded the liquids with a small amount (0.001% w/v) of green fluorescent polystyrene particles with diameter D = 5µm (G0500 Duke Scientific Co.). In the case of silicon oil we transfer the colloidal particles, initially suspended in water, via IPA using consecutive recharges after centrifugation. In order to observe the colloids moving along the vertical menisci we placed a digital microscope (DinoLite AM4113T-GFBW) in front of one of the cuvette’s wall right next to the corresponding orthogonal wall. Using this setup, the motion of the particles do not reflect the full 3D motion but just the projection on the observation plane. For all the liquids we placed the microscope at the same height with respect to the base meniscus (8mm). The visualization window was 1.6 × 1.2mm2 having a resolution of 1.25µm/pixel. All image post-processing was done in ImageJ. After pouring the liquids in the cuvettes we equilibrated approximately 2 minutes before starting the velocity measurements. In this way we ensured to measure the effects of nonhomogeneous evaporation and not the imbibition process itself, which in the case of perfect wetting and open channels last 19 around t ∼ L3 (ρgη)/γ2 ∼ 2s, using L = 4.3cm, η = 1mPa · s, ρ = 1g/cm3 and γ = 20mN/m. In table 1 we report all the relevant physical parameters of the working fluids. The


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surface tension γ and contact angle measurements were done with a goniometer (ramé-hart Instruments Co. Model 290); γ was determined using the pendant drop technique while θ determination was done along the internal walls of the spectrometer cuvettes. The data shown in Table1 indicate that all the liquids except water and water/0.25CMC in quartz will rise spontaneously on the square corners given inequality 3. In practice, we found that in 5 out of 20 corners (a probability of 1/4) these liquids also wetted the square corners spontaneously without any chemical modification on the surfaces. All the measurements were conducted on those corners. In fact, in order to avoid extra variables due to local rugosity and roundedness, we made all the visualizations on the same corners (one for the glass cuvette and the other for the quartz cuvette) at the same height. Table 1: Comparative chart of the relevant physical parameters of the working fluids. For water we considered a mean local relative humidity of RH = 0.59; RH = 0 for IPA and EtOH. The saturated vapor concentration cs was computed using cs = p s /RT , where p s is the saturated vapor pressure of the liquid at 25◦C. The term D(1 − RH)C s is the diffusive evaporation factor and the term γT /ηα came from the Marangoni number, γT being the surface tension temperature coefficient and η, α the viscosity and thermal diffusivity of the liquids, respectively. The values of T boiling , cs , Dair , γT , η and α where obtained from standard tables or other sources. 25,26 γ[mN/m] θglass θquartz T boiling [◦C] cs [mol/m3 ] Dair [mm2 /s] D(1 − RH)C s [mol/(ms)] γT /ηα[1/µm · K]

water 72.2 53.5 ± 3.5 61.8 ± 1.9 100 1.28 26.1 1.369 × 10−5 1.03

water/0.25CMC water/4CMC 57.4 37.2 33.3 ± 6 < 10 53.5 ± 5 < 10

IPA 27 < 10 < 10 82.5 2.44 8.18 1.996 × 10−5 0.53

EtOH 22.6 < 10 < 10 78.4 3.19 10.2 3.253 × 10−5 0.79

sil-oil 19.6 < 10 < 10

0.08

Table 1 also includes the evaporative parameters for pure liquids; for practical purposes we considered sil-oil as a nonvolatile solvent. In the case of the water/SDS solutions, some reports have shown that the evaporation rate of these solutions is marginally equal to pure water, 27 showing just a mild increase of the evaporation rate in the presence of SDS due to the decrease of the contact angle θ. During the surface tension measurements we realized that by plotting the equivalent mean square radius of a pendant drop against time, the resultant curves decrease in a linear fashion, as it must be if diffusivity controls the evaporation process. 27 In figure 1 we show this data for water, water/0.25CMC and EtOH, including the average slope b of the line (R/Ro )2 = 1 − bt, 7 ACS Paragon Plus Environment

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0.9

0.8

2

0.7

o

(R/R )

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0.6

0.5

0.4

0.3

0.2 0

100

200

300

400

500

600

700

800

900

1000

time [s]

Figure 1: Plots of the equivalent square radius of the droplet normalized with the initial radius Ro : water (•), water/0.25CMC (), EtOH (N). The lines denote the average slope in the whole data range. In all the figures the error bars denote the standard deviation; here the samples number m for water, water/0.25CMC and EtOH was 5, 3 and 2, respectively. where Ro is the initial equivalent radius of the droplet. This procedure seems to be very attractive because it opens the possibility to track surface tension and the evaporation process simultaneously. The drawback of this configuration (pendant drop) is that the contact angle is not easily defined; besides, the dispensing needles did not prevent the liquid to climb up the surface in strong wetting liquids. Suffice to say that, for the purposes of the present investigation, we found that diffusive evaporation in water/0.25CMC is slightly higher, 20% on average, than in water. Numerical simulations and model geometries. As mentioned above, most of the diffusive evaporation studies have been conducted in sessile drops having a spherical cap shape. This geometry can be generated by cutting one hemisphere of a sphere, having a normalized radius R = 1, by an horizontal plane at h = cos θ from the base, where θ is the contact angle at which the drop meets the plane. 2 In this work we want to stress out the significance of introducing a third geometrical variable, that is, the angle φ at which two planes meet (φ = 180◦ recovers the plane geometry).


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As a first approach we modelled a droplet with a contact angle θ sitting on a corner having the characteristic angle φ. Unlike the spherical cap shape, in this case we have to make three cuts to the sphere to generate the desired geometry: one to form the spherical cap shape obeying the θ constriction and two more to emulate the folding of the horizontal plane into two planes meeting at the angle φ (see figures 2b & c).

(b) ellipse on the ( xˆ, yˆ )-plane

(a) axisymmetric shape

(c) “cut fruit” shape

(d) tetrahedral shape

Figure 2: Model geometries that emulate non-axisymmetric conditions: (a) corresponds to the referential axisymmetric spherical cap geometry; (b) a droplet sitting on a corner is generated by cutting an spheroid first at the prescribed distance h = cos θ from the equator and then by two oblique walls forming the angle φ, (c) is the resulting shape. (d) emulates the θ = 45◦ , φ = 90◦ condition, which can be generated by drawing a tetrahedron on one corner of a cube.


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The challenge by using the spherical droplet shape sitting on a corner is that cutting a spherical cap shape with any oblique plane will violate precisely the condition of the contact angle on that plane. In order to overcome this geometrical penalty, our strategy was instead to cut an spheroid with one semi-axis larger than the other two: our goal is then to find how large should be this semiaxis in order to obey the θ-constriction and the characteristic angle φ of the corner simultaneously. We started with an spheroid drawn in ( xˆ, yˆ , zˆ) coordinates having yˆ , zˆ semi-axis lengths normalized to 1 (see figure 2b); the larger semi-axis lies on the xˆ axis. Cutting the spheroid at yˆ = h = cos θ from the equator will set the correct θ-angle on the xˆ, zˆ-plane. Once we have the spheroidal cap shape, we proceeded to cut at oblique planes having an angle β = (180◦ − φ)/2 with respect to the horizontal to yield the characteristic folding angle φ (in figure 2b we just showed the cut made on the right quadrant, which by symmetry is the same on the left side). To find out the value of the semi-axis a on xˆ that obey θ and φ constrictions simultaneously, we have to transform the ellipse projecting on the ( xˆ, yˆ )-plane to the new coordinates (x, y). The formula of such ellipse is: xˆ2 + yˆ 2 = 1. a2

(4)

Upon applying translation in yˆ and rotation on the zˆ-axis, the transformation rule become:

xˆ = x cos β − y sin β,

yˆ = x sin β + y cos β + h.

(5)

Substituting eqs.4 in 5 yields the ellipse formula in (x, y) coordinates:

Ax2 + Bx + Cxy + Dy + Ey2 + F = 0,


(6)

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where

A = cos2 β + a2 sin2 β B = 2a2 h sin β C = 2 sin β cos β(a2 − 1) D = 2a2 h cos β E = sin2 β + a2 cos2 β F = a2 (h2 − 1)

(7)

and whose roots are found to be (y = 0):

x=

q −h sin β ± cos2 β(1 − h2 )/a2 + sin2 β cos2 β/a2 + sin2 β

.

(8)

Taking the implicit derivative of eq.6 and solving for y′ at y = 0 gives dy 2Ax + B =− = − tan θ dx Cx + D

(9)

after introducing the θ-constriction, i.e., the droplet interface should meet the oblique lines at the contact angle θ. Substituting eqs.7 and 8 into eq.9 and considering that h = cos θ, we get an equation for the semi-axis a as:

cos2 β sin2 β + a2 sin2 β −

a2 sin2 β sin2 β[a2 (tan β − tan θ) + (cot β + tan θ)]2

= 0.

(10)

We evaluated eq.10 numerically for different θ − β combinations, and from the 6 roots that eq.10 gives for a only the smallest real number constitute the physical solution. For instance, setting β = 35◦ (φ = 110◦ ), the correction found for a is 1.0098 for θ = 89◦ , 1.2119 for θ = 70◦ and 1.4374 for 55◦ . Equation 10 should be taken with care since it has singularities at β → 0 and as β → θ, since physically the curvature of the interphase must change sign as the walls became closer to 11 ACS Paragon Plus Environment

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each other. After all these considerations, the final shape is highlighted with a gray shadow in figure 2b, and resembles a “cut fruit” shape in 3D, as shown in figure 2c. In the supplementary information we have reported as well the shapes of droplets sitting at corners for arbitrary values of the substrate angle β and contact angle θ; we did this by using a surface energy minimization algorithm (Surface Evolver). For the cases when θ > β, we noticed that indeed the resultant droplet shape resembles a section of a solid of revolution or a spheroid. The second interfacial model that we used was when we considered a contact angle of θ = 45◦ and φ = 90◦ . By inequality 3 this means that we are just at the critical angle and the liquid has no yet climbed the square corners of the container. This geometry can be emulated by a tetrahedron circumscribed in the corner of a box (figure 2d). Such pyramidal shaped drop meets the 3 orthogonal planes at θ = 45◦ . The third and final interfacial model that we considered constitute a perturbation of the tetrahedral drop when θ = 45◦ − ǫ, where ǫ is a small angle. Because we still considered that φ = 90◦ , by inequality 3 this condition emulates a drop that just started to climb the square corners. Such a geometry can be modelled by a sphere having a radius L/[2 cos(π/4)] meeting a cube with sides L. Upon merging these two surfaces a curved triangular surface appears (shown in figure 6e), meeting the walls of the container at θ ≈ 45◦ everywhere but approximating the 3 corners asymptotically.

We solved numerically the steady diffusion equation, D∇2 c = 0, on these model geometries using a similar procedure described by Hu & Larson. 2 The gas-liquid interface of the droplet (inner wall of the computational domain) have a concentration value of cs . All the interfacial models have a characteristic size or radius equal to L=R=1mm and were covered by an affine surface having a characteristic size of 20R,L on which the ambient boundary condition was applied, i.e., c = cs RH. With this length scale (1mm) the resultant Bond number, Bo, for water is, for example, Bo = ρgR2 /γ = 0.13. No flux condition is set on the solid planes joining the external wall and the inner interface. In order to ensure that the results were not mesh size dependant, we conducted a mesh refinement study on a 2D axisymmetric geometry. A mesh refinement factor of 1/50 applied


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to a zone 0.5R or 0.5L around the triple contact line or point yields an error of 1% of the total R mass rate transfer −m ˙ = A (J · n) dA compared to the results obtained using a refinement factor of 1/1000. All the computations were done in a 3.5GHz computer with 6 cores and 32GB in RAM

using COMSOL Multiphysics 5.0 which, in turn, has a list of different Boolean operations that facilitate the construction of the model geometries.

Results and discussion Figure 3 shows the paths left by the colloidal particles as they move along the meniscus formed in the corners of the glass and quartz cuvettes for all the working fluids. Figure 4 shows the mean

Figure 3: Superimposed images showing the colloids displacements along the square corners of the glass and quartz cuvettes. The time lapse in all cases was approximately 20s except for the silicon oil case, which was 50.8s. All the measurements were done at the same corners and height (8mm) with respect to the base meniscus. Total meniscus size spans a length of 30mm. longitudinal component of the particle velocities as they rise, or fall, through the vertical menisci. 13 ACS Paragon Plus Environment

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In the supporting information we have included a video showing the motion of colloidal particles along a water meniscus formed at the quartz’s corner cuvette. The typical meniscus thickness δ, as seen from the observation plane, was δ = 280µm, although in EtOH, IPA and sil-oil the triple contact line is not easily defined. The length of the meniscus is, however, very large in comparison to its thickness, spanning a length of 3cm (the superior border of the cuvette) for the liquids with lowest surface tension. An immediate observation after analysing figures 3 and 4 was that the trajectories of the particles, which are essentially unidirectional in the case of water and water with SDS, became more complex (curly shape) in EtOH and IPA, which have the highest evaporation factor D(1 −RH)cs and the lowest surface tension and contact angle. Notice that the silicon oil also has a very low surface energy and wets the corners up to the upper rim, however, the particles do not show relevant ascending/descending motion, confirming that particle drag is due to evaporation induced flows. Next, in figure 4 we observe that the rise velocity in pure liquids increases with the value of D(1 − RH)cs , as shown in Table 1, revealing again that motion is generated by diffusive evaporation. For the water/quartz case, the mean value reached u¯ = 242µm/s; these orders of magnitude in u¯ have been observed as well experimentally in axisymmetric sessile drops using micro-PIV. 11 The main reason why the velocities are higher in the quartz cuvette is because the thicknesses δ of the menisci are smaller there, i.e., the transverse area is smaller so a higher u¯ is needed to achieved the same volumetric or mass transfer rate. For instance, in the case of water, the mean value of δ is 305µm in glass and 131µm in quartz; since θ is close to 45◦ , we can approximate the transverse area as δ2 /2, therefore multiplying the corresponding u¯ values with the transverse area gives the same volumetric flow rate of 2 × 10−3 µL/s in both cases. Another important observation is that when SDS is added to water, a substantial increase in the velocity is apparent, not only in the rising but also in the falling motion of the colloids. An increase up to one order of magnitude is observed, being more predominant in the 0.25CMC solution than in the solution having an excess of SDS (4CMC). We now proceed to extend the discussion on these three main issues.


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(a)

(b)

Figure 4: Bar plots of the longitudinal velocity component of the particles as they rise, or fall, through the menisci. The average m for each value was 30.


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The first observation demonstrated that breaking the axisymmetric condition leads to the formation of 3D stable secondary flows due to the transport along lateral walls. In the supporting information we have included a video showing the motion of the particles in the EtOH/glass configuration. These secondary flows may well be formed by transverse temperature gradients, leading to Marangoni vortical structures, as seen in axisymmetric geometries. 15,28 However, if we look at the values of the factor γT /ηα in Table 1, linked to the Marangoni number, the highest value is for water, which however shows rectilinear particle paths. Hence transverse motion is not uniquely determined by lateral temperature gradients, but also by lateral mass flow, which can occur precisely in fluids having very low surface energies or high evaporation rates. It is worth mentioning that in the case of silicon oil, where evaporation is negligible and therefore the longitudinal motion, the particles move making spirals in the meniscus (see figure 3). Such concomitant action of evaporation/wetting and thermocapillary effects agrees with the previous work of Tsoumpas et al. , 15 which actually proposed an effective Marangoni number Mae f f of the form:

Mae f f ≈

γT hlg Dcs 1 k γ θ

(11)

where k is the thermal conductivity of the liquid and θ is given in radians. This adimensional number weights the contribution of evaporation and wetting properties along the common factor γT /k. The numbers for water, IPA and ethanol are Mae f f = 0.005, 0.1 and 0.18, respectively. Therefore, our best explanation for the spiral motion seen in EtOH and IPA is that it results from the combined ascending motion and lateral circulation induced by wetting/evaporation/Marangoni effects (lateral motion was around 20% of the corresponding longitudinal velocity). We must highlight the point that during these realizations we observed no breaking of the meniscus laying on the corner; hence, the undulatory motion observed in figure 3 is not in strictly sense a dynamical instability characterized by an unstable wavenumber that grows with time. 29 Therefore, the kinematics of the helical motion relies more on the stability of two complementary flows (lateral vortical structures and a longitudinal or axial evaporation flow) rather than on the instability or film breakage.


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Langmuir

Besides thermocapillary effects, we also observed soluto-capillary effects when SDS was added to water. Figure 5a shows a good example of this effect in axisymmetric geometries (the corresponding video is included in the Supporting Information). Here we track a single particle first

Figure 5: Superimposed images showing the displacement of polystyrene particles close to the meniscus tip. The left image (a) shows the form of the meniscus and the path left by a particle tracer in water having 1CMC. At the finger’s tip some accumulation of fluorescent particles is observed. The right image (b) shows the paths formed by the particle tracers in ethanol: longitudinal vortical structures are seen in this case due to axial thermo-Marangoni stresses. rising up to the top of the meniscus finger close to the cuvette’s corner; then it follows down close to the interface. The falling velocity in water/0.25CMC is raised dramatically, up to an average of 1.2mm/s (see figure 4); interestingly, the surface tension gradient effects not only speed up the motion in the falling direction, but also in the whole loop, recalling that this effect cannot be due to a simple increase in the evaporation rate (diffusive evaporation in water/0.25CMC is only 1.2 times larger than in pure water, see figure 1). Due to the saturated concentration effect, the ascending motion in the 4CMC solution is lower compared to the 0.25CMC solution, and we even observed a “salting out” effect of the polystyrene particles (in figure 3 we can see that for the 4CMC solutions


Langmuir

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the particles tend to ascend close to the right, near the cuvette’s corner, but return stuck to the triple contact line). In agreement with the axisymmetric case, 16 here the observed phenomena suggest an increase of the surfactant concentration in the zones of maximum water evaporation fluxes, establishing surface tension gradients along the meniscus and speeding up the falling motion compared to events produced solely by gravity. In this regard Fried, Shen & Gurtin 30 published a mass balance of a nonvolatile surfactant diluted in a volatile solvent and we found their equations to be pertinent in the discussion we are following here. According to these authors, the evolution equation of the number density of a surfactant at the interface, Nint , depends on its interfacial and bulk fluxes and on the term −N U mig , N being the density number of the surfactant in the bulk and U mig the migrational velocity of the interface relative to the normal liquid velocity. Examination of this mass balance equation reveal three main actors contributing to the spatial and temporal variation of N at the interface: the first one is the normal liquid velocity at the interface, which is precisely where nonhomogeneous evaporation fluxes may have a relevant role in setting up nonhomogeneous concentration profiles of the surfactant at the interface; even more, since U mig constitute an interfacial velocity relative to the liquid velocity, the contribution of this term increases with time due to the withdrawal of the gas-liquid interface during the whole evaporation process (the total surfactant concentration increases as the droplet volume decreases). The second contribution is the surfactant flux itself, which as detailed by Fried et al. , 30 can have purely diffusive terms but also thermophoretic effects, i.e., fluxes promoted by temperature gradients. The third factor is the local curvature appearing in the convective interfacial terms. There is still plenty of room in analyzing the pertinence of such equations in the dynamics of general shaped drops having surfactants. In our case, since the liquid bulk constitutes a reservoir of the ionic SDS surfactant, it is straightforward to assume that the evaporation fluxes promote SDS accumulation at the triple contact point.

Before moving to the geometrical models and analyze the evaporation flux profiles in nonaxisymmetric shapes, we shall review first other effects that may influence the longitudinal motion inside the liquid fingers. To begin with, we are going to evaluate the influence of longitudinal


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thermo-Marangoni stresses and convection due to density gradients. In doing this we are going to assume that the evaporation flux at the meniscus tip is maximum and provoke an evaporative cooling ∆T , i.e., thermal-Marangoni stresses or density gradients will tend to pull the liquid towards the meniscus tip. Relying on dimensionless analysis, we can consider a parcel of fluid of volume ∼ δ3 rising a vertical distance z due to thermal-Marangoni stresses or buoyancy forces. In order to induce a convective flow along the meniscus with thickness δ, for the former we should have that

Ma =

γT ∆T (δ/2) z > . αη δ/2

(12)

Similarly, for the case of buoyancy forces we must have that

Ra =

z ρg(δ/2)3 β∆T > αη δ/2

(13)

where Ma, Ra are the Marangoni and Rayleigh numbers, respectively. For the latter, β is the thermal expansion coefficient. An estimation of |∆T | can be given using the fact that ∆T = jhlg λk /k and considering that at the interface temperature gradients occur on a length scale of the order of the Knudsen layer thickness λk = kB T/πd 2 p, where d is the molecular diameter, kB the Boltzmann constant and p the atmospheric pressure. Table 2 shows all the physical data needed to compute Ma, Ra and other relevant parameters and time scales for two selected liquids, water and ethanol, using δ as the length scale. The evaporative flux j ≈ uρ ¯ was computed from the experiments. From Table 2 it is clear that convection effects due to density gradients are small (Ra 1 for both fluids, being particularly large in ethanol. Therefore, we expect in the latter to see axial Marangoni flows at a distance z < 2.7mm far from the meniscus tip. Figure 5b shows superimposed images of the particles moving close to the meniscus tip in ethanol, and unlike the flow paths shown in figure 5a for water, here we see strong vortical structures aligned perpendicular with respect to the meniscus. It is therefore evident that in the case of liquids having high evaporation rates and low contact angles, evaporation in non-axisymmetric geometries is accompanied by thermal-Marangoni effects 19 ACS Paragon Plus Environment

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Table 2: Physical data for water and ethanol. Symbols refer to the same physical quantity as in Table 1; β stands for the thermal expansion coefficient and λk denote the Knudsen layer thickness. tDi f f = δ2 /Dair (1 − RH) denotes the characteristic time of diffusive evaporation, t Ma = (ρδ3 /γT ∆T )1/2 the characteristic time of thermal-Marangoni effects, tthem = δ2 /α the characteristic time of heat conduction, and tevap = ρhlg δ2 /k∆T the characteristic time of the overall evaporation process. ρ[kg/m3 ] η[mPa · s] α[m2 /s] k[W/m · K] β[K −1 ] |γT |[mN/m · K] hlg [J/kmol] λk [m] ∆T [K] Ma Ra tDi f f [s] t Ma [s] ttherm [s] tevap [s]

water 1000 1 14.3 × 10−8 0.6 2.07 × 10−4 0.1477 40.6 × 106 1.7 × 10−7 0.025 3.61 9.7 × 10−4 7.3 × 10−3 7.7 × 10−2 0.54 1.1 × 104

ethanol 789 0.9 7 × 10−8 0.2 7.5 × 10−4 0.0832 38.9 × 106 0.64 × 10−7 0.107 19.7 27 × 10−3 7.6 × 10−3 4.4 × 10−2 1.12 2.4 × 103

in regions close to the meniscus tip, where the thickness δ decreases (notice in Table 2 that even in ethanol the characteristic time for diffusive evaporation, tDi f f , is one order of magnitude shorter faster - than the Marangoni time scale). A third ingredient that may affect the flow in the liquid is convection in the gas phase. Since we are not imposing any external air flow, the Reynolds number in the gas phase is undetermined and hence the boundary layer thickness should be quite large. The other source of convection is that imposed by density gradients in air due to the evaporative currents: humid air is lighter than dried air, for instance, while a mixture of air with alcohol vapors will be heavier than pure air. Employing the Grashof number, Gr, as a measure of natural convection effects, 31 and assuming again that density gradients become important at the tips of the meniscus where maximum inhomogeneous fluxes occur, we have that ! ∆ρ gδ3 ∆ρ ≈

Transport of Colloids along Corners: Visualization ... - ACS Publications

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