Role of the free surface in particle deposition during evaporation of colloidal sessile drops

Transcription

1 Role of the free surface in particle deposition during evaporation of colloidal sessile drops Hassan Masoud and James D. Felske Department of Mechanical and Aerospace Engineering, State University of New York at Buffalo, Buffalo, New York 14260, USA Deposition patterns of particles suspended in evaporating colloidal drops are determined by the flow fields within the drops. Using analytically determined velocities, particle motions are then tracked in a Lagrangian sense. It is found that the majority of particles intersect the free surface as it recedes. Such capture of particles by the free surface is found to be the major mechanism in establishing the deposition pattern. Patterns are calculated for wetting and non-wetting drops whose contact lines are either pinned or freely moving during evaporation. The distribution of evaporative flux which drives the flows is taken to be that engendered by gas-phase diffusion. The theoretical results are found to agree favorably with available experimental data. I. INTRODUCTION Recently, the problem of sessile drop evaporation has found prominence in relation to the deposition of particles that occurs during the drying of colloidal drops. The topic is of high current interest because of the use of evaporating drops in depositing molecules (e.g. DNA or proteins) onto substrates in high through-put genomic or proteomic assays, colloidal particles into ordered structures for potential use directly, or for the templating of ordered structures. In addition, a particular deposition pattern the ring and phenomena related to it have important implications for: drug discovery [1], and the manufacture of novel electronic and optical materials [2,3], including thin films and 1

2 coatings [4-7]. Apart from ring patterns, applications include: buckling instability and skin formation by deposition from polymer solutions [8-10], and evaporation of liquid drops on cool or hot surfaces [11]. The deposition pattern produced depends upon the flow within the drop. In the absence of a surface tension gradient at the free surface, the flow pattern is driven by the combination of the evaporative flux distribution, the shape of the free surface, and the behavior of the contact line. For instance, flow can be towards, away or both towards and away from the contact line under different combinations of these factors. Thermal boundary conditions and surfactant concentration at the free surface are equally important when Marangoni flow is present. Several studies, including our previous analytical analyses [12,13], have been conducted focusing on the flow inside an evaporating sessile droplet without Marangoni flow. In addition, Deegan et al. [14], Popov [15], Fischer [16], Hu and Larson [17] and Widjaja and Harris [18] have investigated the particle deposition during sessile drop evaporation. Deegan [14] and Popov [15] neglected the vertical velocity component and instead considered the behavior of the vertically averaged radial velocity at small contact angles. In their analyses 100 % of the solute particles were swept radially to the contact line of the drop. Fischer [16] obtained the fluid velocity in the limit of the lubrication theory. Neglecting particle diffusion, he calculated the particle concentration distribution due solely to convective mass transfer. Hu and Larson [17] simulated Brownian dynamics to predict deposition. They determined the deposition distribution by freezing the location of any particle that came into contact with the substrate. Widjaja and Harris [18] calculated particle deposition profiles from an Eulerian point of view. They found 2

3 that the deposition pattern is influenced by both the convective and diffusive mass transfer of particles in the bulk liquid as well as by the deposition rate along the substrate. In the present study, deposition patterns resulting from the flow within evaporating colloidal drops are investigated from the Lagrangian point of view when convection is the primary mode of particle transfer. It is shown that the free surface of the drop plays a major role in defining the distribution pattern. Patterns are calculated for wetting and non-wetting drops whose contact lines are either pinned or freely moving during evaporation. II. MODELING AND ASSUMPTIONS The droplets considered in the present study are initially ~1mm from the axis of symmetry to the contact line ( R ). For water drops of this size evaporating under room conditions, the characteristic velocity in the drop is ~ 1 m/ s with a corresponding Reynolds number of 3 ~ 10 [19]. For millimeter size water drops the Bond and capillary numbers are, respectively, Bo ~ 0.04 and Ca ~ Hence, surface tension is the dominant influence on the droplet s shape, and the droplet becomes a spherical cap. The droplets are assumed to contain dilute suspensions of non-volatile molecular or colloidal solutes which are initially uniformly dispersed in the fluid. The solutes are assumed to move at the same velocity as the solvent but, unlike the solvent, they do not evaporate. Particle motion is tracked in a Lagrangian sense. When they are deposited, it is assumed that the deposits do not interfere with the geometry of the drop or the flow field within it. Therefore, in the absence of a surface tension gradient at the free surface, the 3

4 analytical solutions previously derived [12,13] for the velocity distributions are employed. Particle diffusion is neglected with respect to convective transport since, as pointed out by Hu and Larson [17], it takes much longer for a micron size particle to diffuse across the height of a mm-size drop than it takes for the radial flow to carry the particle to the free surface. Also, the particles are considered to be simple spheres which do not interact with one another. Particles may experience the following phenomena: capture by the substrate, capture by the free surface, and transport by the bulk fluid. Regarding capture by the substrate, it is noted that the fluid velocity approaches zero as the substrate is neared. Therefore, without enhancement of diffusion or attractive forces, particles in solution will not be captured by the substrate. On the other hand, it will be seen that capture by the free surface occurs for the majority of particles. Once captured, a particle moves with the velocity of the surface wherein the tangential component carries it towards the contact line. It is assumed that in nearing the contact line, when a particle reaches 2 2 ( rparticle / R 1) ( z particle / R 1) 0.01 it remains fixed at that position. Those particles in the bulk fluid which approach the contact line are treated in the same manner. The deposition thickness is computed at the end of evaporation. Two different models of contact line motion are considered. The first, based on the experimental observation of Hu and Larson [17], takes the droplet contact line to be pinned until the contact angle is reduced to a critical value after which the contact line 4

5 freely recedes. The second model allows the contact line to always freely move (never pinned). FIG. 1: Initial distribution of particles in the drop ( A indicated zone near the contact line stay there. / N 4 10 ). Particles entering the * 4 cs total In comparing particle tracking results for drops of different sizes (contact angles) the area initially allotted to a particle should be the same. Computationally this requires a common value of the ratio of the dimensionless cross sectional area ( A A / R total number of particles ( N total * 2 cs cs ) to the ). In Fig. 1 the uniform initial spacing corresponding to A / N 4 10 is shown for contact angle of 60. This value was used in all of the * 4 cs total present results. That this value was small enough to yield behavior independent of its value was verified in a number of cases by computing for A / N 10. * 4 cs total For a pinned contact line, the rate of change of the contact angle is known as a function of time. In these cases, the simulations were performed by incrementing the contact angle rather than the time. For an unpinned d contact line, the rate of change of the radius is known as a function of time. In this case, simulations incremented this radius rather than time. 5

6 III. RESULTS AND DISCUSSION Figure 2 shows the deposition patterns corresponding to the first model of contact line behavior. This model pins the contact line until the contact angle is reduced to a critical value. When this value is reached, the drop becomes unpinned and the contact radius freely recedes (at fixed contact angle) for the remainder of evaporation. All patterns predicted by this model are ring-like. It is seen that as the initial contact angle increases, the fraction of particles deposited at the contact line (and consequently the thickness of the deposit) increases. This behavior follows from first noting that surface and bulk flows are both toward the contact line when it is pinned. Therefore, since drops having larger initial contact angles provide more time for particle motion, a larger fraction of the particles will finally deposit at the contact line. FIG. 2: Fraction of particles deposited corresponding to different initial contact angles ( 30, 60, 90 and 120 ). First model: contact line pinned until contact angle reaches 3 and subsequently recedes at this angle. 6

7 FIG. 3: For (a) (c), the initial contact angle is 40 ; (a) and (b) show particle depositions corresponding, respectively to viscous and inviscid flow; (c) compares the fraction of particles captured by the free surface as a function of time (equivalently, () t ) in viscous and inviscid flows. (d) Fraction of c particles captured by the free surface in viscous flow for different initial contact angles (30, 60, 90 and 120 ). In all figures the droplet contact line is pinned until the contact angle is reduced to 3 whereupon the contact radius recedes at constant contact angle. Figure 3 compares deposition behavior computed for flows considered to be either viscous or inviscid. From previous analyses [12,13] it was demonstrated that inviscid and viscous flows are similar near the free surface but are somewhat different near substrate (due to the no slip condition in viscous flow). Because of the differences in the near substrate flow behavior, it might be surmised that viscous and inviscid flows will produce distinctly different deposition patterns. This, however, is not the case, as it can be seen from Fig.3 (a) and (b). The deposition patterns are actually quite similar and both agree well with the experimental data of Hu and Larson [17] (Fig. 4). The reason why inviscid and viscous depositions agree so well is that it is not the near substrate flow behavior which controls deposition but rather the flow behavior near the free surface. This stems 7

8 from the behavior illustrated in Fig. 3(d) which shows that the majority of particles are captured by the free surface during the time the contact line is pinned. Of those particles captured by the free surface, most are carried down to the contact line by the tangential fluid velocity on the free surface. Figure 3(c) also illustrates this behavior. Therefore, since the number of particles captured by the free surface is about the same for viscous and inviscid flows, and since near the free surface the flow fields are similar in both cases, then deposition patterns will be similar whether the flow is treated as viscous or inviscid. FIG. 4: Fraction of particles deposited as a function of radial position for two initial contact angles ( 30 and 60 ). Second model: contact line unpinned during evaporation. Figure 4 illustrates particle deposition using the second model of contact line behavior free to move (unpinned) during evaporation. In this case the fractional deposition is slightly decreasing from the axis out to the initial droplet radius. Rings are not produced when contact lines are not pinned. IV. CONCLUSION The present results provide additional understanding of the nature of particle deposition during the drying of colloidal drops. It was shown that the droplet free surface plays a major role in defining the particle distribution pattern. The mechanism of particle capturing by the free surface was found to dominate the transport of particles to the 8

9 substrate. This behavior and the resulting deposition patterns were explored using analytical solutions for the velocity field along with a simple, yet accurate, tracking of particle positions. The shape of the deposition pattern was found to depend on the model used for pinning of the contact line. Finally, deposition patterns were found to be similar for viscous and inviscid models of the flow. This shows that near-substrate transport plays only a minor role in particle deposition. It is therefore possible to employ the more readily calculated inviscid solution in order to obtain reliable, semi-quantitative results. ACKNOWLEDGMENT The importance of this problem was brought to our attention by Prof. R. C. Wetherhold. [1] D. M. Zhang, Y. Xie, M. F. Mrozek, C. Ortiz, V. J. Davisson, and D. Ben-Amotz, Analytical Chemistry 75, (2003). [2] T. Kawase, T. Shimoda, C. Newsome, H. Sirringhaus, and R. H. Friend, Thin Solid Films 438, (2003). [3] D. J. Norris, E. G. Arlinghaus, L. L. Meng, R. Heiny, and L. E. Scriven, Advanced Materials 16, (2004). [4] J. N. Cawse, D. Olson, B. J. Chisholm, M. Brennan, T. Sun, W. Flanagan, J. Akhave, A. Mehrabi, and D. Saunders, Progress in Organic Coatings 47, (2003). [5] M. Kimura, M. J. Misner, T. Xu, S. H. Kim, and T. P. Russell, Langmuir 19, (2003). 9

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