Elimination of the Coffee-Ring Effect by Promoting Particle Adsorption

Sep 9, 2013 - Elimination of the Coffee-Ring Effect by Promoting Particle Adsorption and Long-Range Interaction. A. Crivoi and Fei Duan*. School of Me...
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Elimination of the Coffee-Ring Effect by Promoting Particle Adsorption and Long-Range Interaction A. Crivoi and Fei Duan* School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore S Supporting Information *

ABSTRACT: A Monte Carlo model has been developed to investigate the transition from the coffee-ring deposition to the uniform coverage in drying pinned sessile colloidal droplets. The model applies the diffusion-limited aggregation (DLA) approach coupled with the biased random walk (BRW) to simulate the particle migration and agglomeration during the droplet drying process. It is shown that the simultaneous presence of the particle adsorption, long-range attraction, and circulatory motion processes is important for the transition from the coffee-ring effect to the uniform deposition of finally dried particles. The absence of one of the specified factors favors the coffee-ring deposition near the droplet boundary. The strong outward capillary flow on the latest evaporation stage can easily destroy the entire particle pre-ordering at the early drying stages. The formation of a robust particle structure is required to resist the outward flow and alter the coffee-ring effect.

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action.17 Thus, the previously reported experimental results have shown that adsorption, Marangoni flow, and interparticle interaction play key roles in determining whether the coffeering effect is present or not. We develop a stochastic model, capturing these effects, based on the analytical equations for the vortex flow.18 The finally dried patterns are shown to change from “coffee-ring-like” structure to “uniform-like” style if the interparticle adsorption and aggregation at the air−liquid interface is introduced in the presence of circulatory flow inside the droplet. However, the coffee-ring deposition persists if one of the conditions is not fulfilled.

INTRODUCTION Evaporation of a pinned colloidal droplet on a horizontal surface commonly leaves a residual deposit near the contact line, known as the “coffee-ring” effect.1,2 The effect is caused by a capillary flow toward the pinned edge of the droplet. The investigation to alter the formation of the coffee-stain deposit from the colloidal droplets has been conducted previously.3−12 Among the studies, Yunker et al. raised an important and interesting question in drying of a pinned colloid droplet.3 They stated that the coffeering effect1 can be suppressed by the modification of the polystyrene microparticle shapes. The experimental results demonstrated that the ellipsoidal particles almost uniformly covered the substrate after fully drying a sessile droplet, while the coffee-ring effect was present for the original spherical particles. It is indicated that the long-range interparticle interactions13,14 are stronger for the stretched particles and result in the formation of the arrested interfacial structures,3 which jam the capillary flow2 and prevent the formation of the coffee-ring-shaped deposition. In their experiments,3,15 the clusters migrate at the interface toward the axisymmetric droplet centerline at a rate similar to the rate at which particles are added at the periphery. The particles did not aggregate in the bulk phase but only once they reach the air−liquid interface. The prevention of the ring deposit in existence of surfactant-induced Marangoni flow was also reported recently.8 Still et al. demonstrated that the coffee-ring effect may be eliminated by adding the sodium dodecyl sulfate (SDS) surfactant, which creates a surface tension gradient on the air−water interface and induces a circulatory flow inside the pinned droplet. The adsorption of particles at the air−water interface creating an insoluble monolayer was investigated experimentally with the effects including the temperatureinduced Marangoni flow16 and the capillary particle inter© 2013 American Chemical Society

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MATHEMATICAL MODEL

The mathematical model uses a flat top view of the drying spherical droplet with the particles seeded inside. The particles perform their random moves according to the theoretical equations of the liquid flows inside the evaporating droplet. The changes in the positions of particles are tracked down continuously, and the final morphologies are captured and compared. The model is developed from the diffusion-limited aggregation (DLA) process19 by implementing the concept of the biased random walk (BRW) of particles on a two-dimensional (2D) lattice in a circular domain.20 The BRW equations are subsequently deduced from the analytical expressions for the radial flow velocity inside a drying droplet.18,21,22 The domain shape in the inset of Figure 1 mimics the top view of a pinned sessile droplet. As shown in Figure 1, the diameter (2R) of the circular domain is assigned 300 cells. Note that the non-smooth edge can be neglected if the cell number is large enough. The particles are assumed to have the same size, 2 × 2 cells. On each Monte Carlo step (MCS) of the simulation, each particle performs a random move within Received: January 26, 2013 Revised: September 3, 2013 Published: September 9, 2013 12067

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Figure 1. Schematics of the simulation process. In the inset, the particle flows are shown inside a circular domain defined. Different lengths of the arrows indicate the relative probabilities of the particle move in a given direction. The particles from the different layers are shown in different colors. In the presence of vortex flow, upper and bottom layers move in the opposite directions, while in the absence of vortex flow, the direction is entirely outward. Four side-view images in the inset schematically show the possible combinations of the conditions: (left top) no adsorption in the presence of surface flow results in circulatory motion of particles in liquid bulk; (right top) no adsorption in the absence of surface flow results in purely outward bulk particle motion; (left bottom and right bottom) correspondent cases with the partial adsorption of particles at the air−liquid interface, resulting in the formation of an additional insoluble monolayer. height, λ (=1/2 − θ/π) is a function of the contact angle, θ, and g is the special parameter defining the interfacial inward flow.18 This flow is coupled with the surface tension gradient on the air−liquid interface and might, in turn, be caused by the temperature gradient, surfactant concentration gradient, etc.18,24−28 The equation for g is derived from the previous study by Hu and Larsson21

the circular domain, according to a probability distribution, pmove = {l(x,y),r(x,y),u(x,y),d(x,y)}, for the (left, right, up, and down) directions. The additional no-waiting condition implies that u(x,y) + d(x,y) + l(x,y) + r(x,y) = 100% (no particles are assumed to hold their positions, unless the move is restricted by the domain boundaries), and the symmetric condition requires that u(x,y) + d(x,y) = 50% (the particles might equally likely move in the vertical or horizontal direction on the 2D lattice). The model follows a form of the Fokker−Planck equation for the probability density function (PDF) of a BRW on the 2D lattice23

∂p(x , y) ∂ ∂ (vỹ (x , y)p(x , y)) = − (vx̃ (x , y)p(x , y)) − ∂t ∂x ∂y ⎛ ∂ 2p(x , y) ∂ 2p(x , y) ⎞ ⎟ + + D⎜ 2 ∂y 2 ⎠ ⎝ ∂x

g = − 2h0r (̃ J0 λ(θ)(1 − r 2̃ )−λ(θ) − 1 + 1) − f (Ma , T , ν)

where J0 is the evaporation rate at the top of the droplet and f(Ma,T,ν) is a Marangoni effect function of the cumulative influence of the temperature and surfactant gradients. The parameter J0 is fixed in the model at a non-dimensional value of 100, and f is subject to the parameter study. Here, we assume that the gradient, represented by f, has already been established at the beginning of the simulation run and does not change until the end of the current run. We take into consideration that the model is run entirely on a 2D lattice, but the featured equations use the three-dimensional (3D) coordinate system. Therefore, the following mapping of the third, vertical dimension onto the discrete set of layers is applied. The simulated particles are split into three separate layers: upper layer, bottom layer, and surface layer (see Figure 1). The upper−bottom separation is used to simulate the possible circulatory flow in the droplet bulk. In the modeled cases, the particles from the bottom layer perform BRW corresponding to the ṽr at z̃ = h̃/2 and the particles from the upper layer use the condition z̃ = h̃ in eq 2. Although these conditions do not provide an accurate estimation of the height-averaged velocity of all of the particles moving in a given direction, with the high Marangoni effect, the conditions provide a realistic view of the distinct particle layers moving in the opposite directions. If the Marangoni effect or the simulated contact angle approaches zero, both the upper and bottom layers start to move simultaneously in one (outward) direction. The smooth transition from the two-directional flow toward the onedirectional flow can be proven with time in the simulation run. The positions at z̃ = h̃/2 and z̃ = h̃ in eq 2 are chosen for simplicity, to maximize the difference between outward and inward motions of the simulated particles. Thus, it provides us a sharp view of the circulatory flow. The 2D simulation domain serves as a top view of the drying droplet, and the upper and bottom simulated layers recall the sliced views of the droplet, focusing either on the surface or at the deep liquid

(1)

where p(x,y) is the probability density, the velocity components, ṽx(x,y) and ṽy(x,y), can be derived from pmove,23 and D is the diffusion coefficient. Because it is assumed that the random walk always proceeds with a move in either the horizontal or vertical direction with equal probabilities, D is constant. Equation 1 is used to model the particle migration inside the drying pinned droplet. It is assumed that the diffusion process associated with the Brownian motion is coupled with the transport of particles associated with the inner liquid flows in the droplet. The resultant non-dimensional drift rate, ṽr = (ṽx2(x,y) + ṽy2(x,y))1/2, is linked with the radial component of the circulatory flow velocity18

vr̃ =

⎛ z 2̃ 3 1 z̃ ⎞ [(1 − r 2̃ ) − (1 − r 2̃ )−λ(θ)]⎜ 2 − 2 ⎟ 8 1 − t̃ h̃ ⎠ ⎝ h̃ 2 ̃ g (r ̃, t )̃ h0h ⎛ z ̃ 3 z̃ ⎞ − ⎜ − ⎟ 2R 2 h 2̃ ⎠ ⎝ h̃

(3)

(2)

where 0 ≤ t ̃ ≤ 1 is the range of non-dimensional time (total drying time is set to 1 for simplicity), 0 ≤ r̃ ≤ 1 and 0 ≤ z̃ ≤ 1 are the ranges of the non-dimensional radial and vertical coordinates in the cylindrical system, h̃ (=h/h0), is the non-dimensional thickness of the liquid layer at a given point, h0/R is the maximal initial non-dimensional droplet 12068

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Figure 2. Progression of the simulation run without adsorption of particles. Images are shown from (a) 200 MCS to (y) 2600 MCS. The time gap between the consecutive images is 100 MCS, and the domain has a diameter of 300 cells. Other parameters: f = 3 and J0 = 100. Upper particles are shown in dark gray, and bottom particles are shown in light gray. The full video is available in the Supporting Information. layers. As the droplet evaporates, the top and bottom layers merge with each other and start to move similarly. Because the flow is circulatory and the particles change their motion direction after reaching certain points, the random flips between the top and bottom layers are allowed according to the following rules. The layer switch from bottom to upper is allowed for particles near the three-phase line. It is implemented as a random event with the probability, pbu, having a value of 6/R for the particles distanced from the domain center for at least 5R/6. It means that the bottom layer particles entering a thin boundary region of a R/6 thickness will flip uniformly into the upper layer and change direction in the presence of vortex flow. The upper layer particles are allowed to sink into the bottom layer uniformly throughout the domain, with the probability, pub, equal to 1/R. It is estimated to be consistent with the calculation from the work by Hu and Larsson,18 showing that the vortex trajectories might have different sizes between 0 (stagnation point1) and R (maximal loops). Furthermore, the effect of particle adsorption is applied in the model. The adsorbed particles occupy the third, surface layer in the configuration. In the simulations, particles are allowed to be adsorbed from the upper layer only, with the probability, pus, equal to 1/ (R(1 − r̃)Af), where the adsorption factor, Af, is the additional parameter controlling the adsorption rate and might change from 0 (immediate adsorption of all particles) to infinity (no adsorption). It is assumed that

the adsorption happens more frequently near the three-phase line, where more particles move closer to the air−liquid interface and, hence, pus ∝ 1/(1 − r̃). The adsorbed particles are not allowed to change the layer and, thus, remain on the surface until the end of the simulation run. It follows the assumption of the existing insoluble particle monolayer at the interface.16 The adsorbed particles always move according to the surface velocity value (z̃ = h̃ in eq 2). Summarily, the pmove distribution can be deduced for each layer ⎧ ⎪ r(x , y)= ⎪ ⎪ ⎪ ⎪ l(x , y)= ⎪ ⎨ ⎪ ⎪ u(x , y)= ⎪ ⎪ ⎪ ⎪ d(x , y)= ⎩ 12069

∼ 1 + v∼x ⎫ ⎪ 4 ⎪ ∼⎪ 1 − v∼x ⎪ 4 ⎪ ⎪ ⎬ ∼ 1 + v∼y ⎪ ⎪ 4 ⎪ ∼⎪ 1 − v∼y ⎪ ⎪ 4 ⎭

(4)

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Figure 3. Progression of the simulation run without aggregation of the adsorbed particles (pstick = 0). Images are shown from (a) 0 MCS to (y) 4600 MCS. The time gap between the consecutive images from panels a to e is 100 MCS and from panels e to y is 200 MCS, and the domain has a diameter of 300 cells. Other parameters: f = 3; J0 = 100; and Af = 100. Upper particles are shown in dark gray; bottom particles are shown in light gray; and surface particles are shown in black. The full video is available in the Supporting Information. ∼ ∼ where −1 ≤ v∼x ≤ 1 and −1 ≤ v∼y ≤ 1 are the scaled non-dimensional ∼ ∼ velocity components, v∼ = ṽ /max(ṽ ,ṽ ) and v∼ = ṽ /max(ṽ ,ṽ ). Here, ṽ

x

considered to be more stable. Newly formed clusters continue to move randomly as one unit, following eq 4, and the coordinates of the particles are averaged to calculate the drift velocity from eq 2.

= ṽrx/r̃; ṽy = ṽry/r̃; and the non-dimensional radial velocity ṽr is deduced from eq 2 at z̃ = h̃/2 for the bottom layer and at z̃ = h̃ for the upper and adsorbed layers. Importantly, the particles from the upper and bottom layers are allowed to overlap with each other on the 2D lattice in the model with the multilayer assumption, while the particles from the adsorbed layer are not allowed to overlap (on the air−liquid interface with the monolayer assumption). The adsorbed particles have another condition, while the other particles do not; the adsorbed particles are allowed to aggregate into clusters according to the DLA process. Single, newly adsorbed particles are allowed to stick to the pre-existing clusters (particle−cluster aggregation mode) with the probability, pstick. The influence of this parameter is investigated in the parameter study. Moreover, clusters might stick to each other as well (cluster−cluster aggregation mode) with a small 1% probability. This condition allows for the adsorbed particles to agglomerate into large clusters in the presence of the long-range interactions. Cluster−cluster aggregation is assumed to be significantly weaker, because larger structures are

RESULTS AND DISCUSSION The first case study is conducted without adsorption of particles (1/Af = 0). Figure 2 shows the progression of the system state as the time in MCS increases. Figure 2a displays the initial distribution of the particles, according to the law N ∝ h(r̃), because more particles are placed at the thick part of the droplet. The particles are split initially between the upper and bottom layers. The upper particles tend to move in the inward direction, while the bottom particles migrate toward the three-phase line, where they move into the upper layer (see panels b−e of Figure 2 for details). As the time progresses, outward motion starts to be dominant in the upper layer as well, it follows the governing equation for the radial velocity, defined in eq 2. Consequently, all of the particles tend to concentrate around the stagnation region1 (see panels m−p of Figure 2). As the process approaches the end,

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Figure 4. Progression of the simulation run with intensive aggregation of the adsorbed particles (pstick = 1) in the presence of a weak Marangoni effect ( f = 0.3). Images are shown from (a) 200 MCS to (y) 2600 MCS. The time gap between the consecutive images is 100 MCS, and the domain has a diameter of 300 cells. Other parameters: J0 = 100 and Af = 100. The upper layer particles are shown in dark gray; the bottom layer particles are shown in light gray; and surface particles are shown in black. The full video is available in the Supporting Information.

the later stages, the flow pattern changes dramatically and the surface particles change the direction of their motion. Panels o and p of Figure 3 show that the adsorbed particles start to prefer moving toward the three-phase line. The outward flow intensifies rapidly, and even those particles, accumulated near the center, start to leave the region and contribute to the formation of the coffee ring (panels q−t of Figure 3). Finally, almost no particles are left in the inner part of the circle, and the well-defined monolayer coffee-ring is created, as demonstrated in panels u−y of Figure 3. The final results are somewhat similar to the nonadsorption case, shown in Figure 2, but the intermediate dynamics is different. If the outward flow on the latest stages is not strong enough to dissolve the central patch, then the mixed pattern, including a coffee ring and a central deposition region, might be present (see the later discussion for details). The third simulation case investigates the strong attraction of the adsorbed particles in the presence of a much weaker Marangoni effect (f = 0.3). Figure 4 shows that the adsorbed particles rapidly aggregate into the branched clusters in the

the liquid flow inside the droplet becomes purely outward and pushes all of the particles toward the three-phase line, forming the coffee-ring effect.1,2 The results in Figure 2 are presented for a moderate Marangoni effect (f = 3). However, the characteristic flow pattern is similar in cases of stronger initial inward flows. The outward flow becomes inevitably dominant at the latest stages of the droplet drying (t ̃ → 1). The next simulation, shown in Figure 3, demonstrates the case with particle adsorption but without particle attraction and agglomeration (pstick = 0). The process starts from the thicknessdependent two-layer distribution as well (Figure 3a), but the adsorbed particles continuously start to appear on the surface layer from the early stages (panels b−e of Figure 3). Furthermore, the adsorbed particles move toward the domain center because of the presence of the Marangoni effect (f = 3), as shown in panels f−j of Figure 3. Recalling that the particles from the surface layer are not allowed to sink into the droplet bulk, we observe that the adsorbed particles continue to accumulate around the central region (panels k−n of Figure 3). However, on 12071

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Figure 5. Progression of the simulation run with aggregation of the adsorbed particles (pstick = 0.05) in the presence of circulatory flow ( f = 30). Images are shown from (a) 0 MCS to (s) 7500 MCS, from (t) 8000 MCS to (v) 9000 MCS, and from (w) 10 000 MCS to (y) 20 000 MCS, and the domain has a diameter of 300 cells. Other parameters: J0 = 100 and Af = 100. The upper layer particles are shown in dark gray; the bottom layer particles are shown in light gray; and surface particles are shown in black. The full video is available in the Supporting Information.

surface layer. However, the inward surface motion is quite weak, and most of the adsorbed particles concentrate near the stagnation region (panels g−j of Figure 4), with only several clusters reaching the central part (Figure 4o). Finally, a coffeering pattern of deposition is still dominant, but a significant portion of particles is deposited into branched structures inside the ring (Figure 4y), making it visibly different from the previously shown cases (Figures 2 and 3). Finally, the case study considers a situation with the three effects simultaneously present: particle adsorption (Af = 100), moderate particle aggregation (pstick = 0.05), and Marangoni effect (f = 30). Adsorbed particles move in the inward direction and form a rapidly growing sparse cluster in the surface layer, as shown in panels b−j of Figure 5. When the outward flow starts to become dominant on the later stages, the adsorbed cluster is large enough to cover the whole central region and continues to grow into the boundary region (panels o−t of Figure 5). In the end, the whole domain is finely covered by a single monolayer structure, shown in Figure 5y. The result suggests that the

uniform particle deposition is possible when the cluster is created at the air−water interface and is stable enough to resist the strong outward flow in the final evaporation stage. The inward Marangoni flow is important to pull the adsorbed particles toward the domain center, thus releasing space near the threephase line for newly arriving particles to adsorb. In this case, the uniform deposition may be achieved even for a relatively low particle volumetric concentration, because all of the particles are finally arranged into a monolayer on the droplet surface. However, the present model only makes a qualitative approximation of the real process and does not consider the effects such as cluster deformations, multilayer deposition, etc. Therefore, the realistic approximations of the physical properties required for the transition from the coffee ring to the uniform deposition cannot be provided currently. This is a subject for future work involving quantitative analysis. Nevertheless, a detail parameter study was performed to predict the trends in deposition for changing physical properties (see Figure 6). 12072

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Figure 6. Comparison of the simulation results obtained for the different parameter values. The upper particles are shown in dark gray; the bottom particles are shown in light gray; and the surface particles are shown in black. Panels a−d show the results with varied aggregation of the adsorbed particles (a, pstick = 0.05; b, pstick = 0.1; c, pstick = 0.25; and d, pstick = 0.5) in the presence of a circulatory flow (f = 30). Panels e−h show the results without adsorption of particles and the varied Marangoni effects: (e, f = 0.3; f, f = 3; g, f = 15; and h, f = 30). Panels i−l show the results without aggregation of particles and the varied Marangoni effects: (i, f = 0.3; j, f = 3; k, f = 15; and l, f = 30). Panels m−p show the results with strong aggregation of particles (pstick = 1) in the presence of a weak Marangoni effect ( f = 0.3) and varied adsorption factor parameter: (m, Af = 1; n, Af = 50; o, Af = 200; and p, Af = 500). The diagrams q−t show the corresponding plots of the local particle coverage depending upon the non-dimensional distance from the domain center. The coffee-ring patterns are characterized by sharp rises of the curves near the edge (r/R = 1).

coffee-ring pattern with additional branched structures inside it (see panels o and p of Figure 6). To support the above discussion, the local particle coverage as a function of the nondimensional domain radius is provided in diagrams q−t of Figure 6. Figure 6q is for panels a−d of Figure 6; Figure 6r for panels e− h of Figure 6; Figure 6s for panels i−l of Figure 6; and Figure 6t for panels m−p of Figure 6. Note that the coffee-ring patterns are characterized by sharp rises of the curves near the domain edge.

The results illustrate the possible transformations of the characteristic patterns, shown in Figures 2−5. Panels a−d of Figure 6 show the dependence of the uniform deposition upon the particle sticking probability. Gradually increasing pstick from 5 to 25% does not change the pattern but slightly rearranges the particles inside a cluster. Figure 6d displays more particles closer to the edge, while Figure 6a shows a slightly higher density near the domain center. Panels e−l of Figure 6 illustrate that a higher Marangoni effect favors a higher number of particles near the domain center in cases of no adsorption and adsorption without aggregation. This trend is entirely consistent with the fact that a higher Marangoni effect leads to stronger inward flow on the droplet surface and brings more particles from the upper and surface layers out of the coffee ring. Finally, the influence of the particle adsorption factor, Af, is demonstrated in panels m−p of Figure 6. The trend is shown for the case of low Marangoni effects in the presence of adsorption and strong aggregation (the specific run is described in Figure 4). Figure 6m shows that the fast adsorption favors the whole domain coverage. However, the deposition is still far from uniform, because more particles are deposited near the boundary region in the absence of strong inward flow. The slower adsorption leads to the coexistence of a

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CONCLUSION The newly developed Monte Carlo model investigates the significance of several factors, such as Marangoni flow and particle adsorption and interaction, for the suppression of the coffee-ring effect and transition toward the uniform particle deposition in drying colloidal sessile droplets. The results show that such factors as the particle adsorption on the air−liquid interface, particle agglomeration, and strong surface tension gradient are equally important for the final uniform coverage. The absence of one of the specified properties favors the particle deposition in the coffee-ring-like style near the droplet boundary. The strong outward capillary flow on the latest stage of droplet evaporation can easily sweep away the pre-ordering particle 12073

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(16) Nquyen, V. X.; Stebe, K. J. Patterning of small particles by a surfactant-enhanced Marangoni−Benard instability. Phys. Rev. Lett. 2002, 88 (16), 164501. (17) Yao, L.; Botto, L.; Cavallaro, M., Jr.; Bleier, B. J.; Garbin, V.; Stebe, K. J. Near field capillary repulsion. Soft Matter 2013, 9, 779−786. (18) Hu, H.; Larson, R. G. Marangoni effect reverses coffee-ring depositions. J. Phys. Chem. B 2006, 110, 7090−7094. (19) Meakin, P. Fractals, Scaling and Growth Far from Equilibrium; Cambridge University Press: Cambridge, U.K., 1998. (20) Codling, E. A.; Plank, M. J.; Benhamou, S. Random walk models in biology. J. R. Soc., Interface 2008, 5, 813−834. (21) Hu, H.; Larson, R. G. Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet. Langmuir 2005, 21, 3972−3980. (22) Hu, H.; Larson, R. G. Analysis of the microfluid flow in an evaporating sessile droplet. Langmuir 2005, 21, 3963−3971. (23) Okubo, A.; Levin, S. A. Diffusion and Ecological Problems: Modern Perspectives; Springer: Berlin, Germany, 2001. (24) Xu, X.; Luo, J. Marangoni flow in an evaporating water droplet. Appl. Phys. Lett. 2007, 91, 124102. (25) Duan, F. Local evaporation flux affected by thermocapillary convection transition at an evaporating droplet. J. Phys. D: Appl. Phys. 2009, 42, 102004. (26) Duan, F.; Ward, C. A. Surface excess properties from energy transport measurements during water evaporation. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2005, 72, 056302. (27) Duan, F.; Ward, C. A. Investigation of local evaporation flux and vapor-phase pressure at an evaporative droplet interface. Langmuir 2009, 25, 7424−7431. (28) Vermant, J. Fluid dynamics: When shape matters. Nature 2011, 476, 286−287.

structure at the early drying stages and requires the forming of a macroscopic stable structure to resist it. Additional quantitative estimations and experimental works can potentially provide more accurate estimations for the physical properties required for the described transition.

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ASSOCIATED CONTENT

S Supporting Information *

Animations of the formation of coffee rings and uniform deposition, corresponding to Figures 2−5. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Telephone: +65-67905510. Fax: +65-67924062. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the support of A*Star Public Sector Funding (1121202010).

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REFERENCES

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