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Droplets of Colloids
David Brutin and Florian Carle Mechanical Engineering Department, Aix-Marseille University, Marseille, France
Chapter 19
DROPLETS OF COLLOIDS 19.1 Introduction
While nanofluids are widely studied in several scientific communities (physics, biology, chemistry, etc.), several questions remain regarding their heat transfer efficiency, stability, and toxicity to human health. One important application for nanofluids is inkjet printing (Chhasatia and Sun, 2011), though nanofluids are very promising for two-phase heat transfer as well (Barber et al., 2011). Using metallic nanoparticles, inkjet printer technology is evolving to allow for the printing of metallic inks on soft substrates for photovoltaic and several others applications. These metallic inks are mostly made of silver, and consequently, they are very expensive (about $500 per milliliter). To create controlled patterns using inkjet-printing technology, a chain of droplets, whose volume has been calculated, must create lines of constant diameter following evaporation. The homogeneity of the line thickness is important because it will affect the electrical resistivity of the resulting pattern. If the concentration of nanoparticles is too low, an o-ring pattern will appear, which means that the matter will not have been uniformly deposited on the Droplet Wetting and Evaporation. DOI: http://dx.doi.org/10.1016/B978-0-12-800722-8.00019-9 © 2015 Elsevier Inc. All rights reserved.
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280 Droplet Wetting and Evaporation substrate. If the concentration is too high, then the final thickness will also be too high, exceeding a certain thickness above which the electrical resistivity will no longer change; thus, nanoparticles, which are extremely expensive, will have been wasted. Thus, the search for the optimal concentration that avoids o-ring formation and consequently enables a uniform deposition by minimizing the amount of nanoparticles is of great interest. Several authors are currently looking at the driving parameters that lead to such pattern formation (Brutin, 2013; Askounis et al., 2014). However, because the evaporation of even a pure fluid droplet remains a complex topic, there is much research to be done to understand the evaporation of complex fluids (Conway et al., 1997; David et al., 2007; Kelly-Zion et al., 2011).
19.2 Colloidal system stability Despite the wide range of applications, the major drawback is the fact that nanofluids tend to be unstable and to flocculate. The stability of nanofluids can be greatly improved by choosing the optimal synthesis method. The choice of the method will depend on not only the fluid/particle pair but also the nanofluid stability, which plays a primary role in determining the properties of the nanofluid. To be able to completely understand the following study on the formation of cracks, a brief review of the particle interaction theory and ion neutrality is provided here. The particles will be subject to molecular interactions, such as collision between the fluid and the particle atoms and Brownian motion, and also to unwanted interactions, such as van der Waals forces (short range electrostatic interaction) or other ionic interactions. These latter phenomena will have a disastrous effect on the stability of nanofluids because large packets of material will flocculate, thus quickly reducing the effectiveness of the nanofluid. This phenomenon is well explained by the Derjaguin, Landau and Verwey Overbeek theory (DLVO theory (Verwey et al., 1999)). The DLVO theory suggests that the stability of a dispersion of colloids depends of the total potential energy VT: VT 5 VA 1 VR 1 VS ; where VA and VR are the attractive and repulsive contributions, respectively, and VS is the potential energy due to the solvent. This last contribution plays a negligible role compared to the two other potentials for nanometer-scale separation of the particles. Basically, the suspension stability results from the van der Waals forces (minus the electrical double layer forces) between two particles brought close to each other by Brownian motion. The total potential energy is dependent upon the separation between the particles (Figure 19.1).
Droplets of Colloids 281
Double layer repulsive force
Energy
Secondary minimum Net energy
Primary 0 minimum van der Waals attractive force Particle separation
FIGURE 19.1
Potential energy as a function of the particle separation.
At very small interparticle distances, van der Waals forces are predominant over electrostatic repulsion forces. As a result, the particles are forced to stick together, and the dispersion will then flocculate; this contact is particularly strong and irreversible. The strength of this attractive force decreases inversely with the particle separation. Therefore, repulsive electrostatic forces become dominant, and the particles repulse each other. The net energy admits two different levels of energy minima; all of the particles will tend to stay in these two positions of energy minima. The primary minimum is located at the contact of the particle (flocculation of the nanofluid), and the second one is behind an energy potential barrier. This second minimum is where all of the particles should be to produce a stable nanofluid. Unfortunately, if the particles have sufficient energy to go through this barrier, the nanofluid will flocculate. To avoid flocculation, the repulsive forces must be as high as possible to have the potential barrier be as high as possible. High repulsive forces can be achieved by using surfactants or functional groups to form either a steric barrier (e.g., polymers and plasticizers) or an ionic barrier (e.g., electrostatic stabilization from the electric double layer and from sol-gel precursors) to keep the particles separated and to avoid cluster formation. These additions greatly modify the suspensions but have yet to be investigated for their influence on the crack patterns. The strong effect of surface functional groups hereinafter referred as surface functional groupments on the final crack patterns of nanofluid deposits is described in this section. Surface functional groups have never before been taken into account in studies on drying, crack patterns, or self-assembly of nanoparticles prior to Carle (2014). These functional groups are, however, widely used in several communities—for example, as biological tracers with fluorescent spheres, cancer treatment using their surface functional groups to bond a dedicated protein (Arima and Iwata,
282 Droplet Wetting and Evaporation 2007), or even for inkjet printers, using metallic nanoparticles to create monoatomic layers on various hydrophilic or hydrophobic substrates (Chhasatia and Sun, 2011). For all of these applications, surface functional groups are crucial for nanoparticles to be compatible with a variety of conjugation strategies. Both the convective self-assembly of nanoparticles and the final crack patterns are affected by physicochemical interactions with surface functional groups. Interactions between surface functional groups (Frisbie et al., 1994) and solvents (Halicioglu and Jaffe, 2002) were investigated, but more research is required on the influence of the interaction between surface functional groups itself during the formation of close-packed particle assemblies and also on fracturing during drying.
19.3 Mono-dispersed colloidal systems The evaporation dynamics of a nanofluid sessile droplet follow the same behavior as those of pure fluid sessile droplets in the first stages of evaporation. Figure 19.2 allows for the description, through the example of an aldehydesulfate nanofluid droplet, of colloid evaporation dynamics. 4
Spreading
Mass (µg)
−
3
dm (µg s−1) dt
4 3 2 1 0 0
250
2
500
750 1000 1250
T (s)
Deposition ring Depinning 1
Cracks formation 0
0
250
Final pattern
500
750
1000
1250
Time (s)
FIGURE 19.2 Evaporation of a 4 µL aldehyde-sulfate nanofluid droplet onto a lysine substrate at 24 C and 50% of humidity. Droplet top view pictures show the major evaporation events, along with a schematic representation of droplet interface and particle motion. The inset shows the droplet evaporation flux rate. Figure from Carle (2014).
Droplets of Colloids 283 In the very first instant of the droplet creation, as soon as the fluid touches the substrate, the droplet spreads to reach its maximum wetting diameter. The spreading influence of nanofluids was observed (Chengara et al., 2004; Nikolov et al., 2010), but it was not obvious in all of the experiments performed here. After a few minutes, a circular deposit is clearly visible at the edge of the drops. As we previously observed in Section 5.3, the sessile droplet evaporation profile exhibits a nonuniformity at the triple line; this nonuniformity leads to a radial flow toward the triple line to compensate for the evaporation loss at the contact line (Deegan et al., 1997). Therefore, nanoparticles are dragged outward with this flow in a fairly quick process and then form a deposit in the vicinity of the triple line. Here, the contact line is pinned during most of the evaporation process, whereas the contact angle and the height decrease linearly. As the solvent evaporates, the particles are forced to concentrate near the triple line in a close-packed array (Figure 19.3). Figure 19.3 depicts remarkable cracks that have been found during SEM observations of the dried droplets. In addition to being dramatic nanoscenery, these pictures summarize the physics involved in the crack formation patterns. Picture (A), which was taken at the center of an aldehyde-sulfate droplet on glass, confirms the preferential paths of the cracks in the direction of the evaporation.
(A)
(B)
(C)
(D)
FIGURE 19.3 Remarkable SEM observations of particles deposits. (A) Aldehyde sulfate on glass, (B) carboxylate on lysine, (C) and (D) carboxylate on gold. The scale represents 50 µm for the top pictures and 500 nm for the bottom pictures. Figures from Carle (2014).
284 Droplet Wetting and Evaporation The depinning of the triple line and its receding toward the center induce the cracks to meet at the center of the droplet. Pictures (C) and (D) provide a glimpse of the forcefulness of the formation of the cracks. The compacted layers of nanoparticles are ripped apart by stress due to evaporation. The interesting aspect of these SEM observations is the ability to locally observe the sharpness of the cracks, enabling determination of the rigidity of the material created by the deposition, and the ability to observe the fragility of the deposits, with the broken pieces near the cracks observed in the image (B). Coming back to drying, the pinned evaporation continues for approximately 60% of the total time of the evaporation until the contact line recedes when the critical contact angle is exceeded. The depinned contact line shrinks toward the center. The motion of the solvent induces a stress in all deposits, resulting in the nucleation of cracks. Once all of the solvent has evaporated, the dried particles completely cover the initial wetting area due to their relatively high concentration and form regular crack patterns. Experiments were performed several times for each configuration, and the crack patterns were found to be replicable. Several conditions must be met for the droplets to form a deposition ring on the substrate: first, the solvent must not completely wet the substrate (initial contact angle larger than 0 ); otherwise, the particle deposition will form a layer that will tend to crack, following the direction of evaporation. Then, the triple line must pin a major part of the evaporation to initiate a deposition position that remains the same to accumulate enough particles. Stick slip droplets tend to form irregular concentric rings (Bhardwaj et al., 2009). This phenomenon, when forced to occur regularly, can be used to form patterns for industrial applications: homogeneous rings with a perfect spatial distribution by means of solid spheres in the droplets (Xu et al., 2007) and evaporation onto patterned hydrophobic substrates in the form of pillars (Fan and Stebe, 2004). As an interesting counterexample, the coffee ring effect can be suppressed by using electrowetting (see Section 5.3). Other nanoparticles than polystyrene are used in the literature. Denkov et al. (1992) used latex particles. In the case of mixtures of more complex fluids, at low concentrations, Govor et al. (2004) evidenced the formation of nanoparticle rings following the evaporation of droplets for a liquid matrix of a binary mixture of nitrocellulose, amyl-acetate, and hexane. The authors used 6 nm nanoparticles of CoPt3 at concentrations below 1%, leading to rings measuring 0.6 1.5 µm in diameter. They experimentally observed the phase separation that leads to the formation of a bilayer structure. The CoPt3 nanoparticles located on the contact line assembled along the line. In this case, the accumulation at the drop edge was pronounced. Conway et al. (1997) first investigated the influence of particles on the evaporation dynamics of a drop. Using 200 and 750 nm polystyrene beads, the author observed the formation of an o-ring and a crater for these bead diameters
Droplets of Colloids 285 and monitored the drop height and mass over time. The author provides a linear normalized model of the drop mass and normalized model of the drop height using a root square function. Using 1 µm particles with a disparity going from 0.5 to 2 µm, Marin et al. (2011) demonstrated that the particles that settle first along the contact line of a coffee ring pattern have an ordered, crystalline structure. Toward the center of the drop, a transition to a disordered particle arrangement was observed. The authors propose a model that explains many results described in the literature where large-scale crystalline deposits were observed for particle sizes ranging from 6 to almost 50 nm. In this range of particle sizes, the critical velocity required for a disordered phase is unreachable within the droplet, which explains the absence of a disordered phase. The authors also state that apart from the order-to-disorder transition, there are also transitions within the ordered phase, from square to hexagonal packing and vice versa (Figure 19.1D). In an unconfined system, the most efficient packing lattice is the hexagonal compact. However, in the evaporating drop the particles are confined in a wedge formed by the glass slide on one side and the liquid air interface with contact angle on the other side. Such a confinement indeed leads to a sequence of hexagonal and square packing structures that depend on the most efficient packing for the available space. Indeed, when a new layer is formed, the confinement by the wedge favors the square packing; see Figure 19.1. Away from the step edge, the available space increases and allows for the denser, hexagonal packing structure (Figure 19.4).
19.4 Bidispersed colloidal systems O-ring formation during colloidal drop evaporation was first explained by Deegan (2000). By using sulfate-terminated polystyrene spheres of two sizes, 1 and 0.1 µm, at a maximum volume concentration of 2% dried on glass plates, the author observed the formation of different patterns, depending on the concentration and size of the spheres used. For spheres measuring 100 nm and concentrations ranging from 0.063% to 0.25%, Deegan observed the formation of multiple rings for initial droplets measuring 6 mm in diameter. In all cases, particles remained at the drop center and were not only located at the drop edge. In this study, we observed two different final patterns, depending on the concentration: an o-ring pattern and a nearly uniform deposition pattern on the substrate. Chhasatia and Sun (2011) used bidispersed colloids made of micro- and nanoparticles and observed the interaction between these particles and the contact line. For 100 nm and 1.1 µm particles, they observed that, at the end of the evaporation process, the nanoparticles were located near the triple line, followed by the microparticles (Figure 19.5). For a hydrophobic substrate, the constant contact angle
286 Droplet Wetting and Evaporation (A)
(B)
0.5 mm
0.5 mm (D)
(C)
10 µm
5 µm
FIGURE 19.4 Order-to-disorder transition in the particle stain left by an evaporating drop. (A) A 3 µL sessile water droplet evaporates from a glass substrate. (B) Ringshaped stain of red particles (the coffee stain) left on the substrate after evaporation. (C) A close-up by an optical microscope of the bottom layer of the stain, taken from the white square in (B), shows that the outermost lines of the stain (left) have an ordered, crystalline structure. Toward the center of the drop (right), a transition to a disordered particle arrangement is observed. (D) A top view of the ring stain, taken from the red square in (C) with a scanning electron microscope (SEM), shows that the first lines of particles (left) are arranged in a hexagonal array, while the next lines (brighter in the image) are arranged as square, followed by again hexagonal array. Images from Marin et al. (2011).
mode dominates the evaporation process, leaving little carrier liquid left to rearrange particles according to their sizes at the mixed mode. For a hydrophilic substrate of 0 , θ , 45 , particle separation is incomplete in the final deposition. Three separate regions exist from the outer edges to the center of the drop, including a region with only nanoparticles, a mixture of micro- and nanoparticles, and an inside region with only microparticles. The width of the middle region where micro- and nanoparticles overlap depends on the interactions of the surface tension and friction forces acting on the particles. For a hydrophilic substrate of θ 5 0 , nanoparticles accumulate at the contact line, forming several rows that prevent the contact line from receding. This enhanced pinning by nanoparticles permits radially outward evaporative flow toward the pinned contact line and pushes some of the microparticles to exceed through the liquid vapor interface. Reduced surface tension acting on those exceeded microparticles prevents them from moving inward, whereas most other microparticles move radially inward to
Droplets of Colloids 287
Contact line gets pinned at the beginning of constant contact area mode
Contact line gets pinned at the beginning of constant contact area mode
Nanoparticles move closer to the pinned contact line and delays constant contact angle stage Particles mix during constant contact angle mode
In mixed mode both micro and nanoparticles move closer to liquid–vapor interface
Micro and nanoparticles remain mixed in the deposition
(A)
Contact line gets pinned at the beginning of constant contact area mode
Nanoparticles move closer to the pinned contact line
Particles mix during receding
Nanoparticles move towards contact line in the mixed mode
Nanoparticles get deposited at the outer edge, not clear separation
(B)
Microparticles recede toward center of the drop due to capillary flow
Nano and microparticles from separate rings
(C)
FIGURE 19.5 Schematic of the evaporation process of bidispersed particles (100 nm and 1.1 µm) on three substrates with contact angles of: (A) θ . 45 (mixed), (B) 0 , θ , 45 (partial separation), and (C) θ 5 0 (complete separation). Sketches from Chhasatia and Sun (2011).
the center of the drop. By modifying substrate wettability, the surface tension force acting on the particles is tuned. The control of particle separation according to their sizes can therefore be achieved. The authors have also found that the increase in the size ratio of the bidispersed particles improves particle separation. Because the number of particles affects contact line pinning, the boundaries between mixed, partially separated, and completely separated regimes shift toward a higher contact angle value for a higher particle loading. By controlling the substrate wettability, particle size ratio, particle loading, and relative humidity, the deposition morphology of bidispersed particles can be further controlled. During the drying of droplets containing latex particles measuring from 40 nm to 5 µm, Monteux and Lequeux (2011) observed a rearrangement that led to the formation of rings. The smallest nanoparticles were deposited on the outer side of the ring, while the largest particles were located on the inner side. The authors conclude that this deposition mechanism leads to a tight deposition but leaves a thin liquid film of pure water at the edge of the drop. The deposited particle ring size can be precisely controlled by changing the particle size. The segregation effect may consequently play an important role in polymer and biological fluids, which are composed of polydisperse particles.
288 Droplet Wetting and Evaporation Pattern formation has also been observed for biological fluids, such as whole blood, very recently and shows similar patterns to those of nanofluids (Brutin et al., 2011). The drying dynamics observed for this complex biological fluid show typical regimes that are treated in Section 4.5. Finally, pattern formation is also influenced by the wettability of a droplet on a substrate. The relationship between the spreading of a droplet and the evaporation dynamics has been previously discussed in Section 1.1. The triple line dynamics and the pinning of a drop play a major role in the drying process dealt with in Section 1.3.
19.5 Parameters influencing the patterns Obviously the colloids concentration is the first parameter that influences the pattern. Brutin in 2013 evidenced a critical concentration to observe the classical o-ring formation. Above this concentration a more homogeneous deposition is observed. The author used polystyrene nanoparticles measuring 24 nm that were suspended in deionized water to study the effect of nanoparticle concentration and surrounding humidity on the pattern formation and drying dynamics of sessile droplets. Droplets were deposited on rough Nuflon 16X-coated substrates. The patterns formed were observed to have been influenced mainly by the concentration of nanoparticles but not at all by the drying time. Two patterns were observed: an o-ring formation for concentrations below 0.47% and a flower pattern above 1.15%. The effect of humidity was proportional to the drying rate, but no effect on pattern formation was observed. The experimentally measured evaporative mass flux was slightly higher compared to that predicted by a purely diffusive model, which may indicate a slight enhancement in heat transfer at the triple line due to the local improvement in the substrate heat capacity by nanoparticle deposition. For pure fluids, the effect of the substrates thermal properties was investigated, even at ambient temperature, because the latent heat of evaporation is mostly affected by element with the highest heat capacity, which is the substrate (Sobac and Brutin, 2012). One explanation provided by the author is that the diffusion coefficient of our water-based nanofluid, even at low concentration, in humid air is different from that of pure water in dry air. Experiments are currently underway to address this discrepancy. Another explanation is based on the experimental observation in the literature of an increase in the spreading of nanofluids leading to an increase in the drop perimeter and a simultaneous decrease in the contact angle. The consequence is an improvement in the evaporation rate. This assumption is also under investigation at this time (Figure 19.6). Chhasatia et al. (2010) investigated the effect of relative humidity on the contact angle and nanoparticle deposition during the evaporation of a picoliter droplet of a nanofluid. They observed, for the pinned droplet mode, a spreading effect with the increase in relative humidity. The observed variations in the contact angle
Droplets of Colloids 289 Flower pattern
C = 5.70%
C = 4.80%
C = 2.30%
C = 1.15%
C = 0.47%
C = 0.23%
C = 0.11%
C = 0.05% O-ring pattern
C = 0.01%
FIGURE 19.6 Influence of colloids concentrations ranging from 0.01% to 5.70% on the pattern formation (controlled glovebox humidity of 50% and room temperature of 25.0 6 0.6 C). Image from Brutin (2013).
were linear but exhibited different slopes depending on the relative humidity. The authors conclude that due to the small volume of the droplet, the relative humidity becomes an important parameter driving the evaporation process and consequently the deposition. They also observed an increase in the initial spreading of the drop with increasing humidity. Highly concentrated colloidal suspension exhibits spreading and evaporation instabilities, as observed by Hadj-Achour and Brutin (2014). The authors report a fingering instability that occurs during the spreading and evaporation of a silver metallic ink nanosuspension droplet concentrated at about 20% in mass so far above the low concentrations classically observed. The fingering instability has been widely studied for both Newtonian and non-Newtonian fluids. However, this report describes the first time that a fingering instability is observed for the spreading of a nanosuspension sessile droplet. In this study, the authors demonstrate that in certain cases, the contact line evolves through different spreading regimes according to De Coninck et al. (2001) with an enhancement in the evaporation rate due the formation of the fractal patterns.
290 Droplet Wetting and Evaporation
19.6 Nanoparticles surface charges
Lysine γ = 65.5 mJ m–2 Gold γ = 43.5 mJ m–2
Substrates
Glass γ = 71.4 mJ m–2
Working with nanoparticles, the surface charges exposed at their vicinity cannot be neglected. Carle and Brutin, 2013 studied the influence of this charge on the pattern and the crack periodicity. Working with three substrates of different surface energies and the same nanoparticles but only with different surface charges, they evidenced strong differences in the final pattern. Figure 19.7, if observed vertically, shows the final crack patterns of each nanofluid on three substrates (surfaces are characterized by their increased surface energy) and if monitored horizontally, shows the effect of surface functional groups on cracks (characterized by their increased electrical charge in solution in milliequivalent per gram). Two different sets of cracks are visible on the droplet dryouts: large radial cracks and smaller orthoradial ones with a much shorter characteristic length. In such an
Sulfate −228 mV
Aldehyde-sulfate −237 mV
Carboxylate −377 mV
Type of surface functional groups
FIGURE 19.7 Final crack pattern after evaporation at 24 C and 50% humidity of a colloidal suspension of 24 nm-diameter polystyrene nanoparticles coated with three different surface functional groups with different surface free energies. Image from Carle and Brutin (2013).
Droplets of Colloids 291 isotropic material, cracks tend to form and grow in the direction in which the release of strain energy is the highest (Goehring et al., 2011). In this configuration, the stress in the deposit is due to capillary forces that develop as the solvent evaporates. At the early stages of the evaporation, particles are free to move in the solvent via Brownian motion, and the separation between the particles is subjected to the DLVO theory, where van der Walls forces are balanced by a ionic repulsive forces. As the solvent evaporates, the average distance between particles must be reduced to ensure that all of the particles remain inside the fluid; if a particle becomes exposed to air at the edge of the drop contact line, the surface energy will locally increase (Holmes et al., 2008) and will change the balance of forces. Particle reorganization in the external layer of the deposit can accommodate some of the strain, but the particles inside the lower layer tend to bind to the substrate and resist deformation, thereby preventing any possible relaxation (Chiu et al., 1993). This phenomenon leads to a stress increase, which is subsequently relaxed by plastic deformation and crack nucleation. Large cracks are the first to form, and they do so in the direction of evaporation to release the stress induced by solvent evaporation and by particle adhesion to the substrate. The stress in a gel had previously been quantified for a confined system (Zarzycki, 1988); however, for droplets, the experimental determination of the stress in such an open configuration is almost impossible. As a result, numerical analysis should instead be used. The opening of the material allows the remaining solvent, hitherto confined under the deposit, to evaporate; such drainage increases the stress in large plates of particles. The stress is relieved with the formation of smaller orthoradial cracks between the large cracks. Due to the axisymmetrical aspect of the system, the stress generated by evaporation is identical at each point in the drop. In the vicinity of a crack, the stress is null and increases with distance from the crack. When the stress reaches a maximum and exceeds the fracture strength, a new crack nucleates. The same phenomenon occurs in the deposit in the vicinity of this new crack, which forms another crack, thus inducing crack formation with a regular periodicity. These crack networks, which form a general pattern in the dryouts, are composed of a thick corona on the rim containing the majority of the particles and of a central area containing much fewer particles. Although the particle concentration is too high to form “coffee stain” patterns, the height of the corona is greater than that of the central area. The additional height explains the formation of three crack patterns. As previously explained for confined configurations, the stress increases in the material with distance, but it also increases with the thickness of the deposits. Figure 19.8 confirms this assumption for the droplet configuration and displays the ratio of length wave/deposit height as a function of the initial contact angle. This figure demonstrates that this ratio increases linearly with the contact angle (i.e., with decreasing surface energy). The three clusters of points represent droplets on the three substrates. On glass, the thin deposit and the high
292 Droplet Wetting and Evaporation 50 45 40 35
l H
30 25 20
0.381 Gold
15 10
Lysine Glass
5 0
0
10
20
30
40
50 q i(°)
60
70
80
90
FIGURE 19.8 Crack spacing/deposit height ratio plotted as a function of the initial contact angle. Side and top views show both the behavior of aldehyde sulfate droplets on the substrates and the crack patterns formed. Image from Carle and Brutin (2013).
surface energy of the substrate induced a small wavelength. In this case, the material must relax stress more frequently. However, on gold, the low adhesion and thick deposit allow the material to better resist stress. Thus, crack formation is not required as often as on glass—that is, gold induces a larger wavelength.
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294 Droplet Wetting and Evaporation Verwey, E.J.W., Overbeek, J.T.G., 1999. Theory of the Stability of Lyophobic Colloids. Courier Dover Publications, Mineola, NY, USA. Wasan, D.T., Nikolov, A.D., 2003. Spreading of nanofluids on solids. Nature 423 (6936), 156 159. Xu, J., Xia, J., Lin, Z., 2007. Evaporation-induced self-assembly of nanoparticles from a sphere-on flat geometry. Angew. Chem. Int. Ed. 46 (11), 1860 1863. Zarzycki, J., 1988. Critical stress intensity factors of wet gels. Journal of Non-Crystalline Solids 100 (1-3), 359 363.
Chapter 19
DROPLETS OF COLLOIDS 19.1 Introduction
While nanofluids are widely studied in several scientific communities (physics, biology, chemistry, etc.), several questions remain regarding their heat transfer efficiency, stability, and toxicity to human health. One important application for nanofluids is inkjet printing (Chhasatia and Sun, 2011), though nanofluids are very promising for two-phase heat transfer as well (Barber et al., 2011). Using metallic nanoparticles, inkjet printer technology is evolving to allow for the printing of metallic inks on soft substrates for photovoltaic and several others applications. These metallic inks are mostly made of silver, and consequently, they are very expensive (about $500 per milliliter). To create controlled patterns using inkjet-printing technology, a chain of droplets, whose volume has been calculated, must create lines of constant diameter following evaporation. The homogeneity of the line thickness is important because it will affect the electrical resistivity of the resulting pattern. If the concentration of nanoparticles is too low, an o-ring pattern will appear, which means that the matter will not have been uniformly deposited on the Droplet Wetting and Evaporation. DOI: http://dx.doi.org/10.1016/B978-0-12-800722-8.00019-9 © 2015 Elsevier Inc. All rights reserved.
279
280 Droplet Wetting and Evaporation substrate. If the concentration is too high, then the final thickness will also be too high, exceeding a certain thickness above which the electrical resistivity will no longer change; thus, nanoparticles, which are extremely expensive, will have been wasted. Thus, the search for the optimal concentration that avoids o-ring formation and consequently enables a uniform deposition by minimizing the amount of nanoparticles is of great interest. Several authors are currently looking at the driving parameters that lead to such pattern formation (Brutin, 2013; Askounis et al., 2014). However, because the evaporation of even a pure fluid droplet remains a complex topic, there is much research to be done to understand the evaporation of complex fluids (Conway et al., 1997; David et al., 2007; Kelly-Zion et al., 2011).
19.2 Colloidal system stability Despite the wide range of applications, the major drawback is the fact that nanofluids tend to be unstable and to flocculate. The stability of nanofluids can be greatly improved by choosing the optimal synthesis method. The choice of the method will depend on not only the fluid/particle pair but also the nanofluid stability, which plays a primary role in determining the properties of the nanofluid. To be able to completely understand the following study on the formation of cracks, a brief review of the particle interaction theory and ion neutrality is provided here. The particles will be subject to molecular interactions, such as collision between the fluid and the particle atoms and Brownian motion, and also to unwanted interactions, such as van der Waals forces (short range electrostatic interaction) or other ionic interactions. These latter phenomena will have a disastrous effect on the stability of nanofluids because large packets of material will flocculate, thus quickly reducing the effectiveness of the nanofluid. This phenomenon is well explained by the Derjaguin, Landau and Verwey Overbeek theory (DLVO theory (Verwey et al., 1999)). The DLVO theory suggests that the stability of a dispersion of colloids depends of the total potential energy VT: VT 5 VA 1 VR 1 VS ; where VA and VR are the attractive and repulsive contributions, respectively, and VS is the potential energy due to the solvent. This last contribution plays a negligible role compared to the two other potentials for nanometer-scale separation of the particles. Basically, the suspension stability results from the van der Waals forces (minus the electrical double layer forces) between two particles brought close to each other by Brownian motion. The total potential energy is dependent upon the separation between the particles (Figure 19.1).
Droplets of Colloids 281
Double layer repulsive force
Energy
Secondary minimum Net energy
Primary 0 minimum van der Waals attractive force Particle separation
FIGURE 19.1
Potential energy as a function of the particle separation.
At very small interparticle distances, van der Waals forces are predominant over electrostatic repulsion forces. As a result, the particles are forced to stick together, and the dispersion will then flocculate; this contact is particularly strong and irreversible. The strength of this attractive force decreases inversely with the particle separation. Therefore, repulsive electrostatic forces become dominant, and the particles repulse each other. The net energy admits two different levels of energy minima; all of the particles will tend to stay in these two positions of energy minima. The primary minimum is located at the contact of the particle (flocculation of the nanofluid), and the second one is behind an energy potential barrier. This second minimum is where all of the particles should be to produce a stable nanofluid. Unfortunately, if the particles have sufficient energy to go through this barrier, the nanofluid will flocculate. To avoid flocculation, the repulsive forces must be as high as possible to have the potential barrier be as high as possible. High repulsive forces can be achieved by using surfactants or functional groups to form either a steric barrier (e.g., polymers and plasticizers) or an ionic barrier (e.g., electrostatic stabilization from the electric double layer and from sol-gel precursors) to keep the particles separated and to avoid cluster formation. These additions greatly modify the suspensions but have yet to be investigated for their influence on the crack patterns. The strong effect of surface functional groups hereinafter referred as surface functional groupments on the final crack patterns of nanofluid deposits is described in this section. Surface functional groups have never before been taken into account in studies on drying, crack patterns, or self-assembly of nanoparticles prior to Carle (2014). These functional groups are, however, widely used in several communities—for example, as biological tracers with fluorescent spheres, cancer treatment using their surface functional groups to bond a dedicated protein (Arima and Iwata,
282 Droplet Wetting and Evaporation 2007), or even for inkjet printers, using metallic nanoparticles to create monoatomic layers on various hydrophilic or hydrophobic substrates (Chhasatia and Sun, 2011). For all of these applications, surface functional groups are crucial for nanoparticles to be compatible with a variety of conjugation strategies. Both the convective self-assembly of nanoparticles and the final crack patterns are affected by physicochemical interactions with surface functional groups. Interactions between surface functional groups (Frisbie et al., 1994) and solvents (Halicioglu and Jaffe, 2002) were investigated, but more research is required on the influence of the interaction between surface functional groups itself during the formation of close-packed particle assemblies and also on fracturing during drying.
19.3 Mono-dispersed colloidal systems The evaporation dynamics of a nanofluid sessile droplet follow the same behavior as those of pure fluid sessile droplets in the first stages of evaporation. Figure 19.2 allows for the description, through the example of an aldehydesulfate nanofluid droplet, of colloid evaporation dynamics. 4
Spreading
Mass (µg)
−
3
dm (µg s−1) dt
4 3 2 1 0 0
250
2
500
750 1000 1250
T (s)
Deposition ring Depinning 1
Cracks formation 0
0
250
Final pattern
500
750
1000
1250
Time (s)
FIGURE 19.2 Evaporation of a 4 µL aldehyde-sulfate nanofluid droplet onto a lysine substrate at 24 C and 50% of humidity. Droplet top view pictures show the major evaporation events, along with a schematic representation of droplet interface and particle motion. The inset shows the droplet evaporation flux rate. Figure from Carle (2014).
Droplets of Colloids 283 In the very first instant of the droplet creation, as soon as the fluid touches the substrate, the droplet spreads to reach its maximum wetting diameter. The spreading influence of nanofluids was observed (Chengara et al., 2004; Nikolov et al., 2010), but it was not obvious in all of the experiments performed here. After a few minutes, a circular deposit is clearly visible at the edge of the drops. As we previously observed in Section 5.3, the sessile droplet evaporation profile exhibits a nonuniformity at the triple line; this nonuniformity leads to a radial flow toward the triple line to compensate for the evaporation loss at the contact line (Deegan et al., 1997). Therefore, nanoparticles are dragged outward with this flow in a fairly quick process and then form a deposit in the vicinity of the triple line. Here, the contact line is pinned during most of the evaporation process, whereas the contact angle and the height decrease linearly. As the solvent evaporates, the particles are forced to concentrate near the triple line in a close-packed array (Figure 19.3). Figure 19.3 depicts remarkable cracks that have been found during SEM observations of the dried droplets. In addition to being dramatic nanoscenery, these pictures summarize the physics involved in the crack formation patterns. Picture (A), which was taken at the center of an aldehyde-sulfate droplet on glass, confirms the preferential paths of the cracks in the direction of the evaporation.
(A)
(B)
(C)
(D)
FIGURE 19.3 Remarkable SEM observations of particles deposits. (A) Aldehyde sulfate on glass, (B) carboxylate on lysine, (C) and (D) carboxylate on gold. The scale represents 50 µm for the top pictures and 500 nm for the bottom pictures. Figures from Carle (2014).
284 Droplet Wetting and Evaporation The depinning of the triple line and its receding toward the center induce the cracks to meet at the center of the droplet. Pictures (C) and (D) provide a glimpse of the forcefulness of the formation of the cracks. The compacted layers of nanoparticles are ripped apart by stress due to evaporation. The interesting aspect of these SEM observations is the ability to locally observe the sharpness of the cracks, enabling determination of the rigidity of the material created by the deposition, and the ability to observe the fragility of the deposits, with the broken pieces near the cracks observed in the image (B). Coming back to drying, the pinned evaporation continues for approximately 60% of the total time of the evaporation until the contact line recedes when the critical contact angle is exceeded. The depinned contact line shrinks toward the center. The motion of the solvent induces a stress in all deposits, resulting in the nucleation of cracks. Once all of the solvent has evaporated, the dried particles completely cover the initial wetting area due to their relatively high concentration and form regular crack patterns. Experiments were performed several times for each configuration, and the crack patterns were found to be replicable. Several conditions must be met for the droplets to form a deposition ring on the substrate: first, the solvent must not completely wet the substrate (initial contact angle larger than 0 ); otherwise, the particle deposition will form a layer that will tend to crack, following the direction of evaporation. Then, the triple line must pin a major part of the evaporation to initiate a deposition position that remains the same to accumulate enough particles. Stick slip droplets tend to form irregular concentric rings (Bhardwaj et al., 2009). This phenomenon, when forced to occur regularly, can be used to form patterns for industrial applications: homogeneous rings with a perfect spatial distribution by means of solid spheres in the droplets (Xu et al., 2007) and evaporation onto patterned hydrophobic substrates in the form of pillars (Fan and Stebe, 2004). As an interesting counterexample, the coffee ring effect can be suppressed by using electrowetting (see Section 5.3). Other nanoparticles than polystyrene are used in the literature. Denkov et al. (1992) used latex particles. In the case of mixtures of more complex fluids, at low concentrations, Govor et al. (2004) evidenced the formation of nanoparticle rings following the evaporation of droplets for a liquid matrix of a binary mixture of nitrocellulose, amyl-acetate, and hexane. The authors used 6 nm nanoparticles of CoPt3 at concentrations below 1%, leading to rings measuring 0.6 1.5 µm in diameter. They experimentally observed the phase separation that leads to the formation of a bilayer structure. The CoPt3 nanoparticles located on the contact line assembled along the line. In this case, the accumulation at the drop edge was pronounced. Conway et al. (1997) first investigated the influence of particles on the evaporation dynamics of a drop. Using 200 and 750 nm polystyrene beads, the author observed the formation of an o-ring and a crater for these bead diameters
Droplets of Colloids 285 and monitored the drop height and mass over time. The author provides a linear normalized model of the drop mass and normalized model of the drop height using a root square function. Using 1 µm particles with a disparity going from 0.5 to 2 µm, Marin et al. (2011) demonstrated that the particles that settle first along the contact line of a coffee ring pattern have an ordered, crystalline structure. Toward the center of the drop, a transition to a disordered particle arrangement was observed. The authors propose a model that explains many results described in the literature where large-scale crystalline deposits were observed for particle sizes ranging from 6 to almost 50 nm. In this range of particle sizes, the critical velocity required for a disordered phase is unreachable within the droplet, which explains the absence of a disordered phase. The authors also state that apart from the order-to-disorder transition, there are also transitions within the ordered phase, from square to hexagonal packing and vice versa (Figure 19.1D). In an unconfined system, the most efficient packing lattice is the hexagonal compact. However, in the evaporating drop the particles are confined in a wedge formed by the glass slide on one side and the liquid air interface with contact angle on the other side. Such a confinement indeed leads to a sequence of hexagonal and square packing structures that depend on the most efficient packing for the available space. Indeed, when a new layer is formed, the confinement by the wedge favors the square packing; see Figure 19.1. Away from the step edge, the available space increases and allows for the denser, hexagonal packing structure (Figure 19.4).
19.4 Bidispersed colloidal systems O-ring formation during colloidal drop evaporation was first explained by Deegan (2000). By using sulfate-terminated polystyrene spheres of two sizes, 1 and 0.1 µm, at a maximum volume concentration of 2% dried on glass plates, the author observed the formation of different patterns, depending on the concentration and size of the spheres used. For spheres measuring 100 nm and concentrations ranging from 0.063% to 0.25%, Deegan observed the formation of multiple rings for initial droplets measuring 6 mm in diameter. In all cases, particles remained at the drop center and were not only located at the drop edge. In this study, we observed two different final patterns, depending on the concentration: an o-ring pattern and a nearly uniform deposition pattern on the substrate. Chhasatia and Sun (2011) used bidispersed colloids made of micro- and nanoparticles and observed the interaction between these particles and the contact line. For 100 nm and 1.1 µm particles, they observed that, at the end of the evaporation process, the nanoparticles were located near the triple line, followed by the microparticles (Figure 19.5). For a hydrophobic substrate, the constant contact angle
286 Droplet Wetting and Evaporation (A)
(B)
0.5 mm
0.5 mm (D)
(C)
10 µm
5 µm
FIGURE 19.4 Order-to-disorder transition in the particle stain left by an evaporating drop. (A) A 3 µL sessile water droplet evaporates from a glass substrate. (B) Ringshaped stain of red particles (the coffee stain) left on the substrate after evaporation. (C) A close-up by an optical microscope of the bottom layer of the stain, taken from the white square in (B), shows that the outermost lines of the stain (left) have an ordered, crystalline structure. Toward the center of the drop (right), a transition to a disordered particle arrangement is observed. (D) A top view of the ring stain, taken from the red square in (C) with a scanning electron microscope (SEM), shows that the first lines of particles (left) are arranged in a hexagonal array, while the next lines (brighter in the image) are arranged as square, followed by again hexagonal array. Images from Marin et al. (2011).
mode dominates the evaporation process, leaving little carrier liquid left to rearrange particles according to their sizes at the mixed mode. For a hydrophilic substrate of 0 , θ , 45 , particle separation is incomplete in the final deposition. Three separate regions exist from the outer edges to the center of the drop, including a region with only nanoparticles, a mixture of micro- and nanoparticles, and an inside region with only microparticles. The width of the middle region where micro- and nanoparticles overlap depends on the interactions of the surface tension and friction forces acting on the particles. For a hydrophilic substrate of θ 5 0 , nanoparticles accumulate at the contact line, forming several rows that prevent the contact line from receding. This enhanced pinning by nanoparticles permits radially outward evaporative flow toward the pinned contact line and pushes some of the microparticles to exceed through the liquid vapor interface. Reduced surface tension acting on those exceeded microparticles prevents them from moving inward, whereas most other microparticles move radially inward to
Droplets of Colloids 287
Contact line gets pinned at the beginning of constant contact area mode
Contact line gets pinned at the beginning of constant contact area mode
Nanoparticles move closer to the pinned contact line and delays constant contact angle stage Particles mix during constant contact angle mode
In mixed mode both micro and nanoparticles move closer to liquid–vapor interface
Micro and nanoparticles remain mixed in the deposition
(A)
Contact line gets pinned at the beginning of constant contact area mode
Nanoparticles move closer to the pinned contact line
Particles mix during receding
Nanoparticles move towards contact line in the mixed mode
Nanoparticles get deposited at the outer edge, not clear separation
(B)
Microparticles recede toward center of the drop due to capillary flow
Nano and microparticles from separate rings
(C)
FIGURE 19.5 Schematic of the evaporation process of bidispersed particles (100 nm and 1.1 µm) on three substrates with contact angles of: (A) θ . 45 (mixed), (B) 0 , θ , 45 (partial separation), and (C) θ 5 0 (complete separation). Sketches from Chhasatia and Sun (2011).
the center of the drop. By modifying substrate wettability, the surface tension force acting on the particles is tuned. The control of particle separation according to their sizes can therefore be achieved. The authors have also found that the increase in the size ratio of the bidispersed particles improves particle separation. Because the number of particles affects contact line pinning, the boundaries between mixed, partially separated, and completely separated regimes shift toward a higher contact angle value for a higher particle loading. By controlling the substrate wettability, particle size ratio, particle loading, and relative humidity, the deposition morphology of bidispersed particles can be further controlled. During the drying of droplets containing latex particles measuring from 40 nm to 5 µm, Monteux and Lequeux (2011) observed a rearrangement that led to the formation of rings. The smallest nanoparticles were deposited on the outer side of the ring, while the largest particles were located on the inner side. The authors conclude that this deposition mechanism leads to a tight deposition but leaves a thin liquid film of pure water at the edge of the drop. The deposited particle ring size can be precisely controlled by changing the particle size. The segregation effect may consequently play an important role in polymer and biological fluids, which are composed of polydisperse particles.
288 Droplet Wetting and Evaporation Pattern formation has also been observed for biological fluids, such as whole blood, very recently and shows similar patterns to those of nanofluids (Brutin et al., 2011). The drying dynamics observed for this complex biological fluid show typical regimes that are treated in Section 4.5. Finally, pattern formation is also influenced by the wettability of a droplet on a substrate. The relationship between the spreading of a droplet and the evaporation dynamics has been previously discussed in Section 1.1. The triple line dynamics and the pinning of a drop play a major role in the drying process dealt with in Section 1.3.
19.5 Parameters influencing the patterns Obviously the colloids concentration is the first parameter that influences the pattern. Brutin in 2013 evidenced a critical concentration to observe the classical o-ring formation. Above this concentration a more homogeneous deposition is observed. The author used polystyrene nanoparticles measuring 24 nm that were suspended in deionized water to study the effect of nanoparticle concentration and surrounding humidity on the pattern formation and drying dynamics of sessile droplets. Droplets were deposited on rough Nuflon 16X-coated substrates. The patterns formed were observed to have been influenced mainly by the concentration of nanoparticles but not at all by the drying time. Two patterns were observed: an o-ring formation for concentrations below 0.47% and a flower pattern above 1.15%. The effect of humidity was proportional to the drying rate, but no effect on pattern formation was observed. The experimentally measured evaporative mass flux was slightly higher compared to that predicted by a purely diffusive model, which may indicate a slight enhancement in heat transfer at the triple line due to the local improvement in the substrate heat capacity by nanoparticle deposition. For pure fluids, the effect of the substrates thermal properties was investigated, even at ambient temperature, because the latent heat of evaporation is mostly affected by element with the highest heat capacity, which is the substrate (Sobac and Brutin, 2012). One explanation provided by the author is that the diffusion coefficient of our water-based nanofluid, even at low concentration, in humid air is different from that of pure water in dry air. Experiments are currently underway to address this discrepancy. Another explanation is based on the experimental observation in the literature of an increase in the spreading of nanofluids leading to an increase in the drop perimeter and a simultaneous decrease in the contact angle. The consequence is an improvement in the evaporation rate. This assumption is also under investigation at this time (Figure 19.6). Chhasatia et al. (2010) investigated the effect of relative humidity on the contact angle and nanoparticle deposition during the evaporation of a picoliter droplet of a nanofluid. They observed, for the pinned droplet mode, a spreading effect with the increase in relative humidity. The observed variations in the contact angle
Droplets of Colloids 289 Flower pattern
C = 5.70%
C = 4.80%
C = 2.30%
C = 1.15%
C = 0.47%
C = 0.23%
C = 0.11%
C = 0.05% O-ring pattern
C = 0.01%
FIGURE 19.6 Influence of colloids concentrations ranging from 0.01% to 5.70% on the pattern formation (controlled glovebox humidity of 50% and room temperature of 25.0 6 0.6 C). Image from Brutin (2013).
were linear but exhibited different slopes depending on the relative humidity. The authors conclude that due to the small volume of the droplet, the relative humidity becomes an important parameter driving the evaporation process and consequently the deposition. They also observed an increase in the initial spreading of the drop with increasing humidity. Highly concentrated colloidal suspension exhibits spreading and evaporation instabilities, as observed by Hadj-Achour and Brutin (2014). The authors report a fingering instability that occurs during the spreading and evaporation of a silver metallic ink nanosuspension droplet concentrated at about 20% in mass so far above the low concentrations classically observed. The fingering instability has been widely studied for both Newtonian and non-Newtonian fluids. However, this report describes the first time that a fingering instability is observed for the spreading of a nanosuspension sessile droplet. In this study, the authors demonstrate that in certain cases, the contact line evolves through different spreading regimes according to De Coninck et al. (2001) with an enhancement in the evaporation rate due the formation of the fractal patterns.
290 Droplet Wetting and Evaporation
19.6 Nanoparticles surface charges
Lysine γ = 65.5 mJ m–2 Gold γ = 43.5 mJ m–2
Substrates
Glass γ = 71.4 mJ m–2
Working with nanoparticles, the surface charges exposed at their vicinity cannot be neglected. Carle and Brutin, 2013 studied the influence of this charge on the pattern and the crack periodicity. Working with three substrates of different surface energies and the same nanoparticles but only with different surface charges, they evidenced strong differences in the final pattern. Figure 19.7, if observed vertically, shows the final crack patterns of each nanofluid on three substrates (surfaces are characterized by their increased surface energy) and if monitored horizontally, shows the effect of surface functional groups on cracks (characterized by their increased electrical charge in solution in milliequivalent per gram). Two different sets of cracks are visible on the droplet dryouts: large radial cracks and smaller orthoradial ones with a much shorter characteristic length. In such an
Sulfate −228 mV
Aldehyde-sulfate −237 mV
Carboxylate −377 mV
Type of surface functional groups
FIGURE 19.7 Final crack pattern after evaporation at 24 C and 50% humidity of a colloidal suspension of 24 nm-diameter polystyrene nanoparticles coated with three different surface functional groups with different surface free energies. Image from Carle and Brutin (2013).
Droplets of Colloids 291 isotropic material, cracks tend to form and grow in the direction in which the release of strain energy is the highest (Goehring et al., 2011). In this configuration, the stress in the deposit is due to capillary forces that develop as the solvent evaporates. At the early stages of the evaporation, particles are free to move in the solvent via Brownian motion, and the separation between the particles is subjected to the DLVO theory, where van der Walls forces are balanced by a ionic repulsive forces. As the solvent evaporates, the average distance between particles must be reduced to ensure that all of the particles remain inside the fluid; if a particle becomes exposed to air at the edge of the drop contact line, the surface energy will locally increase (Holmes et al., 2008) and will change the balance of forces. Particle reorganization in the external layer of the deposit can accommodate some of the strain, but the particles inside the lower layer tend to bind to the substrate and resist deformation, thereby preventing any possible relaxation (Chiu et al., 1993). This phenomenon leads to a stress increase, which is subsequently relaxed by plastic deformation and crack nucleation. Large cracks are the first to form, and they do so in the direction of evaporation to release the stress induced by solvent evaporation and by particle adhesion to the substrate. The stress in a gel had previously been quantified for a confined system (Zarzycki, 1988); however, for droplets, the experimental determination of the stress in such an open configuration is almost impossible. As a result, numerical analysis should instead be used. The opening of the material allows the remaining solvent, hitherto confined under the deposit, to evaporate; such drainage increases the stress in large plates of particles. The stress is relieved with the formation of smaller orthoradial cracks between the large cracks. Due to the axisymmetrical aspect of the system, the stress generated by evaporation is identical at each point in the drop. In the vicinity of a crack, the stress is null and increases with distance from the crack. When the stress reaches a maximum and exceeds the fracture strength, a new crack nucleates. The same phenomenon occurs in the deposit in the vicinity of this new crack, which forms another crack, thus inducing crack formation with a regular periodicity. These crack networks, which form a general pattern in the dryouts, are composed of a thick corona on the rim containing the majority of the particles and of a central area containing much fewer particles. Although the particle concentration is too high to form “coffee stain” patterns, the height of the corona is greater than that of the central area. The additional height explains the formation of three crack patterns. As previously explained for confined configurations, the stress increases in the material with distance, but it also increases with the thickness of the deposits. Figure 19.8 confirms this assumption for the droplet configuration and displays the ratio of length wave/deposit height as a function of the initial contact angle. This figure demonstrates that this ratio increases linearly with the contact angle (i.e., with decreasing surface energy). The three clusters of points represent droplets on the three substrates. On glass, the thin deposit and the high
292 Droplet Wetting and Evaporation 50 45 40 35
l H
30 25 20
0.381 Gold
15 10
Lysine Glass
5 0
0
10
20
30
40
50 q i(°)
60
70
80
90
FIGURE 19.8 Crack spacing/deposit height ratio plotted as a function of the initial contact angle. Side and top views show both the behavior of aldehyde sulfate droplets on the substrates and the crack patterns formed. Image from Carle and Brutin (2013).
surface energy of the substrate induced a small wavelength. In this case, the material must relax stress more frequently. However, on gold, the low adhesion and thick deposit allow the material to better resist stress. Thus, crack formation is not required as often as on glass—that is, gold induces a larger wavelength.
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294 Droplet Wetting and Evaporation Verwey, E.J.W., Overbeek, J.T.G., 1999. Theory of the Stability of Lyophobic Colloids. Courier Dover Publications, Mineola, NY, USA. Wasan, D.T., Nikolov, A.D., 2003. Spreading of nanofluids on solids. Nature 423 (6936), 156 159. Xu, J., Xia, J., Lin, Z., 2007. Evaporation-induced self-assembly of nanoparticles from a sphere-on flat geometry. Angew. Chem. Int. Ed. 46 (11), 1860 1863. Zarzycki, J., 1988. Critical stress intensity factors of wet gels. Journal of Non-Crystalline Solids 100 (1-3), 359 363.