Sessile nanofluid droplet drying

    Sessile Nanofluid Droplet Drying Xin Zhong, Alexandru Crivoi, Fei Duan PII: DOI: Reference:

S0001-8686(14)00320-0 doi: 10.1016/j.cis.2014.12.003 CIS 1504

To appear in:

Advances in Colloid and Interface Science

Please cite this article as: Zhong Xin, Crivoi Alexandru, Duan Fei, Sessile Nanofluid Droplet Drying, Advances in Colloid and Interface Science (2014), doi: 10.1016/j.cis.2014.12.003

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Sessile Nanofluid Droplet Drying

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Xin Zhong, Alexandru Crivoi, Fei Duan1

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School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798

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Abstract Nanofluid droplet evaporation has gained much audience nowadays due to its wide applications in painting, coating, surface patterning, particle deposition, etc. In this review, we highlight the evaporative dynamics and particle deposition process by introducing experimental and theoretical works on drying of sessile droplets with suspended insoluble solutes, especially nanoparticles. The effects of the type, concentration and size of nanoparticles on the spreading and evaporative dynamics are elucidated at first, serving the basis for the following introduction of particles motion and deposition process. Stressing on particles assembly and production of preferable residue patterns, abundant experimental interventions, various types of deposits and effects on nanoparticles deposition are then expressed. At last, theoretical investigations, in terms of solutions of the Navier-Stokes equations, the Diffusion Limited Aggregation approach, the Kinetic Monte Carlo method, and the Dynamical Density Functional Theory, are discussed.

Keywords: Nanofluid sessile droplet evaporative dynamics drying patterns

Contents

1 Introduction .................................................................................................................... 2 2 Dynamic Evaporation of Nanofluid Droplets ............................................................. 3 2.1 Effect of Nanoparticles on Spreading Dynamics .................................................... 4 2.1.1 Static Contact Angle ................................................................................ 4 2.1.2 Contact Line Motion................................................................................ 6 2.2 Evaporative Rate of Nanofluid Droplet .................................................................. 8 2.3 Effect of Nanoparticles on Evaporation Regimes .......................................... …. .11 2.4 Flow Dynamics within Nanofluid Droplet .............................................................. 9 3 Self-Assembly and Patterns Formed by Nanoparticles in Droplet .......................... 15 3.1 Experimental Conditions of Nanofluid Droplet Drying ........................................ 15 1

Tel: +65 6790 5510; Fax: +65 6792 4062 Email address: [email protected] (Fei Duan )

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3.2 Drying Patterns on the Substrate ........................................................................... 17 3.2.1 Coffee-Ring Pattern .............................................................................. 17 3.2.2 Uniform Pattern .................................................................................... 18 3.2.3 Stick-Slip Pattern ................................................................................... 18 3.2.4 Fingering Structure ................................................................................ 20 3.2.5 Combined Pattern ................................................................................. 21 3.3 Effects of Parameters on Dried Patterns .............................................................. 22 3.4 Theoretical Studies on Nanofluid Droplet Drying ............................................... 24 3.4.1 Navier-Stokes Equations ...................................................................... 25 3.4.2 Diffusion Limited Aggregation ............................................................ 26 3.4.3 Kinetic Monte Carlo Model.................................................................. 28 3.4.4 Dynamical Density Functional Theory ................................................ 30

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4 Conclusion ..................................................................................................................... 22

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5 Acknowledgements ...................................................................................................... 32

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1 Introduction

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References .......................................................................................................................... 33

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It is a universal phenomenon that a liquid droplet evaporates when it is surrounded by its unsaturated vapor. The commonly observed cycle of water in nature is an example of the water vapor condensation and liquid evaporation of droplets or other forms [1]. When a droplet contains insoluble particles, a deposition pattern would be formed on the solid surface after evaporation. The movement of particles is correlated with the mode of droplet evaporation and the dynamics of the three-phase line. The coffee ring, for instance, was formed by solutes or suspensions driven by an outward flow in a pinned droplet [2], [3], [4]; the uniform pattern can be produced by controlling the evaporation kinetics and particle interactions at the liquid/air interface [5]; the concentric rings were composed as a result of alteration of pinning and depinning behaviors of the three-phase line [6], [7], [8]. Thus, it is crucial to study the evaporative dynamics of a sessile droplet and the influence resulting from the presence of the solutes. Droplet shape, spreading dynamics, evaporation rate and evaporation regimes are the factors affected by the addition of particles. Controlling the organization of particles and the final deposit patterns can be achievable from further understanding of droplet evaporative dynamics. Changing a number of intrinsic parameters and external conditions could generate diverse deposition patterns, including the central-congregated bump [9], combined structures and highly ordered organization of particles as well as the three aforementioned patterns. These distinctive patterns are presented in different applications. The coffee-ring effect, for example, is shown in the ink-jet printing [10], while a stretched individual particle is desirable in the DNA deposition process [11]. In this review, we primarily stress on the evaporation of droplet containing nano-sized particles, known as the nanofluids. Nanofluids are a kind of engineered colloidal suspensions that exhibit distinctive properties, such as thermal conductivity [12], [13], heat capacity [14] and viscosity [15], [16], as compared with the relative pure liquids. The dynamic drying process and

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the evaporative-induced self-assembly in a sessile droplet significantly depend on the type, size and shape of nanoparticles. In addition to highlight nanoparticles role, we mainly emphasize the general concepts of sessile droplet and the major mechanisms in controlling the liquid flow. Spreading dynamics, evaporation rate, evaporation regimes and flow dynamics are four parts exhibiting varieties due to the presence of nanoparticles. Afterwards, a series of experimental approaches to control the self-assembly of nanoparticles, types of deposition patterns and parameters effecting patterns formation are explicated. Finally, four main theoretical methods are depicted for the evaporation of macro-scale droplets and micro-scale films and the process of particle aggregation. It is worth noticing that some investigations studying the effect of colloidal particles which are not nanoparticles are included as well, as they shed light on the dynamics and the complex evaporation and deposition processes in this review.

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2 Dynamic Evaporation of Nanofluid Droplets

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In comparison with pure liquid droplets, the droplets containing nanoparticles show a great variation in spreading, evaporation and liquid motion. These variations caused by nanoparticles are mainly revealed here in static contact angle, motion of the contact line, evaporation rate, evaporation regimes and flow dynamics in a drying droplet.

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2.1 Effect of Nanoparticles on Spreading Dynamics

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The spreading of a sessile nanofluid droplet on a solid surface constitutes two primary stages. Once the droplet is touched with the solid surface, it undergoes a rapid spreading driven by capillary force and gravity force, until it reaches an equilibrium state. The process is relatively rapid and the droplet could soon reach balance. Then it comes into the second stage when a wedged film ahead of the bulk droplet has formed. Nanoparticles tend to arrange orderly in the film, resulting in an additional pressure, known as the disjoining pressure which enhances the spreading behavior. To elaborate the effect of nanoparticles on the spreading process, we should pay attention on both static wetting and dynamic wetting of the droplet. To reflect them, the static contact angle and contact line motion are investigated. 2.1.1 Static Contact Angle When a droplet sits on a smooth and homogeneous surface without any fluid motion, its contact angle in Figure 1, is the static contact angle, which is normally used to define static wetting behavior of liquid on a substrate. The Young’s equation in Eq. (1) was proposed to describe the static contact angle and the surface tensions at the contact line of a droplet under static state [17] .

γ LV cos θ = γ SV − γ SL

(1) where γ LV , γ SV , and γ SL are the liquid-vapor, solid-vapor, and solid-liquid surface tensions respectively. Noted that Young’s equation describes an ideal situation with a smooth, rigid, and homogeneous surface. The contact angle hysteresis is neglected and the effects of droplet size and possible depositing ways are not considered. In reality, the defects on a solid surface are normally inevitable, thus the observed contact angle on a non-ideal solid surface is in a range of value

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frequently instead of solely one value. However, the contact angle in Young’s equation can still be used locally for both smooth and non-smooth solid surfaces, in order to reflect the relative strength of the molecular interaction, known as the wettability among liquid, solid, and vapor phases. Strong wettability usually results in a relatively small contact angle while poor wettability leads to a larger contact angle. One key factor primarily determining the degree of wettability and contact angle is whether the solid surface is hydrophilic or hydrophobic. Förch, et al. defined that the solid surface is hydrophilic if the contact angle is smaller than 90
; otherwise, it is hydrophobic [18]. The addition of nanoparticles into droplets have been widely found to vary the static contact angle considerably. One key factor is the deposition of nanoparticles that could vary the superficial characteristics of the solid surface, such as the local roughness and topography. Moreover, the adsorption of nanoparticles in the vicinity of the three-phase line was experimentally found to modify the liquid-solid surface tension evidently [19]. The modified superficial characteristics and the liquid-solid surface tension were considered to be the main reasons for greatly enlarged contact angles in the experiments of droplets with added titanium oxide (TiO 2 ) [20] and aluminium (Al) [19] nanoparticles at low concentrations as compared to the pure liquid droplets. The case of TiO 2 nanoparticles demonstrated the TiO 2 clusters are more like to deposit at the location with a distance away from the contact line. The non-uniform deposition of TiO 2 clusters could form a heterogeneous surface: the area covered by TiO 2 clusters is more hydrophilic, while the contact line region is more hydrophobic due to less TiO 2 residuals. The hydrophilic area has higher surface energy than as the hydrophobic contact line region, thus the contact line tends to absorb towards the hydrophilic area, accordingly shrinks the wetting area and enlarges the contact angle [20]. This enlargement in contact angle due to nanoparticles was also observed by Vafaei, et al. [21]. Besides the altered surface tension and nanoparticles assembly near the contact line, they concerned that the enlarged contact angle might also result from a repulsive force between the negative charged carboxylic groups of nanoparticles and the hydroxyl groups on the silicon wafer substrate. The electrostatic-interacting force depended on the nature of nanoparticles as they could be charged either positive or negative in suspensions. Notably herein the contact angle increases only when the nanofluid concentration is relatively low. At a relatively high nanofluid concentration, agglomeration and clustering of nanoparticles have a higher chance to occur, which have been corroborated by Radiom, et al. in the experiment [20] that the larger effective size of nanoparticles could prevent the nanoparticle aggregate from accessing the wedged film region, so the number of available nanoparticles taking effect in varying the contact angle could be reduced. It is in consistent with the experimental outcome obtained by Vafaei, et al. [21], that smaller-sized nanoparticles led to greater change in contact angle, and that as the nanoparticle concentration increased to a point the contact angle reached to peak and then started to decrease. The first ascending and then descending trend was observed as well by adding sub-micron polystyrene beads at differen sizes [22], and bismuth telluride (Bi 2 Te3 ) [23]. 2.1.2 Contact Line Motion Contact line shown in Figure 1 for a sessile droplet is a three-phase line where the liquid, vapor and solid surfaces coexist. The dynamic motion of the contact line reveals the evolution of the droplet shape, and the deposition patterns if the droplet contains particles. Pinning, depinning, and stick-slip are three primary behaviors of the contact line, resulting in a coffee ring, a relatively

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into motion [27].

∏ , the ∏ can be

comprises three components: the structural component

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∏(h) molecular component ∏

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uniform pattern, and concentric rings respectively [24], [6]. In order to detect liquid movement during droplet spreading, Dussan and Davis [25] added a dye into the upper surface of the wedge-shaped film of a droplet, and found that the dye transported to the edge first, and then get stuck to the solid surface, representing a rolling trajectory. Therefore, the energy burns in a way of viscous friction mainly in three regions [26]: (1) the wedged region representing the nominal contact line; (2) the precursor film ending up with a real contact line; and (3) the real contact line. The addition of nanoparticles was found to vary the energy in Regions (1) and (3) through the particles deposition, and changed surface tension and viscosity accordingly [20]. The effect of nanoparticles on energy dissipation in the precursor film is primarily resulted from nanoparticles self-assembly in the thin film ahead of a droplet. Due to the narrow space of the film confined by liquid-solid and liquid-vapor interfaces, there is an induced excess pressure, defined as the disjoining pressure, ∏(h) , driving the liquid in the film and the electrostatic component

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induced by nanoparticles layer-by-layer assembly in the thin film, within which

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exponential decay with the increasing film thickness, which is in corresponding to nanoparticles self-assembly in the thin film [28], [30], [31], [32], as shown in Figure 2. The function of the energy in the precursor film, T ∑ f , can be expressed as, T ∑ = US ,

(2)

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∏(h)dh

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where U is the liquid velocity; S is the spreading coefficient, identifying whether a droplet can spread or not; ∏0(h0 ) is the sum of capillary pressure and hydrostatic pressure; and h0 is the equilibrium thickness of the film. Equation (3), known as the Frumkin-Derjaguin equation, indicates that any change in disjoining pressure would lead to the change in T ∑ f , and thus vary the spreading behavior of the droplet. The induced structural disjoining pressure and the varied superficial characteristic of the solid surface due to nanoparticles deposition have been confirmed to promote the pinning effect of the droplet. Compared to that of a pure liquid droplet, the contact line of a nanofluid droplet might be prevented from receding since it would pass through nanoparticle residuals on the way of sliding back. Joanny and de Gennes [33] proposed a model to describe this phenomenon by considering a deformation of a contact line due to the presence of a single circular defect. It was found that the contact line deformed around the defect but stayed intact away from it. Many experimental studies have verified the enhanced pinning effect led by nanoparticles [20], [34], [35], [36], [32], [37], [38]. In a study of microliter-sized droplets containing 250 nm gold particles, the droplet drying exhibited a second pinning stage rather than the behavior of the pure liquid

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droplet, revealing a significantly pronounced pinning effect [36]. This was possibly due to an increase in the disjoining pressure as a result of particle-ordering in the meniscus-film region [32]. One essential factor determining the pinning effect of the contact line is the geometry of the particle which affects the way the particles interact and organize. The commonly employed spherical particles are able to form a coffee ring effortlessly, as they are easy to transport to the periphery and pin the contact line. Marin, et al. investigated the orderly-formed coffee ring by spherical microspheres and found that the ring region displayed an order-to-disorder transition, as shown in Figure 3. Spheres arrived at the outermost part of the ring organized as square, followed by a hexagonal arrangement, and then a disorder assembly [39]. This is attributed to the accelerated velocity of the spheres in the proceeding of evaporation of a pinned droplet. This transition was also observed by using hard spheres at microscale [40], silicon dioxide (SiO 2 ) nanoparticles [41], [42], and polystyrene latex [43]. The transition of self-organization at the contact line region was also manifested in the droplets containing spindly carbon nanotubes (CNTs) [44], [45], [46]. CNTs, in particular the single-wall carbon nanotubes (SWNTs), were observed to be driven by a capillary flow in the sessile droplet, and tended to align in parallel to the periphery to form a monodomain. In the region away from the contact line, nanotubes deposited more randomly [46]. Similar outcomes were found by Li, et al. The transition from parallel to random organization was related to the proceeding of the evaporation. At an early stage, CNTs preferred to direct themselves along the contact line due to a strong edge effect; as the deposition proceeded, the edge effect was less strong, thus CNTs began to orient along the direction of the radial outward flow [44]. Zeng, et al. ascribed the parallel and perpendicular assemblies to capillary force and fingering instability at the contact line respectively. They obtained the desirable assemblies of CNTs based on manipulation of these two mechanisms with the appropriate configuration of wedge. [45]. In the above reports, the coffee-ring effect was present during the entire evaporation. With the employment of ellipsoidal particles, however, the coffee-ring effect was suppressed reported by Yunker, et al. An aspect ratio, α , was applied to represent the shape of the particles. The spherical particles, α = 1, produced a typical coffee-ring deposition. When α is 3.5, the ellipsoids particles tended to form a uniform deposition. The elongated shape of the particles was found to enhance the long-ranged particle-particle interactions, and thus led to loosely-packed aggregates at a relatively large scale which prevented particles from transporting to the droplet edge. Besides, the ellipsoidal particles were easy to adhere to and thus deformed the liquid-vapor interface [47]. Although the droplets containing both kinds of the particles remain pinned on the solid surface during evaporation, the amount of the spherical particles depositing at the contact line region is way larger. We could extrapolate that the droplet containing spheres display a stronger pinning effect.

2.2 Evaporative Rate of Nanofluid Droplet In most calculations of evaporation rate for pure liquid droplets, two assumptions are made: (1) the evaporation process is treated as a quasi-steady state, and (2) vapor is transported solely by diffusion (convection is neglected). These assumptions stem from the well-known Maxwell-assumption [48]. Under specific conditions, such as the liquids with large saturation vapor concentration values, transient and convection terms should be taken into account as they contribute an nontrivial portion to the evaporation [49]. Evaporation was found to occur non-uniformly along the droplet interface, resulting in varied temperature field and flow dynamics

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π − 2θ 2π − 2θ

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r (4) j = j0 (1 − ( ) 2 ) − λ (θ ) R where j0 is the evaporation flux at the centerline of the sessile droplet surface, R is the initial radius of the droplet, r is the distance from the droplet center, and θ is the contact angle. λ is a function of θ and is expressed as,

(5)

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As illustrated in Figure 4, Deegan, et al. shows that the evaporation flux increased along the surface from the centerline to the edge of the droplet, thus an outward flow was induced to replenish the evaporation loss there [2], [3], [52]. Most of the subsequently developed formulas of evaporation rate are with different assumptions from Deegan, et al. [2], [52]. By using the finite element method (FEM), Hu and Larson [53] numerically predicted the evaporation flux by applying the vapor concentration above the droplet as a boundary condition. The numerical solution was comparable with their experiments with 750 nm fluorescent particles in their experimental droplets. Nanoparticles in a sessile droplet is considered to influence evaporation in three primary aspects: (1) the pinning effect of the contact line can be enhanced by nanoparticles. Different from that in a pure liquid droplet, such a pinned state in a drying nanofluid droplet was prolonged conspicuously on the planar substrate [54], rough heated substrate [19] or superhydrophobic surfaces [36]; (2) a higher nanofluid viscosity than its pure liquid in most cases could slow down the outward flow mentioned by Deegan [3] in sessile droplets, and reduce the evaporation primarily occurring at the edge. The limiting effect due to the increased viscosity was verified in the investigations on the gold nanofluid droplet [36] and the Al nanofluid droplet [19]. The evaporation rate of the nanofluid droplet at its pinned stage was slower than the relative pure liquid droplet with the same base diameter. But nanoparticle helped spreading the droplet base diameter and extending the pinning period, which will be discussed in the next section. The relative larger surface area would make the overall evaporation rate of the nanofluid droplet faster [19]; and (3) nanoparticles could cover the liquid-vapor interface, resulting in a varied liquid-vapor surface tension and thus an affected evaporation rate. Therefore, with different types of nanoparticles, the evaporation rate could be either enhanced or attenuated as compared to the base fluid droplet. In the experimental investigation of Moghiman and Aslani [55], iron oxide (Fe 2 O3 ), zirconium dioxide (ZrO 2 ), or nickel/iron (Ni/Fe) nanoparticle addition into droplets was observed to reduce evaporation, while the presence of clay nanoparticles increased evaporation . The viscosity and surface tension of Fe 2 O3 nanofluids are both increased. Contrarily, the clay nanofluid has a reduced surface tension. Another experimental study with employment of diverse types of nanoparticles showed different trends of evaporation rate as well [56]. Based on the D 2 -Law ( D 2 = D02 − Kt , in which D is the instantaneous diamter, D0 is the initial diameter, K is the evaporation rate constant, and t is the consuming time), the nanofluid droplets exhibited various K in comparison with the pure liquid droplet. They deduced that the value of K was basically

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inversely proportional to the latent heat of vaporization. The presence of nanoparticles was found to enlarge the latent heat of vaporization, thus the K value reduced. During evaporation, the nanofluids was concentrated, and the K value exhibited a transition as the nanofluid concentration reached a critical value, which was observed in both gold and Fe 2 O3 nanofluid droplets. Combined with nanoparticles, the factors involving humidity, fluid convection, and radiation have been taken into account as well. In respect to the effect of humidity, its variation along the droplet free surface was created by making droplets sit close to each other so the humidity was higher in the neighboring regions. It turned out that arches could be formed after evaporation. In the area with higher humidity the evaporation rate was slower while the particles agglomerated in the places where the drying was faster. It is suggested that the effect of humidity on the evaporation rate cannot be neglected [57]. Compared to the influence of humidity, the imposed force-convection affects droplet evaporation more evidently. In the study of Gan and Qiao, the evaporation followed the D 2 -law for the ethanol-based nanofluids under weak convection, but a deviation was observed likely at a higher particle concentration and for the base fluid with a higher boiling point under the natural convection at a relatively low temperature. Therefore the evaporative lifetime might be an important reference to whether the deviation of the D 2 -law happens. The long evaporative lifetime could increase the feasibility of the particle aggregation, which would inhibit diffusion and slow down the evaporation rate [57], [58]. Radiation is the other thermal factor exerting influence on droplet evaporation. As well as the pure liquid droplet, the nanofluid droplet with Al, aluminum oxide (Al 2 O3 ), CNTs, multi-walled carbon nanotubes (MWCNTs), and carbon nanoparticles (CNPs) which have different radiation absorption levels were compared [14], [59]. The evaporation rates of the ethanol-based nanofluids containing MWCNTs and CNPs exhibited higher values than the others [59]. The radiation energy was discussed to be absorbed by nanoparticles, leading to an increased particle temperature and a temperature difference between nanoparticles and the surrounding liquids. Thus the convection in nanoscale took place and the radiation energy was dissipated accordingly. However, with the loading of Al 2 O3 nanoparticles, the enhancement of evaporation rate was mitigated, which might result from a larger possibility of nanoparticle aggregation that could suppress the energy dissipation and diffusion. Different from Al nanoparticles, Al 2 O3 nanoparticles exhibited a less increase of evaporation rate due to a lower radiation absorption level [14].

2.3 Effect of Nanoparticles on Evaporation Regimes During evaporation, a sessile droplet often follows two evaporation modes or a combination of them: (1) the constant contact radius (CCR) mode with a pinned contact line; (2) the constant contact angle (CCA) mode during which the contact area keeps reducing while the contact angle remains unchanged [60]. Hong, et al. [61] experimentally observed both the CCR and the CCA modes during evaporation of a water droplet deposited on a narrow planar solid surface. The droplet with the initial volume of 11 µ l kept pinning with the CCR mode until it reduced to 6 µ l. From this moment on to 3 µ l, the droplet shrank following the CCA mode. Chon, et al. showed a sequence of evaporation phases of a strong pinned droplet containing 11 nm Al 2 O3 nanoparticles. As shown in Figure 5, the droplet was pinned just on a micro-heated

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substrate at first. Induced by liquid evaporation, the strong pinning behavior drove the nanoparticles to the edge. With further evaporation of the base fluid, the contact angle reduced to a critical value, and the thin core liquid region began to break away from the rim, known as the depinning process. The depinned core liquid then shrank towards the center as it dried out further. Finally, the dried ring-shaped nanoparticle stain was formed along the rim [62]. The frequently observed sequence of CCR mode followed by CCA is interesting to many researchers. Zhang, et al. established a model based on the pinning force of the contact line to study this sequence. The pinning force kept the droplet evaporating in the CCR mode before the contact angle decreased to its critical value [63]. Another simulation predicted the CCA and CCR modes of nanofluid droplets on textured surfaces. It showed that the pinning force was of the molecular origin, and it was the local liquid/solid interaction in the vicinity of the contact line dominating the wetting behavior. A wider characteristic width of the inhomogeneity of the textured surface would generate a larger energy barrier to avoid a hydrophilic pattern, and result in a larger pinning force. The pinning of the contact line or the CCR mode was found at the beginning of evaporation with a large pinning force. While on textured surfaces with a smaller characteristic width of the inhomogeneity, the CCA and mixed modes happened [64]. Besides, the capillarity, Marangoni effects, evaporation and disjoining pressure were all indicated to influence the droplet evaporative regimes in the simulation based on the lubrication theory. The numerical results showed that droplets exhibited a variety of different behaviors including spreading, evaporation-driven retraction, contact line pinning, and “terrace" formation [65]. A key point to analyze the droplet evaporation is the moment of transition from pinning to depinning. The pinning time or the period, before the contact line starts to recede, is affected by the factors involving the concentration and size of nanoparticles, droplet sizes and thermal effects. The addition of Al nanoparticles to ethanol droplets, for instance, made the droplet recede at a lower critical contact angle and thus lead to a longer pinning stage in comparison with the pure ethanol droplets [19]. The evaporation process could be quite complex, with more than one switch of the pinning and depinning behaviors of the contact line, i.e. the stick-slip behavior. Fukai, et al. [66] experimentally investigated the effects of droplet sizes and particle concentrations on the depinning behavior. For the macroscale xylene–polystyrene droplets, the contact angle at which the contact line began to recede decreased with increasing the initial solute concentration, while the relation between them was not well observed for the microscale droplets. The three-phase line receded even when the contact angle was above the receding contact angle. What is more, the dot-like and the ring-like deposits were formed from different initial particle concentrations. This fact showed that either the pinning time or the receding distance was an important factor to determine the shape and dimensions of the film. The evaporation rate and surface hydrophobicity were found as the other two factors controlling the depinning behavior of the contact line. In the experiments with Au nanofluid droplets on the substrates bearing different hydrophobicity, the dependence of the evaporation regimes on the evaporation rate at a superhydrophobic surface was weak. However, a second pinning of the contact line emerged as the evaporation rate was significantly reduced on a planar hydrophobic surface [36].

2.4 Flow Dynamics within Nanofluid Droplet During droplet evaporation, the main mechanisms controlling liquid flow have been widely studied as they determine particle movement and the final deposit profiles to a great extent. The capillary flow and Marangoni flow are two common flow regimes frequently observed in an

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evaporating sessile droplets. Once a droplet is pinned on the solid surface, surface tensions initiate a radially outward flow to replenish the evaporation liquid loss at the periphery, known as the capillary flow (Figure 6) [2], [67]. As a result, the particles in the liquid are driven outward and adsorbed at the three-phase line. The formation of the coffee-ring pattern from a pinned colloidal droplet is ascribed to the capillary flow [2]. But in many applications, a uniform deposition is preferred instead of a coffee-ring style, so the capillary flow needs to be suppressed or even eliminated. A Marangoni flow with a reverse direction might work. Marangoni convection is driven by an uneven distribution of the surface tension along the liquid-vapor interface. The nonuniform distribution of liquid-vapor surface tension can be resulted from temperature gradient [68] or concentration gradient [67], [69] along the free surface of droplet. For thermally induced Marangoni flow, or thermocapillary flow, the direction of convection is found to be determined by the nonuniform temperature distributions at the sessile droplet which arise from the non-uniformity of evaporation rate along the droplet and the non-uniformity of heat transfer from the substrate. The balance between these two sources of temperature distribution determines the direction of Marangoni flow, and a concept of k R (the ratio of substrate conductivity k s to liquid

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thermal conductivity k L ) is introduced for identification of the flow direction [70], as illustrated in Figure 7. Compared to the thermocapillary flow, the one arising from concentration gradient has been more frequently observed, which was achieved by using the binary base fluid [71] or adding surfactant in the droplets [67], [69]. At the initial time, due to the capillary flow, surfactant molecules tended to gather at the droplet edge, so the local concentration of the surfactant increased there, reducing the surface tension and creating a Marangoni flow towards the centerline of the sessile droplet. Therefore, the particles were brought to the central region and a relatively uniform deposition pattern was formed ultimately [69]. Passively driven by liquid flows inside a droplet, the velocity of the particles also depends greatly on the capillary flow and Marangoni flow. An simplified formula of a dimensionless ~ vertically-averaged velocity, ur , was developed by Deegan, et al. in the absence of the Marangoni flow [52], 1 1 1 ~ ut ~ 2 −λ (θ ) − (1 − ~ ur = r f = r 2 )) (6) ~ ((1 − r ) R 4 1− t ~ r ~ where ur is the vertically-averaged velocity, t f is the time for the droplet to evaporate, t and ~ r are the dimensionless time and the dimensionless distance from the sessile droplet centerline, and the expression for λ (θ ) is expressed in Eq. (5). The variation of the radial velocity can be calculated from Eq. (6) for a sessile droplet with a certain contact angle, θ , at one ~ t . Afterwards, Hu and Larson took the Marangoni effect into consideration and developed both non-dimensional radial and vertical velocities [72]. Other than the analytically derived formulas of velocity, the numerical and simulating investigations have been intensively carried out, please refer to the following section of “Navier-Stokes Equation". In addition to being driven by the flows in the liquid phase, the particles are acted by the forces such as drag, van der Waals, surface tension forces or the others. The magnitudes of these forces were compared using the scale analysis by Jung, et al. [73]. At the beginning of evaporation, at which the contact line was pinned, the motion of a single particle suspended in liquid was mostly affected by drag force. Then, as the contact line receded, it was the surface tension force that controlled the single particle motion [73]. Instead of studying a single nanoparticle, Kajiya, et al. combined the fluorescent microscopy with a lateral observation of droplet profile to visualize the solute concentration within the droplet [74]. It was observed that

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much higher concentration of suspension was produced in the vicinity of the contact line at the early evaporation stage, while the concentration in the central area remained almost constant until the late stage. The solvent in the central area was possibly removed mainly by a radial flow instead of evaporation. It was the possible reason that the layers became very thin in the central area after evaporation finished. The radial flow was controlled by the evaporation rate whose reduction mitigated it. Kim, et al. also recorded the solute profile in an evaporating droplet under a laser absorbance monitoring system [75]. It can be seen that the polymers moving inside evaporating droplets at the late stage of evaporation altered their directions because of the alteration of the substrate temperature. The droplet profile also differed depending on the temperatures imposed. The study identified that controlling substrate temperature was an effective way to manipulate the flow dynamics inside the evaporating droplets.

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3 Self-Assembly and Patterns Formed by Nanoparticles in Droplet

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Based on the fluid motion in a sessile nanofluid droplet, it becomes achievable to manipulate particles transport and deposition processes. Experimental approaches to control the forming of the final patterns are introduced initially. A series of deposition patterns, the effects on nanoparticles transport and a number of theoretical works primarily focusing on the prediction of pattern formation are then explicated.

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3.1 Experimental Conditions of Nanofluid Droplet Drying

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To obtain desirable deposition patterns, designing experiments to control the motion of nanoparticles is a major challenge in the study of nanofluid droplet evaporation. Majumder, et al. designed an ethanol-vapor atmosphere based on the assumption that ethanol would evaporate dominatingly until the vapor was saturated and then the drying of water would began [76]. Another intensively manipulated direction is the base fluids which can have multiple combinations of diverse liquids or additions of the effective surfactants. Binary mixtures were frequently adopted as the base fluids since only the intrinsic parameters were required to be controlled [77] . Due to the different boiling points, different components of the solvent did not evaporate simultaneously but in a sequence, leading to a surface tension gradient along the droplet surface. On the other hand, adding surfactants into the base fluids was as well an effective approach to modify surface tension in the reported studies. Truskett and Stebe [78] and Nguyen and Stebe [79] showed that the surfactant tended to spread on the droplet surface non-uniformly, resulting in a surface tension gradient. Besides manipulation with the liquid and vapor phases, and the aforementioned nanoparticle species, solid substrate is another aspect which has been modified in respect to superficial topography, different level of hydrophobicity, roughness, etc. The substrate surface can be fabricated with the diverse topographies, such as sharp-tip latching nanostructures of micro-pillars (Figure 8) [80], [81], protruded topographies in microscale [9], and porous structures with different sizes [82], etc. In addition, controlling the temperature profile of the substrate [83] and employing different coating temperatures [84] were confirmed to be capable to alter the transport of the stains and dynamics of the contact line. In the above experiments, the droplets were allowed to evaporate freely and the resulting configurations were from natural self-assembly. A more radical and efficient way to modify deposit profiles is to apply voltage on a droplet. For example, electrowetting on dielectrics

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(EWOD) requires one of the electrodes to be connected with the droplet conventionally. EWOD used the applied voltage to vary, control, and move liquid droplets [85]. AC (alternating current) and DC (direct current) are two kinds of currents imposed on droplets, but the underlying mechanisms are totally different. AC modified the liquid flow of the droplet, and it could induce a depinning behavior of the contact line observed in the reports [86], [87], [88]. Eral, et al. employed 0.1 and 5 µ m polystyrene particles to test the effect of AC. The eliminated coffee stain is seen as a result of internal flows induced by high AC frequencies. Besides, an oscillation of the contact line was produced, and the contact line failed to fully pin on the substrate, and thus particles did not arrive at the contact line [87]. Both the generated internal flows and the unpinned contact line counteracted the indispensable conditions for the formation of a coffee-ring stain. In the other experimental study, with the applied AC, the contact line was receding continuously, together with an induced circulating flow, and deposition patterns with different degree of homogeneity under various AC frequencies were generated [88], as shown in Figure 9. Dissimilar to AC that modifies hydrodynamic flows, capillary flows, or behaviors of the contact line, DC could be imposed on particles to direct their movement governed by an electrokinetic mechanism [85]. The particles were found to be driven toward the electrode with opposite charge [89]. Based on this mechanism, Orejon, et al. [85] stressed on the motion of TiO 2 nanoparticles in a droplet imposed with DC conditions. They found that with the applied DC, TiO 2 preferred to deposit on the solid surface instead of moving to the droplet edge resulting from the electrophoretic forces. Accordingly, the typical “stick-slip" behavior was suppressed with a DC voltage, and a more uniform deposition was produced accompanied with the continuously receding contact line. An more radical intervention was reported by Shao, et al. where the rest of the fluid before the completion of the droplet drying was removed on purpose [90]. The imposed geometric constraints on droplets were also investigated. A confined geometry with a sphere on a flat substrate [91], [92] could produce highly-ordered and regular patterns, as shown in Figure 10. The capillary held solution, shown by the “liquid bridge" in Figure 10 (a), followed by a consecutive stick-slip behavior which resulted in gradient concentric rings (Figure 10 (b)) [92].

3.2 Drying Patterns on the Substrate 3.2.1 Coffee-Ring Pattern As mentioned above, a typical uniform deposit can be formed by ellipsoidal polystyrene particles (Figure 11 (a) and is distinctively different from the dried coffee ring formed by spheres (Figure 11 (b)) [47]. Deegan, et al. theoretically predicted the flow velocity, the growth rate of the ring, and the distribution of solute within the drop [52]. Later, Hu and Larson [83] showed that only pinning of contact line and continuous evaporation of solvents were not enough for coffee-ring formation, but a Marangoni flow should be suppressed. Based on the experiments on organic fluids they found that stains were left at the center of droplet, as a result of a recirculating flow driven by the surface-tension gradient arising from evaporation. Therefore, to ensure the production of coffee-ring stains, the Marangoni flow should be reduced, which is possible by manipulating temperature profiles on substrates. Another factor that may determine the formation of a coffee ring is the droplet size. Shen, et al. [93] found that the competition between the time scales of the liquid evaporation and the particle movement determined the resulting ring

ACCEPTED MANUSCRIPT formation. If the liquid of the droplet evaporated too fast, the particles had insufficient time to travel to the periphery, thus the coffee-ring effect was reduced. Therefore, they proposed that there existed a lower limit of droplet size, D c , for the formation of a coffee ring pattern. As the particle

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concentration was above the threshold value, D c could be estimated by concerning the combined effects of the liquid evaporation and the particle motion. For the particles with the size at around 100 nm, D c was found to be at 10 µ m. Moreover, the coffee-ring effect was approved can be either pronounced by using active agents. With the addition of the surfactant of cetyltrimethyl ammonium bromide (CTAB), the Al 2 O3 nanofluid droplets left more distinct ring-like patterns, while a uniform distribution was created from a droplet without the surfactant [94]. The coffee-ring drying patterns were also observed from the graphite-CTAB-water nanofluid droplets [95], as shown in Figure 12 (a). To look at the coffee ring closer, an atomic force microscopy was used to measure the width and height of ring-stains from the sessile nanofluid droplet evaporation. The structures were found to reach a peak with about 5 µ m in width and 1 µ m in height and to exhibit a gentle slope gradually increasing in height in the direction of contact line motion [96]. 3.2.2 Uniform Pattern

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Non-uniform patterns, such as coffee rings or bumps of stains in the interior, are the results of far-from-equilibrium processes such as the fast liquid movement and fluctuations at the last stage of droplet drying. However, some counterbalance effects of the far-from-equilibrium behavior are able to make particles self-organize orderly, thus forming a uniform structure. As discussed above, the environmental conditions, interior flows including the capillary flow and Marangoni flow, and the particle-particle interactions determine the liquid drying process. The factors might compete to each other to produce the final deposition patterns. As an example studied by Bigioni, et al. [5], a highly uniform, long-range-ordered compact monolayer of nanocrystals was formed in a large scale area. They demonstrated that the uniform formation was controlled by the evaporation kinetics and the particle interactions with the liquid-gas interface. A uniform pattern is also possible to be produced by suppressing the coffee-ring effect. As shown in Figure 8, after evaporation a uniform distribution of latex spheres has been generated by an induced “Wenzel state" resulting from a porous gel foot formed by the spheres at the rim of the droplet [81]. Moreover, with the help of an attractive particle-interface interaction, a rapid early stage evaporation dynamically produced a two-dimensional (2D) assembly of nanoparticles at the liquid-gas interface. The rapid oriented particle moving was prevented in the last stage of the evaporation. In the experiment of the surfactant effect on the drying patterns, the graphite nanoparticle aggregations had a tendency to promote the uniform pattern of deposition in the nanofluids free of surfactant, as shown in Figure 12 (b) [95]. 3.2.3 Stick-Slip Pattern If a nanofluid droplet drying on a non-perfect surface followed the stick-slip behavior, several rings with one outside another would be left on the solid surface after the liquid was evaporated completely [6], as shown in Figure 13. But for the pure ethanol droplet, it behaved almost ideally with a continuously receding contact line at a constant contact angle. A higher concentration of nanoparticles enhanced the stick-slip behavior due to the accumulation of nanoparticles at the contact line. The hysteretic energy barriers that were close to the line tension

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induced the “slip" behavior. Once the hysteretic energy barrier was surpassed by the depinning force, the contact line was unable to maintain on the solid surface but to recede back. An unexpected phenomenon of three-phase line drift was observed, named as “pseudo-pinning". It may be resulted from the small scale pinning of local contact line by the deposited particles, or by the increased viscosity due to intense accumulation of nanoparticles [6], [7], [97]. In order to explain this oscillatory motion, Adachi, et al. built a mathematical model from which the mechanism of the stripe-stain formation was obtained as well [8]. It was indicated that the contact line bore a friction force when the particles transported from the interior to the edge of the droplet. Due to the competition between the friction force and surface tensions at the contact line, the droplet oscillated during evaporation and generated a striped film composed of particles. Therefore, no matter if the contact line was under pinned or depinned, the forces acting on it determined its behavior to a great extent. From the experiments coupling evaporation-induced particle deposition and contact line movement, the pinning force on the contact line was found as a unique function of the deposit volume that was a product of deposition rate and pinning time. However, the pinning time was more greatly dependent on the interval contact line shrinking speed thus that could be used to control the wavelength of the patterns [98]. An unexpected formation of a disordered region at the exterior of a ring stain was found during the stick-slip process. The behavior of the pinning and stick-slip of the contact line was significantly affected by several factors, especially the ambient pressure. Crystallinity of nanoparticles was enhanced when the flow velocity was increased by reducing the environmental pressure. The quantified particle velocity and comparison with the measured results revealed their contribution to the formation of the disordered region through the interaction between the particle velocity (main ordering parameter) and the disjoining pressure (main disordering parameter). Another disordered region was observed in the interior, resulting from the rapid contact line movement during the slip process that immobilized the particles. The hexagonally packed structures mainly composed by the resting particles were occasionally found to be suppressed by the square-packed structures when each particle layer was formed, which was a result of the particles eagerness to get the highest volume fraction within the limited wedged space. For the sake of quantification of the pinning barriers acting on the contact line, the evolution of the excess free energy was computed, which increased to a peak just before each jump and was equivalent to the pinning forces [99]. 3.2.4 Fingering Structure A series of structures, including polygonal networks, elongated structures, spinodal-like, worm-like and ribbon-like islands and fractal cavities were observed during the drying of a nanofluid droplet with an unpinned contact line [100], [101], [102]. In addition to these patterns, a branched structure, known as the fingering structure, has been investigated experimentally [102], [103], [104], [105]. The pattern was induced by the front instability occurring at the contact line with a dewetting behavior. The structured instability was early proved to depend on the capillary flow, the Marangoni effect and some other external exerted forces like gravity, centrifugal force or surface shear stress [106], [107]. Weon, et al. showed that in a narrow range of the ratio of the volume fraction of microparticles to that of nanoparticles, indicated by φL /φS , the fingering structure emerged in the coffee ring from droplets containing bidispersed particles [108], as shown in Figure 14. Both symmetric and asymmetric patterns of fingering structure were observed after the nanofluid droplet dried out [102], as shown in Figure 15. The occurrence of the

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asymmetric patterns was explained to be resulted from the uneven distribution of evaporation rate along the droplet free surface or the dynamics of the contact line. Another interesting observation of the fingering structures was achieved through placing a drop of surfactant on a film of water [109]. A thinner water film, on which the surfactant droplet was placed, led to the fingering instability. Besides, the substrate morphologies, the chemical features of solutes, and the other factors exhibited the influence on the formation of fingering structures as well. For example, with the use of poly-sodium carboxymethyl cellulose polyelectrolyte complexes (PECs) as the particles and the aqueous sodium hydroxide (NaOH) as the solvent, a branched pattern was shown emerging with the nucleation and growing under the mechanism of diffusion limited aggregation (DLA). The factors, including PEC composition, chemical structures, concentration, evaporation temperature, and pH value of solvents were examined for their effects on the structure formation [110].

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3.2.5 Combined Pattern

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In addition to those common patterns formed naturally from drying of droplets, many other unique structures were formed by nanoparticles produced in given conditions. With combination of the factors, involving droplet profile (contact angle, diameter, etc.), properties of the solvent, nanoparticle and solid substrate, the ambient atmosphere and so on, a simultaneous emergence of different patterns that often appeared solely from the droplet drying was observed, such as a pattern with a coffee ring outside and a monolayer pattern inside [111]. A combined structure with a fingering structure inside the coffee ring was shown up in some experiments of nanofluid droplet drying [111]. As illustrated in Figure 16, the finger-like structures are connected with a ring, which is explained to be resulted from the contact line movement of water droplets suspended with copper (Cu) nanoparticles on the silicon wafer substrate. This combined structure was also considered as a result of the competition between the Marangoni flow and the capillary flow, especially when the former one was surpassed by the latter one. These patterns were possible only when the critical contact angle was large in a certain colloidal mixture. It was the geometric constraint that determined and controlled the final patterns [108]. Another pattern, named the inner coffee ring deposits (ICRDs), is shown in Figure 17. One structure enclosing the other was found in the evaporation of nanofluid droplets with SiO 2 . nanoparticles and organic pigment nanoparticles on the nanoscale smooth hydrophobic surfaces. The ICRDs was formed when the particles precipitated at the later stage of pinning instead of the early pinning of the contact line. The ICRDs diameter was smaller than the diameter of the rings formed when the droplet was just pinned, and it increased with an increase of the contact angle hysteresis. The various dendrite structures inside the ICRDs were observed. These patterns were the consequence of a second pinning of the contact line, at which the forces acting on the particles were balanced. The size and patterns of ICRDs and the evaporation kinetics of nanoparticle sessile droplets were strongly dependent on the contact angle hysteresis, the particle concentration, and the colloidal interaction forces such as the electrical double-layer forces [112]. A coffee ring and a petal-like pattern in Figure 18 were observed from dried iso-density water-based nanofluid droplets by Brutin [113]. A transition was shown from the coffee-ring pattern to the petal-like pattern when the nanoparticle concentration, C, increased above a critical value. When C was over 1.15 % the pedal-like pattern showed up. While the coffee ring pattern was produced with C smaller than 0.47 % . Humidity was controlled and changed to examine its effect on the evaporation dynamics and the pattern composition. Unexpectedly, the pattern was

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3.3 Effects of Parameters on Dried Patterns

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identical regardless of the humidity, and the measured evaporation fluxes were slightly higher than the expected flux, following a classical purely diffusive model for the pure fluids. It was explained that the diffusion coefficient of the nanofluids in humid air was different from that of the pure water in dry air. As a result of an increase in the drop diameter, the contact angle decreased and the evaporation rate was enhanced greatly.

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Different patterns forming from nanofluid droplets, as discussed above, are significantly affected by the motion of contact line. Besides, the nanoparticles deposition process was dependent on parameters in the bulk region, such as liquid motion within the droplet and the forces acting on nanoparticles [114]. Based on DLVO theory named after Derjaguin, Landau, Verwey and Overbeek, the transition between a ring-pattern to a uniform pattern was explained by the electrostatic and van der Waals forces that altered the particle deposition process. As illustrated in Figure 19, in terms of fluid movement, a phase diagram including three flow modes were proposed related to the final patterns: a radial flow driven by evaporation at the wetting front (Figure 19(a)), transport of particles attracted to the substrate induced by the DLVO interactions (Figure 19(b)) and a Marangoni-driven recirculating flow due to surface tension gradients (Figure 19(c)). The phase diagram accounted for the three commonly observed deposits: a coffee-ring pattern, a uniform distribution, and a small central bump. A plenty of studies demonstrate different movements of particles with different sizes during droplet drying. For example, one found that microparticles and nanoparticles moved separately, that smaller particles tended to move outward while larger particles tended to move inwards. The congregation of nanoparticles at the contact line enhanced droplet pinning, and under the surface tension gradient, the microparticles were triggered to the droplet center [115]. Weon and Je stated that it was the capillary force that drove the larger particles to transport inwards [116]. The capillary force near the contact line was found positively proportional to particle size while negatively proportional to contact angle. The thermal stability of aggregated structures and hence the occurrence of more complex patterns could be enhanced by smaller particles because assemblies of the smaller nanoparticles had much higher cohesion strength than those made of larger ones [117]. By applying the particle-size effect, Wong et al. succeeded in separating organic colloidal particles with different sizes [118]. They obtained patterns composed by proteins, micro-organisms, and mammalian cells which have different sizes. It showed that the particle with a smaller size located near the contact line (Figure 20). The findings have direct implications for developing low-cost technologies for the disease diagnostics. Similarly, Hendarto and Gianchandani sorted out the particles with different sizes by manipulating substrate temperature to control the Marangoni effect [119]. Glass cenospheres with the different diameter range at the microscale were separated by evaporating the isopropyl alcohol (IPA) droplets on a smooth glass substrate. When the substrate temperature was at 55
C, larger spheres tended to deposit in the droplet center, but smaller spheres were found everywhere throughout the dried region. At the substrate temperature of 80
C (higher than the boiling point of IPA), most of the smaller spheres were found close to the droplet periphery. Also based on the particle-size effect, Sommer, et al. created the unique circular deposition patterns from binary suspensions containing two different nanospheres in size [120]. It was found that the slow evaporation produced well-ordered structures on a smooth substrate, while the large particles self-assembled in dense hexagonal structures, forming apparently an external ring around the massive inner ring.

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However, an opposite motion of particles with the diverse sizes was observed in the experiments by using four nanofluids containing 2 nm Au, 30 nm copper oxide (CuO), 11 nm Al 2 O3 , and 47 nm Al 2 O3 nanoparticles. The smaller nanoparticles had a tendency to form wider edge aggregation and more uniform central deposition while the larger particles were more likely to congregate at the contact line. To better understand the pattern formation, a schematic illustration of nanofluid droplet evaporation and drying process depending on the particle sizes is presented in Figure 21. Smaller nanoparticles with a higher density and a viscosity counteracted to the thermally driven nanoparticle motion, resulting in a thicker and more uniform pattern in the central area with a loosely increased width of a ring around. On the other hand, larger nanoparticles with a lower viscosity tended to move to the rim due to the more active evaporation there, and thus led to a highly distinctive ring-shaped stain [62]. Nanoparticle concentration is another factor exhibiting a strong influence on nanoparticles assembly. As the concentration increases above a critical value, the uniform patterns are more likely to be formed, while the concentric rings are emerged below this critical value. Wong, et al. developed a model to demonstrate the competence between the evaporation rate and particle deposition rate to predict the thickness and length of the rings and to control the patterned deposition [121]. As seen in Figure 22, a dot-like deposit was formed at initial concentration at c [66]. The nanoparticle i = 0.5 wt % , while a ring-like pattern was formed at c i = 3.0 wt % concentration was shown to be related to the cracks in the final drying patterns in a study of polystyrene water-based nanofluids. The dynamic evaporation was recorded by an accurate electronic balance. The pattern formation was analyzed in relation with the mass evolution. The key parameters, including the lifetime of droplet, the diameter of the contacting area, the final deposition diameter, and the number and the spacing of the cracks were determined. A ring pattern was confirmed to form when the particle concentration was above a critical value. The pattern formation was influenced by the liquid evaporation dynamics but only depended on the concentration of nanoparticles [122].

3.4 Theoretical Studies on Nanofluid Droplet Drying The theoretical investigation of the sessile nanofluid droplet evaporation mainly stresses on two parts: one is related to the evaporation dynamics around and within a sessile droplet, the other is the motion or deposition behavior of nanoparticles. In the previous sections we briefly mention several theoretical results of evaporation flux and liquid velocity [53], [60], [72]. Now we only introduce the theoretical studies of the evaporation process and the drying patterns of nanofluid droplet. The approaches are the Navier-Stokes (N-S) equations, the Diffusion Limited Aggregation (DLA) approaches, the Kinetic Monte Carlo (KMC) methods, and the Dynamic Density Function Theory (DDFT). In most case studies, the initial droplet shape was treated as a spherical cap to simplify the simulation. While just before dryout of the droplet, the left solvent was instead treated as an ultra-thin film in which the particles were in rapid motion. Normally solving N-S equations was applied to describe macroscale droplet evaporation process; The DLA approach was often used to simulate nanoparticles aggregation process in a droplet while the KMC and DDFT models were used for both film and droplet with an unpinned contact line. 3.4.1 Navier-Stokes Equations The analytical and numerical methods are two primary ways to obtain solutions from the

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N-S equations. In analytical methods, the approximation method and the lubrication theory are generally employed. In the classic description from Deegan, et al. [2] of a small sessile droplet, the advective velocity, evaporation flux, and deposit growth with time were solved on the basis of an electrostatic analogy. Extensive works were conducted with comprehensive considerations. Hu and Larson described the flow dynamics within a sessile droplet under quasi-steady state [123]. They obtained a time-dependent full velocity field including radial and vertical velocities [72], and also obtained a numerical result by using FEM. Based on the solutions, Hu and Larson took the Marangoni effect into account. The energy equation was simplified to the Laplace equation and solved with combining with the boundary conditions related to force balance at the droplet interface and the finite element analyzed temperature profile along the droplet surface. The velocity affected by the Marangoni effect and the streamlines of the flow field were obtained, as shown in Figure 23. The results also indicated that when the surfactant concentration at surface was about 300 molecules/ µm 2 , the Marangoni effect induced by surfactant concentration gradients could suppress the outward radial flow [123]. On the basis of the solutions of Deegan, et al. [2], [52] and Hu and Larson [53], [72], [83], [123], Jung, et al. adopted the scale analysis to elucidate the drag, electrostatic, van der Waals and surface tension forces on a single particle [73]. At the pinning stage, the drag force mostly affected the particle motion. Once the depinning stage started, most of the particles have deposited on the solid surface and the dominate force was the surface tension force. Duun, et al. proposed a formula of the evaporation flux as a function of the vapor concentration and temperature at the saturation point [124]. The formula took account of the evaporation-induced cooling on the saturation vapor at the droplet free surface. Similarly to that in the study of Hu and Larson [53], the vapor concentration was solved by FEM with the assistance of COMSOL Multiphysics software. The numerical methods can be applied to discretize the droplet domain and calculate the equations with iterations. Through the numerical methods both evaporative dynamics and particles motion can be predicted. With the application of FEM, for instance, the motion of particles and the formation of different deposition patterns could be described [125]. Combined with the mass, N-S and energy equations, the particles concentration profile was obtained by solving a continuum advection-diffusion equation. The transition between a ring-like and an uniform deposit was explained by a FEM-based study of Bhardwaj, et al. [125], with the consideration of DLVO interactions such as the electrostatic and van der Waals forces. A phase diagram was developed as well to explain the competitions among the radial flow, the Marangoni flow and the DLVO interactions. The Galerkin method [126] was used in a study combined with FEM to discrete the spatial domain and an adaptive finite difference method (FDM) was applied to discretize the time domain. The ring-like deposit and the uniform pattern were predicted by this model. Types of the deposition patterns relied on on the convective and diffusive mass transfer in the bulk liquid phase and on the deposition rate along the solid surface. Maki and Kumar described a less numerically explored phenomenon of the formation of a layer of particles at the surface of the droplet [127]. Based on a thin colloidal droplet, the precursor films were considered instead of the pinned contact line, for the sake of being fully consistent with the lubrication theory. Evaporation has been considered limited by the speed of the molecules escaping from the liquid phase rather than the vapor diffusion. The coupled equations were solved by FDM with a moving overset grid. The obtained contours of the particle volume fraction and velocity field at two times are shown in Figure 24. The particle aggregation at the droplet surface resulted from the presence of evaporation and a large Peclet number. Reducing the outward capillary flow, the Marangoni effect could

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pronounce the skin formation. Moreover, Sumardiono and Fischer used the molecular dynamics simulation to investigate the dynamics of a droplet when it was heated from its warm vapor side [128]. They indicated that the radial velocity had a trend to increase at first and then decrease, and the velocity value reduces with a greater number of time steps. Instead of only stressing on the pinned droplet for most cases, Petsi and Burganos derived velocity fields for both pinned and unpinned sessile droplets [129]. For pinned droplets, the liquid flow was oriented towards the edges, and therefore inducing the coffee-ring deposits. While for an unpinned droplet and contact angle less than π/2 , the flow was oriented towards the droplet centerline, as shown in Figure 25.

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3.4.2 Diffusion Limited Aggregation

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DLA is a process by which a fractal can be formed by solutes undergoing a random walk induced by Brownian motion in a solution. It was proposed by Written Jr and Sander [130], in which diffusion was the primary means of transport, but affected by the aggregation. DLA was combined with the other models to describe particles movement and aggregation [111], [131] while it could be used in such a plenty of systems as electro-deposition, Hele-Shaw flow, mineral deposits, and dielectric breakdown [132]. In the process, the irreversible aggregation of the particles occurred due to sticking. The two effects competed with each other in the process. In order to achieve a fractal in the aggregation process, the concentration of particles should be low enough so that particles could not come in contact with each other simultaneously, but could slowly assemble together. In the drying of suspensions, the particles have possibility to meet and aggregate so the arms of the cluster may grow longer and produce branches. During the DLA process, the sticking coefficient, a chance of a particle to stick to a cluster, is an important factor taken into account. The nondimensional sticking coefficient is 1 if the particle always sticks to the existing cluster once the particle is the structure, otherwise the value is less than 1. Based on the nature of nanoparticle in the base fluids in a drying droplet, the diffusion limited cluster-cluster aggregation (DLCA) process has been studied recently [133], [134], [135] , [136], [137]. The DLCA model has been employed to investigate fractal aggregation in nanofluids, during which the aggregation between the clusters was considered. A formed cluster grew and got in touch with the other clusters or nanoparticles. The value of fractal dimension of formed clusters was of critical importance as the different aggregation processes had the different value of it. The theoretical results demonstrated that it was the thermal restructuring induced by their low bonding energies leading to a large fractal dimensions of clusters from small nanoparticles. But it is still unknown for the small fractal dimension value of clusters from large particles [133]. By coupling the 2D DLA method, the experimentally observed transition from the uniform pattern to the coffee ring from both graphite nanofluid droplets and Al 2 O3 nanofluid droplets as a result of adding surfactant was simulated [94], [95]. The simulation demonstrated that it was the sticking force between nanoparticles controlling the final deposition patterns. The simulated results were statistically in a good agreement with the images of the ultimate self-assembled structures recorded experimentally, as shown in Figure 26. Additionally, the 2D model has been developed into a three-dimensional (3D) layers model [138] and full 3D model [139] for the transition from the coffee ring to the uniform deposition. By coupling DLA with the biased random walk to simulate microparticle motion and aggregation during droplet evaporation [47], the simulation [138] presented the particle adsorption, long-range attraction, and circulatory motion for the transition. In the simulation, the capillary flow at the late stage of evaporation was

ACCEPTED MANUSCRIPT shown strong enough to destroy particles organization at the initial stage. 3.4.3 Kinetic Monte Carlo Model

where

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The classical KMC model was proposed by Rabani, et al. [140]. It has been extended to simulate the particles assembly and organization with a 2D square lattice domain. Based on the interaction between drying of base fluid and diffusion of nanoparticles, different nonequilibrium patterns are able to be produced. In the 2D model in Figure 27(a), the solvent was divided into an array of 2D square lattices with size dictated by nanoparticles. Each cell was considered occupied either with a nanoparticle, liquid or vapor, which could be effectively described by two variables n and l. The possible states are: liquid ( l = 1, n = 0 ), nanoparticle ( l = 0, n = 1 ), and vapor ( l = 0, n = 0 ). This model was applied in a closed domain firstly [140] and was used to simulate inner part of the wetted domain. The Hamiltonian corresponding to energy expression for an entire configuration can be expressed as,

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indicates a summation of adjacent pairs of cells, among which, ε nn , ε nl and ε ll

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are the interaction energies for particle-particle, liquid-particle, and liquid-liquid cells respectively. The Hamiltonian energy function describes the energy profile and a balanced state of the system. The interaction energy values are calculated by the first three terms on the right side respectively. The extra-energy for the phase transition was expressed by the last term with the chemical potential. The energy calculated from this function for the i-th lattice cell represents the energy it contributes to the system. In the closed-domain KMC model [140], the area was surrounded by wet substrate so the studied region was the inner part of the wetted domain that was far from the wetting front (“Closed domain" in Figure 27(b)). Based on this, Pauliac-Vaujour, et al. [105] simulated the experimentally observed fingering structures that formed behind the macroscopic wetting front. Different from the closed domain, an open domain method was proposed to simulate the condition of the final drying stage of thin layer [141], in which the edge was treated as the wetting front to describe square liquid surrounded by the empty substrate, as shown in the “Open domain" in Figure 27(b). From the method, the formation of fractal-like microstructures from nanofluid was investigated. The particles described in the model were isotopically deposited into the fractal-like structure that remained immobilized after the base fluid totally dried [141]. The formed fractal-like structures were not the same as the fractal “holes” obtained in the closed domain model. Afterwards, the 2D model was improved to simulate the evaporation of a nanofluid sessile droplet with a circular depinned moving contact line. The 2D circular domain was selected to describe the top view of the spherical-cap-shaped sessile droplet. Coupling with the non-uniform chemical potential function and the thickness profile of a droplet, a pseudo-3D Monte Carlo simulation model was introduced. A fractal-like structure was formed due to the evaporation of nanofluid and was comparable to the recorded images of nanoparticle-assembled structures from the experiments [142]. The KMC model was also used to simulate the observed symmetric and asymmetric drying patterns formed during the drying of sessile nanofluid droplets suspended with nanoparticles, as shown in Figure 27 (c) [102]. Figure 28 shows a predicted sequence of the drying process of a droplet, in which the receding drying front induced particles into fingering

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structures toward the centerline of the droplet. The parameters, such as domain diameter, chemical potential distribution, particle interaction energy, particle coverage, moveability, etc., were shown to have great effects on the ultimately formed patterns [102]. The simulation was comparable with the experiments. Then, the KMC model was used to simulate the process of formation of a well-distinguished fingering structure inside a coffee-ring pattern from a millimeter-sized sessile nanofluid droplet on a silicon substrate [111]. The simulation indicated that the nucleation and growth process that started inside the ring resulted in the observation of emergence of branched structures connected to the inner side of the ring [102]. Although the KMC model is effective in description of various patterns formation from evaporation of nanofluids, the limitations brought by a 2D assumption without considering liquid thickness should be amended. In an investigation of a local controlling of dewetting-induced 2D pattern formation from nanofluids, Vancea, et al. [143] revealed that certain kinds of structures could not be simulated and which were of significant importance to the study of directed-dewetting by the purely 2D KMC approach. They made a change to the original KMC model by coupling the chemical potential with the global solvent coverage. The results showed that the evaporation rate of solvent was inversely proportional to its coverage and agreed well with the experimental images. Sztrum and Rabani [144] and Stannard [145] extended the 2D KMC model to 3D under different conditions. Sztrum and Rabani found that the assembled structures in the 3D model were in a good agreement with those of the 2D model when liquid evaporated in a manner of layer by layer and solvation forces were strong [144]. The researchers also confirmed that the 3D model was indispensable to predict the formation of nano-stalagmites. Furthermore, in order to produce dual-scale structures resulting from the nonequilibrium nanoparticles-assembly, Stannard simulated it with a pseudo-3D lattice gas-based Monte Carlo model with the coupling chemical potential to solvent density [145]. Dual-scale cellular networks, ring and fingering structures with small-scale patterning were produced by this extended model. 3.4.4 Dynamical Density Functional Theory Dynamical Density Functional Theory (DDFT) was applied to predict the behavior of an ultra-thin precursor film by coupling the density profile of the liquid and the nanoparticles in the nanofluids [146]. In the simulation, the free energy function was first derived, while the dynamic equations for the liquid and nanoparticle density profiles followed. The free energy function, f, for a continuous system can be written as, f ( ρ l , ρ n ) = kT [ ρ l ln ρ l + (1 − ρ l ) ln (1 − ρ l )] + kT [ ρ n ln ρ n + (1 − ρ n ) ln (1 − ρ n ) − 2ε ll ρ l2 − 2ε nn ρ n2 − 4ε nl ρ n ρ l

(8)

where ρ l andρ n are the densities of the liquid and the nanoparticles respectively, they denote the possibilities to fill a given cell with liquid or a nanoparticle; and µ , k and T denote the liquid chemical potential, the Boltzmann constant and the temperature, respectively. A free energy in Eq. (9) was calculated by taking a gradient expansion of the free energy function of a continuous system [146]. Thiele, et al. made the description in the KMC system [147]. From the derivative of the free energy function, the chemical potential of nanoparticles was demonstrated as µ n = δF [ ρ l , ρ n ]/δρ n (r ) . The value of µ n was set constant, but it might change with space in a

ACCEPTED MANUSCRIPT nonequilibrium system, defined as µ n (r , t ) . The nanoparticles movement were proposed to be

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determined by the thermodynamic force, ∇µ n , and the the nanoparticle current, J = − M n ρ n∇µ n , in which M n was defined as the mobility coefficient, dependent on ρ . An evolution equation of nanoparticle density field was formulated by combining the current and continuity equations, (9)

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∂ρ n δF [ ρ n , ρ l ] = ∇ ⋅ [ M n ρ n∇ ] δρ n ∂t

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Marconi and Tarazona [148], Archer and Rauscher [149], Archer and Evans [150] and Monson [151] derived Eq. (10) as well by considering that nanoparticles had over-damped random equations of motion.

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∂ρ l δF [ ρ n , ρ l ] δF [ ρ n , ρ l ] = ∇ ⋅ [ M lc ρ l ∇ ] − M lnc ∂t δρ l δρ l

(10)

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where the coefficients of M lc and M lnc were assumed to be constants. The DDFT approach has been used to predict the evolution of the evaporation process of nanofluid film. Exemplified by the dewetting front during film evaporation as shown in Figure 29, the initial wetting front had about 16 wiggles, indicating a short-wave instability. But after it, only 7 wiggles were left, which suggested a higher level of stability. Afterward, the number of the fingers remained constant through the balance between the creation of new wiggles and joining of old ones. Since the diffusion constant was independent of the nanoparticle concentration, the particles were excluded from the driven force of the instabilities [147].

4 Conclusion

In this review, we highlight sessile nanofluid droplet evaporation primarily from experimental and theoretical investigations. The role of nanoparticles presence, concentration, size and shape in influencing the spreading, evaporation rate, the evaporation mode and the flow dynamics of a sessile droplet have been elucidated by reviewing the experimental and theoretical investigations. Aiming to control particles motion and deposition process, abundant experimental methods have been adopted to guide particles assembly and create preferable residue patterns. Studies stressing on parameters effect on particles motion and aggregation serve as the foundation for better manipulation of the transport of nanoparticles. The existing theories then are briefly introduced. The analytical and numerical solutions for the N-S equations, the DLA method centering on particles aggregation process, and the KMC and DDFT approaches for liquids with a receding contact line were discussed. The flow dynamics in a sessile droplet, particles motion and different patterns formed are well predicted for the particular cases, most of which agree well with the relative measured results. However, the droplet evaporation is very complicated, more universal models considering the real experimental evaporating conditions and nanoparticles assembly could facilitate our understanding in the field of nanofluid droplet evaporation. The continuous studies might be on mixing particles with different shape, size and properties, making base fluids by using liquids of various thermal properties, designing novel confined geometries, controlling contact line motion by using substrates with different level of wettability, etc.

ACCEPTED MANUSCRIPT 5 Acknowledgements

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

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Figure captions

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Figure 1: The contact line and contact angle of a sessile droplet. Figure 2: Structural disjoining pressure over the film thickness of a droplet. Nanoparticles self-assemble orderly in the film [28]. Figure 3: Particle ordering in the coffee-ring stain. (a) Overview of the different patterns in the ring. (b1)–(b3) A close-up reveals the sequence of patterns in the stain: square packing close to the contact line (b1), followed by hexagonal packing (b2), and finally disordered packing (b3) [39]. Figure 4: Evaporation flux j increases along the droplet interface from the center to the edge of the droplet [2]. Figure 5: Evolution of 11 nm Al 2 O3 nanofluid droplet evaporation with sequential photographs and schematic sketches. Once the droplet is placed on the microheater substrate, it goes through stages of (a) pinning, (b) liquid dominant evaporation, (c) depinning, (d) dryout progress and (e) formation of nanoparticles [62]. Figure 6: Description of the capillary flow from the center to the edge of a droplet, in which particles are driven outwardly [67]. Figure 7: Droplets with different ranges of the relative thermal conductivities of the substrate and liquid k R = k s /k L , and the consequent Marangoni flows and temperature gradients

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along the droplet surface when (a) k R > k Rcrit and (b) k R < k Rcrit . Arrows outside the droplets indicate directions of increasing temperature; arrows inside the droplets indicate directions of the Marangoni circulations [70]. Figure 8: Schematic “Wenzel state" of droplet evaporation on a patterned hydrophobic silicon surface with pillar arrays and a uniform distribution of the latex spheres after complete evaporation [81]. Figure 9: Deposition patterns formed without and with applied AC under at 6 Hz, 1 kHz, 100 kHz, and 100 kHz + 100 Hz modulation [88]. Figure 10: (a) Schematic cross section of a liquid bridge formed in a confined geometry with a spherical lens on a silicon surface. (b) Left: The digital image of entire gradient concentric ring patterns formed by solutes deposition in the geometry in (a). Right: A small zone of the ring patterns. As the solution front moves inwards, rings become smaller and the height decreases as illustrated in lower left schematic [92]. Figure 11: (a) A uniform pattern from ellipsoid polystyrene particles with α at 3.5 and (b) a typical coffee ring from spherical polystyrene particles with α at 1.0. α is the aspect ratio [47]. Figure 12:(a) Coffee-ring patterns from droplets with 0.1 wt% CTAB surfactant with graphite nanoparticles concentration at 2g/L and 5g/L; (b) uniform patterns from droplets without surfactant with graphite nanoparticles concentration at 2g/L and 5g/L [95]. Figure 13: Stick-slip pattern on solid surface after evaporation of ethanol droplet containing TiO 2 nanoparticles [6]. Figure 14: Fingering structure inside the coffee ring. φs is the volume fraction of small colloids with radius of 0.1 µm ; φ L is the volume fraction of large colloids with radius of 1.0 µm ; [108].

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Figure 15: (a)Symmetric and (b) asymmetric drying patterns of fingering structure from copper nanofluid droplets [102]. Figure 16: A fingering structure inside the coffee ring from drying of copper nanofluid droplet [111]. Figure 17: Inner coffee-ring pattern. Black color indicates particles and the rest part is the empty area [112]. Figure 18: Coffee-ring pattern and pedal-like pattern generated at different nanoparticles concentration C. The coffee-ring patterns emerge when C equals to 1.15% , 2.3% , 4.8% and 5.7% . The pedal-like patterns are generated when C equals to 0.01% , 0.05% , 0.11% , 0.23% and 0.47% [113]. Figure 19: Three convective mechanisms compete to form the deposit. In (1), a ring forms due to radial flow caused by a maximum evaporation rate at the pinned wetting line; in (2), a uniform deposit forms due to an attractive DLVO force between the particles and the substrate; in (3), a central bump forms due to a Marangoni recirculation loop [114]. Figure 20: (a) Schematics showing the concept of size-dependent particle separation near the contact line of an evaporating colloidal droplet. (b) Optical fluorescence image showing the separation of 40 nm (green), 1 µ m (red), and 2 µ m (blue) particles after evaporation.(c) A semi-log plot showing the dependence of the coffee-ring separation distance (edge-to-edge) on particle concentration.(d) Optical fluorescence image showing the separation of antimouse IgG antibodies (blue), Escherichia coli (green), and B-lymphoma cells (red) in a dried liquid drop [118]. Figure 21: Schematic illustration of nanofluid droplet evaporation and dryout process depending on particle sizes: (a) a uniform distribution formed by nanoparticles with smaller size and (b) a coffee-ring pattern formed by nanoparticles with larger size [62]. Figure 22: Cross section of the formed polymer film for each initial concentration c i and diameter d 0 [66]. Figure 23: Streamline plots of the flow field from the finite element model for the Marangoni-stress boundary condition at contact angle of 40
and Marangoni number of 841, from (a) FEM solution and (b) analytical solution. The inset is the corresponding velocity field [123]. Figure 24: Contours of the particle volume fraction in the left part of the droplet and velocity field in the right part of the droplet at two times ((a) and (b)) with consideration of a monolayer formed by particles on the droplet surface [127]. Figure 25: Vector representation of the internal flow field in a kinetically controlled evaporating droplet.(a) a pinned droplet and (b) an unpinned droplet [129]. Figure 26: Comparison of the experiments and simulation. The coffee rings display the particle distribution profiles in the simulated patterns (a, b), captured after 2500 MCS with the initial particle coverage at 5%, and the finally dried experimental graphite structure from 2% graphite nanofluid with CTAB (c); The mixed patterns display the particle distribution profiles in the simulated patterns (e, f), captured after 2500 MCS and the initial particle coverage at 10%, and the finally dried experimental graphite structure from 5% graphite nanofluid with CTAB (g) [95] . Figure 27: (a) Schematics of the two-dimensional lattice-gas KMC model; (b) closed domain [140] and open domain [141] in KMC model; (c) symmetrical and asymmetrical drying patterns obtained by using KMC model [102]. Figure 28: Intermediate steps of the simulation process for the uniform droplet

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evaporation. The images are captured after (a) 2000, (b) 4000, (c) 6000, (d) 8000 and (e) 10000 MCS, respectively [102]. Figure 29: Density profiles for the situation where the substrate is covered by nanoparticles with average density ρ nav = 0.3 and with the liquid excluded from the region y < 0 . This row shows the nanoparticle density profiles at the times t/tl = 1000 (a), 10000 (b) and 30000 (c), where tl = 1/kTM lncσ 2 . The parameters are KT/ε ll = 0.8 , ε nl /ε ll = 0.6 , ε nn = 0 ,

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Highlights Influence of nanoparticles for droplet spreading and evaporation is introduced.

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Types of drying patterns from nanofluid droplets are demonstrated.

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Theoretical investigations for drying droplets are briefed.