Effect of a single nanoparticle on the contact line motion

*Corresponding author, Tel: +86-551-6360-1238, E-mail: [email protected]. 8. 9 ... the driving force for the depinning of the contact line and the na...
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Effect of a Single Nanoparticle on the Contact Line Motion YingQi Li, HengAn Wu, and FengChao Wang* CAS Key Laboratory of Materials Behavior and Design of Materials, Department of Modern Mechanics, CAS Center for Excellence in Nanoscience, University of Science and Technology of China, Hefei, Anhui 230027, China S Supporting Information *

ABSTRACT: In this paper, we use a single nanoparticle (NP) to achieve active control of the droplet contact line. When the droplet is out of equilibrium, the resulting excess free energy provides the driving force for the depinning of the contact line and the NP. There are three ways to increase the energy barriers to be surmounted and to realize the pinning of the contact line, namely, the enhancement of the interactions between the NP and the substrate, the increase in substrate hydrophilicity, and the reduction in the NP hydrophilicity. On this basis, we obtained three styles of contact line motion including complete slipping, alternate pinning−depinning, and complete pinning and theoretically interpreted them. The basic theory presented in this paper can be applied to explain and regulate the dynamics of the contact line involved in many physical processes such as evaporation and spreading.

and controlled ink-jetting.18 By manipulating the pinning behaviors of NPs during evaporation, patterned depositions and structures formed by NPs can be derived.19,29,30 These controllable pinning and deposition of NPs are significant to numerous applications such as 3D photonic crystals31 and printed electronics on flexible substrates.32 Zhong et al. reviewed previous studies on sessile nanofluid droplet drying and the deposition patterns,25 including the analyses of NP properties such as the NP size and concentration and the theoretical analyses such as the internal fluid flow and NP selfassembly. Adjusting the substrate wettability can control the evaporation process of nanofluid droplets containing mixed NPs with different sizes. The smaller particles remain at the droplet rim, whereas the larger particles move radially inward.33,34 NP separations based on size differences are achieved.35,36 The dragging force imposed on the NP because of the liquid outward flow is the main reason for the pinning of the NPs and NPs that remain at the contact line during sessile droplet evaporation on the hydrophilic substrate.35,37 Whether the NPs remain at the contact line or move radially inward depends on the competition between the attraction forces among the NPs and the capillary surface tension force exerted on the NPs.34 In practice, the forces among the NPs are very small and in different directions relative to the NP−substrate interactions, which is not enough to cause the pinning of the NPs.35 Weon and Je proposed a self-pinning mechanism in which the colloid particles in the contact line region reached a critical packing fraction to satisfy pinning and formed a layered structure.38

1. INTRODUCTION Nanoparticles (NPs) can modify the wetting and spreading of nanofluids on a solid substrate.1−8 Unlike pure droplets that spread rapidly up to the maximum contact radius on the solid substrate,9−15 the spreading of nanofluid droplets can be controlled by adjusting the properties and mobility of the NPs inside of the droplet.1,3−5,7,16 There have been many studies on the aggregation and self-organization of NPs in the vicinity of the contact line.1,3−5,16 The dynamic behaviors of NPs in the nanofluids, especially in the contact line region, have implications for various industrial and biological processes including coatings,17 controlled ink-jetting,18 patterned assembly,19 oily soil removal,1 detergents,3,5,6,8 and enhanced oil recovery.20,21 For example, Nikolov et al.8 proposed a detergent mechanism for oily soil removal by indicating that the selflayering of NPs caused by the NP confinement at the wedge film region results in a structural disjoining pressure at the wedge. This enhanced the spreading of the nanofluids and further enhanced the detachment of the oil droplet from the solid surface.5,6 Accumulation and pinning of the NPs at the contact line region have also left a ringlike stain at the droplet rim after the nanofluid droplet dries out, which is the well-known coffeering.22−25 Deegan et al. reported that the outward capillary flow can carry all dispersed NPs to the droplet rim. Deposition and self-pinning of NPs at the droplet rim are independent of the suspended particles, the substrate, and the carrier fluid.22 A few experiments, however, have discovered that the coffee-ring can be eliminated by controlling the NP properties such as shape and size.26−28 This is because NPs can be depinned from the contact line and be reversed toward the droplet center via the capillary surface force27 or Marangoni flows.28 The detachment of NPs at the droplet rim is significant in uniform coatings17 © 2016 American Chemical Society

Received: October 1, 2016 Revised: October 31, 2016 Published: November 3, 2016 12676

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focused on the effect of single or two NPs on the contact line motion and the force condition on the single NPs, through both simulations and experiments.39,40 These studies show that it is significant and reasonable to focus on the effect of a single NP on the contact line. The standard 12−6 Lennard−Jones (LJ) pairwise potential was adopted to describe the nonbonded intramolecular and intermolecular interactions

Sangani et al. indicated that larger particle volume fraction, lower contact angle, and smaller particle size will make the contact line easier to pin.16 The pinning of the NPs at the contact line is a complex issue involving multiple force analyses across scales, such as capillary force, surface tension, disjoining pressure, van der Waals force, electrostatic force, and friction.16,27,35,37,38 The pinning and deposition of NPs at the contact line depend on many factors including the NP size, bulk concentration, concentration near the contact line, droplet radius, and wetting characteristics of the substrate.16,35,38 Basic theories of NP pinning near the contact line are essential to the control and applications of nanofluids suspended with NPs. Force analyses on single NPs at the contact line are important and the basis for studying the contact line pinning and its mechanism.35−38 There have been numerous studies on manipulating the dynamic behaviors of single NPs at the contact line, which is a vital method for insights into the pinning mechanism of NPs and the control of contact line motion.16,35,37−40 In this study, we use the theory of excess free energy to interpret the pinning of single NPs in the contact line and the different movement patterns of it. We noted that when the droplet is not in equilibrium, the corresponding excess free energy will overcome the energy barriers and provides the driving force for the depinning of the NP. Two terms give rise to the energy barriers: the potential energy between the NP and the substrate and the work of adhesion of liquid droplet to the surface. By adjusting the energy barriers through the NP, liquid, and substrate properties, the contact line motion can be changed from complete slipping to alternate pinning/ depinning, then to complete pinning. This will provide a theoretical basis for the contact line pinning issues in many nanofluid-related physical phenomena such as spreading and evaporation.

⎡⎛ ⎞12 ⎛ ⎞6 ⎤ σij σij Uij = 4Cijεij0⎢⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥⎥ r ⎝ rij ⎠ ⎦ ⎣⎝ ij ⎠

(1)

where rij is the distance between atom types i and j, σij is the distance at which the pair potential reaches zero. LJ parameters obey the mixing rules represented in eq 2

εij0 =

rijmin = (rimin + r jmin)/2

εi · εj

(2)

The coupling coefficient, Cij, enables us to adjust the relative affinities between atom types i and j using εij = Cijεij0. The cutoff distance of LJ potential rc is set to be rc = 12 Å. The LJ parameters for the nonbonded interactions are listed in Table 1.

Table 1. LJ Parameters for Standard Nonbonded Interactions45,46 atom number

atom types

ε (kcal/mol)

σ (Å)

1 2 3 4 5 6

CH3 CH2 SH Au liquid atom substrate atom

0.0934 0.2260 0.3974 0.7717 0.6975 0.5962

3.930 3.930 4.450 2.737 3.500 3.500

During the simulations, the Au−Au LJ interaction was strengthened to be 35.85 kcal/mol to ensure the cluster integrity.47 The thiol−gold (SH−Au) interactions approximated as a nonbonded m−n potential,48 whose parameters are compatible with the SH−Au-bonding interaction49,50

2. METHODS AND SIMULATION DETAILS The simulations were conducted using molecular dynamics (MD) simulation package LAMMPS.41 Single surfactant NPs were placed at the contact line, as shown in Figure 1. The liquid droplet was

U (r ) =

E0 ⎡ ⎛ r0 ⎞n ⎛ r0 ⎞m⎤ ⎢m⎜⎝ ⎟⎠ − n⎜⎝ ⎟⎠ ⎥ n − m⎣ r r ⎦

(3)

where E0 = 9.2240 kcal/mol, n = 8, m = 4, and r0 = 2.9 Å.49,50 The bonds of the liquid molecules were described using the finite extensible nonlinear elastic potential51,52

⎧ ⎡ ⎛ r ⎞2 ⎤ 1 ⎪ ⎪− kR 0 2 ln⎢1 − ⎜ ⎟ ⎥ ⎢⎣ ⎝ R 0 ⎠ ⎥⎦ E FENE(r ) = ⎨ 2 ⎪ ⎪ ∞ r ≥ R0 ⎩

r < R0 (4)

where k = 30ε55 and R0 = 1.5σ55. The intramolecular interactions of octanethiol chains were modeled using constraints for the bonds, harmonic potentials for angular degrees of freedom,

Figure 1. Descriptions of the MD simulation model. The surfactant NP is Au core with the well-dispersed chains of octanethiol covering its surface. The free liquid edge sticks to the virtual wall. By applying the constant velocity to the virtual wall on the left, the droplet and the contact line intend to move.

Uangle(θ) =

1 k′(θ − θ0)2 2

(5)

and dihedral angle potentials for the torsional interactions composed of 3120 eight-atom chains, which is an appropriate liquid model in previous droplet-related issues.42 The substrate was constructed based on the face-centered-cubic (FCC) lattice structure. The NP was composed of the gold core and the surfactant, which has been widely adopted in previous simulation studies.43,44 The gold core was a truncated octahedral motif consisting of 405 FCC Au atoms, with 116 well-distributed octanethiol (C8H18S) molecules covering its surface. The CH2, CH3, and SH groups were represented using pseudoatoms,45 as illustrated in Figure 1. Several previous studies have

Udihedral(ϕ) =

1 1 a1(1 + cos(ϕ)) + a 2(1 − cos(2ϕ)) 2 2 1 + a3(1 + cos(3ϕ)) 2

(6)

The interaction parameters used in eqs 3 and 4 are listed in Table 2. MD simulations were performed in the constant volume and constant temperature ensemble. Fixed boundary conditions were applied to y and z directions, and periodic boundary conditions are 12677

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Langmuir Table 3. Alternate Parameters for Different Modelsa

Table 2. Surfactant Intramolecular Interactions45,46 interaction groups rigid bond CH3−CH2 rigid bond CH2−CH2 rigid bond CH2−SH harmonic angle CH2−CH2−CH3 harmonic angle CH2−CH2−SH dihedral angle CH2−CH2−CH2−SH

dihedral angle CH2−CH2−CH2−SH

a

parametersa 1.540 1.540 1.540 k′ = 62.109 θ0 = 114.400 k′ = 62.109 θ0 = 114.400 a1 = 5.9046 a2 = −1.1340 a3 = 13.1068 a1 = 5.9046 a2 = −1.1340 a3 = 13.1068

number

liquid−substrate (lsx, x = 1, 2, 3)

1

C56 = 0.4623

hydrophilic substrate

2 3 1

C56 = 0.3000 C56 = 0.1572 C15 = 2.55, C25 = 2.00 C15 = 1.20, C25 = 1.20 C16 = 1.00, C26 = 0.40 C16 = 1.00, C26 = 1.00 C16 = 1.50, C26 = 1.50 C16 = 2.50, C26 = 2.00 C16 = 2.75, C26 = 2.20 C16 = 3.20, C26 = 2.20 C16 = 5.00, C26 = 3.00

hydrophobic substrate

NP−liquid (lny, y = 1, 2)

2 NP−substrate (snz, z = 1, ..., 7)

1 2 3 4

Distance in Å, angles in degree and energy in kcal/mol.

5

used to the x-axis. The width of the simulation box in x direction is L = 6 nm. The length in the y-axis is large enough to make sure that the liquid can always move along the y direction. The liquid temperature was kept constant at 300 K by employing a Nose−Hoover thermostat with a temperature damping coefficient of 0.1 ps. Velocity Verlet algorithm with a time step of 1 fs was adopted to integrate Newton’s equations of motion. A virtual wall was put at the liquid edge, and the wall gained a constant speed of v = −5 nm/ns. The liquid edge sticks to the virtual wall and was driven to move together with it toward the negative y-axis. The droplet was then stretched, and the contact line tends to move. This is one method to move the contact line. Contact line dynamics can be investigated in many physical processes such as evaporation, spreading, and the self-driven contact line motion on substrates with wettability gradients. Especially the universally adopted evaporation model in previous simulation studies53−55 is intuitive to observe the contact line motion and the important phenomena such as “coffee-ring”. In this study, however, we focus on making clear of how properties such as NP wettability and substrate wettability will result in the contact line pinning and providing theoretical analyses on the pinning mechanism. This method can reflect the contact line motion caused by many conditions such as spreading, evaporation, and selfdriven contact line by wettability gradients. Thus, the findings and mechanism in this study can be applied to these physical processes. The results of nanofluid evaporation in the Supporting Information part 2 show that the conclusions in this study can be applied to further studies of evaporation and to control the deposition patterns such as coffee-ring or coffee-eyes on different substrates. Interaction parameters Cij were adjusted to study the pinning and depinning of the NPs. The selected values and the corresponding properties are listed in Table 3. As shown in Figure 1, symbols l, s, and n represent the liquid, the substrate, and the NP, respectively. As illustrated in Table 3, three important factors are included, the substrate wettability, the NP−substrate interactions, and the NP wettability. The substrate wettability is controlled conveniently by varying the liquid−substrate interaction parameters, instead of using the real materials. We know that the definition of substrate wettability, especially the hydrophobicity, depends not only on the equilibrium contact angle but also on the contact angle hysteresis, which relates to the chemical or physical heterogeneity on the real surface. In MD simulation, an ideal substrate with chemical and physical homogeneity everywhere was adopted; we distinguish the substrate hydrophobicity by measuring the equilibrium contact angle of the droplet on the substrate. The equilibrium contact angle is an ideal contact angle that depends on the geometrical shape and can be measured directly through the MD simulation (actually the apparent contact angle θap).56 In the Supporting Information part 1, we have shown the measurement of equilibrium contact angle by fitting the averaged droplet density profile. As shown in Table 3, substrates with θap > 90° are defined as hydrophobic substrates. The substrates with θap ≈

interaction parameters

interaction groups

6 7

corresponding properties

NP is completely wetted NP is partially wetted NP−substrate interaction strengths from ns1 to ns7

a

Other coupling coefficients that are not mentioned in the table are the default of 1.00.

136.7° (C56 = 0.1572) are defined “more hydrophobic” than the substrates with θap ≈ 91.9° (C56 = 0.30).

3. RESULTS AND DISCUSSION 3.1. Three Motion Modes of NPs at the Contact Line. In this simulation, we stretched the free liquid edge to achieve the motion of NPs and the contact line. The main resistance force exerted on the NP is the friction force Ff that is parallel to the substrate. It originates from the total attraction force between the NP and the substrate, Fa,35,37,38 which relates to the NP−substrate interactions. Fa increases with an increase in the NP−substrate interaction. According to the relationship that Ff = f Fa ( f is the friction coefficient between the substrate and the NP), the friction force also increases, resulting in the enhancement of NP pinning.35,37,38 This means that enhancement of the interactions between the NP and the substrate will result in the pinning of NPs on the substrate.39 Figure 2 shows the NP COM versus time of ls2ln1. The NP is completely wetted, and the substrate is hydrophobic. As the NP−substrate interaction increases from ns1 to ns6, the NP sticks more and more tightly to the substrate. The motions of the NP and the contact line on the smooth substrate were progressively suppressed. Three kinds of contact line movement were observed including (I) complete slipping without hysteresis. When the NP−substrate interaction ns is small, as the results of ls2ln1ns1 in Figure 2a and the snapshots in Figure 2b, the NP slips at a constant speed, which approximates the speed of the liquid edge v = 5 nm/ns. This means that there was no pinning of NPs or the contact line. (II) Alternate pinning and depinning. When the NP−substrate interaction increases to ns6, alternate pinning and depinning were observed. The stick-slip of the contact line has been experimentally observed in previous studies of evaporation or spreading.57,58 It showed that this phenomenon appears on the weakly wetted substrate. This is because of the weak interactions of the droplet with the substrate, resulting in the weak pinning of the contact line.57 As illustrated in the 12678

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Figure 2. Dynamic process of ls2ln1 that contains completely wetted NPs on a hydrophobic substrate. (a) Variations in NP center-of-mass (COM) over time. Three motion patterns, complete slipping, alternate pinning−depinning, and complete pinning are obtained by changing the NP− substrate interactions. (b) and (c) Snapshots of the complete slipping of ls2ln1ns1 and the complete pinning of ls2ln1ns7, respectively. The dashed lines represent the NP center-of-mass (red) and contact line (black) positions; (d) detailed snapshots of the alternate pinning−depinning patterns of ls2ln1ns6 in (a), the tagged depinning stage (1) and pinning stage (2). The details of the droplet profile changes shown in (d) are obtained by outlining the corresponding MD snapshots on its right side.

effectiveness of the theoretical simplification, we have provided the comparison between the above-mentioned θap (measured from the MD simulations directly) and the calculated Young’s contact angle θY in the Supporting Information part 1. It has been reported that on an ideal surface, θap (ideal contact angle, actually) is identical to θY.56 It turns out that the two sets of data match well. The difference between them comes from the errors in the fitted measurement and from the simplification of the theoretical model. We defined them as equilibrium contact angle θe uniformly for simplicity and to avoid confusion. The liquid volume Vdrop is constant. According to the geometric relationship

snapshots and droplet profiles in Figure 2d, at the pinning stage [the snapshots of stage (2) from 2.08 to 2.48 ns], the NP and the contact line get completely pinned. At the depinning stage, as shown in the inset stage (1) from 1.80 to 1.96 ns, the contact line and the NP move consistently. The NP is completely immersed in the liquid and cannot protrude from the liquid−air interface because of the hydrophilicity of the NP.59 (III) The complete pinning mode. When the ns interaction is very large, the resulting friction force exceeds the maximum value of the driving force. The NP cannot depin from the substrate and complete pinning occurs, as depicted in the snapshots in Figure 2c. 3.2. Pinning and Depinning Mechanisms of NPs at the Contact Line. Stretching the free liquid edge causes the droplet to no longer be in equilibrium. According to the thermodynamic theory proposed by Shanahan,60 the changes in the droplet shape lead to the excess free energy δG. When δG is sufficient to overcome the energy barriers, the contact line will depin to achieve the next equilibrium position. This is the mechanism of contact line jumping during evaporation and is the cause for the stick-slip evaporating pattern.57,58,60,61 Experiments have shown that the stick times of the contact line during evaporation coincide well with the results predicted using the excess free energy theory.57 We simplified the droplet using a crow-shaped model within the y−z plane, the thickness along the x-axis is L = 6 nm, as schematically illustrated in the inset of Figure 3a. This enables us to evaluate the excess free energy of the unequilibrated droplet compared with that at equilibrium. We applied the equilibrium contact length le measured from the MD simulations to the theoretically simplified model to calculate the theoretical equilibrium contact angle θY (the Young’s contact angle, which is used to deduce the excess free energy in the following). To verify the

Vdrop =

Alv =

l 2L ⎡ θ 1 ⎤ − ⎥ ⎢ 2 2 ⎣ (sin θ) tan θ ⎦

θlL sin θ

(7)

A sl = lL

(8)

where l is the contact length of the liquid on the substrate. The associated Gibbs free energy, G, is given by59 ⎛ θ ⎞ G = γlvAlv + (γsl − γsv)A sl = γlvLl⎜ − cos θe⎟ ⎝ sin θ ⎠

(9)

where the Young’s equation has been used to replace (γsl − γlv) by −γlv cos θe. γlv, γsl, and γsv represent the liquid−air, solid− liquid, and solid−air interface tension, respectively. The contact length l and the contact angle θ vary depending on the motion of the contact line. According to eq 7, at constant liquid volume dθ 1 sin θ(θ − sin θ cos θ ) =− dl l sin θ − θ cos θ 12679

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Langmuir δG̃ =

γlv(δl)2 sin 2 θe(θe − cos θe sin θe) 2l sin θe − θe cos θe

(13)

We defined κ = l/le to describe the elongation of the contact length. During the overall stretching process, there is κ ≥ 1. Thus δG̃ =

γlvle(κ − 1)2 sin 2 θe(θe − cos θe sin θe) 2κ sin θe − θe cos θe

(κ ≥ 1) (14)

Figure 3a shows that the excess free energy increases with an increase in κ (we investigated that κ is less than 2 for any model during the 4 ns simulation). This implies that the excess free energy accumulates at the pinning stage and is released at the depinning stage. When the excess free energy is lower than the energy barriers, the NP and the contact line get pinned. When the excess free energy is sufficient to overcome the energy barriers, they depin and move. Several factors contribute to the energy barriers, for example, we have shown in Figure 2 that the adhesion of the NP to the substrate is one of the factors. The force analyses are shown in the inset of Figure 3b. The forces acting on the NP and parallel to the substrate include two terms: the driving force Fd on which the liquid acts and the resistance friction force Ff imparted by the substrate. The stretching velocity v = 5 nm/ns is gentle, to ensure that Fd is balanced with Ff during the motion. Maximum values of the driving forces Fd were calculated in the MD simulations. The available excess free energy per unit length of the contact line is consistent with the driving force Fd, both having the same magnitude of 10−10 N. This confirmed that the excess free energy provides the driving force for the NP. Previous studies have described the capillary force, Fs = 2πrNPγlv cos2 θ, as the primary force for the NPs near the contact line.16,27,38 Here, γlv is the liquid−air interface tension, rNP is the NP radius, and θ is the contact angle. We evaluated the value of Fs and found that it has the same magnitude of 10−10 N. This proved that the thermodynamic-based driving mechanism of the contact line in this study is consistent with the previous experimental analyses. However, this does not mean that the driving force Fd exerted on the droplet will continually increase as the excess free energy accumulates. Assuming the situation where the NP gets pinned, and the liquid phase was stretched until it detaches from the NP. The driving force is zero once the detachment occurs, at which time the excess free energy is very large. The driving force Fd exerted on the NP also relates to the substrate wettability. According to Figure 3b, Fd exerted on the NP on hydrophilic substrate ls1 is much smaller than that on the hydrophobic substrates ls2 and ls3. In Figure 2, we explained that the NP and the contact line move synchronously. To move them, the excess free energy needs to overcome not only the energy barriers arising from the adhesion of the NPs to the substrate but also the energy barriers arising from the adhesion of the liquid droplet to the substrate. 3.3. Energy Barriers Resulting in the Pinning of the NP and the Contact Line. 3.3.1. Energy Barriers Arising from the Adhesion of the NPs to the Substrate. We have demonstrated in Figure 2 that the adhesion of the NPs to the substrate relates to the NP−substrate interaction. In this section, we will interpret that it is also connected to the NP wettability. Before this, we defined the pinning number of the NP to quantitatively reflect the pinning degree of the NP.

Figure 3. (a) Excess free energy available per unit length varies with the elongation of the contact line κ on substrates having different wettabilities, the hydrophilic substrate ls1, the hydrophobic substrate ls2, and more hydrophobic substrate ls3. The inset is the schematic of the simplified theoretical model. (b) The maximum driving force Fd exerted on the NPs on the substrates with various wettabilities.

Upon stretching the free liquid edge, the droplet was slightly out of equilibrium, such that l = le + δl and θ = θe − δθ. By Taylor’s theorem60 G(l) = G(le + δl) ⎡ dG ⎤ (δl)2 ⎡ d2G ⎤ = G(le) + δl⎢ ⎥ + ⎢ 2 ⎥ + Ο[(δl)3 ] ⎣ dl ⎦l = l 2 ⎣ dl ⎦l = l e e

(11)

Because le and θe are the initial contact length and contact dG angle, respectively, and ⎡⎣ dl ⎤⎦ = 0, we can evaluate the excess l=l e

free energy, δG, versus that of the initial equilibrium state and make use of eqs 9−11 δG = G(l) − G(le) ≈

(δl)2 γlvL sin 2 θe(θe − cos θe sin θe) 2l sin θe − θe cos θe (12)

The excess free energy available per unit length of the contact line, δG̃ 12680

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np = 1 −

D Dt

between the NP and the liquid. This will lead to a decrease in the driving force imparted on the NP by the liquid phase. Figure 4b verifies that a reduction in the NP hydrophilicity will enhance the pinning degree of the NP. At the same NP− substrate interaction, the pinning degree of the partially wetted NP is much larger than that of the completely wetted NPs. This implies that the energy barrier arising from the adhesion of the partially wetted NPs to the substrate is larger than that arising from the completely wetted NPs. The lifting effect of the liquid on the NP illustrates this phenomenon.39 The negative potential energy Ep(sn) between the NP and the substrate is evaluated in Figure 5 to reveal the lifting effect. The value of

(15)

in which D is the actual moving distance of the NP during the 4 ns simulation. Dt = 20 nm is the moving distance when the NP moves together with the liquid in v = 5 nm/ns without any pinning during the 4 ns simulation. In addition, there is 0 ≤ np ≤ 1. np = 0 represents the situation where there is no pinning during the movement, and np = 1 corresponds to the complete pinning. Figure 4a shows the motion of the NP that was partially wetted. In contrast to ls2ln1 shown in Figure 2, where the NP is

Figure 5. NP−substrate potential energy varies with the NP−substrate interaction for the pure NP, completely wetted NP ls2ln1, and partially wetted NP ls2ln2.

potential energy Ep(sn) increases with an enhancement of the NP−substrate interactions. This is in consistence with the results of Figure 2 that the contact line transfers from complete slipping to complete pinning by increasing the NP−substrate interactions. At the same ns interaction, ls2ln1 has a NP that is totally immersed in the liquid. The value of Ep(sn) is much lower than that of the pure NP on the substrate. Previous studies have shown that this is due to the hydrophilic nature of the NP. The liquid phase lifts up the totally immersed NPs from the substrate surface versus that without liquid. The friction force Ff generated between the NP and the substrate will decrease or disappear.39 This lifting effect of the liquid phase on the NP reduces the energy barrier originating from the adhesion of the NP to the substrate. The lifting effect is determined by the NP wettability. The Ep(ns) value between the substrate and the partially wetted NP is lower than the Ep(ns) value between the substrate and the pure NP. The lifting effect of the liquid phase on the partially wetted NPs cannot be ignored. On the other hand, the Ep(ns) value between the substrate and the partially wetted NP is much larger than that for a completely wetted NP. By reducing the NP wettability, the lifting effect is weakened. Two aspects contribute to the pinning enhancement of the partially wetted NP versus the completely wetted NP. On the one hand, the protruding of the NP from the contact line reduces the contact area between the liquid and the NP. Driving forces exerted on the NP by the liquid phase therefore decrease. On the other hand, the lifting effect of the liquid on the partially wetted NP is reduced versus that on the

Figure 4. (a) NP COM changes with time for models containing partially wetted NPs: ls2ln2ns1−ls2ln2ns6. The inset of MD snapshots shows the complete pinning of ls2ln2ns6. (b) Comparison of the pinning number between ls2ln1 containing a completely wetted NP and ls2ln2 containing a partially wetted NP.

completely wetted and immersed in the liquid, the NP of ls2ln2 protrudes out of the contact line, as illustrated in the inset snapshots in Figure 4a. This is due to the relative hydrophobic property of the NP. Even at low NP−substrate interactions ls2ln2ns1 and ls2ln2ns3, the NP represents obvious alternate pinning−depinning. Complete pinning of the NP was observed at a moderate NP−substrate interaction of ls2ln2ns4. According to the inset snapshots of ls2ln2ns6, the NP gradually protrudes out of the liquid−air interface, which reduces the contact area 12681

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Figure 6. Comparison of the NP motions on substrates with different wettabilities ls1−3. The NP pinning is enhanced with an increase in substrate hydrophilicity from (a) in the case of low NP−substrate interactions ns1; (b) under the situation of relatively large NP−substrate interactions ns4.

Figure 7. (a) Variations in the pinning number for different substrate wettabilities (determined by the ls interaction); (b) variations in the excess free energy vs time on three kinds of substrate wettabilities, ls1−3ln1ns1 containing completely wetted NP and having low ns interactions. The inset plot shows the variation in κ vs time on three substrate wettabilities.

even when the NP is subjected to a very small frictional resistance force Ff from the substrate. According to the force analyses of NP in the inset of Figure 3b, that is, in the case of a hydrophilic substrate, the driving force Fd imparted on the NP by the liquid is very small. At large NP−substrate interactions ns4 in Figure 6b, pinning enhancement via increasing the substrate wettability was investigated. In the case of ls3ln1ls4 where the substrate is very hydrophobic, and the NP moves with only short pauses during the process. Under the situation of a moderate hydrophobic substrate ls2ln1ns4, significant alternate pinning and depinning were observed. Continuously enhancing the substrate wettability to ls1ln1ns4 resulted in the complete pinning. A comparison of the pinning degree between NPs on different substrate wettabilities is shown in Figure 7a. It shows the trends in which the pinning degree of the NP increases with the enhancement of the substrate hydrophilicity. The pinning number of NPs on the hydrophilic substrate exceeds 0.8, even at low NP interaction ns1, where the substrate almost provides no frictional pinning force for the NP. This is much larger than the cases on hydrophobic substrates ls2 and ls3. Distinctions between the pinning degrees due to the substrate wettability are

completely wetted NP. The adhesion of partially wetted NP to the substrate is enhanced. Increasing the NP−substrate interactions increases the energy barrier originating from the adhesion of the NP to the substrate. By contrast, increasing the NP wettability will reduce this energy barrier. The energy barrier originating from the adhesion of the NP to the substrate is determined by both features. 3.3.2. Energy Barrier Arising from the Adhesion of Liquid Phase to the Substrate. The substrate wettability and the resulting contact angle hysteresis have significant effects on the motion of the contact line, the dispersion of the NPs inside of the droplets, and pinning of the nanofluid droplet during evaporation.35 Manipulation of single particles in the liquid droplet is also controlled by the substrate wettability.39,40 We conclude from Figure 6 that the pinning of the NP was enhanced by an increase in the substrate hydrophilicity. At low NP−substrate interactions ns1, NPs in ls2ln1ns1 and ls3ln1ns1, which are on the hydrophobic substrates, slipped about 20 nm during the 4 ns simulation, without any pinning. By contrast, the NP in ls1ln1ns1, which is on a hydrophilic substrate, moves slightly for about 2.5 nm. The average speed is about 0.625 nm/ns, which is far less than that of the liquid edge v = 5 nm/ ns. The NP exhibits strong pinning on the hydrophilic substrate 12682

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free energy is consumed by the liquid−substrate interfacial adhesion energy. This results in the pinning of the NP. (II) Alternate pinning−depinning occurs only on the substrate with moderate hydrophobicity, which is θe ≈ 90° in our study. The energy barrier of the adhesion of the liquid to the substrate is small but cannot be ignored. As shown in Figure 8, during the

particularly significant because the lifting effect of the liquid on the completely wetted NPs can effectively reduce the pinning energy barrier. According to the Young−Duper equation that describes the work of adhesion of the liquid to the solid surface, Wa = γlv(1 + cos θe),62 smaller equilibrium contact angle θe results in larger Wa. This means that on a hydrophilic substrate, it is hard to move or detach the liquid phase from the solid substrate. Previous studies have reported the pinning of contact lines on the hydrophilic substrates, the extended constantcontact-area evaporating pattern, and the coffee-ring it generated.35 This result is consistent with the experimental studies on the contact line motions during evaporation and spreading.57,58 It turns out that on a weakly wetted polymer substrate, the contact line is featured by stick-slip sliding. On the strongly wetted metal surface, however, the contact line gets pinned and shows giant contact angle hysteresis.57 The energy barrier arising from the work of the adhesion of the liquid on the solid substrate contributes to the pinning of the contact line. According to Figure 3b, at low NP−substrate interaction ns1, the energy barrier arising from the adhesion of the NP on the substrate is small and can be ignored. This allows us to evaluate the energy barrier arising from the adhesion of the liquid to the substrate. As shown in Figure 7b, in cases of hydrophobic substrate ls2ln1ns1 and more hydrophobic substrate ls3ln1ns1, the excess free energy is nearly zero during the stretching. The inset plot in Figure 7b explained that on the hydrophilic substrate, the value of κ keeps increasing because of the complete pinning of contact lines. κ values of less than 1.2 were observed on the hydrophobic substrate ls2 and the more hydrophobic substrate ls3. This means that the energy barrier arising from the adhesion of the liquid to the substrate is small in cases of hydrophobic substrates. That is, a slightly out of equilibrium of the contact length will result in the depinning of the contact line on the hydrophobic substrates. In the case of hydrophilic substrate ls1ln1ns1, however, the excess free energy continually accumulates the overall stretching process, with complete pinning of the NP and the contact line. Considering that NP−substrate energy potential Ep(sn) is very small at ns1, the energy barrier caused by the adhesion of the liquid to the hydrophilic substrate is significant. Most of the excess free energy was consumed to overcome the liquid−substrate interfacial energy barrier. Thus, small driving forces Fd, as shown in Figure 3b, are exerted on the NPs, which results in high pinning degree of NPs on the hydrophilic substrate. 3.4. Excess Free Energy-Based Mechanism of the Three Motion Modes. On the basis of the above analyses, the driving force Fd exerted on the NP is determined by the amount of the free energy distributed on overcoming the NP− substrate potential energy. On the hydrophilic substrate, the work of adhesion of the liquid droplet to the substrate is large and consumes most of the excess free energy. This is why small driving forces acting on the NPs were investigated in the case of hydrophilic substrates ls1 in Figure 3b. The three motion modes can be interpreted according to the excess free energy theory. (I) Complete pinning occurs when the substrate is hydrophilic or the NP−substrate interactions are large. On the one hand, on the hydrophilic substrate, the energy barrier arising from the work of adhesion of the liquid droplet to the substrate is large and is difficult to overcome. This results in the pinning of the contact line. Besides, a small driving force is imparted on the NP because most of the excess

Figure 8. Alternate pinning and depinning motion mode of ls2ln1ns6 containing a hydrophobic substrate. The excess free energy accumulates and releases during the pinning and depinning stages, respectively. The depinning stage (1) and pinning stage (2) correspond to that in the above Figure 2.

depinning stage (1), the contact line shrinks to minimize the free energy of the system. When the excess free energy is lower than the energy barriers, the NP and the contact line get pinned again. Once the NP gets pinned, being similar to stage (2), the excess free energy accumulates until it is sufficient to overcome the energy barriers. Regular alternate pinning and depinning then occur as a result. Of note, owing to the energy dissipation of contact line and NP displacement on the substrate, the depinning of the contact line and the NP from a pinning stage requires more absolute excess free energy than that from the previous pinning stage. This dissipation energy resulting from the contact line displacement has been included in the additional energetic barrier.58 This is why the excess free energy in Figure 8 cannot return to the initial minimum value after experiencing a depinning stage. (III) Complete slipping occurs on the substrate ls3 of high hydrophobicity with θe much larger than 90°. Energy barriers caused by the work of liquid− substrate interfacial adhesion and the NP−substrate attractive potential energy are both very small owing to the significant lifting effect and the hydrophobic nature of the substrate, respectively. The excess free energy arising from a slightly out of equilibrium of the contact length, or the contact angle, is sufficient to overcome the energy barriers and leads to the depinning of the contact line and the NP.

4. CONCLUSIONS In this paper, we regulated the contact line motion using a single NP. We explained the driving mechanism of the contact line. The nonequilibrium states of the droplet lead to the excess free energy, which overcomes the energy barriers and provides driving forces for the NP and the contact line. The energy barriers consist of two terms: the work of adhesion of the liquid droplet to the substrate, which increases with the substrate hydrophilicity, and the attractive potential energy between the 12683

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(3) Wasan, D.; Nikolov, A.; Kondiparty, K. The wetting and spreading of nanofluids on solids: Role of the structural disjoining pressure. Curr. Opin. Colloid Interface Sci. 2011, 16, 344−349. (4) Han, J.; Kim, C. Spreading of a suspension drop on a horizontal surface. Langmuir 2012, 28, 2680−2689. (5) Kondiparty, K.; Nikolov, A. D.; Wasan, D.; Liu, K.-L. Dynamic spreading of nanofluids on solids. Part I: Experimental. Langmuir 2012, 28, 14618−14623. (6) Liu, K.-L.; Kondiparty, K.; Nikolov, A. D.; Wasan, D. Dynamic spreading of nanofluids on solids Part II: Modeling. Langmuir 2012, 28, 16274−16284. (7) Li, Y.; Wang, F.; Liu, H.; Wu, H. Nanoparticle-tuned spreading behavior of nanofluid droplets on the solid substrate. Microfluid. Nanofluid. 2015, 18, 111−120. (8) Nikolov, A.; Kondiparty, K.; Wasan, D. Nanoparticle selfstructuring in a nanofluid film spreading on a solid surface. Langmuir 2010, 26, 7665−7670. (9) de Gennes, P. G. Wetting: Statics and dynamics. Rev. Mod. Phys. 1985, 57, 827−863. (10) Mechkov, S.; Cazabat, A. M.; Oshanin, G. Post-Tanner stages of droplet spreading: The energy balance approach revisited. J. Phys.: Condens. Matter 2009, 21, 464131. (11) Mechkov, S.; Cazabat, A. M.; Oshanin, G. Post-Tanner spreading of nematic droplets. J. Phys.: Condens. Matter 2009, 21, 464134. (12) Popescu, M. N.; Oshanin, G.; Dietrich, S.; Cazabat, A.-M. Precursor films in wetting phenomena. J. Phys.: Condens. Matter 2012, 24, 243102. (13) He, G.; Hadjiconstantinou, N. G. A molecular view of Tanner’s law: Molecular dynamics simulations of droplet spreading. J. Fluid Mech. 2003, 497, 123−132. (14) Yuan, Q.; Zhao, Y.-P. Precursor film in dynamic wetting, electrowetting, and electro-elasto-capillarity. Phy. Rev. Lett. 2010, 104, 246101. (15) Yuan, Q.; Zhao, Y.-P. Multiscale dynamic wetting of a droplet on a lyophilic pillar-arrayed surface. J. Fluid Mech. 2013, 716, 171−188. (16) Sangani, A. S.; Lu, C.; Su, K.; Schwarz, J. A. Capillary force on particles near a drop edge resting on a substrate and a criterion for contact line pinning. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2009, 80, 011603. (17) Layani, M.; Gruchko, M.; Milo, O.; Balberg, I.; Azulay, D.; Magdassi, S. Transparent conductive coatings by printing coffee ring arrays obtained at room temperature. ACS Nano 2009, 3, 3537−3542. (18) Zhang, Z.; Zhang, X.; Xin, Z.; Deng, M.; Wen, Y.; Song, Y. Controlled inkjetting of a conductive pattern of silver nanoparticles based on the coffee-ring effect. Adv. Mater. 2013, 25, 6714−6718. (19) Choi, S.; Stassi, S.; Pisano, A. P.; Zohdi, T. I. Coffee-ring effectbased three dimensional patterning of micro/nanoparticle assembly with a single droplet. Langmuir 2010, 26, 11690−11698. (20) Wang, F.-C.; Wu, H.-A. Enhanced oil droplet detachment from solid surfaces in charged nanoparticle suspensions. Soft Matter 2013, 9, 7974−7980. (21) Ehtesabi, H.; Ahadian, M. M.; Taghikhani, V.; Ghazanfari, M. H. Enhanced heavy oil recovery in sandstone cores using TiO2 nanofluids. Energy Fuels 2014, 28, 423−430. (22) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Capillary flow as the cause of ring stains from dried liquid drops. Nature 1997, 389, 827−829. (23) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Contact line deposits in an evaporating drop. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2000, 62, 756−765. (24) Ristenpart, W. D.; Kim, P. G.; Domingues, C.; Wan, J.; Stone, H. A. Influence of Substrate Conductivity on Circulation Reversal in Evaporating Drops. Phys. Rev. Lett. 2007, 99, 234502. (25) Zhong, X.; Crivoi, A.; Duan, F. Sessile nanofluid droplet drying. Adv. Colloid Interface Sci. 2015, 217, 13−30.

NP and the substrate, which can be enhanced by an increase in the NP−substrate interactions or by a decrease in the NP hydrophilicity. The enhancement of the NP−substrate interactions, the increase in substrate hydrophilicity, and the reduction in the NP hydrophilicity will lead to the pinning of the contact line and the NP. We obtained three motion modes of the NP and interpreted them accordingly. (I) On the hydrophobic substrate, the contact line and the NP get completely pinned. (II) In the case of substrates of high hydrophobicity with a contact angle much larger than 90°, the contact line and the NP slip without hysteresis. (III) On the hydrophobic substrate with θe ≈ 90°, alternate pinning− depinning was observed. This study explained the driving mechanism of the NP confined in the contact line, which made clear of how properties such as NP wettability will result and affect the pinning of the contact line and the NP. The basic mechanism presented in this study can be applied to many physical processes and to important phenomena such as evaporation and “coffee-ring”. We have provided the preliminary results of studies on nanofluid evaporation (see the Supporting Information part 2) based on the findings of this study, and we will complete the analyses of multiple deposition patterns further.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b03595. The measuring method of the contact angle; definition of the substrate wettability; preliminary MD simulation results of the nanofluids evaporation, and the patterned depositions (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +86-551-6360-1238. ORCID

FengChao Wang: 0000-0002-5954-3881 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (11572307, 11525211, 11302218, and 11472263), the Anhui Provincial Natural Science Foundation (1408085J08), the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB22040402), and the Fundamental Research Funds for the Central Universities of China. The numerical calculations have been performed on the supercomputing system in the Supercomputing Center of University of Science and Technology of China.

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REFERENCES

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