“Stick–Slip” Behavior of Evaporating Droplets ... - ACS Publications

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Comments on the Energy Barrier Calculations during “Stick−Slip” Behavior of Evaporating Droplets Containing Nanoparticles Melik Oksuz and H. Yildirim Erbil* Gebze Institute of Technology, Department of Chemical Engineering, Cayirova, Gebze 41400, Kocaeli, Turkey ABSTRACT: The application of three different potential energy barrier equations which were developed by Shanahan and co-workers for the “stick−slip” motion of droplets during evaporation of nanosuspensions to experimental data was investigated, and the results were compared. In theory, these potential energy barrier equations were assumed to be equivalent to each other, and thus the excess Gibbs free energy calculation results should be very close to each other. However, the calculated potential energy barrier results were found to be much different from each other when these equations were applied to two recently published experimental data. The details of the calculation procedures and reasons for these differences in the results are discussed, and eight new slightly modified equations are given in two sets to obtain more reliable, comparable and easily plotted potential energy barrier values.

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INTRODUCTION The evaporation of droplets containing micro- and nanoparticles is an important topic of interest because of the need to control the deposition geometry of particles dispersed in the drops to form specific patterns on solid surfaces.1 Cheap and high-quality nanoparticles can be produced by recent technological advances, and the application of nanofluid dispersions containing these nanoparticles is under investigation in many fields such as drug delivery, biomedicine, microfluidics, electrical or heat conducting patterns, solar heat harvesting, detergency, and evaporative cooling.2,3 When a drop of liquid containing dispersed microand nanoparticles evaporates on a solid, it generally leaves a nonuniform ring-shaped stain, where most of the particles in the drop are deposited. This phenomenon is known as the “coffeering effect”, and the control of the morphology and size of stains after the evaporation of a dispersion drop is a subject of intensive research in the past two decades after the pioneering publications of Deegan and co-workers.4−6 Drop evaporation of aqueous dispersions plays a crucial role in many novel applications such as DNA or RNA microarray formation,7 inkjet printing,8 and substrate patterning.9−14 In general, drop evaporation is a quasisteady process controlled by the diffusion of vapor into the air where the rate of drop evaporation can be related to its contact angle,1,15−17 diffusion coefficient,1,18,19 and vapor pressure of liquid.20−22 The mechanism to form a coffee-ring deposit by evaporation of a droplet containing particles was well explained previously: flat droplets evaporate faster at the droplet edge than the center, and the radial capillary flows from the drop center to its edge carry the suspended or dissolved solutes to the perimeter as evaporation proceeds after the contact line around the perimeter of the droplet pins on the solid.4−6,23,24 These radial capillary flows from its center to the edge of the drop are formed mainly due to the Marangoni flows which are produced by gradients of © 2014 American Chemical Society

the surface tension along the surface of the droplet from regions of low to high surface tensions. There are two types of Marangoni flows: thermal and solutal.23 Heat conductivity of the substrate is an important factor, and thermal Marangoni flows carry particles either toward the edge of the droplet or toward the center according to the balance of the drop evaporation and heat conduction rates.25 On the other hand, if surfactants are present in a suspension drop, solutal Marangoni flows may occur since surface tension gradients are produced from the concentration differences of solute along the droplet surface. The formation of surfactant monolayers at the water/air interface could both induce Marangoni flow and act to block evaporation, depending on the surfactant phase state, and the deposition patterns are affected.26−29 In general, there are two distinct drop evaporation modes: at constant contact angle with decreasing wetting contact radius, drop height; and at constant wetting contact radius with decreasing contact angle, drop height.15−17,30 A third drop evaporation mode, named “stick−slip”, has been observed for some liquid−substrate pairs31 especially during the evaporation of colloidal nanosuspensions where the nanoparticles cause the self-pinning of the three-phase contact line of the drop and form a set of concentric rings.32−44 Drop evaporation cycle follows two distinct and repetitive phases in the “stick−slip” mode: Initially, the droplet evaporates, retaining a constant contact radius while contact angle and height of the droplet decrease during the “stick” (pinning) phase. After the evaporation proceeds and the system reaches a threshold, minimum contact angle, then the three-phase contact line “slips” (depins) to a more energetically favorable position leading to a new, smaller contact radius and an Received: January 28, 2014 Revised: April 3, 2014 Published: April 7, 2014 9228

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THEORY If we neglect the gravity effect and assume that a droplet obeys the spherical cap geometry, then the volume (V) and liquid/ vapor interfacial area (A) of a droplet are given as

increase in both contact angle and drop height. This cycle is repeated until complete evaporation of the drop liquid. In general, the “slip” phase occurs very rapidly and the “stick” phase accounts for the majority of the evaporation time. Shanahan proposed a simple theory for “stick−slip” drop evaporation mode in 1995.31 This theory is the basis of the subsequent energy barrier equations which were derived by Shanahan and co-workers to explain the drop evaporation behavior of nanosuspensions.32−36 Many articles were published in the past decade on the topic of nanosuspension drop evaporation37−42 or pure liquid drops which are strongly pinned on substrates,42,43 and some of them used the equations derived by Shanahan and co-workers.30−44 In the 1995 paper, Shanahan assumed that when the volume of a pure liquid drop decreased by evaporation, the decrease of contact angle of a droplet pinned on a substrate with a constant contact radius leads to the lack of capillary equilibrium and the absence of the capillary equilibrium is the sole source of excess free energy giving rise to subsequent triple line motion to increase the contact radius with a jump. He also assumed that the pinning effect can be attributed to a potential energy barrier of value, U, per unit length of the triple line. The Gibbs free energy of the drop, G, was due entirely to the interfacial free energies when the spherical cap form is retained although the droplet loses liquid, and Shanahan ignored the effects of local features leading to contact angle hysteresis and also line tension for simplicity. He then defined the excess Gibbs free energy per unit length of the three-phase contact line (triple line), δG̅ . When δG̅ attains to U, this means that sufficient energy is available to overcome the potential energy barrier effect and then the triple line jumps to its next equilibrium position.31 Later, Shanahan and co-workers derived two different sets of new equations to calculate the free energy barriers during drop evaporation of pure liquids and nanosuspensions.32−36 In the first set, they derived the dependence of δG̅ on the change of contact radius, δr, and also on the change of contact angle, δθ, during the stick−slip motion of the droplet.32−34 In the second set, they derived a new equation giving δG̅ based on the volume constancy of droplets obeying spherical cap geometry at a given instant, in order to eliminate the equilibrium contact radius term, ro.35,36 The derivation of these two sets of equations is given in Theory for comparison. At the end, we have three different equations derived by the same group which should be equivalent to each other since all obey the same assumptions: the spherical cap geometry and the increase of excess Gibbs free energy per unit length of the three-phase contact line due to the lack of capillary equilibrium. However, this is not the case when experimental data is applied to calculate δG̅ and U values, since the results are considerably scattered. The investigation of the reasons of these calculation discrepancies is important because these equations and the related concepts are recently under use by many researchers in the field.32−44 In this work, we calculated the numerical results of time dependent δG̅ and Ui parameters by applying three different equations to two sets of experimental results already supplied by Shanahan and co-workers,31−33,35,36 and we compared our results with the results reported in these publications. In some cases, we found large differences in the U values which are reported in Results and Discussion. Then, we tried to explain the details of the calculation procedures and investigated the reasons for these differences and proposed 8 slightly modified new equations in order to calculate δG̅ and Ui values for further drop evaporation studies.

V=

πr 3 (1 − cos θ )2 (2 + cos θ ) 3 3 sin θ

(1)

A=

2πr 2 (1 + cos θ )

(2)

where r is the contact radius of the droplet with the substrate and θ is the contact angle. Within an additive constant, the interfacial Gibbs free energy, G, of the droplet may be written as G = AγLV + πr 2γSL − πr 2γSV = AγLV + πr 2(γSL − γSV ) (3)

where γLV, γSL, and γSV are interfacial tensions between liquid/ vapor, solid/liquid, and solid/vapor respectively and πr2 is the contact area of the droplet with the substrate as shown in Figure 1.

Figure 1. Schematic representation of a liquid drop deposited on a solid surface. γ’s are interfacial tensions.

For the ideal and equilibrium conditions, Young’s equation is given as γSV − γSL = γLV cos θo

(4)

where θo is the equilibrium contact angle. By combining eqs 2−4, one obtains 2πr 2γLV

− πr 2(γLV cos θo) (1 + cos θ ) ⎡ ⎤ 2 = γLVπr 2⎢ − cos θo⎥ ⎣ (1 + cos θ ) ⎦

G=

(5)

Shanahan proposed that only an equilibrium contact radius, ro, and contact angle, θο, exist for a given drop volume. But, if the drop is slightly out of equilibrium, for example with the increase of ro, as happens during the “slip” phase of drop evaporation, then the contact radius can be defined as ro =r − δr and the corresponding contact angle can be given as θ = θo − δθ. The excess Gibbs free energy of the droplet, δG = G(r) − G(ro), can be calculated by Taylor’s expansion theorem:31 9229

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−sin θ(2 + cos θ ) dθ = dr r

G(r ) = G(ro + δr ) 2⎡ 2

⎡ dG ⎤ (δr ) d G ⎤ = G(ro) + δr ⎢ ⎥ + ⎢ ⎥ ⎣ dr ⎦ r = r 2 ⎣ dr 2 ⎦ r = r o

and when (δr)2 is eliminated from eqs 8 and 10, one obtains eq 9. Shanahan and co-workers explained that when the droplet pins on a substrate, δG̅ increases during evaporation, since δθ physically increases (δθ = θo − θ) until it just exceeds the potential energy barrier, U. Three-phase contact line may depin and recede at this moment, leading to a smaller new equilibrium value of ro corresponding to the recently diminished drop volume. In this simplified model, after the jump occurs the original θο may be regained, the new ro may be smaller than the previous one, the height of the drop, h, will increase with the jump, and δG̅ should again become zero. It was also assumed that the three-phase contact line movement occurs very rapidly, and mass loss by evaporation and explicit time dependence were neglected in this model.32 Shanahan and Sefiane suggested that there are two values of θo, depending on whether the substrate has previously been in contact with the drop liquid or not.32 They defined the initial value of θo as θo,1 which corresponds to the γSV condition where the liquid contacted a dry substrate. However, the newly exposed solid surface has previously been in contact with the drop liquid for the subsequent stick−slip cycles, and therefore some liquid may remain, due to either slight physical retention within the surface roughness or adsorption. Thus, a second value of θo, which is denoted θo,2, is more appropriate for later calculations. They found a primary maximum of δG̅ , followed by secondary maxima where the primary maximum was markedly greater than all of the secondary peaks and the first secondary maximum was greater than most of the following peaks.32 They measured that slip distances (δr) diminish approximately with the square root of contact radius and calculated a δG̅ value of the order of 10−7 J/m for the first cycle based on θo,1 and 10−8 J/m for subsequent cycles based on θo,2 and pointed out that the unit of δG̅ (J m−1) is the same as those of line tension. Shanahan and co-workers derived a new equation in 2013 by assuming that the contact radius, ro, remains constant and the Gibbs free energy change with the change of contact angle can be given using eq 5 as35

o

+ 0[(δr )3 ]

(6)

Since ro corresponds to equilibrium, then [dG/dr]r=ro = 0 and the excess Gibbs free energy was evaluated as31 δG = G(r ) − G(ro) =

(δr )2 ⎡ d2G ⎤ ⎢ ⎥ 2 ⎣ dr 2 ⎦ r = r

o

2

2

≈ πγLV sin θo(2 + cos θo)(δr )

(7)

The quantity of the excess Gibbs free energy, δG, depends on the size of the droplet and the assumption that an exact spherical cap geometry is valid, and Shanahan reported that the excess Gibbs free energy per unit length of the three-phase contact line can be written as δG̅ ≈ ≈

πγLV sin 2 θo(2 + cos θo)(δr )2 2πr γLV sin 2 θo(2 + cos θo)(δr )2 2r

(8)

On an “ideal”, chemically homogeneous and atomically flat surface, the droplet must prefer to maintain its θo with a smoothly receding three-phase contact line (i.e., receding contact radius, r) to minimize its free energy for its volume at a given moment. However, when three-phase contact line is pinned due to the presence of the chemical heterogeneity, surface roughness, or presence of the dispersed particles within the droplet, then conversely a droplet prefers to maintain its ro with a decreasing θ. Shanahan proposed that this pinning effect can be quantified with a potential energy barrier of value U per unit length of the triple line. He assumed that the radius of the droplet, r, is slightly out of equilibrium and when the droplet approaches equilibrium of the same volume, then δr = r − ro. When δG̅ attains the value of the potential energy barrier, U, this indicates that sufficient energy is available to overcome this barrier effect (δG̅ max = U) and the triple line jumps (slip phase) to its next equilibrium position at ro (a new stick phase starts). This jump occurs sufficiently rapidly for the process to be considered as taking place at constant volume. He also implicitly assumed that inertial effects are sufficient to allow the triple line to ride over any possible metastable states existing for contact radii between r and ro, but insufficient to carry the triple line further which would mean raising θ above θo. Shanahan pointed out that he has assumed the contact angle after a jump to be the Young angle, θo in his analysis, but a metastable value of θ would be more appropriate. He neglected the possible effects due to the presence of line tension and also the potential influence of liquid viscosity.31 In 2009, Shanahan and Sefiane derived a new equation to calculate δG̅ by using a Taylor expansion in powers of δθ, rather than δr as in eq 5, to obtain32 δG̅ =

⎧ ⎡ ⎤ 2 − cos θo⎥ δG = G(θ ) − G(θo) = γLVπ ⎨r 2⎢ ⎦ ⎩ ⎣ (1 + cos θ ) ⎪

⎪

⎡ ⎤⎫ 2 − ro 2⎢ − cos θo⎥⎬ ⎣ (1 + cos θo) ⎦⎭ ⎪

⎪

(11)

where θo is the equilibrium (Young) contact angle, and θ is the decreasing contact angle during drop evaporation. The authors indicated that the term ro corresponds to the radius of a drop of equal volume to the actual drop, but at equilibrium (r ≥ ro) and ro decreases as evaporation continues, although θo does not, since volume, V, decreases. The volume of the droplet at a given instant is given by eq 1, and for the equilibrium conditions it can be shown as

γLVr(δθ )2 2(2 + cos θo)

(10)

V=

(9)

Equation 9 can be derived from eq 8 easily because, for a constant spherical cap drop volume, the derivative of the change of contact angle with the contact radius is given as

πro3 3 sin 3 θo

(1 − cos θo)2 (2 + cos θo)

(12)

By combining eqs 1 and 12, Shanahan and co-workers isolated the ro parameter as 9230

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r sin θ0 (1 − cos θ )2/3 (2 + cos θ )1/3 sin θ (1 − cos θo)2/3 (2 + cos θo)1/3

Article

(13)

and they inserted it into eq 11 to obtain δG =

γLVπr 2 (1 + cos θ )

[2 − cos θo(1 + cos θ)

− (1 − cos θ )1/3 (2 + cos θ )2/3 (2 + cos θo)1/3 (1 − cos θo)2/3 ]

(14)

Then, they calculated the excess free energy, per unit length of three-phase contact line, δG̅ , at any instant during evaporation from δG̅ = ∂G/2πr as γLVr δG̅ = [2 − cos θo(1 + cos θ) 2(1 + cos θ) − (1 − cos θ )1/3 (2 + cos θ )2/3 (2 + cos θo)1/3 (1 − cos θo)2/3 ]

(15)

As seen in eq 15, no δr and δθ data is required and the initial equilibrium contact angle, θo, is used at any time to calculate δG̅ . Shanahan and co-workers published an Erratum36 in 2013 to correct their miscalculated δG̅ values given in ref 35.

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METHOD Time dependent contact angle and drop radius results which were reported in two published papers were used (after the wrong data was corrected by the authors) for the calculations of potential energy barriers of stick−slip motion during drop evaporation. The first paper reported the evaporation of water droplets having a volume of 3 μL containing TiO2 (% 0.1 by weight) nanoparticles on a PTFE (polytetafluoroethylene) surface.33 In the second one, water droplets having a volume of 3 μL containing SiO2 (% 0.125 by weight) nanoparticles were evaporated on a silicone substrate (covered with a native oxide layer).35,36 Both of the substrates were reported to be very flat. The authors of these publications sent us the time dependent ri, θi, ro, θo, δr, and δθ data for every jump, and these data are given in Figures 2 and 3 and Tables 1 and 2. (The authors informed us that some of the data given in ref 33 was incorrect and the corrected (revised) data was given in the PhD thesis of Daniel Orejon.45 Some of the data given in ref 35 was also incorrect, and an Erratum was published giving the correct data in 2013.36 We only used the correct data in our calculations.) Potential energy barriers (U = δG̅ max) which were calculated by using eqs 8, 9, and 15 were also given in Tables 1 and 2. The γLV value of water was taken as 0.0728 N/m in all of these calculations since it was reported that the addition of nanoparticles does not change the surface tension of the liquid.33

Figure 2. (a) Contact radius, r (mm), and (b) contact angle, θ (deg), vs time, t (s), plot of experimental drop evaporation data of 0.1% TiO2 (by weight) containing water nanosuspensions on a PTFE surface performed by Orejon et al.33

absolutely constant during the “stick” phase, and ro,o diminishes to rmin‑1 just before the first jump and then the equilibrium contact radius after the first jump is shown as ro,1 in Figure 2a. Similarly, in most of the published experimental data, ro,o was not constant during the “stick” phase but decreased slightly to rmin‑1 and later rapidly to ro,1. This process repeated from ro,1 to rmin‑2. The equilibrium contact radius after the second jump is shown as ro,2, and it also decreases down to rmin‑3 as shown in Figure 2a. The same notation applied to other jumps as well. The initial contact radius, ro,o, equals 0.936 mm in Figure 2a, and the initial contact angle, θo,o, equals 110.5° in Figure 2b, indicating a low interaction with the droplet and the Teflon surface. θo,o decreases down to a minimum contact angle of θmin‑1 by the evaporation of the nanosuspension droplet, and the equilibrium contact angle after the first jump is shown as θo,1 in this figure. Similarly, the second minimum contact angle is shown as θmin‑2 and equilibrium contact angle after the second jump as θo,2 and so on as shown in Figure 2b. There were 5 considerable jumps during the evaporation of the TiO2 containing nanosuspension droplet, and all the experimentally reported δri and δθi values are given in Table 1. We realized that both δri and δθi values are large, and the mean δri was 7% of ro,o and mean δθi was 7.8% of θo,o. This contradicts the underlying derivation assumptions of Shanahan’s original treatment where he assumed that the associated δri and δθi were very small. Despite that, these large δri and δθi values were used in eqs 8 and 9 by Shanahan and co-workers without any reservation,32−34,37,38 and we applied the same procedure as well.

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RESULTS AND DISCUSSION The change of contact radius and contact angle of evaporating droplet containing 0.1% TiO2 (by weight) nanosuspension on a PTFE surface which was performed by Orejon et al.33 with time is given in Figure 2a,b. We calculated δri, δθi, and potential energy barrier (U = δG̅ max) values by using eqs 8 and 9 for this data, and the results are given in Table 1. First of all, we discriminated the notations for contact radius and contact angles during all the steps of the evaporation process and defined the initial contact radius as ro,o. This value is not 9231

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Table 2. θi and Potential Energy Barriers (U = δG̅ max) Which Were Calculated by Using Eq 15 for the Experimental Evaporation Data of Drops containing 0.125% SiO2 (by Weight) Aqueous Nanosuspensions on a Silicone Wafer Surface Performed by Askounis et al.35,36a Ui from eq 15 (10−7 J m−1) jump no.

θmin‑o,i (deg)

1 2 3

13.5 12.7 8.4

a

rmin‑i (10

−3

1.537 1.138 1.021

m)

our value

published value35,36

74.87 58.29 67.89

∼75 ∼59 ∼68

θo,o = 47.9°.

experimental δri values when eq 8 was used and arithmetic mean of experimental δθi values when eq 9 was used. We reported their Umean values in the last two columns at right in Table 1 as 2.44 × 10−7 and 4.76 × 10−7 J m−1. We compared our and Orejon et al.’s results and found that our result is exactly equal to 2.44 × 10−7 J m−1 for eq 8 as theirs, but slightly different from 4.76 × 10−7 J m−1 (between 4.65 and 4.80 × 10−7 J m−1) for eq 9, indicating that we repeated the same procedure as Orejon et al. did. However, we noted that the potential energy barriers (U = δG̅ max) which were calculated by using eqs 8 and 9 are too different from each other and eq 9 resulted in nearly twice the values obtained by using eq 8. This discrepancy between the results obtained when eqs 8 and 9 are used is an important point since both equations are equivalent to each other mathematically as shown in Theory. On the other hand, the change of contact radius and contact angle of evaporating droplet containing 0.125% SiO2 (by weight) nanosuspension on a silicone wafer surface which was performed by Askounis et al.35,36 with time is given in Figure 3a,b. ri and θi values during the jumps and potential energy barriers (U = δG̅ max) which were calculated by using eq 15 for this data are given in Table 2. The same notations for ro,o, rmin‑i, ro,i, θo,o, θmin‑i, and θo,i parameters given above were also applied to Figure 3a,b. The initial radius was ro,o = 1.578 mm, and initial contact angle was θo = 47.9°, which indicates a strong interaction with the droplet and the surface. There were 3 jumps during the evaporation of this droplet. Similar to Askounis et al., we calculated the potential energy barriers (U = δG̅ max) by using eq 15 and found that our results are very close to the results reported by Askounis et al.36 as given in the last two columns at right in Table 2. No δr and δθ data is required to be used in eq 15 as given in Theory, and the initial equilibrium contact angle, θo,o, is always inserted into eq 15 along with ri and θi values at each jump time to calculate the (U =

Figure 3. (a) Contact radius, r (mm), and (b) contact angle, θ (deg), vs time, t (s), plot of experimental drop evaporation data of 0.125% SiO2 (by weight) containing water nanosuspensions on a silicone wafer surface performed by Askounis et al.35,36

However, the other question remains, how did they calculate δri and δθi values when all the experimental ro,o, rmin‑i, ro,i, θo,o, θmin‑i, and θo,i data are all present. We repeated the various forms of possible calculation procedures and determined that Orejon et al. calculated δri values by using the δri = rmin‑i - ro,i equation and δθi values by using δθi = θo,i − θmin‑i and did not use the ro,o value in δri, and θo,o value in δθi calculations. They also used r = ro in the denominator of eq 8 and in the numerator of eq 9. These important details were lacking in their original publications.33,34,37,38 Orejon et al. also did not report Ui values for every jump, and they preferred to report only the mean Ui value which was calculated by using the arithmetic mean of

Table 1. δri, δθi, and Potential Energy Barriers (U = δG̅ max) Which Were Calculated by Using Eqs 8 and 9 for the Experimental Evaporation Data of Drops Containing 0.1% TiO2 (by Weight) Aqueous Nanosuspensions on a PTFE Surface Performed by Orejon et al.33a Umean (10−7 J m−1) Ui (10−7 J m−1) jump no.

δri (10−3 m)

δθi (deg)

from eq 8

from eq 9

1 2 3 4 5

0.0665 0.0685 0.0445 0.0755 0.0740

8.1 8.1 6.2 10.2 10.4

2.49 2.64 1.11 3.21 3.08

4.13 4.13 2.42 6.54 6.80

published value33

our value from eq 8

from eq 9

using eq 8

using eq 9

2.44b 2.51d

4.65c 4.80d

2.44b

4.76c

ro,o = 0.936 × 10−3 m; θo,o = 110.5°. bUmean calculated from mean of (δri) values. cUmean calculated from mean of (δθi) values. dUmean calculated from mean of Ui values. a

9232

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jumps ranged between 4.26 and 74.87 × 10−7 J m−1 and individual values of three single jumps varied between 1.87 and 74.87 × 10−7 J m−1 (40 times difference). In conclusion, these results show that eqs 8, 9, and 15 neither are equivalent nor use the same contact radius and contact angle parameters in their calculations, and some novel systematic calculation procedures are required to determine the potential energy parameter, U. Another issue is the time dependence of the excess Gibbs free energy parameter, δG̅ . According to the underlying assumptions of Shanahan, δG̅ increases with time during drop evaporation, and when it attains the value of U, sufficient energy is available in the drop to overcome the barrier effect (U = δG̅ max) and the triple line jumps to its next position where δG̅ becomes zero again and it further increases by time.31,32,35 We tried to plot the time dependence of δG̅ to calculate the (U = δG̅ max) value. It is clear that eq 15 is suitable for this plot since it contains time dependent parameters of ri and θi. However, eq 8 cannot be used because it contains constant ro, θo, and also time independent δri values. In contrast with this, eq 9 can also be used for the time dependent plots because it contains time dependent δθi and ri values. Then, we plotted Askounis et al.35,36 data and obtained the same figure given in Erratum36 as Figure 1, however, this plot seems to be wrong conceptually because δG̅ values start around 30 × 10−7 J m−1 after the second and third jumps, instead of 0. The error arises from the fact that Askounis et al. used the initial contact angle θo,o = 110.5° in all of their calculations, however, θo,i must be used instead of θo,o for every jump to obtain δG̅ = 0 just after the jump. Thus, there is a need to modify both of the eqs 15 and 9. We write δG̅ for any given time until to the first jump of the droplet with our notations, instead of eq 15, in order to obtain δG̅ = 0 just after the jump as γLVri δGi̅ = 2(1 + cos θi) ⎡ 2 − cos θ (1 + cos θ ) − (1 − cos θ )1/3 ⎤ o,o i i ⎢ ⎥ ⎢ ⎥ 2/3 1/3 ⎢(2 + cos θi) (2 + cos θo,o) ⎥ ⎢⎣ (1 − cos θo,o)2/3 ⎥⎦ (16)

δG̅ max) parameter. We realized that Askounis et al. used ri = rmin‑i and θi = θmin‑i in their calculations. The good fit with our and Askounis et al.’s results for the three jumps (59−75 × 10−7 J m−1) indicates that we repeated the same procedure as they did. However, the use of θo,o for all of the U calculations is a problem to prevent zeroing of the δG̅ value, as we will see later. As a second step, we tested the applicability of eq 15 to Orejon et al.33 data in order to check whether the U values will agree with the results obtained from eqs 8 and 9. The results are given in Table 3. As imposed by Askounis et al. the initial contact angle θo Table 3. θi and Potential Energy Barriers (U = δG̅ max) Which Were Calculated by Using Eq 15 for the Experimental Evaporation Data of Drops Containing 0.1% TiO2 (by Weight) Aqueous Nanosuspensions on a PTFE Surface Performed by Orejon et al.33a

a

jump no.

θi (deg)

ri (10−3 m)

Ui from eq 15 (10−7 J m−1)

our Umean from eq 15 (10−7 J m−1)

1 2 3 4 5

93.3 95.2 97.4 94.1 93.8

0.924 0.855 0.787 0.742 0.666

14.47 10.84 7.52 10.66 9.89

10.67b

θo,o = 110.5°. bUmean calculated from mean of Ui values.

= 110.5° is always used in eq 15 along with the ri = rmin‑i and θi = θmin‑i values at the jump time for these calculations. It was determined that eq 15 resulted in much larger Ui values than those calculated by eq 8 (2.3−5.8 times) and eq 9 (2.0−3.7 times), and Umean = 10.67 × 10−7 J m−1 was found from eq 15 for Orejon et al.33 data. In summary, Umean values for these five jumps ranged between 2.44 and 10.67 × 10−7 J m−1, and individual values of five single jumps varied between 1.11 and 14.47 × 10−7 J m−1 (13 times difference). As a third step, we tested the applicability of eqs 8 and 9 to Askounis et al.35,36 data in order to check whether the U values will agree with the results obtained by eq 15. The results are given in Table 4. As imposed by Orejon et al., we calculated δri values Table 4. δri, δθi, and Potential Energy Barriers (U = δG̅ max) Which Were Calculated by Using Eqs 8 and 9 for the Experimental Evaporation Data of Drops Containing 0.125% SiO2 (by Weight) Aqueous Nanosuspensions on a Silicone Wafer Surface Performed by Askounis et al.35,36a Ui (10−7 J m−1)

Then, the first potential energy barrier, U1, for the first jump can be calculated from eq 16 as γLVrmin‐1 U1 = δGmax ̅ ‐1 = 2(1 + cos θmin‐1) ⎡ 2 − cos θ (1 + cos θ ) ⎤ o,o min‐1 ⎢ ⎥ 1/3 ⎢ − (1 − cos θmin‐1) ⎥ ⎢ ⎥ ⎢(2 + cos θmin‐1)2/3 (2 + cos θo,o)1/3 ⎥ ⎢ ⎥ 2/3 ⎣ (1 − cos θo,o) ⎦ (17)

Umean (10−7 J m−1)

jump no.

δri (10−3 m)

δθi (deg)

from eq 8

from eq 9

from eq 8

from eq 9

1 2 3

0.312 0.193 0.080

10.2 8.4 5.6

33.01 12.63 2.17

6.82 4.23 1.87

12.90b 15.94d

4.26c 4.31d

ro,o = 1.578 × 10−3 m; θo,o = 47.9°. bUmean calculated from mean of (δri) values. cUmean calculated from mean of (δθi) values. dUmean calculated from mean of Ui values. a

Similarly, the δG̅ value for any given time between the first and the second jump can be written as γLVri δGi̅ = 2(1 + cos θi) ⎡ 2 − cos θ (1 + cos θ ) − (1 − cos θ )1/3 ⎤ i i o,1 ⎥ ⎢ ⎥ ⎢ 2/3 1/3 ⎥ ⎢(2 + cos θi) (2 + cos θo,1) ⎥⎦ ⎢⎣ (1 − cos θo,1)2/3 (18)

by using the equation δri = rmin‑i − ro,i and δθi values from δθi = δθo‑i − δθmin‑i. We also used r = ro in the denominator of eq 8 and in the numerator of eq 9 similar to their approach. We obtained Umean = 12.90−15.94 × 10−7 J m−1 from eq 8 and 4.26−4.31 × 10−7 J m−1 from eq 9, which are much smaller than the 59−75 × 10−7 J m−1 value which was calculated by applying eq 15 to Askounis et al.’s data. In summary, Umean values for these three 9233

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Figure 4. Excess Gibbs free energy per unit length of triple line (J m−1) vs time, t (s), calculated by using eqs 16 and 18 derived from eq 15 (triangles), using eqs 20 and 22 derived from eq 9 (circles), and using directly eq 15 (squares) for the experimental drop evaporation data of 0.125% SiO2 (by weight) containing water nanosuspensions on a silicone wafer surface performed by Askounis et al.35,36

and the second potential energy barrier, U2, for the second jump

δGi̅ =

can be calculated as U2 = δGmax ̅ ‐2 =

γLVrmin‐2

2(2 + cos θo,o)

(20)

Then, the first potential energy barrier, U1, for the first jump can be calculated from eq 20 as

2(1 + cos θmin‐2)

⎡ 2 − cos θ (1 + cos θ ⎤ o,1 min‐2) ⎢ ⎥ 1/3 ⎢ − (1 − cos θmin‐2) ⎥ ⎢ ⎥ ⎢(2 + cos θmin‐2)2/3 (2 + cos θo,1)1/3 ⎥ ⎢ ⎥ 2/3 ⎣ (1 − cos θo,1) ⎦

2 γLVr( i θo,o − θi)

U1 = δGmax ̅ ‐1 =

γLVrmin‐1(θo,o − θmin‐1)2 2(2 + cos θo,o)

(21)

The δG̅ value for any given time between the first and the second jump can be written as

(19)

This procedure may be applied to the other jumps by changing the rmin‑i, θo,i, and θmin‑i parameters. On the other hand, time dependent calculation of the δG̅

δGi̅ =

parameter is also possible by using eq 9 if it is modified. We write

2 γLVr( i θo,1 − θi)

2(2 + cos θo,1)

(22)

The second potential energy barrier, U2, for the second jump can be calculated from eq 22 as

δG̅ for any given time until to the first jump, instead of eq 9, as 9234

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Figure 5. Excess Gibbs free energy per unit length of triple line (J m−1) vs time, t (s), calculated by using eqs 16 and 18 derived from eq 15 (triangles) and using eqs 20 and 22 derived from eq 9 (circles) for experimental drop evaporation data of 0.1% TiO2 (by weight) containing water nanosuspensions on a PTFE surface performed by Orejon et al.33

U2 = δGmax ̅ ‐2 =

γLVrmin‐2(θo,1 − θmin‐2)2 2(2 + cos θo,1)

two jumps. However, Ui values of the second and third jumps are close to the Ui values calculated by eqs 8 and 9 as given in Table 4. We may state that our procedure maintains the δG̅ = 0 condition just after the jump and gives exactly the same U1 value as expected and clarifies the error introduced by the presence of θo,o data in all of the calculations. As a last step, we applied eqs 16, 18, 20, and 22 to Orejon et al. data and plotted the δG̅ results in Figure 5. The δG̅ parameter became zero after every jump as required. The potential energy barrier, Ui, obtained from Ui = δG̅ max‑i resulted in values between 1.7 and 18.4 × 10−7 J m−1, which is between the values obtained using eqs 8, 9, and 15 given in Tables 1 and 3. Only the U1 value of the first jump fits the value calculated by the original eq 15 as expected. U2 to U5 values are more close to the values calculated by eqs 8 and 9 as given in Table 1. The first jump resulted in the largest Ui values as seen in both Figures 4 and 5. A possible reason is that the drop pinning is the strongest after we place the suspension droplet on a dry surface, because after the first jump the droplet is on a previously wetted surface, the pinning energy is decreased, and δG̅ cannot reach the U1 value of the first jump again. There is another reason: the

(23)

This procedure may be applied to the other jumps by changing the rmin‑i, θo,i, and θmin‑i parameters. Then, we plotted the excess Gibbs free energy per unit length of triple line versus time in Figure 4 which was calculated by using eqs 16 and 18 derived from eq 15 (triangles), using eqs 20 and 22 derived from eq 9 (circles), and using directly eq 15 (squares) for the experimental drop evaporation data of 0.125% SiO2 (by weight) containing water nanosuspensions on a silicone wafer surface performed by Askounis et al.35,36 As seen in Figure 4, the results obtained from eqs 9 and 15 are close to each other if the required data is carefully selected. Ui values calculated by both methods give very close results to each other as can be seen in Figure 4: 75.5 × 10−7 for the first, 5.2 × 10−7 for the second, and 6.2 × 10−7 J m−1 for the third jump. When we compare these results with the results given in Table 2, it was found that only the result of the first jump fits the results calculated by original eq 15 and decreased sharply for the next 9235

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Table 5. Deviations on the First Potential Energy Barrier for the First Jump (U1 = δG̅ max‑1) Which Were Calculated by Using Eqs 17 and 21 When Hypothetical Errors of 1, 2, an 5% Are Applied into the Experimentally Measured θ0,0, θi‑min, and ri‑min Parameters of the Evaporation Data of Drops Containing 0.1% TiO2 (by Weight) Aqueous Nanosuspensions on a PTFE Surface Performed by Orejon et al.33 and Evaporation Data Performed by Askounis et al.35,36 param

θ0,0 (deg)

θi‑min (deg)

ri‑min (10−3m)

θ0,0 (deg)

θi‑min (deg)

ri‑min (10−3m)

error % in param

θ0,0 (deg)

0 1 2 5 1 2 5 1 2 5

110.50 111.61 112.71 116.03 110.50 110.50 110.50 110.50 110.50 110.50

0 1 2 5 1 2 5 1 2 5

47.9 48.38 48.86 50.30 47.90 47.90 47.90 47.90 47.90 47.90

ri‑min (10−3 m)

θi‑min (deg)

U1 from eq 21 (10−7 J m−1)

U1 deviation %

Experimental Evaporation Data Performed by Orejon et al.33 93.30 0.9240 18.4 93.30 0.9240 21.0 −12.7 93.30 0.9240 23.9 −23.2 93.30 0.9240 33.9 −45.8 94.23 0.9240 16.4 11.8 95.17 0.9240 14.6 25.8 97.97 0.9240 9.8 88.3 93.30 0.9332 18.6 −1.0 93.30 0.9425 18.7 −2.0 93.30 0.9702 19.3 −4.8 Experimental Evaporation Data Performed by Askounis et al.35,36 13.50 1.5370 75.5 13.50 1.5370 77.8 −2.9 13.50 1.5370 80.2 −5.8 13.50 1.5370 87.4 −13.6 13.64 1.5370 74.9 0.8 13.77 1.5370 74.3 1.6 14.18 1.5370 72.6 4.0 13.50 1.5524 76.3 −1.0 13.50 1.5677 77.0 −2.0 13.50 1.6139 79.3 −4.8

(24)

and not from δθ = θo,1 − θmin‐1 = θo,2 − θmin‐2 = θo,i − θmin‐i

U1 deviation %

14.5 16.3 18.2 24.4 13.1 11.8 8.1 14.6 14.8 15.2

−11.2 −20.5 −40.8 10.6 23.0 77.8 −1.0 −2.0 −4.8

74.9 77.0 79.1 85.7 74.2 73.6 71.7 75.6 76.4 78.6

−2.7 −5.4 −12.6 0.9 1.7 4.4 −1.0 −2.0 −4.8

data35,36), then U1 deviations are around 1.0−2.5 times the contact angle deviations. Both of eqs 17 and 21 resulted in very close U1 deviation values, indicating the similar applicability of both equations. When we inspected eq 21, we found that the source of the large deviation comes from the (θo,o − θmin‑1)2 part of this equation, since the square of the difference of contact angles in radians causes large numerical differences. This shows that very precise contact angle measurements are required for further drop evaporation studies containing nanoparticles. However, this does not mean that the exclusion of the eq 20−23 set is required in the U1 calculation procedure, since it is possible to obtain very close plots by using both the eq 16−18 and eq 20−23 sets as seen in Figures 4 and 5.

deposition of nanoparticles alters the substrate surface free energy during the initial stick phase and the first jump is different from the rest due to this reason. It seems that the use of eqs 16−23 and plotting δG̅ with time gives somewhat more reliable and easily understandable results of the Ui values. In order to calculate these values, the use of both sets of eqs 16−19 and eqs 20−23 is possible. However, one should be careful enough on the difference of the δθ parameter calculation for the use of the eq 20−23 set: δθ should be calculated as δθ = θo,o − θmin‐1 = θo,1 − θmin‐2 = θo,(i − 1) − θmin‐1

U1 from eq 17 (10−7 J m−1)

■

(25)

CONCLUSIONS Three-phase contact line is pinned during the evaporation of colloidal nanosuspension drops where the deposition of nanoparticles causes the self-pinning of the drop and forms a set of concentric rings. This drop evaporation mode is named as “stick−slip”, and the calculation of the mean potential energy parameter, Umean, to quantify the sufficient energy which must be available to overcome this barrier effect is the subject of this work. It was expected that the calculation of Ui is a straightforward process when all three different equations derived by Shanahan and co-workers were applied to the experimental drop evaporation results. However, this is not the case, and we found large differences in Ui values when we calculated and compared the numerical results of Ui and time dependent excess Gibbs free energy per unit length, δG̅ , for each jump during “stick−slip” motion of droplets containing nanosuspensions by applying these different equations to two sets of experimental results already supplied by Shanahan and co-workers.31−33,35,36 This unexpected result indicates that these three equations are

32,33

as initially proposed by Shanahan and co-workers. We also checked the effect of the magnitude of errors of θ and r parameters on the energy barrier calculation results since the uncertainty on measuring the angle is not the same as the uncertainty on measuring the contact radius. The deviations on the first potential energy barriers for the first jump (U1 = δG̅ max‑1) which were calculated by using eqs 17 and 21 for the application of hypothetical errors of 1, 2, and 5% into the experimentally measured θ0,0, θi‑min, and ri‑min parameters of Orejon et al. data33 and Askounis et al. data35,36 are given in Table 5. It is realized that the effect of the deviations of the contact radius, ri‑min, on the resultant U1 values is very small and proportional with the % of the deviation of ri‑min. However, the deviations of the initial contact angles of θ0,0 and θi‑min on the U1 values is large and approaches up to 17 times that of the % deviation of the θ0,0 and θi‑min values. If θ0,0 or θi‑min values are large (i.e., around 93−110° as given in Orejon et al. data33), then U1 deviations are around 8−17 times of the contact angle deviations, but when θ0,0 or θi‑min values are small (i.e., around 14−48° as given in Askounis et al. 9236

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(16) Erbil, H. Y.; McHale, G.; Newton, M. I. Drop Evaporation on Solid Surfaces: Constant Contact Angle Mode. Langmuir 2002, 18, 2636−2641. (17) Hu, H.; Larson, R. G. Evaporation of a Sessile Droplet on a Substrate. J. Phys. Chem. B 2002, 106, 1334−1344. (18) Erbil, H. Y.; Dogan, M. Determination of Diffusion CoefficientVapor Pressure Product of Some Liquids from Hanging Drop Evaporation. Langmuir 2000, 16, 9267−9272. (19) Erbil, H. Y.; Avcı, Y. Simultaneous Determination of Toluene Diffusion Coefficient in Air from Thin Tube Evaporation and Sessile Drop Evaporation on a Solid Surface. Langmuir 2002, 18, 5113−5119. (20) Erbil, H. Y. Determination of Peripheral Contact Angle of a Sessile Drop on Solids from the Rate of Drop Evaporation. J. Phys. Chem. B 1998, 102, 9234−9238. (21) Erbil, H. Y.; McHale, G.; Rowan, S. M.; Newton, M. I. Analysis of Evaporating Droplets Using Ellipsoidal Cap Geometry. J. Adhes. Sci. Technol. 1999, 13, 1375−1391. (22) Erbil, H. Y. Determination of Peripheral Contact Angle of Sessile Drops on Solids from the Rate of Evaporation. J. Adhes. Sci. Technol. 1999, 13, 1405−1413. (23) Larson, R. G. Re-Shaping the Coffee Ring. Angew. Chem., Int. Ed. 2012, 51, 2546−2548. (24) Shao, F. F.; Neild, A.; Alan, T. Controlled Particle Self-Assembly in an Evaporating Droplet. Colloids Surf., A 2012, 398, 64−68. (25) Hu, H.; Larson, R. G. Marangoni Effect Reverses Coffee-Ring Depositions. J. Phys. Chem. B 2006, 110, 7090−7094. (26) Truskett, V. N.; Stebe, K. J. Influence of Surfactants on an Evaporating Drop: Fluorescence Images and Particle Deposition Patterns. Langmuir 2003, 19, 8271−8279. (27) Crivoi, A.; Duan, F. Effect of Surfactant on the Drying Patterns of Graphite Nanofluid Droplets. J. Phys. Chem. B 2013, 117, 5932−5938. (28) Still, T.; Yunker, P. J.; Yodh, A. G. Surfactant-Induced Marangoni Eddies Alter the Coffee-Rings of Evaporating Colloidal Drops. Langmuir 2012, 28, 4984−4988. (29) Doganci, M. D.; Sesli, B. U.; Erbil, H. Y. Diffusion-Controlled Evaporation of Sodium Dodecyl Sulfate Solution Drops Placed on a Hydrophobic Substrate. J. Colloid Interface Sci. 2011, 362, 524−531. (30) Bourges-Monnier, C.; Shanahan, M. E. R. Influence of Evaporation on Contact Angle. Langmuir 1995, 11, 2820−2829. (31) Shanahan, M. E. R. Simple Theory of “Stick-Slip” Wetting Hysteresis. Langmuir 1995, 11, 1041−1043. (32) Shanahan, M. E. R.; Sefiane, K. In Contact Angle,Wettability and Adhesion; Mittal, K. L., Ed.; Brill Academic Publishers: Leiden, NL, 2009; Vol. 6, pp 19−31. (33) Orejon, D.; Sefiane, K.; Shanahan, M. E. R. Stick-Slip of Evaporating Droplets: Substrate Hydrophobicity and Nanoparticle Concentration. Langmuir 2011, 27, 12834−12843. (34) Askounis, A.; Sefiane, K.; Koutsos, V.; Shanahan, M. E. R. The Effect of Evaporation Kinetics on Nanoparticle Structuring Within Contact Line Deposits of Volatile Drops. Colloids Surf., A 2014, 441, 855−866. (35) Askounis, A.; Sefiane, K.; Koutsos, V.; Shanahan, M. E. R. Structural Transitions in a Ring Stain Created at the Contact Line of Evaporating Nanosuspension Sessile Drops. Phys. Rev. 2013, E 87, 012301−8. (36) Askounis, A.; Sefiane, K.; Koutsos, V.; Shanahan, M. E. R. Erratum: Structural Transitions in a Ring Stain Created at the Contact Line of Evaporating Nanosuspension Sessile Drops. Phys. Rev. 2014, E 88, 049903 (E). (37) Moffat, J. R.; Sefiane, K.; Shanahan, M. E. R. Effect of TiO2 Nanoparticles on Contact Line Stick - Slip Behavior of Volatile Drops. J. Phys. Chem. B 2009, 113, 8860−8866. (38) Moffat, J. R.; Sefiane, K.; Shanahan, M. E. R. Nanofluid Droplet Evaporation Kinetics and Wetting Dynamics on Flat Substrates. J. Nano Res. 2009, 7, 75−80. (39) Bodiguel, H.; Doumenc, F.; Guerrier, B. Stick−Slip Patterning at Low Capillary Numbers for an Evaporating Colloidal Suspension. Langmuir 2010, 26, 10758−10763.

not equivalent to each other and there is a need to improve the calculation methods of Ui. Then, we investigated the details of the calculation procedures and derived 8 slightly modified new equations in two sets given as eqs 16−19 and 20−23 which can be used successfully in order to plot δG̅ with time to obtain more reliable and easily comparable results of the Ui values from the experimental data. The reasons for the discrepancies are also discussed.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +90 (262) 605 2114. Fax: +90 (262) 605 2105. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank to Professors M. E. R. Shanahan and K. Sefiane, Drs. D. Orejon and V. Koutsos, and PhD student A. Askounis for sending us the correct experimental data of their publications given in refs 33, 35, and 36.

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