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Deposition Patterns of Two Neighbouring Droplets: Onsager Variational Principle Studies Shiyuan Hu, Yuhan Wang, Xingkun Man, and Masao Doi Langmuir, Just Accepted Manuscript • Publication Date (Web): 15 May 2017 Downloaded from http://pubs.acs.org on May 20, 2017
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Deposition Patterns of Two Neighbouring Droplets: Onsager Variational Principle Studies Shiyuan Hu,1 Yuhan Wang,1 Xingkun Man∗, ,1, 2 and Masao Doi1, 2 1
School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China
2
Center of Soft Matter Physics and its Applications, Beihang University, Beijing 100191, China
Abstract When two droplets containing nonvolatile components are sitting close to each other, asymmetrical ring-like deposition patterns are formed on the substrate. We propose a simple theory based on the Onsager variational principle to predict the deposition patterns of two neighbouring droplets. The contact line motion and the interference effect of two droplets are considered simultaneously. We demonstrate that the gradients of evaporation rate along two droplets is the main reason for forming asymmetrical deposition patterns. By tracing the relative motion between the contact line and the solute particles, we found that the velocities of solute particles have no cylindrical symmetry anymore because of the asymmetrical evaporation rate, giving the underlying mechanism of forming asymmetrical patterns. Moreover, controlling the evaporation rate combined with varying the contact line friction, fan-like and eclipse-like deposition patterns are obtained. The theoretical results of pinned contact line cases are qualitatively consistent with the pervious experimental results.
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INTRODUCTION
Deposition patterns after drying of droplets have drawn wide interest because of its importance of various fabrication process, including film coating [1, 2], printing [3–5], fabrication of nano-structures [6, 7], two dimensional crystallization [8], and medical diagnostics [9, 10]. Various deposition patterns have been reported, including coffee-ring [11, 12], mountainlike [13–15], volcano-like [16] and multi-ring patterns [17–19]. Ordered stripe structures and concentric rings can be created when droplet evaporates in restricted geometries (e.g., curves-on-flat geometries) [20–23]. Even more complex structures such as crack patterns [24], cellular and lamellar patterns [25] are observed. If solute particles of different shapes is considered, the deposition pattern can be much more complex [26]. It is clear that we are far from discovering all the possible deposition patterns, and the full understanding of the underlying mechanism of controlling the deposition pattern are still lacking [27]. Previously, most theoretical works focused on the single droplet case. Deegan et al. [28] explained that the solid ring is formed because the solute particles are convected toward the pinned contact line by the evaporation-induced outward flow. Freed-Brown [29] proposed a theoretical model with receding contact line, and mountain-like deposition pattern is obtained. Hu and Larson [30–32] showed that besides the friction of contact line, the Marangoni effect is also important in the formation of various deposition patterns. Frastia et al. [19] developed a dynamic model, in which by assuming the solution viscosity as a function of solute concentration to mimic the stick-slip motion of the contact line, they obtained the multi-ring pattern. Recently, Kaplan et al. [33] demonstrated a transition from the uniform pattern to the ring-like pattern by studying a multiphase model of colloidal particles in the solvent. More generally, Man et al. [34] proposed a simple Onsager variational principle model for drying droplet. They showed that the evaporation rate and friction of contact line are crucial to the final deposition patterns. By varying the two properties of droplets, they demonstrated the transition from coffee ring to volcanolike and then to mountainlike patterns. Compared with the case of single droplet, much less work has been done for the deposition patterns of many droplets. Previous works show that droplets with pinned contact lines deposited together have vapour-mediated interaction with each other, resulting deposition patterns different from the single droplet case. Deegan et al. [11] experimentally shows ACS Paragon 2 Plus Environment
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an asymmetrical deposition ring formed after drying of two neighbouring droplets, where the deposition density is lower in the inner side region than that in the outer side region. Chen and Evans [35] observed arched structures when colloidal droplets are sitting close to each other. Such asymmetrical ring patterns have also been studied by Pradhan et al. [36] recently. Their experimental and simulation results showed that the symmetry of the fluid convection inside one droplet is broken by another adjacent droplet due to the non-uniform evaporation flux [37], leading to the asymmetrical ring patterns. These works noticed that the increased vapour concentration in the inner side of droplets leads to less evaporation. Therefore, the evaporation rate is lower in the inner side region than that of the outer side region, resulting different deposition patterns compared with the single droplet case. Cira et al. [38] demonstrated such asymmetrical evaporation rate can cause spontaneous droplets movement over a distance. Kobayashi et al. [39] numerically demonstrated asymmetrical deposition patterns of nine neighbouring droplets with pinned contact lines. However, all the previous works on deposition patterns of many droplets only focused on pinned contact line case. It is clear that the deposition problem becomes much more complex when both the asymmetrical evaporation rate and moving contact line effect are taken into account. In this paper, we propose a simple model of two neghbouring droplets that accounts for the contact line motion and the asymmetrical evaporation simultaneously. The evolution equations of this system are determined by the Onsager variational principle. By assuming a linear asymmetrical evaporate rate between two droplets, fan-like and eclipse-like deposition patterns are obtained. The underlying mechanism of deposition is demonstrated by tracing the relative motion between the contact line and the solute particles.
THEORETICAL FRAMEWORK
Evolution equations of two neighbouring droplets
We consider two droplets placed on a substrate, as schematically shown in Fig. 1 (side view). We set R(t) as the radius of the the contact line circle, H(t) as the height of the droplets at the center and xc as the center position of the right droplet. Due to the mirror symmetry, we hereafter consider the right droplet only. We assume that the contact angle ACS Paragon 3 Plus Environment
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(a)
(b)
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(c)
Volcano
Coffee-ring
Mountain
FIG. 1. Schematic of two neighbouring droplets and three different deposition patterns (side view). Relevant parameters are the radius of the contact line R(t), the height of the droplet at the center H(t), the contact angle θ(t), the center of the droplet xc , and the profile of liquid/vapor interface h(r, t). (a) coffee-ring deposition pattern, (b) volcano-like deposition pattern, (c) mountain-like deposition pattern.
is very small (R(t) H(t)) and therefore the profile of the droplets can be written as a parabolic function (x − xc )2 + y 2 h(r, t) = H(t) 1 − . R2 (t)
(1)
The contact angle is given by θ(t) = 2H(t)/R(t), and the droplet volume V (t) is V (t) =
π H(t)R2 (t). 2
(2)
The volume V (t) of the droplet decreases in time due to solvent evaporation. It has been shown that the volume change rate, V˙ (t), is proportional to the radius of the contact circle [39, 40], and is weakly dependent on the contact angle. We therefore write V˙ (t) as R(t) V˙ (t) = V˙ 0 R0
(3)
where V˙ 0 (< 0) and R0 are the initial values of V˙ (t) and R(t), respectively. We further use the Onsager variational principle [41–44] to determine the evolution equa˙ tion, R(t), by minimizing a Rayleighian function defined as < = Φ + F˙
(4)
where F˙ denotes the time derivative of the free energy, and Φ is the energy dissipation function of the system. ACS Paragon 4 Plus Environment
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The free energy F and the energy dissipation function are defined in the same way as in the previous work [34]. The free energy is the sum of the interfacial energy. Z R p 2 F = (γLS − γSV )πR + γLV 2πr 1 + h0 (r)2 dr 2 0 4V πR2 θe2 = γLV + πR4 2
(5) (6)
where θe is the equilibrium contact angle defined as θe = [2(γLV + γLS − γSV )/γLV ]1/2 with γLV , γLS and γSV being the interfacial energy density at the liquid/vapor, liquid/substrate and substrate/vapor interfaces, separately. Then the time derivative of the free energy is # " 2 ˙ 16V 8V V 2 . (7) F˙ = γLV − + πθe R R˙ + πR5 πR4 We use the lubrication approximation to calculate the energy dissipation function. Let v(r, t) be the height-averaged velocity. The energy dissipation function Φ is written as Z 1 R 3η Φ= dr2πr v 2 + πξcl RR˙ 2 . 2 0 h
(8)
where η is the viscosity of the fluid, and ξcl is a phenomenological parameter representing the friction of the contact line: ξcl is infinitely large for pinned contact line, and is zero for freely moving contact line. The velocity v(r, t) in eq. (8) is obtained from the solvent mass conservation equation h˙ = −∇ · [v(r, t)h(r, t)] − J
(9)
where J(r, t) is the evaporation rate (the volume of solvent evaporating per unit time per unit surface area) at position r and time t. In our model, J(r, t) consists of two terms, the symmetric term J0 [34] and asymmetric term Ja V˙ (t) x − xc J(r, t) = J0 + Ja = − 2 + Je , πR (t) xc
(10)
where Je is an input parameter characterizing the asymmetry of the evaporation rate. We assume that Ja is a linear function of x, and decreases with the increase of the centerto-center distance of the two droplets. It is clear that we can split v(r, t) into two parts, v(r, t) = v0 (r, t) + va (r, t), and can be solved from eq. (9), respectively.
v0 (r, t) = [(x − xc )i + yj ]
R˙ H˙ − 2R 4H
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! .
(11)
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va (r, t) =
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Je R 2 i 2xc H
(12)
In this paper, we ignore the center mass movement, i.e. x˙c = 0, and v(r, t) as " ! # ! R˙ H˙ Je R2 R˙ H˙ v(r, t) = r cos ϕ − + i + r sin ϕ − j 2R 4H 2xc H 2R 4H
(13)
where ϕ = arctan [y/(x − xc )]. Inserting the expression of v(r, t) into eq. (8), the energy dissipation function Φ is calculated as 3πηR4 Φ= 8H
R˙ H˙ − R 2H
!2
R 3πηJe2 R6 R ln −1 + + πξcl RR˙ 2 ln 2ε 8x2c H 3 2ε
(14)
where ε is the molecular cutoff length, which is introduced to remove the divergence in the energy dissipation at the contact line. Hereafter, we set ε = 10−6 R0 for all calculations. The evolution equations of R(t) is determined by the condition ∂(Φ+F˙ )/∂ R˙ = 0, resulting the same equation as the single droplet case. More details can be found in ref. [34]. We define two time scales to simplify the equation, the evaporation time τev , which represents the characteristic time for the droplet (of initial size V0 ) to dry up, and the relaxation time τre , which represents the relaxation time needed for the droplet (initially having contact angle θ0 ) to have the equilibrium contact angle θe . 1
τev
V0 = , |V˙ 0 |
ηV03 τre = γLV θe3
(15)
By such a definition, we have 1
V 3 θ (θ2 − θe2 ) V0 R2 + 0 (1 + kcl ) τev R˙ = − 4R0 V 6Ckev θe3
(16)
where C = ln(R/2) − 1, and kcl is defined by kcl =
ξcl θ 3Cη
(17)
which is the ratio of the extra friction constant ξcl of the contact line to the normal hydrodynamic friction ξhydro = 3Cη/θ [45]. Contact line is pinned for large kcl , and for small kcl , contact line can move more easily. kev is defined as kev =
τre τev
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(18)
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characterizing the mean evaporation rate. For pure water or dilute polymer solutions of macroscopic size (diameter 1mm), kev is less than 10−3 . While, for concentrated polymer solutions with high viscosity kev can be larger than 10−1 [46, 47]. The evolution equation of V (t) after rescaling is written as τev V˙ = −V0
R(t) . R0
(19)
Finally, θ is related to R and V by 4V . (20) πR3 Equations (16), (19) and (20) are the set of equations which determine the time evolution of θ=
the system, while kcl , kev and Je are the three key parameters in determining the evaporation process.
Deposition pattern distribution
Based on the above model, we can calculate the distribution of the deposition density. We assume that the solute particles move with the same velocity as the fluid flow [48]. The density ρ(r, θ) of the deposited solute particles is obtained by the following way. We randomly generate solute particles which are distributed homogenously inside the droplet at the beginning of evaporation. According to the assumption, the equation of motion of these solute particles is the same as eq.(13), " x˜˙ = (˜ x − xc )
V˙ R˙ − R 4V
y˜˙ = y˜
!
R˙ V˙ − R 4V
R4 + Je C1 xc V !
# (21)
(22)
where r˜(t) = (˜ x, y˜) is the height-averaged position of one solute particle at time t, and C1 is defined as C1 = π/4. This motion equation gives the position of each solute particle at time t, facilitating that we can trace and compare the position of solute particles with R(t). Once r˜(t) > R(t), which means that the solute particle is already outside the droplet, we do not evolve the position of this particle anymore and record it. After the droplet dries out, we count the number of solute particles n located in the small unit volume, rdrdθh(r, θ). Finally, ρ(r, θ) is written as ρ=
n Ntotal rdrdθ
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(23)
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where Ntotal is the total number of solute particles. Here, we divide the x − y plane into N by N grids in the polar coordinate, and take N = 100 for all calculations.
RESULTS AND DISCUSSION
In this section, we first show a typical example of the formation process of an asymmetrical deposition pattern, and then discuss the effects of the gradient of evaporation rate, Je , the contact line friction parameter, kcl , and the mean evaporation rate, kev on the deposition patterns. For simplicity, we only consider the situation that the initial contact angle is equal to the equilibrium contact angle, i.e. θ0 = θe = 0.2. When θ0 6= θe , spreading or dewetting of the droplet will happen, resulting a complex situation of deposition.
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0 .0
x
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-0 .5
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x
FIG. 2. Time evolution of the deposition during drying process shown in contour plot for kev = 1 (a fast evaporation rate) and kcl = 5. (a) t = 0τev , (b) t = 0.5τev , (c) t = 0.8τev , (d) t = 1τev , (e) t = 1.04τev . For all the cases, θe = θ0 = 0.2 and ∆t/τev = 10−5 . The color code corresponds to the four intervals of local solute density as is depicted in the figure.
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Deposition pattern of two neighbouring droplets
Fig. 2 is the time evolution of the density distribution of solute particles during drying process of two neighbouring droplets. Here, we just show the deposition patterns of the right droplet. In this calculation, kev = 1 which is set for a fast evaporation, and the contact line friction parameter kcl = 5. Note that all time is in unit of τev . Fig. 2(a) is the initial status, in which we assume that the solute particles are distributed homogeneously inside the droplet with an average number density ρ0 . As evaporation goes on, the contact line shrinks and causes the deposition of solute particles. In Figs. 2(b)-(e), it is obvious that solute particles accumulate to the right part of the droplet, forming asymmetrical ring patterns. The density of deposited solute particles is weakest at the inner side region of the two droplets, which is consistent with previous experimental findings [11, 35–37]. It is worth to note that in the later evaporation process (Fig. 2(d)), there are still some solute particles inside the droplet. After a short time (∆t = 0.04 τev ), almost all solute particles deposit at the contact line, indicating the fluid flow velocity is very large in the end of evaporation.
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J e= 0 .3
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J e= 0 .5
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-0 .5 0 .0
x
0 .5
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FIG. 3. Asymmetric mountain-like deposition patterns: (a)-(c) are the contour plots of the distribution of the deposited solute particles while (d) is the corresponding side views. For all calculations, kcl = 0, kev = 0.001, θe = θ0 = 0.2 and ∆t/τev = 10−5 , but (a) Je = 0, (b) Je = 0.3, (c) Je = 0.5. It is clear that increasing the value of Je , the deposition patterns become more and more asymmetrical. Fan-like deposition pattern is formed Je = 0.5 in (c).
Previous work [34] showed that by changing the contact line friction, the deposition pattern continuously changes from mountain-like to volcano-like, then to coffee-ring pattern. We therefore investigate the effect of the asymmetrical evaporation rate on these three ACS Paragon 9 Plus Environment
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deposition patterns.
(a )
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(c )
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J e= 0 .5
2
ρ
y
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x
0 .5
-0 .5 0 .0
x
0 .5
-0 .5 0 .0
0 .5
- 0 .5
0 .0
x
0 .5
x
FIG. 4. Asymmetric volcano-like deposition patterns: (a)-(c) are the contour plots of the distribution of the deposited solute particles while (d) is the corresponding side views. (a) Je = 0, (b) Je = 0.3, (c) Je = 0.5. In all calculations, kcl = 10, kev = 0.001, θe = θ0 = 0.2 and ∆t/τev = 10−5 . More and more solute particles deposit in the outside region when increasing the asymmetry of the evaporation rate.
Fig. 3 shows the effect of Je on the mountain-like pattern for the case of freely moving contact line (i.e., kcl = 0). Fig. 3(a) shows the case of Je = 0. In this case since there is no interaction between the droplets, the final deposition pattern is exactly the same as that of single droplet case. Solute particles deposit near the original center of droplet forming a symmetrical mountain-like pattern. Fig. 3(b) and (c) show the cases of asymmetrical evaporation, where Je = 0.3 and 0.5, respectively. For both cases, solute particles deposit mostly in the outer side region of the droplet. With the increase of Je , less solute particles deposit in the inner side region of the two droplets. Particularly, for large value of Je (Je = 0.5), a fan-shaped deposition pattern is formed as shown in Fig. 3(c). Fig. 3(d) is the corresponding side views of the deposition patterns shown in (a)-(c), where ρ is the number density of deposited solute particles along the x−axis. Such side views is presented here to give better indication of the change of deposition patterns when increasing Je . The effects of Je on the volcano-like and coffee-ring pattern are shown in Figs. 4 and 5, which are quite similar to the mountain-like case. The contact line friction is set as kcl = 10 and kcl = 50 in Fig. 4 and 5, respectively. For the later case, the contact line nearly does not move (pinned contact line case), resulting the coffee-ring pattern. The former case, kcl = 10, is chosen to mimic the situation between freely moving and pinned case, of which volcanoACS Paragon10 Plus Environment
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FIG. 5. Asymmetric coffee-ring deposition patterns: (a)-(c) are the contour plots of the distribution of the deposited solute particles while (d) is the corresponding side views. (a) Je = 0, (b) Je = 0.3, (c) Je = 0.5. For all cases, kcl = 50, kev = 0.001, θe = θ0 = 0.2 and ∆t/τev = 10−5 . Eclipse-like deposition patterns are obtained while there is interference between two neighbouring droplets. The maximum value of deposited solute particles is different from the one in Fig. 2, in order that the modified color code will give a better representation of the deposition pattern.
like pattern is expected. By increasing the value of Je from 0 to 0.5, both volcano-like and coffee-ring patterns change from isotropic ring to asymmetrical ring. Moreover, almost all solute particles deposit in the outer side region forming the eclipse-like deposition patterns for both cases. 3
(a )
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9 (b )
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k
e v
= 1
k
e v
= 0 .0 0 1
6 1 2
ρ
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FIG. 6. Effects of kev on the deposition patterns (side view). (a) kcl = 0, (b) kcl = 10, (c) kcl = 50. For all calculations, Je = 0.5 θe = θ0 = 0.2 and ∆t/τev = 10−5 .
Fig. 6 shows the effects of the mean evaporation rate kev on the deposition patterns. We compare the deposition patterns of a slow evaporation case, kev = 0.001, and a fast evaporation case, kev = 1, for mountain-like (kcl = 0), volcano-like (kcl = 10) and coffee-ring ACS Paragon11 Plus Environment
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(kcl = 50) deposition patterns in non-uniform evaporation rate situation Je = 0.5. Results show that for all cases, the peak position of the deposited density distribution shifts outward accompanying an increasing of the magnitude. Fast mean evaporation rate enhances the effect of Je on the deposition patterns.
The mechanism of deposition
In order to understand the deposition mechanism during evaporation, the relative motion between the contact line and the solute particles is studied by tracing their positions at each iteration time. We first study a simple case that the evaporation rate is symmetrical, i.e. Je = 0. In this case, the droplet has axial symmetry in a cylindrical coordinate system, we therefore investigate half (x-axis) part of the droplet by tracing four solute particles which have initial coordinates (0.8,0), (0.6,0), (0.4,0) and (0.2,0). 1 .2
C L
(a )
(b )
x 0= 0 .8
(c )
= 0 .6 = 0 .4 = 0 .2
0 .8
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t / τe v
0 .8
1 .2 0 .0
0 .5
t / τe v
1 .0
FIG. 7. Motion of right intersection points of the contact line (CL) and the four points of solute particles. The initial positions of this four points are xi (0) = 0.2, 0.4, 0.6, 0.8, separately, and yi = 0 for all points. In all calculations, Je = 0 and kev = 0.001, while (a) kcl = 0, (b) kcl = 10, (c) kcl = 50 representing mountainlike, volcanolike, and coffee ring, separately.
Fig. 7 shows the motion of the four points and the contact line for various values of kcl (different contact line friction cases). Solute particles deposit at the position where they meet the contact line. For the freely moving case (Fig. 7(a), kcl = 0), these four points move toward the center of the droplet slightly while the contact line recedes. As we assume that solute particles are distributed homogeneously inside the droplet at the beginning of ACS Paragon12 Plus Environment
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evaporation, the deposition pattern is similar to its initial distribution, which is mountainlike. Oppositely, for the pinned contact line case (Fig. 7(c), kcl = 50), R(t) is nearly unchanged, and all particles are convected to the boundary to meet the contact line, forming the coffeering pattern. For the intermediate friction case (Fig. 7(b) with kcl = 10), the contact line moves inward while the solute particles move outward. Therefore, the intersection points are located in the region between the center of droplet and the original position of the contact line, resulting the volcano-like pattern. It has been shown that the density peak of the deposition patterns is determined by kcl and kev simultaneously [34].
(a )
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(b )
C L (R ) x 0= 0 .8 0 .6 0 .4 0 .2 -0 .2 -0 .4 -0 .6 -0 .8 C L (L )
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FIG. 8.
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t / τe v
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1 .2
0 .0
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1 .2
Motion of right and left intersection points of contact line and the tracing
eight points of solute particles.
The initial positions of this eight points are xi (0) =
−0.8, −0.6, −0.4, −0.2, 0.2, 0.4, 0.6, 0.8, separately, while yi = 0 for all points. In all calculations, kcl = 10 and kev = 0.001, θe = θ0 = 0.2 and ∆t/τev = 10−5 , but (a) Je = 0 and (b) Je = 0.5.
We now discuss the drying of droplet with asymmetrical evaporation rate cases, i.e. Je 6= 0. Here, we choose eight solute particles, of which four are in the left part and another four are in the right part. Their initial positions are xi (0) = −0.8, −0.6, −0.4, −0.2, 0.2, 0.4, 0.6, 0.8, respectively, and yi = 0 for all particles. To understand the effects of Je on the final deposition patterns, relative motion between solute particles and the contact line for both Je = 0 and Je 6= 0 are shown in Fig. 8. Fig. 8(a) shows the case of symmetrical evaporation rate (Je = 0, kcl = 10 and kev = 0.001). As shown in Fig. 7 (b), this corresponds to the volcano-like pattern, where solute ACS Paragon13 Plus Environment
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particles move outward while the contact line move inward. Fig. 8(b) shows asymmetrical evaporation rate case. All parameters are the same as in Fig. 8(a), except Je = 0.5. As shown in eq. (13), Je has an extra contribution to the x-component velocity in the positive x-direction and breaks the cylindrical symmetry. The fluid velocity induced by the symmetrical evaporation rate (J0 ) is always outward from the center of droplet. However, the extra velocity is always toward the positive x direction. So, the velocity of the solute particles in the right part increases, but the velocity of the solute particles in the left part decreases or even reverses. Fig. 8(b) clearly shows such a phenomena, where the four points located on left part move toward the right part. As a result, the solute particles, which deposit on the left contact line when Je = 0, are convected to the right contact line. We therefore obtain an asymmetrical pattern, of which the density is lowest in the inner side and highest in the outer side, resulting the eclipse-like deposition patterns. 1 .0
(a )
(b )
fin a l
0 .5 0 .0
x
x
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fin a l
= x 0
- 0 .5 - 1 .0
- 1 .0
- 0 .5
0 .0
x
k
e v
= 1
k
e v
= 0 .0 0 1
0 .5
1 .0
- 1 .0
- 0 .5
0 .0
x 0
0 .5
1 .0
0
FIG. 9. The effects of kev on the finial deposited position of nineteen solute particles that are initially located on x-axis at −0.9, −0.8, ..., 0.8, 0.9 for (a) kcl = 0 and (b) kcl = 10. The dash line is to guide that the deposited position of particles locate at their initial positions (xf inal = x0 ). For all calculations, Je = 0.5.
Fig. 9 shows the mechanism of the effect of kev on the deposition patterns. We fucus on the asymmetrical evaporation case Je = 0.5 for both kcl = 0 and kcl = 10. Nineteen points were selected on x−axis with initial coordinates −0.9, −0.8, ..., 0.8, 0.9. We then track the motion of these nineteen solute particles and record their deposited positions. As ACS Paragon14 Plus Environment
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we can see, when increasing the mean evaporation rate from kev = 0.001 to kev = 1, the deposited positions shift outward to the right droplet boundary. This indicates that the solute particles are convected with the outward fluid flow to a longer distance and deposited inside a narrower region. Thus, more solute particles are deposited in the outer side region, leading to the fact that the peak of deposited distribution shifts outward accompanying an increasing magnitude. For the case of coffee-ring deposition pattern (kcl = 50), the solute particles already deposit at the edge of droplet, and the shifting of the peak position of the deposit density distribution is small and negligible, thus the deposition pattern for this case is dominated by kcl .
CONCLUSION
In this paper, we have proposed a simple model for the drying of two neighbouring droplets accounting for the vapour-mediated interaction and the moving contact line simultaneously. The evolution equations of the contact line and the motion of solute particles are determined by the Onsager variational principle. We show that the asymmetrical evaporation rate (Je ) and the contact line friction (kcl ) are the two key factors in determining the final deposition patterns. The former factor causes asymmetrical density distribution of deposited solute particles, while the later one induces the transition from mountain-like to volcano-like then to coffee-ring pattern. Their combined effects result in fan-like and eclipse-like deposition patterns. It is worthwhile to note that the eclipse-like deposition patterns are also found in the previous experimental results [11, 35, 36]. We show a clear deposition mechanism of drying droplets containing nonvolatile solute. The relative motion between solute particles and the contact line are crucial to the final deposition pattern. For a droplet with freely moving contact line, the solute particles moves toward the center slightly while the contact line is receding, resulting a density peak near the original center of droplets. Oppositely, when the contact line is pinned, solute particles are convected to the boundary forming the coffee-ring pattern. In the intermediate region, the contact line recedes inward while the solute particles move outward from the droplet center. Therefore, a density peak of deposited solute particles forms between the center and the contact line, leading to the volcano-like pattern. For two neighbouring droplets, increased vapour concentration between them leads to ACS Paragon15 Plus Environment
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less evaporation. Such gradients of the evaporation rate, Je , break the cylindrical symmetry of the velocity of solute particles. The corresponding extra velocity, Je R2 /(2xc H), is along the x-axis. Therefore, solute particles are convected to the side with larger evaporation rate. In other words, more solute particles deposit in the outside region than in the inner region. We noticed that much more complicated deposition patterns than ring-like can be formed after the evaporation of droplets. As in this preliminary model we only assume an asymmetrical evaporation along the x−axis, multi-ring [49], faceted and dendritic [50] deposition patterns are not found here. We hope these patterns can be addressed in the future studies. Another interesting direction of extending the present study is to consider many droplets deposited together to further explore more possible deposition patterns. Acknowledgement.
This work was supported in part by grants No. 21404003 and
21434001 of the National Natural Science Foundation of China (NSFC), and the joint NSFCISF Research Program, jointly funded by the NSFC and the Israel Science Foundation (ISF) under grant No. 51561145002 (885/15).
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for Table of Contents use only Deposition Patterns of Two Neighbouring Droplets: Onsager Variational Principle Studies Shiyuan Hu, Yuhan Wang, Xingkun Man∗ , and Masao Doi∗ A s y m m e tric a l e v a p o ra tio n ra te
D ro p le t
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