Drying of Droplets of Colloidal Suspensions on Rough Substrates

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Drying of Droplets of Colloidal Suspensions on Rough Substrates Truong Pham, and Satish Kumar Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02341 • Publication Date (Web): 22 Aug 2017 Downloaded from http://pubs.acs.org on August 23, 2017

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Drying of Droplets of Colloidal Suspensions on Rough Substrates

Truong Pham and Satish Kumar1 Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455

Abstract - In many technological applications, excess solvent must be removed from liquid droplets to deposit solutes onto substrates. Often, the substrates on which the droplets rest may possess some roughness, either intended or unintended. Motivated by these observations, we present a lubrication-theory-based model to study the drying of droplets of colloidal suspensions on a substrate containing a topographical defect. The model consists of a system of one-dimensional partial differential equations accounting for the shape of the droplet and depth-averaged concentration of colloidal particles. A precursor film and disjoining pressure are used to describe the contact-line region, and evaporation is included using the well-known one-sided model. Finite-difference solutions reveal that when colloidal particles are absent, the droplet contact line can pin to a defect for a significant portion of the drying time due to a balance between capillary-pressure gradients and disjoiningpressure gradients. The time-evolution of the droplet radius and contact angle exhibits the constant-radius and constant-contact-angle stages that have been observed in prior experiments. When colloidal particles are present and the defect is absent, the model predicts that particles will be deposited near the center of the droplet in a cone-like pattern. However, when a defect is present, pinning of the contact-line accelerates droplet solidification, leading to particle deposition near the droplet edge in a coffee-ring pattern. These predictions are consistent with prior experimental observations, and illustrate the critical role contact-line pinning plays in controlling the dynamics of drying droplets. 1

E-mail address for correspondence: [email protected]

1

ACS Paragon Plus Environment

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1

Introduction

After a liquid droplet containing colloidal particles finishes drying, it deposits the solutes in a pattern that reflects the coupled heat, mass, and momentum transport processes occurring during drying. A pattern often encountered in everyday life is the ‘coffee-ring’ deposit left by a drying coffee droplet [1]. Being able to control and predict the deposition of solutes from drying droplets is crucial for many technological applications such as ink-jet printing [2], spray coating [3, 4], microfabrication [5–7], and bio-assays [8–10]. Pinning of the droplet contact line is thought to be a key factor in determining solute deposition patterns. The coffee-ring pattern, for example, was argued to be the result of contact-line pinning, with solvent evaporation driving a flow that carries particles to the droplet edge [1]. When the contact line is not pinned, different deposition patterns can arise such as concentric rings [8,11] and conical shapes [12]. Whereas in some studies the contactline was observed to successively pin and depin from solute deposits on the substrate [8,11,13], other studies have shown that contact-line motion can be actively controlled by patterning the substrates chemically [14, 15] or topographically [16–18]. Contact-line pinning can also occur due to the natural roughness of a substrate [19–22]. In the absence of particles, a commonly observed observed mode of contact-line motion in evaporating droplets is a constant-radius stage followed by a constant-contact-angle stage (e.g., [13, 16, 17, 23–25]). An example of this is shown in Fig. 1, where we have replotted data from Orejon et al. [13] for water droplets evaporating on silicon and parylene substrates. Three distinct stages are observed, with the beginning of each stage marked with a black arrow: (I) constant radius, decreasing contact angle, (II) decreasing radius, constant contact angle, and (III) decreasing radius and contact angle. The third stage occurs toward the end of drying and typically involves more rapid decreases than in the earlier stages. The work described in the present paper has two objectives. First, we seek to better un2


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(a)

(b)

Figure 1: Replotted experimental data from Orejon et al. [13] for evolution of (a) droplet radius and (b) contact angle for pure-water droplets on silicon and parylene substrates. The beginning of each drying regime is marked with an arrow. The drying regimes shown are: (I) constant radius, decreasing contact angle, (II) decreasing radius, constant contact angle and (III) decreasing radius and contact angle.

derstand the connection between substrate roughness and the three drying stages illustrated in Fig. 1. Second, we wish to better understand the connection between surface roughness and some of the particle deposition patterns described above. To address these objectives, we consider a model problem in which a droplet laden with colloidal particles evaporates on a substrate containing a topographical defect. Many previous studies have examined mathematical models of the drying of pure-solvent droplets on perfectly flat substrates [26–34], but these models do not predict the three drying stages illustrated in Fig. 1. In addition, droplet spreading on rough substrates in the absence of evaporation has also been considered [35–38]. However, the influence of substrate roughness on the drying of droplets of remains an open issue. Some recent papers have examined the evaporation of pure-solvent droplets on rough substrates [39–41], but the models presented in these papers treat the constant-radius and constant-contact-angle stages separately. The model developed in the present paper describes the transition between these 3


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two stages naturally. We will show that the former stage arises when the droplet is pinned to a topographical defect, and that the latter stage arises when the droplet depins from the defect. Before proceeding, we note two important aspects about the mathematical model developed in this paper. First, we use the lubrication approximation as the basis for our model because in many cases of interest the droplets are long and thin. Apart from this assumption, no other restrictions are placed on the droplet shape. Whereas many studies assume that the droplet shape is a spherical cap [42–44], the advantage of our model is that it can readily be extended to much more general situations (e.g., droplets on inclined surfaces) where the spherical-cap assumption is no longer valid [32,45,46]. Second, although real surfaces contain a distribution of topographical features of varying size, shape, and location, we focus here on the presence of a single topographical defect. This approach allows us to gain insight into how individual topographical features influence droplet dynamics, and provides a rational basis for understanding more complex situations where multiple defects are present. The rest of the paper is organized as follows. The mathematical model is developed in Section 2, and a reference case of an evaportaing droplet of pure solvent on a perfectly flat (defect-free) substrate is treated in Section 3. We then consider the influence of surface roughness in Section 4, and the presenece of colloidal particles in Section 5. Concluding remarks are offered in Section 6.

2

Mathematical model

In this section, we derive the evolution equations governing droplet shape and colloidal particle concentration (Eqs. (24) and (26)). We also discuss the evaporation model, calculation of contact angles, and solution procedure.

4


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2.1

Hydrodynamics

We consider an axisymmetric liquid droplet consisting of a Newtonian solvent and colloidal particles on a horizontal substrate (Fig. 2). The vertical and radial coordinates are denoted by z and r, respectively. The liquid-vapor interface is located at z = H(r, t) and the substrate is located at η(r). A precursor film is present at the droplet edge, and its thickness far from the droplet is denoted by b. The velocity field in the droplet is v = uˆ r + wˆ z, with u and w representing the radial and vertical velocity components, respectively.

Figure 2: Schematic of problem geometry In this work, we will consider substrate topography η(r) in shape of a small Gaussian bump to study the influence of a surface defect on droplet evaporation. Further details about our choice of η(r) are given in Section 4.1. Before evaporation begins, the droplet is characterized by an initial maximum thickness h0 and radius r0 . The precursor film thickness b is typically 102 to 104 times smaller than h0 [47]. On length scales comparable to h0 and r0 , it appears that the substrate is perfectly flat, and the liquid-vapor interface intersects the liquid-solid interface to form a circular contact line. On length scales comparable to b, topographical defects on the substrate are 5


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revealed, and the interfaces do not intersect. The droplet is assumed to be thin so that its characteristic height h∗ = h0 and radius r∗ = r0 satisfy ǫ = h∗ /r∗ ≪ 1, allowing us to apply lubrication theory. The characteristic time scale is taken to be that for capillary-driven spreading, which is the capillary time, t∗ = 3µ0 r∗ /σ0 ǫ3 ≡ tc , where µ0 and σ0 are the viscosity and surface tension of the pure solvent. The horizontal velocity scale is u∗ = r∗ /t∗ , and the pressure scale is p∗ = h∗ σ/r∗2 . With these scales, we define the following dimensionless variables (denoted with primes): r = r∗ r′ , z = h∗ z ′ , u = u∗ u′ , w = ǫu∗ w′ ∗ ′

(1)

∗ ′

t=t t, p=p p.

Assuming that gravitational effects are negligible relative to surface-tension effects, the leading-order equations governing momentum and mass conservation in the droplet are

−3p′r′ + (M u′ )z′ z′ = 0,

(2)

p′z′ = 0,

(3)

1 ′ ′ (r u )r′ = 0, r′

(4)

wz′ ′ +

where the subscripts r′ and z ′ denote spatial derivatives, and M is the viscosity of the droplet. Here, we use the well-known Krieger-Dougherty relationship [48] to relate the viscosity to the particle volume fraction c,

M=

µ c −2 . = 1− µ0 0.64

(5)

At the random-packing limit of c = 0.64, the viscosity of the colloidal suspension diverges and the droplet is assumed to be solidified. Particle sedimentation is assumed to be negligible. The liquid-vapor interface is defined by z ′ = H ′ (t′ , r′ ) = h′ (t′ , r′ ) + η ′ (r′ ), and the timeevolution of the droplet thickness h′ can be described using the leading-order kinematic 6


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boundary condition, h′t′ = w′ − u′ (h′ + η ′ )r′ − Ej ′ ,

(6)

where j ′ is the evaporation flux and the subscript t′ denotes a time derivative. Here, E = tc /te is the evaporation number, the ratio between the capillary and evaporation time scales; the evaporation time te will be defined in Section 2.3 when we discuss heat transport within the droplet. To obtain expressions for the velocity components u′ and w′ , we need to apply the no-slip and no-penetration boundary conditions at the substrate-liquid interface z ′ = η ′ (Eq. (7)), and tangential and normal stress balances at the liquid-vapor interface z ′ = h′ + η ′ (Eqs. (8)-(9)), u′ = w′ = 0,

(7)

u′z′ = 0,

(8)

p′ − p′v = −h′r′ r′ −

h′r′ r′

− Π′ ,

(9)

where p′v is the pressure in the vapor phase. The first two terms on the right-hand side of Eq. (9) account for capillary pressure in the droplet, which arises from the curvature of the liquid-vapor interface. The final term Π′ of Eq. (9) represents the disjoining pressure [49], which will determine the wetting behavior of the droplet. Thermal Marangoni effects are neglected in this work, as calculations including them indicate that they do not qualitatively change our results (see also Section 4.3). Since the pioneering work of Schwartz et al. [50, 51], disjoining pressure and precursor films have been used to study droplet dynamics in numerous contexts, including evaporation of pure-solvent droplets on heated substrates [26–28], droplet spreading on substrates with topography [52], and drying of droplets of colloidal suspensions [29, 31,32,53]. The principal advantage of this approach over having an explicit contact line and a slip law is that the 7


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resulting equations are much easier to solve [52]. This is because the contact-line position is not an explicit unknown; it can be extracted from the droplet height profile (as will be discussed in Section 2.5). The tradeoff is that careful choices must be made of the disjoining pressure and precursor-film thickness. As have previous works [28, 38, 50, 51, 54, 55], we will use a two-term disjoining-pressure expression to describe the behavior of the contact-line region,

′

Π = A1

A2 h′

n

−

A2 h′

m

,

(10)

where A1 > 0 is the dimensionless Hamaker constant, which describes the magnitude of the energy of intermolecular interactions, and A2 is a parameter that controls the precursor-film thickness. For a non-volatile droplet, A2 is equal to the equilibrium thickness of the precursor film b [26, 38, 50]. The first and second terms of Eq. (10) describe repulsion and attraction between liquid-vapor and liquid-solid interfaces, respectively [56]. We choose n = 3 and m = 2 because previous works have shown that these values provide an adequate description of the contact-line region at a reasonable computational cost [38, 54, 55]. We note that when colloidal particles are present in the contact-line region, ‘structural’ effects can become important and the disjoining pressure may depend on particle concentration in a more complicated way. Such effects have been considered theoretically and experimentally in studies on the spreading and drying of droplets of colloidal suspensions [29,35,57–60]. In the present work, we use the simple expression in Eq. (10) to isolate the influence of concentration-dependent viscosity and diffusivity.

2.2

Evaporation

Previous studies have employed two different approaches to describe evaporation of drying droplets. The first approach assumes that evaporation is controlled by diffusion of solvent vapor in air [1, 44, 61–63]. In this approach, a diffusion equation for the vapor concentration 8


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in the gas phase is simultaneously solved with the transport equations in the liquid droplet. To reduce the computational complexity, these studies often assume that the droplet has a spherical-cap shape. This assumption comes from energy-minimization considerations, and is expected to be reasonable when the substrate is horizontal and sufficiently flat, and when surface-tension forces dominate over other forces (e.g., viscous, gravitational). However, if these conditions do not hold, then the spherical-cap assumption may no longer be valid. As a consequence, the problem becomes considerably more difficult computationally since the droplet shape is now an unknown along with the flow field in the droplet and the vapor concentration in the gas phase [45, 64–69]. The second approach to describe evaporation is to assume that evaporation is limited by how fast the solvent molecules escape the liquid phase and enter the vapor phase [26,27,30,32, 35,63,70–73]. The evaporation flux used in this approach is proportional to the temperature and pressure differences across the interface. For this approach, only transport equations in the liquid phase need to be considered, which reduces the computational cost without making any assumptions regarding droplet shape. Because the solvent vapor concentration in the gas phase does not need to be tracked, this approach to describing evaporation is known as the one-sided model. Notably, in the absence of thermal Marangoni effects, the solutions obtained using this approach share many of the features exhibited by droplets undergoing diffusion-limited evaporation [30] since both evaporation models predict that the evaporation flux is largest near the droplet contact line. Because of the simplicity and versatility of the one-sided model, we adopt it in the present paper. Further discussion about the different approaches to modeling evaporation can be found in Refs. [30] and [74]. The expression for the evaporation flux of the one-sided model was first derived by Schrage using the kinetic theory of gases to describe evaporation of a liquid into its saturated vapor [75]. Since then its linearized form has been used in studies of drying thin films [70–74,76,77]

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and droplets [26, 27, 29–33], p ˆ ˆ sat La 1 2π RT j = (p − pv ) + (Ti − Tsat ), ρv ρl Tsat

(11)

ˆ is the ideal gas constant per unit mass, La ˆ is the latent heat of vaporization per unit where R mass, Ti is the temperature at the liquid-vapor interface, and ρl and ρv are the densities of the solvent in liquid and vapor forms. The partial pressure of the solvent in the gas phase is pv , and the saturation temperature Tsat can be calculated from the Clasius-Clapeyron equation using pv as the saturation pressure [28]. From Eq. (11), it can be seen that deviations from pv and Tsat drive evaporation. We note that because Eq. (11) was originally developed for a one-component system, it may not necessarily provide a quantitatively accurate expression for a droplet ladden with colloidal particles. Nevertheless, Eq. (11) has been used to describe the qualitative behavior of drying multicomponent thin films and droplets in previous studies [29, 31–33, 71].

2.3

Energy transport

The substrate is held at a constant temperature Tb , and temperature is scaled as T ′ = (T − Tsat )/(ǫ2 Tsat ). The leading-order energy transport equation is

Tz′′ z′ = 0.

(12)

When the density, viscosity, and thermal conductivity of the vapor are much less than those of the liquid, it can be shown that all of the heat conducted from the substrate to the droplet-vapor interface is used to vaporize the liquid [70]. After balancing the conductive heat within the droplet with the heat dissipated through evaporation, we arrive at a scaling ˆ where k is the thermal conductivity of the droplet. From factor for j, j ∗ = kTsat /h0 La, this scaling, we can obtain an expression for the evaporation time, te = h0 ρl /j ∗ . As have 10


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previous works, we assume that the thermal conductivity of the solvent is the same as that of the colloidal particles [29,31–33,71]. The dimensionless temperature boundary conditions are

(13)

T ′ = 1 at z ′ = η ′ , − Tz′′ = j ′ at z ′ = η ′ + h′ . The non-dimensional expression for the evaporation flux is

Kj ′ = δ(p′ − p′v ) + T ′ ,

(14)

p ˆ sat kTsat σ0 2π RT K= ,δ= . 2 ˆ ˆ h 0 ǫ2 ρl Lar0 ǫ ρv La

(15)

where

Parameters K and δ in Eq. (15) describe the dependence of the evaporation flux on the interfacial temperature and pressure differences. By solving Eq. (12) subject to the boundary conditions in Eq. (13), we obtain

j′ =

1 + δ(p′ − p′v ) . K + h′

(16)

To prevent the precursor film from evaporating, it is necessary to set j ′ = 0 in the precursor-film region and combine Eqs. (9), (10) and (16) to solve for the steady-state precursor-film thickness,

b′ = −2

2.4

r

δA1 A22 sinh 3

1 3A2 asinh − 3 2

s

3 δA1 A22

!!

.

Colloidal particle transport and evolution equations

A convection-diffusion equation governs the transport of colloidal particles, 11


(17)

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ct + v · ∇c = ∇ · (D∇c),

(18)

where the diffusion coefficient D is related to the particle volume fraction c [48, 78, 79],

D = D0 (1 − c)

6.55

d dc

1.85c 0.64 − c

.

(19)

Here, D0 is given by the Stokes-Einstein expression [48],

D0 =

kB Tsat , 6πµ0 rp

(20)

where kB is Boltzmann’s constant and rp is the particle radius. At low particle concentrations, the diffusion coefficient D decreases with particle concentration due to hydrodynamic interactions between the particles. At high particle concentrations, osmotic pressure becomes significant and D diverges as the particle concentration approaches the random-packing limit of 0.64 [48, 80]. Eq. (18) is complemented by no-flux boundary conditions at the liquid-solid and liquidvapor interfaces, −D∇c + (v − vI )c = 0,

(21)

where vI is the velocity of the interface. After non-dimensionalizing, Eq. (18) becomes c′t′

+

u′ c′r′

+

w′ c′z′

1 = 2 ǫ Pe

′

(Dc )z′ z′

1 ′ ′ + ǫ ′ (Dr cr′ )r′ , r 2

(22)

where P e = u∗ r0 /D0 is the Peclet number, describing the ratio between convective and diffusive transport of colloidal particles in the radial direction. To simplify Eq. (22) further, we assume that transport of particles in the vertical direction is dominated by diffusion, which is quantitatively equivalent to having ǫ2 P e ≪ 1. Following Jensen and Grotberg [81], we asymptotically expand c in terms of ǫ2 P e, 12


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c′ = c′0 (r′ ) + ǫ2 P ec′1 (r′ , z ′ ).

(23)

We substitute Eq. (23) into Eq. (22) and apply the vertical-average operator, •¯ = η ′´+h′ ′ • dz ′ , to both sides of Eq. (22) to obtain a transport equation for the vertically 1/h η′

averaged concentration c0 , which will henceforth be denoted as c. Afterward, we apply Eq. (21) to obtain an evolution equation for c,

ct + u¯cr =

1 Ejc (Dhrcr )r + , hrP e h

(24)

where u¯ is the vertically averaged radial velocity component. In Eq. (24), we have dropped the prime superscripts to simplify the notation, a convention we will follow for the rest of the paper. With the concentration of colloidal particles depending only on the radial coordinate, the viscosity of the droplet is constant in the vertical direction. This allows us to integrate Eq. (2) twice in z and apply Eqs. (7) and (8) to derive an expression for the radial velocity component u,

pr u= M

3 2 3 2 z − 3(h + η)z − η + 3(h + η)η , 2 2

(25)

where pr is calculated using Eq. (9). From Eqs. (4) and (25), we can obtain the vertical velocity component and rewrite the evolution equation (6) for droplet thickness h as 1 ht = r

rh3 M

1 −Π − (h + η)rr − (h + η)r r

r

− Ej.

(26)

r

Here, M is the concentration-dependent viscosity (Eq. (5)), Π is the disjoining pressure (Eq. (10)), and j is the evaporation flux (Eq. (16)). Equations (24) and (26) are the evolution 13


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equations we will solve numerically to obtain the droplet height and particle concentration profiles at different times during drying.

2.5

Contact angles

(a)

(b)

Figure 3: (a) Contact-line region of a droplet on a flat substrate. (b) Contact-line region of a droplet near a topographical defect. Here, the defect is taken to be a Gaussian bump.

For a droplet on a perfectly flat substrate, the apparent contact angle θa is defined as the largest angle between the tangent to the droplet-vapor interface and the substrate, which coincides with the horizontal plane of our coordinate system (Fig. 3a). When the droplet is non-volatile and the intial apparent contact angle θa is greater than the equilibrium contact angle θe , the droplet will spread until θa = θe . When θa < θe , the droplet will retract until θa = θe . For the disjoining-presure model given by Eq. (10), a small equilibrium contact angle θe can be related to A1 and A2 through the following expression [38, 50]:

θe ≈ tan(θe ) ≈ ǫ tan(θe′ ) ≈ ǫ tan( 14

p

A1 A2 ) ≈ ǫ


p

A1 A2 .

(27)

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Here, θe and θe′ are the actual (lab-frame) and scaled (based on lubrication theory) contact angles, respectively. It can be seen from Eq. (27) that as A1 approaches 0, the liquid becomes perfectly wetting. In presenting our results, we will report actual contact angles rather than scaled ones using a representative value of ǫ = 0.1. If the contact-line region of the droplet is located near a topographical defect (Fig. 3b), it is useful to define the mesoscopic contact angle, θm , which is the angle between the tangent to the droplet-vapor interface and tangent to the substrate (at the point where the tangent to the droplet-vapor interface intersects the substrate) [36, 38]. The difference between θm and θa can be expressed as (Fig. 3b)

θm − θa = θs ,

(28)

where θs is the angle the tangent to the topography η(r) makes with the horizontal plane. Note that on a perfectly flat substrate, θs = 0 and θm = θa . The mesoscopic contact angle can be calculated using the following formula [38]:

tan(θm ) =

hr . 1 + (hr + ηr )ηr

(29)

Analogous to case of a perfectly flat substrate, the position of the apparent contact line on a rough substrate is determined by finding the location along the droplet-vapor interface where the mesoscopic contact angle θm (Eq. (29)) is largest, and extrapolating the tangent at this point to the substrate. The apparent contact angle θa is obtained from the slope of this tangent.

2.6

Parameter values and numerical methods

Table 1 provides order-of-magnitude estimates for various physical parameters. Using the values given in Table 1 and ǫ = 0.1, we can estimate the the typical capillary time tc = 15


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O(10−2 s) and evaporation time te = O(102 s), which corresponds to an evaporation number E = O(10−4 ). From the values in Table 1, we can also estimate K = O(10−3 ) − O(10−1 ). As have previous studies [26, 31] , we choose δ such that a desirable steady-state thickness for the precursor film can be obtained from Eq. (17). For this study, a precursor film thickness b ∼ 10−3 h0 is used, which provides a good balance between computational convenience and physical realism [26, 38, 47, 50]. Table 1: Order-of-magnitude estimates of physical parameters. Data are from Refs. [13,17,47,49,70] .

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Parameter

Definition

Order-of-magnitude estimate

ρl (kg m−3 )

mass density of liquid phase

103

ρv (kg m−3 )

mass density of vapor phase

1

h0 (m)

characteristic droplet height

10−5 − 10−3

r0 (m)

characteristic droplet radius

10−3 − 10−2

µ0 (Pa s)

characteristic dynamic viscosity

10−3

σ0 (N m−1 )

characteristic surface tension

10−3 − 10−2

k (W K−1 m−1 )

thermal conductivity of liquid phase 10−1 − 1

Tsat (K)

saturation temperature

298 − 323

La (J kg−1 )

latent heat of evaporation

106

Rg (J mol−1 K−1 )

ideal gas constant

8.314

A1 (Pa)

Hamaker constant

10 − 104

A2 (m)

equilibrium precursor-film thickness

10−8 − 10−6

rp (m)

particle radius

10−8 − 10−7

Using Eq. (20), the diffusivity D0 of a particle with radius rp = 10−8 m is around 10−10 m2 /s at the saturation temperature given in Table 1. The Peclet number is then O(104 ) − O(106 ), which indicates that transport of particles in the radial direction is dominated by convection.

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Equations (24) and (26) are numerically solved on the domain 0 ≤ r ≤ 5 subject to the boundary conditions

hr (0, t) = hr (5, t) = hrrr (0, t) = cr (0, t) = cr (5, t) = 0,

(30)

h(t, 5) = b,

(31)

where b is the steady-state thickness of the precursor film given by Eq. (17). The initial condition for the droplet shape is given by a fourth-order polynomial that satisfies the boundary conditions in Eqs. (30) and (31) at r0 = 1. When the droplet contains colloidal particles, the initial concentration of particles is given by

c=

   c

0

  0

r ≤ r0

(32)

r > r0 ,

which describes an experimental situation where a droplet laden with colloidal particles is deposited on a pure-solvent precursor film [29]. On the computational domain 0 ≤ r ≤ 5, the spatial derivatives in Eqs. (24) and (26) are approximated with second-order centered finite differences. We typically use between 2000 and 5000 spatial nodes, depending on the thickness of the precursor film. Time-integration is performed with the DDASPK iterative solver, which integrates the discretized equations in time using an implicit scheme [82]. The equilibrium contact angle θe for each droplet was first obtained by letting the droplet spread on a flat substrate without evaporation. We waited until the droplet achieved an equilibrium shape and recorded its contact angle as described in Section 2.5. These values are then rescaled with ǫ = 0.1 to give the actual contact angle (Section 2.5). In all cases we considered, the values of the equilibrium contact angle obtained in this way agree with those predicted from Eq. (29) to within 1 to 3 degrees. Each simulation with evaporation is

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stopped when the total volume of the droplet is less than 10−3 , or when the particle volume fraction is within 98.4% of the maximum random-packing limit. Lubrication theory predicts that when a pure-solvent droplet is perfectly wetting (i.e., θe ≈ 0), the contact line spreads at a speed ∼ t1/10 [56, 83]. We have verified this scaling behavior with our model by setting A1 = 10 and A2 = 10−3 , which corresponds to a small equilibrium contact angle, θe ≈ 0.5o . For brevity, the scaling-law comparison is not shown here.

3

Evaporation of a pure-solvent droplet on a perfectly flat substrate

Before proceeding, it is useful to consider as a reference case the drying of a pure-solvent droplet on a perfectly flat substrate. The liquid-solid interface in this case coincides with the horizontal plane at z = 0. Fig. 4 shows the time-evolution of the apparent contact angle and droplet radius (i.e., the position of the apparent contact line). Inspection of Fig. 4 reveals three distinct stages as the droplet evaporates. The beginning of each stage is marked with a downward arrow. In stage I, the droplet quickly spreads under the action of capillary pressure gradients. During this period, the radius of the droplet increases and the contact angle decreases. The droplet height profile and evaporation flux along the interface near the end of stage I are plotted in Fig. 5. The evaporation flux is maximum in the contact-line region, and as a result, evaporation starts to dominate and halts the spreading of the droplet at t ≈ 100. The contact line of the droplet appears to be pinned for a short duration, when evaporationinduced flow is balanced by capillary flow to the contact-line region. After this short pinning period, the droplet dynamics enter stage II, when evaporation becomes stronger than the outward capillary flow, and the droplet retracts due to strong 19


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Figure 4: Evolution of droplet radius and apparent contact angle for a pure-solvent droplet on a perfectly flat substrate. Values of parameters are: θe = 1.6◦ (A1 = 102 and A2 = 10−3 ), b = 3.9 × 10−4 , K = 10−1 , δ = 10−3 , and E = 10−4 .

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evaporation at the contact line. The droplet radius decreases during this time, but the contact angle remains almost constant. In stage III, when most of the liquid has evaporated, both the contact angle and radius of the droplet quickly drop to zero.

Figure 5: Droplet height profile and evaporation flux profile of a pure-solvent droplet on a flat substrate at t = 100. Evaporation is fastest near the contact line. Values of parameters are: θe = 1.6◦ (A1 = 102 and A2 = 10−3 ), b = 3.9×10−4 , K = 10−1 , δ = 10−3 , and E = 10−4 .

Previous models have demonstrated the same three stages [26,29]. However, those models considered perfectly wetting liquids and only included the repulsive term of Eq. (10) in the disjoining pressure. For the case shown in Figs. 4 and 5, the droplet is partially wetting. The qualitative agreement between our results and those for perfectly wetting droplets suggests that partial wetting does not play a significant role in the drying of thin pure-solvent droplets on perfectly flat substrates. In both cases, the temporary pinning of the droplet is primarily controlled by a balance between capillary pressure gradients that drive flow to the contact-line region, and mass loss 21


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Page 22 of 46

through evaporation, which is largest in that region. Disjoining-pressure gradients, which drive flow from the contact-line region toward the droplet interior, appear to play a secondary role. However, as will be discussed in the next section, the presence of a topographical defect on the substrate can greatly amplify the influence of disjoining-pressure gradients, leading to significant changes in droplet dynamics.

4

Evaporation of a pure-solvent droplet near a topographical defect

We now turn to the influence of substrate roughness on the evaporation of a pure-solvent droplet. As discussed in Section 1, although real surfaces exhibit a complex surface topography, we focus here on the presence of a single topographical defect. As we will see, accounting for the presence of a single topographical defect will allow us to naturally describe the transition from the constant-radius stage to the constant-contact-angle stage that has been observed in experiments (Fig. 1). To describe the topographical defect, a convenient (albeit arbitrary) choice of the substratetopography function η(r) is a Gaussian bump. Thus, we choose η(r) = η0 exp(−(r − robs )2 /2s2d ), which describes a defect centered at robs with height η0 and width sd . To place the defect near the initial location of the contact line (r0 = 1), we take robs = 1.3, and to keep the defect relatively small, we set η0 = sd = 10−2 . After a very short initial spreading stage, the droplet becomes pinned at the defect (As noted in previous work [38], this should be interpreted as an apparent pinning since the contact line may still move very slowly.) Below, we discuss the time-evolution of the droplet, the underlying physical mechanisms, and the influence of several problem parameters.

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4.1

Time-evolution

The time-evolution of the apparent contact angle, θa , and the corresponding contact-line position (droplet radius) are shown in Fig. 6. Recall that θa is the angle between the droplet-vapor interface and the horizontal plane (Fig. 3). Figs. 7a and 7b show the droplet shape and total pressure profiles within the droplet at different times.

Figure 6: Evolution of droplet radius and apparent contact angle for a pure-solvent droplet on a substrate with defect centered at r = 1.3. Values of parameters are: θe = 5.3◦ (A1 = 103 and A2 = 10−3 ), b = 6.8 × 10−4 , K = 10−1 , δ = 10−3 , E = 10−4 , and η0 = sd = 10−2 . The beginning of each drying stage is marked with an arrow: (I) constant radius, decreasing contact angle, (II) decreasing radius, constant contact angle and (III) decreasing radius and contact angle. Fig. 6 shows three distinct drying stages. The beginning of each stage is marked with a downward black arrow. In stage I, the droplet is pinned at the defect, and the contact line remains there for a significant portion of the drying time, from t ≈ 0 to t ≈ 550. During this period, the contact angle decreases due to evaporation. Stage II starts at t ≈ 550, when the droplet depins from the defect and the contact line abruptly retracts. This fast 23


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(a)

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(b)

Figure 7: (a) Droplet height profiles at several different times for the case shown in Fig. 6. (b) Pressure distributions inside the droplet near the start of pinning and right before depinning.

pinning-depinning transition can be seen in the steep decrease in the radius and increase in the contact angle in Fig. 6. The transition can also be seen if we compare the droplet profile before depinning at t = 545 and the profile after depinning at t = 560 in Fig. 7a. From t = 1.6 to t = 545, the droplet contact line remains fixed to the defect site, at r = 1.3. From t = 545 to t = 560, however, the contact line quickly retracts to a new location closer to the droplet center, and the apparent contact angle increases during this transition. During stage II, the contact line keeps retracting while the contact angle remains almost constant. Stage III begins near the end of the drying process, when both the contact angle and droplet radius quickly decrease. The three distinct drying stages seen in our simulations (Fig. 6) are similar to those observed experimentally (Fig. 1). Even though we do not have information about the detailed surface topography of the substrates used in the experiments, our simulations show that interaction of a droplet with a single topographical defect is enough to account for the transition from the constant-radius stage to the constant-contact-angle stage. 24


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Interestingly, the contact-angle increase during the depinning transition at t ≈ 550 in Fig. 6 is comparable in magnitude to the contact-angle increases at depinning shown in Fig. 1b, even though the contact angles in the experiments are much larger and in a regime where the lubrication approximation would not be expected to be accurate. These observations imply that while lubrication-theory-based models are unlikely to provide quantitatively accurate predictions the behavior of drying droplets having a relatively large contact angle, they can predict many of the key qualitative features.

4.2

Physical mechanisms

The physical mechanism for the constant-radius stage (stage I) is revealed by examining the pressure profile inside the droplet at t = 1.6 in Fig. 7b. This pressure profile, represented by the dashed-line, has a minimum at the defect location. This means that the pressure gradient at the defect site is zero, and, according to Eq. (25), the fluid velocity here is zero. As discussed in Section 2.1, capillary pressure and disjoining pressure contribute to the total pressure inside the droplet. From the center of the droplet to near the contact line, negative capillary-pressure gradients dominate and drive flow outward to the contact-line region. Near the defect, there exist steep positive disjoining-pressure gradients that oppose the capillary flow and pin the droplet contact line. As the solvent evaporates, the contact angle decreases, and the driving force for capillary flow weakens. The pressure profile at t = 545, represented by a dotted-line in Fig. 7b, shows that the positive disjoining-pressure gradients now become stronger than the capillarypressure gradients, and the minimum in pressure at the defect disappears. At the onset of depinning, the pressure profile at t = 545 indicates a pressure that monotonically decreases from the contact line to the droplet center, which drives flow from the defect toward the center. As a result, the contact line retracts quickly and the droplet depins. In both Figs. 4 and 6, there exists a pinning-depinning transition when the droplet 25


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radius starts to decrease. This transition can be seen at t ≈ 100 for a perfectly flat substrate (Fig. 4) and t ≈ 550 when a defect is present (Fig. 6). Whereas the contact line on the perfectly flat substrate undergoes a smooth pinning-depinning transition, the contact line on the rough substrate undergoes a more abrupt transition. This discrepancy is due to different mechanisms controlling the pinning-depinning transition. The transient pinned stage on the perfectly flat substrate is a result of a balance between radially outward capillary flow and evaporation-induced mass loss (Fig. 4) . Therefore, the pinning-depinning transition is caused by mass-loss becoming stronger than capillary spreading. In contrast, pinning by a defect on the substrate lasts longer relative to the overall drying time, and is caused by a balance between the capillary- and disjoining-pressure gradients (Fig. 7b). Thus, as noted at the end of Section 3, the presence of a topographical defect on the substrate can greatly amplify the influence of disjoining-pressure gradients, leading to significant changes in droplet dynamics. It can also be seen in Figs. 4 and 6 that during the pinning-depinning transition, the apparent contact angle decreases to a constant value for a perfectly flat substrate whereas it increases to a constant value when a defect is present. To help explain this difference in behavior, we shown in Figs. 8a and 8b the contact-line region before and after depinning from the defect occurs. The arrows in these figures represent the direction of contact-line motion. Fig. 8c shows the time evolution of both the mesoscopic and apparent contact angles. As mentioned in Section 2.5, the difference between the apparent contact angle and the mesoscopic contact angle is θs = θm − θa , the angle the tangent to the substrate topography makes with the horizontal plane. When the contact line is pinned on side of the defect having positive slope (Fig. 8a), θs > 0 and the apparent contact angle θa is less than the mesoscopic contact angle θm . This difference can be seen in Fig. 8c, from t ≈ 0 to around t ≈ 550. When the contact line depins and moves to a flat region on the substrate where θs = 0, the

26


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apparent contact angle increases to the mesoscopic value (Fig. 8b and Fig. 8c). Based on the discussion in Section 2.5, the droplet is expected to retract when the angle between the tangents to the droplet-vapor interface and the substrate becomes less than the equilibrium contact angle θe . When this happens, the disjoining-pressure contribution to the flow becomes greater than the capillary-pressure contribution. The evolution of the mesoscopic contact angle θm in Fig. 8c fits in this physical framework. The equilibrium contact angle θe for the droplet here is around 5◦ . Depinning starts occurring when the mesoscopic contact angle (i.e., the angle between the tangents to the droplet-vapor interface and the substrate) becomes smaller than 5◦ (Fig. 8c). Because the value of the apparent contact angle (i.e., the angle between the tangent to the droplet-vapor interface and the horizontal plane) strongly depends on θs , a measure of the local substrate roughness, it cannot be used to predict when depinning occurs on a rough substrate. Indeed, as seen in Fig. 8c, θa is smaller than θm and θe before depinning.

4.3

Parametric study

We now discuss the influence of several different problem parameters. Figs. 9a and 9b show the time-evolution of the droplet radius and apparent contact angle when we increase the defect height η0 from 0.01 to 0.02 while keeping the width sd constant. It can be seen that the pinned stage lasts longer for the substrate with a higher defect amplitude: the droplet is now pinned up to t ≈ 800 when η0 = 0.02 (Fig. 9a). Because of the longer pinning time when the defect amplitude is larger, the apparent contact angle reaches smaller values and undergoes a larger jump after depinning. When the defect width is increased for a fixed amplitude, the pinning time decreases because the substrate becomes flatter. We have also considered the case where the defect is a depression (η0 < 0) instead of a bump (η0 > 0), and observe qualitatively similar results. For brevity, the results of these studies are not shown here. 27


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(a)

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(b)

(c)

Figure 8: (a) Contact-line region before pinning. (b) Contact-line region after pinning. (c) Evolution of apparent (θa ) and mesoscopic (θm ) contact angles for a pure-solvent droplet on a substrate with defect centered at r = 1.3. Values of parameters are: θe = 5.3◦ (A1 = 103 and A2 = 10−3 ), b = 6.8 × 10−4 , K = 10−1 , δ = 10−3 , E = 10−4 , and η0 = sd = 10−2 .

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(a)

(b)

Figure 9: Evolution of (a) droplet radius and (b) apparent contact angle for a pure-solvent droplet on a substrate with defect centered at r = 1.3 for different defect heights. Values of parameters are: θe = 5.3◦ (A1 = 103 and A2 = 10−3 ), b = 6.8 × 10−4 , K = 10−1 , δ = 10−3 , E = 10−4 , and sd = 10−2 .

Besides substrate roughness, Orejon et al. [13] pointed out that substrate wettability also plays an important role in contact-line pinning. For the same level of substrate roughness, contact-line pinning during evaporation might be expected to last longer on a more wettable substrate since it is more energetically favorable for the the droplet to maintain as much contact area with the substrate as possible. However, Orejon et al. [13] were not able to separate these effects since the average roughness of their substrates was not kept constant as wettability was varied. Indeed, for the experimental data shown in Fig. 1, the droplet on the more wettable substrate (silicon) is pinned for a shorter time. Our model can provide insight into this issue if we keep the same defect dimensions while changing the equilibrium contact angle θe by adjusting the Hamaker constant A1 in Eq. (27). The time-evolution of the droplet radius and contact angle for this case is shown in Figs. 10a and 10b. Here, the substrate wettability was varied by decreasing A1 from 1000 to 500, which decreases the equilibrium contact angle from 5.3◦ to 3.0◦ .

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(a)

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(b)

Figure 10: Evolution of (a) droplet radius and (b) apparent contact angle for a pure-solvent droplet on a substrate with defect centered at r = 1.3 for different substrate wettabilities. Values of parameters are: A2 = 10−3 , K = 10−1 , δ = 10−3 , E = 10−4 , and η0 = sd = 0.01. For θe = 5.3◦ , A1 = 103 , and for θe = 3.0◦ , A1 = 5 × 102 .

It can be seen in both Figs. 10a and 10b that contact-line pinning lasts longer for the more wettable substrate, a result consistent with the conjecture discussed above. The physical mechanism behind this trend can be explained by considering the balance between capillary- and disjoining-pressure gradients. When the Hamaker constant A1 is decreased, disjoining-pressure gradients are weakened and cannot oppose as strongly the outward flow driven by capillary-pressure gradients. As a result, pinning lasts longer for a more wettable substrate (i.e., lower A1 ). Finally, we briefly comment on the influence of thermal Marangoni effects. Because the droplet is thinner in the contact-line region, the temperature is higher there relative to the droplet center. The resulting surface-tension gradient drives liquid toward the droplet center and causes depinning to occur faster. However, these Marangoni flows do not qualitatively change the behavior we observe.

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5

Drying of a droplet laden with colloidal particles

We now compare the drying of droplets containing colloidal particles on rough and perfectly flat substrates. Initially, we set a uniform particle concentration of c0 = 10−3 inside the droplet (0 ≤ r ≤ r0 ), with no particles in the precursor-film region. This relatively low initial value of the particle concentration is chosen so that the random-packing limit is not reached too quickly (i.e., before interesting dynamics occur). Figs. 11a and 11c show the droplet height and particle concentration profiles at equal time intervals for a droplet drying on a perfectly flat substrate. There is a very short initial spreading stage which is not shown here, and these profiles are recorded when the contact line starts moving inward. Fast evaporation of solvent in the contact-line region quickly increases the concentration of particles there. This high-concentration front moves inward with the contact line as the droplet evaporates. When the particle concentration reaches the random-packing limit of 0.64, the viscosity diverges, and the contact line retraction is completely halted at tf = 1898. Figs. 11b and 11d show the droplet height and concentration profiles at equal time intervals for a droplet drying on a substrate with a topographical defect. The substrate topography function η(r) used here is a Gaussian centered at r = 1.3 (cf. Section 4). The amplitude of the defect η0 has been increased to 0.05 (with the same defect width) so that the droplet remains pinned long enough for a significant number of particles to get transported to the defect location. There is again a very short initial spreading stage (not shown), after which the droplet becomes pinned at the defect site. During the pinning, the particle concentration increases in the contact-line region due to the fast evaporation there. Since the contact line is pinned and cannot move inward until relatively late times, the particle concentration reaches the random-packing limit at tf = 887, a much earlier time than in the case of a perfectly flat substrate.

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(a)

(b)

(c)

(d)

Figure 11: Droplet height and particle concentration profiles at different times for a droplet drying on (a), (c): a perfectly flat substrate, and (b), (d): a substrate with a defect centered at r = 1.3 with η0 = 0.05 and sd = 0.01. Values of parameters are: θe = 5.3◦ (A1 = 103 and A2 = 10−3 ), b = 6.8 × 10−4 , K = 10−1 , δ = 10−3 , c0 = 0.001, P e = 104 , and E = 10−4 . The time it takes to reach the random-packing limit is tf = 1898 for the perfectly flat substrate, and tf = 887 for the substrate with the defect.

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The final particle concentration profile in Fig. 11d has two local maxima: one at the defect site, r = 1.3, and one at the final droplet contact-line position, r ≈ 0.6. In addition, the height of the droplet near the center appears to increase at later times (Fig. 11b). These phenomena can be understood if we examine the the droplet height and particle concentration evolution at the later times, shown in Fig. 12.

(a)

(b)

Figure 12: Evolution of (a) droplet height and (b) particle concentration profiles for a drying droplet on a substrate with a defect centered at r = 1.3. The profiles are taken at 5 time points: t = 817, 839, 845, 886, and 887 (labeled 1 through 5, respectively). Values of parameters are: θe = 5.3◦ (A1 = 103 and A2 = 10−3 ), b = 6.8 × 10−4 , K = 10−1 , δ = 10−3 , c0 = 0.001, P e = 104 , E = 10−4 , η0 = 0.05, and sd = 0.01.

In Fig. 12, droplet height and particle concentration profiles at five different times from t = 817 (labeled as 1) to t = 887 (labeled as 5) are plotted. Fig. 12a shows that at t = 817, the droplet height has two maxima. However, there is only a single maximum in the particle concentration profile, and it is located in the contact-line region near the defect. At the next time (labeled 2; t = 839), the droplet breaks up, with a large droplet near the center of the original droplet and a smaller droplet near the defect. Because of the axisymmetry, the smaller droplet corresponds to a rim around the larger droplet. The breakup is due to the attractive interaction between the droplet-vapor and droplet-substrate 33


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interfaces (this corresponds to the second term of Eq. (10)). Similar breakup phenomena have been observed in previous studies of both non-volatile and volatile thin films where the disjoining pressure contains an attractive term [84, 85]. During the breakup, disjoiningpressure gradients drive liquid into the larger droplet, causing an increase in its height. As the droplet breaks up, the only peak in the particle concentration profile 1 in Fig. 12b splits into two peaks, as seen in the concentration profile 2, which correspond to the two newly formed contact lines at r ≈ 1.3 and r ≈ 1.1. At the third time (labeled 3; t = 845), the height of the larger droplet has increased, and a third peak in the particle concentration profile appears which is located near the contact line of the larger droplet (r ≈ 0.6). At later times (labeled 4 and 5), evaporation causes the height of the larger droplet to decrease. The high evaporation flux at the contact lines of the smaller cause the concentration peaks to keep growing (time 4). The concentration peaks near the contact-lines of the smaller droplet merge together at late times as the rim region disappears and the random-packing limit is reached (time 5). When the droplets in Fig. 11 reach the random-packing limit, there is still some solvent left. As a consequence, a low final particle concentration near the droplet center does not necessarily mean that the absolute number of particles there is less compared to the contactline region. As have previous studies, we also plotted the particle area density, defined as the product between particle concentration and the droplet thickness, ch, to provide a more complete description of the final particle distributions [1, 33, 53, 71]. The final particle area density profiles for the perfectly flat and rough substrates are shown in Figs. 13a and 13b. In Fig. 13a, it can be seen that most particles are brought to the droplet center for a perfectly flat substrate. This is due to the continuous inward motion of the droplet contact line: as the contact line retracts, it drags the particles along with it. The final particle concentration profile in Fig. 11c may give one the impression that most particles are located near the droplet rim. However, this may be misleading if the region near the droplet center

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(a)

(b)

Figure 13: Final particle area density for a droplet laden with colloidal particles on (a) a perfectly flat substrate (b) a substrate with a defect located at r = 1.3. Values of parameters are: θe = 5.3◦ (A1 = 103 and A2 = 10−3 ), b = 6.8 × 10−4 , K = 10−1 , δ = 10−3 , c0 = 0.001, P e = 104 and E = 10−4 .

has a lower concentration simply because there is more solvent there. Fig. 13a suggests that after a droplet finishes drying on a perfectly flat substrate, the final particle deposit will be cone-like, with more particles located toward the center of the original droplet. Cone-like deposits have been observed in previous experiments where the contact line moves continuously inward during drying [12, 86, 87]. In these experiments, contact-line pinning is negligible and the substrates behave like the perfectly flat ones in our model. Fig. 13b shows that for a substrate with a defect, most particles are found near the defect site, and the rest are found near the center of the droplet. Because the droplet is pinned at the defect for a significant portion of the drying time, particles have time to travel to the defect site. As the droplet breaks up, additional particles get pushed to the defect site, making the particle area density here higher than anywhere else inside the droplet. The formation of a region with a high particle area density near the defect is similar to what is observed in the coffee-ring phenomenon [1, 42]. However, this does not imply that

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droplet breakup is necessary for producing a coffee-ring pattern. In experiments, particles can adsorb to the substrate, meaning that a particle deposit could remain at the defect site even if the droplet depins without breaking up. It is worth noting that the particle area density profiles shown in Fig. 13 only describe the particle distribution at the time when the contact-line region becomes solidified (i.e., the viscosity diverges). Since the suspension can still flow near the center of the droplet, Fig. 13 is not expected to give a quantitatively accurate description of the particle distributions after all of the solvent has evaporated. However, since the volumes of solvent at the point of contact-line solidification are small compared to the original volumes in both cases in Fig. 11, we expect that the particle area densities in Fig. 13 will qualitatively reflect the particle distributions after all the solvent has evaporated. We note two other observations from additional calculations we have performed which are not shown here. First, for faster evaporation rates, the random-packing limit can be reached in the contact-line region even before the droplet depins. However, the resulting particle area density profiles have a cone-like shape similar to that in Fig. 13a. Second, replacing the concentration-dependent diffusivity with a constant value does not change the qualitative behavior reported here. One conclusion of practical interest that we can draw from examining the plots in Fig. 13 is that when the contact line freely moves inward, most particles are drawn to the center of the droplet, and the resulting deposition pattern is expected to be a cone-like structure centered around the axis of symmetry of the droplet. When the contact line is pinned for a significant time period, however, most particles are transported to the defect site, resulting in the well-known coffee-ring pattern. This implies that to achieve a uniform deposition of particles, a mixed mode of contact-line motion that follows a series of pinned and depinned stages may be particularly helpful..

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6

Conclusions

By accounting for the interaction of a drying droplet with a topographical defect on a substrate, the mathematical model presented in this work is able to qualitatively describe several phenomena that have been observed experimentally: • The model reproduces the three drying stages shown in Fig. 1, including the transition from the constant-radius stage to the constant-contact-angle stage (Sec. 4.1). The former stage arises when the droplet is pinned to the defect, and that the latter stage arises when the droplet depins from the defect. • The model describes the contact-angle increase that occurs during depinning (Fig. 1), and provides insight into how this can be understood by considering (i) the interplay between capillary- and disjoining-pressure gradients and (ii) the behavior of the mesoscopic contact angle (Sec. 4.1 and 4.2). Indeed, the presence of the defect greatly amplifies the influence of disjoining-pressure gradients, leading to significant changes in droplet dynamics. The model also allows us to clearly delineate the influence of surface topography and surface wettability on droplet pinning (Sec. 4.3). • The model shows that when droplet pinning is significant, particles can form a coffee-ring pattern, whereas when pinning is unimportant, particles can form a cone-like structure near the droplet center (Sec. 5). These observations are consistent with experimentally observed particle deposition patterns. Although our model is formally applicable to droplets that are long and thin, it appears to be able to predict many of the key qualitative features of drying droplets whose shapes may not satisfy the restrictions of lubrication theory. The model can readily be extended to incorporate the more complex surface topography characteristic of real rough surfaces, as well as chemical heterogeneities that give rise to wettability gradients. Thermal and solutal Marangoni flows, diffusion-limited evaporation, and a host of other phenomena affecting 37


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droplet drying can also be described systematically within the framework of the model. While the model cannot account for phenomena at the scale of individual particles (e.g., contactline pinning on particles) and flow after the random-packing limit is reached, it would be particularly instructive to consider in future work conditions that lead to sequential pinning and depinning of the contact-line. Such a mixed mode of contact-line motion may be useful for generating uniform particle deposition patterns, which are desirable for the technological applications described in Sec. 1.

7

Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. CBET-1449337.

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