liquid interface capture competes with coffee-ring effect

Chemical Engineering Science 167 (2017) 78–87

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Manipulating colloidal residue deposit from drying droplets: Air/liquid interface capture competes with coffee-ring effect Tuan A.H. Nguyen ⇑, Simon R. Biggs, Anh V. Nguyen ⇑ School of Chemical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia

h i g h l i g h t s

g r a p h i c a l a b s t r a c t


Lagrangian particle tracking model of

colloids within drying droplets.
Relative roles of air/liquid vs. solid/

a r t i c l e

i n f o

Article history: Received 7 November 2016 Received in revised form 21 February 2017 Accepted 2 April 2017 Available online 4 April 2017 Keywords: Particle assembly Coffee ring Particle deposit Uniform deposit Evaporating droplet DLVO forces Microhydrodynamic interaction

[1]

+ [2]

100

100

80

80

60

60

40

40 1500 kg/m3

20

20

2000 kg/m3

Relave humidity (%)

Air/liquid capture efficiency (%)

liquid interfaces on the coffee ring effect.
Critical line separates colloids captured first by free interface or by substrate.
The descending air-liquid interface could capture a large portion of colloids.
Regulating dried deposit by tuning the colloidal forces with air-liquid interface.

ρs = 2500 kg/m3

0

0 0.0

0.1

1.0

Dimenonless number, τ = Tf / TG

a b s t r a c t Significant efforts have been undertaken in recent years to understand, and ultimately control, the drying patterns of colloidal particle filled droplets on solid substrates. The ubiquitous coffee-ring effect has long been attributed to the drag of radial liquid flow towards the droplet edge, driven by diffusive evaporation. Here we report new results showing that there is a temporal competition between migrating colloidal particles, dispersed randomly inside a droplet, and the air/liquid interface in reaching the solid surface. Using the Lagrangian modelling approach to track the motion of colloidal particles within the evaporating droplets, we reveal a boundary line that separates colloidal particles into two groups: above it colloids are captured by the air/liquid interface prior to reaching the solid surface and vice versa. This critical line allows us to quantify the effectiveness of the two emerging routes for manipulating the coffee-ring effect, which work by controlling the interaction between colloids and either the substrate or the air-liquid interface. Our modelling results affirm recent experimental evidence that the air-liquid interface of sessile droplets plays a vital role in the formation of residual deposits. This improved fundamental understanding provides versatile strategies to control droplet deposition morphology, especially the most desired uniform deposit as well as an unusual ‘‘overhang” shape of the cross section of the ring-like deposits. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction

⇑ Corresponding authors. E-mail addresses: [email protected] (T.A.H. Nguyen), anh.nguyen@eng. uq.edu.au (A.V. Nguyen). http://dx.doi.org/10.1016/j.ces.2017.04.001 0009-2509/Ó 2017 Elsevier Ltd. All rights reserved.

The ability to create uniform deposits of colloidal particles from drying droplets is highly desirable for many applications and has led to many attempts to eliminate coffee-ring type deposits. Generally, the self-assembly of colloidal particles inside an evaporating

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sessile droplet is strongly affected by the physical and chemical properties of particles, solvents and solid surfaces. Many important factors and parameters including capillary, thermal and concentration driven Marangoni flows (Hu and Larson, 2006; Nikolov et al., 2002), colloidal interactions among particles and between particles and surfaces, contact line motion, surface hydrophobicity and particle orientation have been examined and applied in controlling the particle deposit patterns. The focus has been on manipulating the convective fluid flows inside the sessile droplets. Accordingly, the assembly mechanism of the well-known coffee-ring deposits is assigned to the evaporation-driven radial capillary flow which drags colloidal particles from the bulk solution toward the threephase contact line (TPCL) of the droplet edge (Deegan et al., 1997). On the other hand, the thermal Marangoni flow normally reverses the coffee-ring effect and ultimately forms a central bump of particles (Hu and Larson, 2006). However, these assembly mechanisms cannot explain many other particle deposit structures, especially, the uniform deposit patterns (Bigioni et al., 2006; Yunker et al., 2011) and the colloidal skin (Bigioni et al., 2006; Maki and Kumar, 2011) of particles formed along the air/liquid interface of the sessile droplets. Bhardwaj et al. (2010) proposed that a uniform deposit could be obtained if the transport of particles towards the solid substrate, controlled by attractive DerjaguinLandau-Verwey-Overbeek (DLVO) forces dominates the particle motion to the edge by the radial outward flow and the Marangoni convection. This requirement is only satisfied under the condition of very slow evaporation rate or very small droplet and/or small contact angle since the DLVO forces are effective only within very short inter-surface distances (100 nm). However, the recent literature shows that a uniform deposit can be obtained for large drops (mm in diameter) evaporating under ambient (room) conditions. For instance, Yunker et al. (2011) found that ellipsoidal particles could form a uniform deposit rather than a coffee-ring deposit which was commonly obtained for the spherical particles. They proposed that the formation of a loosely packed structure of these ellipsoidal particles along the air/liquid interface could hinder the formation of the coffee-ring deposits. Similar results were reported earlier for a non-spherical sessile drops containing ligated gold nanoparticles (Bigioni et al., 2006). The transition from the coffee rings to uniform deposits was due to the jamming of particle monolayer on the air/liquid interface of the droplet. This uniform particle ‘‘skin” initially formed on the air/liquid interface could finally be deposited onto the solid substrate when the evaporating solvent was depleted, i.e., when the air-liquid interface disappeared on the solid-liquid interface. However, the physics of the particle accumulation along the air-liquid interface still remains poorly quantified. Direct evidences of particle accumulation at the air/liquid interface of drying sessile droplets have been obtained by many advanced experimental techniques, including three-dimensional confocal microscopy (Yunker et al., 2011), optical coherence tomography (Trantum et al., 2013), optical microscopy (Bigioni et al., 2006) and fluorescence microscopy (Berteloot et al., 2012). However, the resolution limits of these optical techniques used to visualize the three-dimensional motions of liquids and particles inside sessile droplets have not allowed for accurate determination of the temporal and spatial changes in position of particles inside the droplets. This study aims to fill this gap of knowledge by applying the Lagrangian modelling approach to describe the particle motion inside a sessile droplet during the course of evaporation. The condition for particle deposition onto the solid/liquid and the air/liquid interfaces is established for different evaporation rates, initial droplet contact angles, particle densities, DLVO forces and microhydrodynamic resistance. By understanding transport and fate of colloidal particles within drying droplets, we then aim to compare the effectiveness of the two emerging routes for manipu-

lating the coffee-ring effect, which work by controlling the interaction between colloids and either the substrate or the air-liquid interface. 2. Mathematical modelling The geometry used to develop a mathematical model for particle motion inside an evaporating sessile droplet is shown in Fig. 1. The droplet edge normally becomes pinned at the TPCL and evaporation cannot easily shrink the droplet base. In order to keep the droplet edge unchanged the liquid must flow towards the edge to refill the liquid lost at the edge by evaporation. If the sessile droplet contains suspended particles, they are dragged towards the solid surface by the liquid flow moving outwards to the droplet edge, leading the deposition of particles on the surface and at the edge known as the coffee-ring effect. The sessile droplet is considered rotationally symmetric about the axis of symmetry of the spherical cap. A schematic representation of the expected trajectory of a particle inside a drying sessile droplet is depicted in Fig. 1. When the particle, dragged by the liquid flow, is close to the substrate, it will experience colloidal interaction forces and may deposit onto the surface if the attractive components are dominant (i.e. van der Waals force). However, if the particle is far away from the solid surface, the air-liquid interface descending towards to the solid surface may capture the particle before it can reach the surface. We can describe, therefore, an initial (virtual) boundary in the droplet; above this boundary we expect the particles to interact with the descending air-liquid interface before they can reach the solid substrate. It is our aim to quantify this boundary by applying the particle trajectory analysis and calculate the efficiency of particle deposition onto the solid surface as well as the efficiency of particle entrapment by the air/liquid interface. Here, the focus is on particle motion (non-Brownian) towards the solid substrate in competition with the descending air/liquid interface. The interactions between colloidal particles and the solid/liquid and air/liquid interfaces are based on the basic DLVO theory. More complex interactions with the air/liquid interface are not dealt with explicitly in this paper although we acknowledge their possible importance to a more complete understanding eventually. 2.1. Particle motion equation within an evaporating sessile droplet We consider the migration of colloidal spheres under the influence of a liquid flow drag force, colloidal forces and gravitational forces. The effect of Brownian (thermal) diffusion (significant in quiescent fluids) on the particle motion is neglected as are the effects of inertial forces on the motion of colloidal particles. The Stokes drag force on colloidal spheres is usually described !

!

!

!

by 6plRp ðV  W Þ, where V and W represent the particle and liquid velocities, respectively, l is the liquid viscosity and Rp is the particle radius. The commonly known Stokes drag is valid for particle motion far away from a solid surface. In the vicinity of a solid surface, however, the particle will experience an increased drag force due to the micro-hydrodynamic interactions caused by the drainage of an intervening liquid film. The Stokes drag force is then !

!

described by 6plRp ðRv  V Rw  W Þ, where Rv and Rw are the tensors of microhydrodynamic (MHD) resistance of the particle motion and the fluid flow in the directions normal and parallel to the substrate surface, respectively (Nguyen and Schulze, 2004). The drag force is generally balanced by a number of non-inertial external forces including the gravitational force F g , the van der Waals force F v dw and the electrical double-layer (EDL) force F edl (this assumption is valid when the time constant for the drag force

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T.A.H. Nguyen et al. / Chemical Engineering Science 167 (2017) 78–87

Fig. 1. Schematic of particle trajectory inside a drying sessile droplet of the spherical cap shape and the critical positions (green solid line), above which particles are captured by the descending air-liquid interface first (dash-line) and below which particles deposit onto the solid substrate. These ‘‘successful” captures are defined by the ‘‘infinite sink” conditions whenever the shortest separation distances between the particle and the interfaces are smaller than k - a small but nonzero distance (of the order of one nanometer) (Nguyen et al., 2006b). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

is small enough which is the case here since the Stokes number is O !

!

!

(10 )). The force balance gives: Rv  V ¼ Rw  W þF ext =ð6plRp Þ. In the rotationally symmetric cylindrical coordinate system in Fig. 1, 11

!

!

!

the velocity components are defined as: V ¼ i r ðdr=dtÞ þ i z ðdz=dtÞ !

!

!

and W ¼ i r W r þ i z W z . Decomposing the force balance equations into the orthogonal directions gives: In the vertical z direction,


 dh 1 4 f ¼ F DLVO s  pR3p Dqg  F DLVO a cos a þ 2 W z dt 6plRp f 1 3 f1

ð1Þ

In the radial r direction,

dr F DLVO a sin a f 4 ¼ þ Wr dt 6plRp f 3 f3

ð2Þ

In Eqs. (1) and (2), Dq is the particle density less the liquid density, g is the acceleration due to gravity, h ¼ z  Rp is the shortest separation distance between the particle and the solid substrate, a ¼ arctan½r=ðh þ R cot hÞ is the angular distance from the z-axis to the particle, h is the droplet contact angle at time t, F DLVO s and F DLVO a are the particle/solid substrate and particle/air-liquid interface DLVO forces, respectively, and f i ði ¼ 1; 2; 3; 4Þ are the universal microhydrodynamic resistance functions which are described below. 2.2. Microhydrodynamic functions The microhydrodynamic resistance functions present the universal corrections to the Stokes drag force by taking into account the hydrodynamic interactions in presence of the intervening liquid film between the particle and the solid substrate. When the particle is far away from the solid surface, these functions approach unity. Otherwise, they are a function of the separation distance between the particle and the solid surface (Adamczyk et al., 1983; Brenner, 1961; Nguyen and Evans, 2002). The functions are described as the infinite series of inter-surface separation distance h. Semi-analytical expressions obtained by asymptotic analysis have also been obtained (Nguyen and Schulze, 2004) and are computationally suitable for modelling exercises. The semi-analytical equations for the MHD functions for particles approaching solids surfaces are described as follows (Nguyen and Schulze, 2004): 0:89 1:124

f 1 ¼ ½1 þ ðRp =hÞ f2 ¼



ð3Þ

2:022 þ h=Rp 0:626 þ h=Rp

f 3 ¼ f1 þ 0:498fln½1:207ðRp =hÞ

ð4Þ

0:986

1:027 0:979

þ 1g

g

ð5Þ

f4 ¼

1:288 þ h=Rp 0:724 þ h=Rp

ð6Þ

Eqs. (3)–(6) account for the deviation of drag forces from the Stokes resistance on particle in the vicinity of the solid surface. These equations show that the MHD functions are equal to 1 when separation distance h is large and increase with decreasing h. Physically, the particle experiences greater resistance in the vicinity of the surface than in the bulk, far away from the surface. The changes in the resistance on particle also depend on the particle velocity and liquid velocity and the directions of the velocities. Therefore, there are four MHD functions which account for the changes in the drag forces by liquid flow and particle motion in two directions parallel and perpendicular to the solid substrate. 2.3. Colloidal forces The colloidal forces operating between the particle and the solid surface (F DLVO s ) and the air-liquid interface (F DLVO a ) are predicted based on the DLVO theory. The attractive van der Waals force between a sphere and a planar surface is derived from the Lifshitz theory for van der Waals interaction energy between two flat surfaces and the Derjaguin approximation as follows:

F v dW ¼ Aðj; hÞ

Rp 2

6h

ð7Þ

In Eq. (7), Aðj; hÞ is the (Hamaker) function of the separation distance and the Debye constant j which accounts for the screening effect of salts on the van der Waals interaction (Supporting Materials). When a particle approaches another solid substrate their diffuse electrical double layers (EDL) must overlap, giving rise to the EDL interaction. The EDL force can be predicted using the PoissonBoltzmann equation for the electrostatic potential in an ionic solution as a function of position relative to the surfaces and is accurate down to separation distance of a few nm. Since the surface potentials usually remain constant during the interaction, the wellknown Hogg-Healy-Fuerstenau (HHF) expression for EDL force in a symmetric (z : z) salt solution can be used, giving (Hogg et al., 1966)

F edl ¼ 2pee0 jRp

2fp fs expðjhÞ  f2p  f2s expð2jhÞ  1

ð8Þ

where f with the subscripts ‘‘p” and ‘‘s” describes the zeta potentials of particle and substrate, respectively (the subscripts are dropped if the potentials are equal), e0 is the permittivity of the vacuum, e is the dielectric constant of the medium. The Debye constant (reciproP cal length) is described by j2 ¼ f z2i e2 N A 1000ci =ðee0 kB TÞg, where e is the charge of an electron, ci is the molar concentration of salt ions of type i in the bulk solution with the valence zi , and N A is the

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Avogadro number. Eq. (8) is obtained by the Debye-Hückel linearization of the Poisson-Boltzmann (PB) equation and is only valid for small surface potentials. Comparing with the numerical solution of the PB equation shows quantitatively that the linearized HHF expression for 1:1 salt solutions is a good approximation for the absolute value of surface potentials up to 50 mV. For high surface potentials, a number of complicated approximate predictions are available (Nguyen et al., 2006a). Eqs. (7) and (8) are also used to describe the interaction between particle and the air-liquid interface, in which the Hamaker constant for (SiO2) particle interacting with air across water and the air-liquid potential are chosen as

2.5. Transient contact angle and evaporation time Under the diffusion-control conditions, the transient contact angle needed in calculating the liquid velocities described by Eqs. (9) and (10) can be obtained from the available theory for the constant contact base radius evaporation mode as follows (Nguyen and Nguyen, 2012b; Nguyen et al., 2012):

dh DðC s  C 1 Þ ¼ gðhÞ dt q R2

ð12Þ

AH ¼ 1020 J (Butt et al., 2004) and fa ¼ 40 mV (Nguyen et al., 2006b).

where g is a transcendental function and can be approximately described using a semi-analytical function as follows (Nguyen and Nguyen, 2012a):

2.4. Liquid flow inside an evaporating sessile droplet

gðxÞ ¼

Convective fluid flow inside a drying sessile droplet has attracted great interest since it was unveiled as the key factor that shapes drying patterns of colloidal droplets (Deegan et al., 1997, 2000; Masoud and Felske, 2009a). For slow (diffusion-controlled) evaporation of small (R ¼ 1 mm) spherical-cap droplets, a quasisteady state can be assumed (Hu and Larson, 2005) and the liquid flow inside an evaporating droplet with pinned contact line (constant base radius) can be obtained from the Stokes equations by employing a numerical computation technique and/or an analytical method. The semi-analytical solution of the liquid velocity components derived from the lubrication approximation by Hu and Larson (2005) is used as the input to the solution of particle motion equations:

  2 bþ1  1 ~z2 2~z ~ r ¼ 3=8 ð1  ~r Þ  W ~2 H ~ 1  ~t ~r ð1  ~r 2 Þb H  bþ1  2 ~ 2 ~z 3~z2 ~ Ho Jb þ ð1  ~r Þ  þ ~rH 2 bþ1 ~ 2H ~2 R H ð1  ~r 2 Þ   2 bþ1 ~z2 þ b ~z3 ~ z ¼ 3=4 ð1  ~r Þ W  ~ ~2 H 1  ~t ð1  ~r 2 Þbþ1 3H   bþ1 ~z3 3=2 ð1  ~r2 Þ  1 ~z2 ~o H  þ b ~ 2 3H ~3 1  ~t 2H ð1  ~r2 Þ ! bþ1 ~ o ~z3~r H2o ð1  ~r 2 Þ þ ~Jb 2 ~z3 H ~z    2 bþ1 ~2 ~ H H R ð1  ~r 2 Þ   2 ~ ~z3 H Jbðb þ 1Þ~r ~z2   2o ~ R ð1  ~r 2 Þbþ2 H

0:1139x4



1:0009x3

ðp  xÞ3 þ 3:4838x2  6:0419x þ 6:0233 ð13Þ

Total evaporation time (or droplet lifetime) t f is a function of initial contact angle, ho , and dynamics of the contact line. The triple contact line is typically pinned during the droplet evaporation. For simplicity, this constant contact base radius evaporation mode is considered in this modelling paper and the available theory gives (Nguyen and Nguyen, 2012a):

tf ¼

qR2 DðC s  C 1 Þ

Z 0

h0

dh

ð14Þ

gðhÞ

When the approximate expression described by Eq. (13) for gðhÞ is used, the integral in Eq. (14) can be calculated analytically (Nguyen and Nguyen, 2012a). 2.6. Dimensionless number

ð9Þ

As described by Eqs. (1) and (2), the key physical factors affecting the particle motion inside evaporating droplet include the viscous flow of evaporating liquid, gravitational forces and colloidal forces. Dimensionless analysis can be performed by scaling the forces. Since J 0 ¼ DðC s  C 1 Þ=R is the characteristic evaporative flux of a sessile droplet, J 0 =q can be chosen as the characteristic velocity of the flow and R as the characteristic size of the system. Therefore, the first term on the right hand side of Eq. (14) can be R chosen as the characteristic time, T f ¼ DðCqs C , of the viscous flow 1Þ 2

ð10Þ

In Eqs. (9) and (10), the non-dimensional variables are defined ~ r ¼ W r t f =R, W ~ z ¼ W z tf =H0 , ~t ¼ t=t f , ~r ¼ r=R, ~z ¼ z=H0 , as W ~ ~ H ¼ L=H0 , H0 ¼ L0 =H0 , b ¼ 0:5  h=p, where L0 ðr; 0Þ and Lðr; tÞ are the local droplet height and the local height of the air-liquid interface at time t, respectively, H0 ¼ L0 ð0; 0Þ is the initial height of the droplet, h is the droplet contact angle measured in radians and tf is the droplet lifetime. The non-dimensional evaporative flux, ~J, at the gas-liquid interface induces convective liquid flow inside the evaporating droplet. The numerical results for ~J were approximated by Hu and Larson as follows (2005):


 2  ~J ¼ DðC s  C 1 Þ t f ð0:27h2 þ 1:3Þ 0:6381  0:2239 h  p qR H0 4

of evaporating liquid. The characteristic gravitational settling time of particles inside the droplet is defined as the particle settling velocity divided by the characteristic length of the system, giving T g ¼ 2R2 ð9qlRqÞg. The ratio of these two times can be used as the scalp

s

ing factor for examining the physics involved. One obtains

s¼

2R2p Rðqs  qÞqg Tf ¼ TG 9lDðC s  C1Þ

ð15Þ

For the typical inputs described in Table 1 (qs ¼ 2500 kg=m ), the dimensionless time ranges from 6:060 (at 99% relative humidity or C 1 =C s ¼ 0:99) to 0:061 (at 1% relative humidity). By changing s, the role of environment conditions like relative humidity and evaporation rates can be integrated into the solution of the governing equations. 3

2.7. Interfacial entrapment efficiency

ð11Þ where D is the vapor diffusion coefficient, R is the droplet base radius, q is the liquid density, C s and C 1 are the vapor concentrations at the droplet surface and in the ambient environment, respectively.

As the droplet evaporates, there is a temporal competition between the solid particles, randomly distributed inside the droplet, and the air-water interface in reaching the solid surface. Some of the particles are initially close to the solid surface and are able to reach the solid surface ahead of the air-water interface. Conversely,

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Table 1 Input parameters used in the simulation. Parameters

Value

Droplet base radius, R Particle radius, Rp Water density, ql Water viscosity, l Particle density, qs Parameters for van der Waals force

1 mm 0.1 mm 1000 kg/m3 0.8903  103 Pa s 1500 and 2500 kg/m3 (quartz) n2s ¼ n2p ¼ 2:359, es ¼ ep ¼ 3:8

Surface potentials Salt background, KCl Temperature (ambient)

fs;p ¼ 0 and  30 mV, fa ¼ 40 mV 0.1 mM 298 K

there will be a population of particles that are initially far away from the solid surface and which cannot reach that solid surface before the air-water interface catches them. This population of particles may then be captured by the air-water interface before reaching the solid surface. Evidently, there exists a (virtual) boundary of initial positions for the particles dividing the initial particle population into two groups; one group being deposited onto the solid surface (mainly because these particles were initially placed in the suspension closer to the solid surface than other particles of the second group) and the other being captured by the airwater interface, as shown in Fig. 1. This critical line allows us to calculate an efficiency, E, for a particle being entrapped by the air-liquid interface. This approach is analogous to that used in quantifying the efficiency of particle separation in froth flotation processes (Elimelech et al., 1997; Nguyen and Schulze, 2004). Specifically, the interfacial entrapment efficiency of particles is calculated as the ratio between the number of particles initially contained in the volume enclosed by the critical line and the initial airliquid interface, and the total number of particles contained inside the droplet. Assuming that the particles are initially distributed uniformly within the droplet, the entrapment efficiency is equal to the ratio of the volume above the critical line to the entire initial droplet volume. The particle capture by the air-liquid interface can be checked by the (shortest) separation distance, d, between the particle surface and the interface. By considering the simple geometry of the spherical cap of droplet, the separation distance can be described qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as follows: d ¼ R= sin h  r 2 þ ðz þ R cot hÞ2  Rp , where ðr; zÞ are

successfully applied to numerically define the particle capture by the air-water interface. This consideration was indeed equivalent to the acceptance of the ‘‘infinite sink” conditions applied to the particle capture by the solid surface and air-water interface. Typical input parameters for the numerical solution are given in Table 1. 3. Results and discussions 3.1. Particle trajectories In order to study the effects of the colloidal forces and the MHD resistance on the assembling of particles we first consider the behavior of particles moving in the vicinity of the solid substrate. Fig. 2 depicts trajectories for particles which are initially located at 2 mm above the substrate and near the droplet center. The particles are first transported downward and toward the droplet edge under the effects of hydrodynamic and external forces. When the separation is smaller than a certain value, i.e. about 100 nm, the total force is dominated by the EDL force. An attractive EDL force between oppositely charged surfaces can quickly pull the particle to the substrate, while a repulsive EDL force hinders the particle from approaching closer to the substrate. In the latter case, particles are transported radially further (outward) by the fluid flow before been captured by the descending air-liquid interface. The MHD resistance will also result in the particle being suspended for longer in the bulk phase. With the presence of these drag correction functions (MHD-model), the radial displacement of a particle is generally one order higher than without f i in the conventional model (dashed lines - Fig. 2). Consequently, particles have a good chance to be captured by the air-liquid interface if the attractive forces are not sufficiently strong to overcome the MHD resistance and the repulsive EDL force. Given the significant effect of MHD on the particle motion within the confined space of evaporating droplets, the results shown below are obtained with the consideration of MHD unless indicated otherwise. The governing Eqs. (1) and (2) are integrated at different initial radial and vertical locations, ðr o ; zo Þ, to numerically track particle motion inside an evaporating droplet. Some typical trajectories in Fig. 3 show that particles can be captured by either the solid substrate or the descending air-liquid interface depending on their ini-

the actual coordinates of particle position. 2.8. Computational methodology The particle motion inside an evaporating droplet is described by three ordinary differential Eqs. (1), (2) and (12), which are nonlinearly coupled together with a number of algebraic equations described by Eqs. (3)–(11), (13) and (14). This system of differential-algebraic equations can be solved numerically from a specified initial position ðr0 ; z0 Þ. These differential equations together with the initial conditions were first solved using a Matlab ODE solver (ODE45) (Release 2015b, MathWorks, Inc., USA). The output of the ODE solver for each step was used as input to sequentially update the algebraic equations and the right sides of the differential equations. Then the differential equations were solved for the next step and the loop continued until the desired targets were met. A small but nonzero value for the smallest separation distance was normally used to avoid division by zero when calculating the functions and colloidal forces (Nguyen et al., 2006b). Typically the small distance of the order of one nanometer was found to be sufficient. The numerical results were indeed insensitive to the selection. Similar selection of a small but nonzero distance between the particle surface and the air-water interface was also

Fig. 2. Trajectories of particles located in the vicinity of the solid substrate as predicted by the conventional theory (without MHD functions - dashed lines) and the theory with MHD functions included (Eqs. (1) and (2) - symbols). The blue curves are calculated without the DLVO forces, while the red and green curves show the important roles of DLVO forces including an attractive () and repulsive (+) EDL force, respectively, and the van der Waals force. An electrolyte background of 0.1 mM KCl, fp ¼ 30 mV; fs ¼ 30 mV, qs ¼ 2500 kg/m3, and s ¼ 0:1 (at 40% relative humidity) are used in these calculations. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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3.2. Residence time of particles in the bulk phase

0.6 0.4

EDL(+) EDL(-)

0.2

1

Travelling time / tf

tial locations. The integration is, therefore, terminated when particle is either captured by the solid substrate or the air-liquid interface. Generally, initial locations close to the substrate and the droplet center line end with the particles being captured by the solid substrate easier than the other initial locations. The steepness (slope) of a particle trajectory decreases when the particle gets closer to the TPCL. This is consistent with previous experimental results showing that the radial velocity component of particles that are close the TPCL increases significantly as evaporation time progresses (Todorova et al., 2012). This is because the capillaryinduced fluid velocity, W, increases when the contact angle decreases with time (Hu and Larson, 2005). Additionally, for a certain contact angle, the evaporation rate is higher near the TPCL than in the center, leading to a higher fluid flow velocity at a shorter distance from the TPCL (Deegan et al., 1997).

0.8

A θₒ = 60° ro = 0.5R τ= 0.1 I = 0.1 mM

0

0.8

θₒ = 60° τ = 0.1 EDL(-) I = 0.1 mM

B

0.6 0.4

r r r r

0.2 0 1

Travelling time / tf

Fig. 3. Typical particle trajectories at different initial radial (r o =R ¼ 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7) (left) and vertical (zo =L ¼ 0.1, 0.3, 0.5, 0.7, 0.9) locations (right) inside an evaporating droplet, ho ¼ 60o , s ¼ 0.1 (blue lines) and 0.3 (red lines), qs ¼ 2500 kg/m3, and an attractive EDL with fp ¼ 30 mV and fs ¼ 30 mV. L is the local droplet height at radial distance r o . The trajectories are terminated when the particle is either captured by the solid substrate or by the air-liquid interface. The background with cyan color presents the initial sessile droplet with the scaled base radius of 1 mm. Of these trajectories, only particles whose initial vertical positions are close to the solid/liquid substrate is actually captured by the substrate. The other trajectories are captured by the air-water interface. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Travelling time / tf

1

0.8

= 0.2R = 0.4R = 0.6R = 0.8R

θₒ = 60° ro = 0.5R EDL(-) I = 0.1mM

C

0.6 0.4

RH=40%, ρ=1500 RH=80%, ρ=1500 RH=80%, ρ=2500 RH=40%, ρ=2500

0.2 0 0.001

0.01

0.1

1

zo/L In this section, we examine the ‘‘residence” time of a suspended particle which can travel in the bulk phase before being captured by either the solid substrate or the air-liquid interface. A relationship between this residence time, or equivalently the particle displacement, and particle’s initial location is useful in experimentally elucidating the role of fluid dynamics inside an evaporating droplet (Bodiguel and Leng, 2010; Trantum et al., 2013) as well as in designing the final assembly structure of the particle. As can be seen in Fig. 4A, although the colloidal forces can significantly change the particle trajectory (Fig. 2), they impact on the particles which are located very close to the solid substrate. This conclusion can be drawn by observing the dependence of the residence time and the particle initial vertical position. Accordingly, when a particle is far away from the substrate, i.e., zo =L P 0:05 (Fig. 4A), its residence time is the same regardless the presence or absence of the EDL force applied to the models. The curves of residence time only start to deviate from each other for particles whose initial vertical locations are very small. Specifically, when a particle and the substrate possess opposite electric charges, the curve of residence time starts to decrease when zo =L < 0:03. In contrast, the particle is remained suspended in the bulk phase to almost the end of the droplet lifetime if there is a repulsive EDL force between the particle and the substrate. Our modelling also shows that the residence time of a suspended particle strongly depends on its initial location. Interestingly, at each radial distance, there is a critical vertical position at which the residence time reaches a maximum (arrows in Fig. 4). From there, the residence time decreases when the location moves vertically towards either the solid substrate or the air-liquid

Fig. 4. Residence time of a particle in the bulk phase before being captured by either the solid substrate or the air-liquid interface: (A) effects of the EDL forces, (B) effects of particle radial location, and (C) effect of relative humidity (RH) (different evaporation rates) and gravitational force (of a light and a heavy particle, qs ¼ 1500 and 2500 kg/m3, respectively). An attractive EDL force was calculated using fp ¼ þ30 mV and fs ¼ 30 mV. The arrows divide the curves into two critical parts discussed in the text.

interface. A possible reason for this observation is that particle motions are confined between the two interfaces so that the short displacement (short residence time) is made when the particle is close to either of these two interfaces. Similarly, the residence time is also short for particles located radially close to the TPCL, i.e., at large values of ro =R, as shown in Fig. 4B (arrows). This critical point becomes even closer to the solid substrate when the particle and the solid substrate possess the same sign of surface charges (Fig. 4A). According to the numerical results depicted in Fig. 4C, evaporation rate and particle density are also shown to influence the residence time. A high particle density corresponds to a fast gravitational sedimentation rate, while the descending rate of the air-liquid interface is proportional to the evaporation rate (inversely proportional to the relative humidity). The ratio between these two rates determines when a particle is captured by the air-liquid interface. The relative positions of resident-time curves in Fig. 4C reveal a competition between the receding rate of the air/liquid interface and the sedimentation rate of colloidal particles. Generally, a combination of high humidity and a high particle density (RH 80% and qs ¼ 2500 kg=m3 , red curve) allows a particle

3.3. Interfacial entrapments The existence of the peaks in particle residence time in Fig. 4 suggests that there is a vertical critical position for each of radial distances. These critical positions collectively form a critical curve which divides the area inside a droplet into two regimes as shown in Fig. 5. A particle may be captured by the air-liquid interface if its initial location is above this critical curve; otherwise, it reaches the solid substrate before being captured by the air-liquid interface. At slower evaporation rate (higher s), the critical curve locates further from the solid substrate. Assuming that particles are uniformly distributed inside a droplet at the reference time t ¼ 0, one can calculate the collection efficiency of the air/liquid interface (E) based on position of the critical line. Interestingly, it is found that a significant portion of particles moving outward intersects the descending air-liquid interface (Fig. 6). For instance, more than 80% of particles actually have chance to reach the air-liquid interface when a droplet is drying under room conditions, i.e. at 25 °C and 55% RH. Additionally, E is insensitive to difference in particle density when the relative humidity is less than 50%. This result, again, confirms our discussion before on the competition between the receding rate of the air/liquid interface and the sedimentation rate of colloidal particles. The capture efficiency can be enhanced by either increasing the receding rate of the air/liquid interface or decreasing the sedimentation rate of the colloidal particles. Accordingly, under the same relative humidity, smaller particle density, or equivalently slower downward migration due to gravity, will enhance the air/ liquid interface capture efficiency. The same effect can be obtained by simply conducting droplet evaporation at lower relative humidity conditions. Our simulation results could explain the quantitative results of the experimental observations on the interfacial entrapment of particles during evaporation process (Bigioni et al., 2006; Masoud and Felske, 2009b; Trantum et al., 2013; Yunker et al., 2011). More interestingly, the relationships between key process parameters (relative humidity and particle density) on E can be simultaneously reflected by using the dimensionless number s. All data points are now collapsed into a linear relationship, i.e. E ¼ 97:62  70:59s (for conditions listed in Fig. 6). Accordingly, more particles are trapped at the air/liquid interface at lower s values.

0.06

Vercal distance, z/R

τ=0.1

τ=0.2

   

ˮV NJPu ˮV NJPu ˮV NJPu



EDL(-), θo=30°, I=0.1 mM

 











Relave humidity (%) 









 

 1500



kg/m3



2000 kg/m3 ρs = 2500

Relave humidity (%)

to be transported for a longer distance (long residence time) before being captured by the receding air-liquid interface, comparing with the combination of low RH (40%) and low particle density (1500 kg=m3 ) (cyan curve).

Air/Liquid capture efficiency, E (%)

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Air/Liquid capture efficiency, E (%)

84

kg/m3



 



Dimenonless number, τ = Tf / TG



Fig. 6. Variation of the collection efficiency, E, of particles by the air-liquid interface versus dimensionless number s. The simulations were performed for ho = 30°, np ¼ 30 mV and ns ¼ 30 mV. The solid line represents the linear relationship between the interfacial entrapment efficiency and the dimensionless number.

We have found that the Lagrangian modelling approach adopted in this study is useful to better understand how suspended particles are migrated within an evaporating droplet. This model allows us to point out the existence of the critical position, which then helps quantitatively compare the roles of colloid/substrate and colloid/air-liquid interface interactions (the collection efficiency E) in controlling the coffee ring effect. However, it is computationally expensive to use this approach to describe the scenarios when the interaction forces between the colloids and interfaces are both repulsive. Therefore, the Eulerian approach would be a better method to further explore those cases.

τ=0.3

0.05

4. Discussion

0.04 0.03 0.02 0.01 0 0

0.2

0.4

0.6

0.8

1

Radial distance, r/R Fig. 5. Lines of critical initial particle positions, determined for different evaporation rates, s ¼ 0:1; 0:2 and 0:3. The simulations were conducted for a droplet of ho ¼ 30o , using an attractive EDL (fp ¼ þ30 mV and fs ¼ 30 mV), and qs ¼ 2500 kg/m3.

Significant knowledge about coffee-ring deposits has been gained during the last two decades, but shape variations for cross-sections of the deposits have not been thoroughly examined, to date. Although there are a few studies (Askounis et al., 2011; Xie et al., 2013) concerned with the outer shape of ring deposits formed on the receding path of TPCL, the shape and structure of the inner side of the ring deposits is much less known. The inner surface of the ring deposits was simply considered ending with an abruptly vertical wall (Popov, 2005) or with an inner wall formed on the TPCL descending path whose curvature strongly depends on the receding contact angle (Bhardwaj et al., 2009). The latter case has been experimentally reported quite often in the literature using optical microscopy, SEM (Hampton et al., 2012) and AFM (Askounis et al., 2011; Hodges et al., 2010) from

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a uniquely top-view perspective. Recently, Berteloot et al. (2012) used fluorescence microscopy to study the assembly of fluorescent particles inside a sessile droplet. They raised an interesting concern of whether the inner side (descending part) of the deposit surface grows along the solid substrate or along the air-liquid interface. The latter phenomenon, if it occurs, cannot be discriminated when analyzing post-drying from the top-view perspective. Inspired by that idea, we conducted evaporation experiments with a sessile droplet containing 0.01 wt% of a 40 nm SiO2 sample onto a silicon wafer under ambient (room) conditions (The experimental details were previously reported elsewhere (Hampton et al., 2012)). From the top-view perspective, a ‘‘normal” coffee ring structure was observed, as shown in Fig. 7. However, when the ring was removed from the substrate and turned upside down, an ‘‘overhang” crosssection of the ring was apparent. Interestingly, this structure supports the idea that the inner part of the ring may grow along the air-liquid interface. This phenomenon cannot be explained using a conventional assembly mechanism in which the coffee-ring is believed to be built up from particles being dragged from the bulk phase to the droplet edge by evaporative convection. Thus, this conventional assembly mechanism needs to be revised. Given that there is a significant portion of particles whose trajectories cross the descending air-liquid interface (Fig. 6), their transportation along the interface should play a vital role in establishing the deposit patterns. We propose that coffee-rings are not exclusively formed from particles being transported from the bulk phase by the liquid flow to the droplet edge, but rather must include particles that are attached to the air-liquid interface. The overall result of the two particle transport processes is the formation of the ‘‘overhang” structure of the ring deposits as shown on the right of Fig. 8. A ring structure with interior fully filled by particles is obtained if the particles are transported from the bulk to the edge as conventionally suggested (the left picture in Fig. 8). A possible concern with this proposed assembly mechanism arises when the particles do not stick or ‘‘float” on the free surface. In that case, the particles may drop back into the bulk phase as considered by Petsi et al. (2010). Even if this drop back occurs, these particles still remain in the vicinity of the descending air-

Top-view

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liquid interface. Then, the particles can be brought back to the free surface again. Therefore, the proposal that the air-liquid interface plays a vital role in the deposit formation still holds even if the particles cannot strongly attach to the free surface. This helps explain for the overhang structure obtained from the experimental conditions given in Fig. 7, from which we estimated that only less than 1% of particle were captured by the air/liquid interface when the Brownian motion was ignored in the calculation. However, the real capture efficiency of the air/liquid interface should be significantly larger as Brownian motion supplies particles enough energy to overcome the EDL repulsive force. Thus, there is not enough ‘‘truly” captured particles to form an overhang structure, but should be more than enough of ‘‘almost” captured ones near the air/liquid interface. The above discussion is only applicable for zero shear stress at the air-liquid interface and the interfacial particles are transported to the droplet edge primarily by the tangential component of the fluid flow at the interface. Obviously, this outward interfacial flow creates the ring-like deposit, whereas an inward interfacial flow can form a central bump structure. Another important factor is the interparticle interaction. A strong capillary attraction between particles located on the free surface prevents them from rearrangement while being transported to the droplet edge. Consequently, a packed layer of particles can be formed along the air-liquid interface as time goes by, which inhibits the movement of those particles in either direction. A uniform structure should be expected in this case even though the outward interfacial flow still exists in the system. This phenomenon was demonstrated by the experimental works of Yunker et al. (2011) and Bigioni et al. (2006). This principle may open a new and effective approach to regulate the residual deposit since particles can be easily engineered to stick to the free surface according to the desired deposit patterns. Additionally, the interfacial flow can be control not only by the phenomena happened inside a droplet but also by applying different external stresses along the air-liquid interface. It is possible that there is only a small portion of particles that directly reach the solid substrate from the bulk phase (under given conditions in Fig. 6), meaning that only this small population of par-

Up-side-down view

Fig. 7. SEM images of the underside of the coffee ring to show and confirm an ‘‘overhang” structure of the ring’s cross section. The coffee ring deposit was made of SiO2 (Rp ¼ 57 nm, fp ¼ 47:45 mV and q ¼ 1130 kg=m3 ) from a drying sessile droplet on a hydrophobised silicon wafer (fs ¼ 30:18 mV) under 55% RH and room temperature (25 °C) (Hampton et al., 2012).

Mass flux

Fig. 8. Illustration of the proposed assembly mechanism by which a coffee ring is formed mainly from the particles coming down from the air-liquid interface (right) rather than from the bulk phase as in the conventional theory (left).

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ticles will experience the effects of DLVO forces. Also, these colloidal forces can be effective only when the separation distance is shorter than about 100 nm. This means that controlling the residual deposit by the DLVO forces between particles and the solid substrate as previously reported could be an ineffective approach unless the contact angle is very small (i.e., these thin sessile droplets are similar to liquid films being strongly affected by colloidal forces). The concepts of the particle capture efficiency by the wall, and division of the droplet to account for whether particles will be captured by the wall or by the air-liquid interface are novel and well developed in this study. These concepts help us quantitatively compare the roles of colloid/substrate and colloid/air-liquid interface interactions in controlling the coffee ring effect. In other words, the critical line (the collection efficiency E) will inform us how much of a chance we have to manipulate the deposit pattern by controlling these interactions. However, it is worth noting that the final deposit pattern cannot solely be predicted based on this collection efficiency E. One of the reasons is that our calculations (integration of Eqs. (1) & (2)) stop when a particle reaches either the substrate or the air-liquid interface. However, in reality, detachment or re-attachment of particles on these interfaces is also possible after ‘‘successful” attachment events. 5. Conclusions The Lagrangian particle tracking model has been used to analyze the self-assembly of colloidal particles inside an evaporating sessile droplet. Convective motion and migration by external forces, including gravitational forces, double-layer interaction, van de Waals forces and microhydrodynamic resistance forces were considered at different evaporation rates. Particle trajectory and particle residence time (displacement) were found strongly influenced by its initial location. Critical lines were introduced to separate the droplet volume into two parts with two distinctive regimes of particle capture: the particles in the droplet close to the solid surface are captured by the solid substrate and particles in the droplet away from the solid surface are captured by the droplet free surface first. The modelling shows that a large portion of particles is captured by the descending air-liquid interface. The capture efficiency of the air/liquid interface is inversely proportional to the dimensionless number s, which reflects the completion between sedimentation rate and the evaporation rate. Large particle density, and low evaporation rate were found to decrease the likelihood of particles being captured by the free surface. For those particles located below the critical lines, their trajectories were found to be mainly influenced by the colloidal forces. In the immediate vicinity of the substrate, i.e., below 100 nm, the EDL forces on particle migration toward the solid substrate dominated the other forces. A repulsive EDL force and the MHD prevented the particle from approaching the solid substrate. However, not all the particle population actually has the chance to experience the forces operating at very short separation distances. Regulating the residual deposit by tuning the colloidal forces, therefore, should not be as effective as controlling the interaction between the particles and air-liquid interface. Base on the simulation results, a new assembly mechanism by interfacial particles to form the coffee-ring deposits was proposed. The interaction between the particles and the airliquid interface is the key of the new particle assembly mechanism and open a new effective approach to control the residual deposits. General principles for obtaining uniform residual deposits from sessile droplets were also proposed. Notes The authors declare no competing financial interest.

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