Effect of Salt Concentration on the Motion of Particles near the

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Effect of salt concentration on the particle motions near the substrate in drying sessile colloidal droplets Guozhi Xu, Wei Hong, Weixiang Sun, Tao Wang, and Zhen Tong Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b03899 • Publication Date (Web): 03 Jan 2017 Downloaded from http://pubs.acs.org on January 5, 2017

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Effect of salt concentration on the particle motions near the substrate in drying sessile colloidal droplets Guozhi Xu1, Wei Hong1, Weixiang Sun1,3*, Tao Wang1, Zhen Tong1,2* 1

Research Institute of Materials Science, South China University of Technology, Guangzhou

510640, China 2

State Key Laboratory of Luminescent Materials and Devices, South China University of

Technology, Guangzhou 510640, China 3

State Key Laboratory of Pulp and Paper Engineering, South China University of Technology,

Guangzhou 510640, China *Corresponding authors E-mail addresses: [email protected] (W. Sun), [email protected] (Z. Tong)

1

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ABSTRACT:

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The motions of the particles on the substrate of a drying sessile colloidal droplet

of water were measured by multi-particle tracking. 250 mM) of sodium chloride (NaCl) were compared.

Droplets with different concentrations (0 ~ Several statistical quantities were

proposed to characterized the heterogeneous behaviors of the particles and distinguish the effects of the flow field and the substrate interactions.

For the salt-free droplet, most of the particles

are non-adsorbed and mobile without friction.

With the presence of salt, the fraction of the

adsorbed particles increase with increasing evaporation time and the initial salt concentrations, which was explained by the Derjaguin-Landau-Verwey-Overbeek (DLVO) interaction. fraction of mobile particles is mostly frictionless for all samples.

The

At low salt concentration,

the velocity of mobile particles increase with the evaporation time to a peak and then decrease. The velocity is lower for higher salt concentrations.

The effect of salt on the non-adsorbed

particles was attributed to the electrokinetic effect.

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Introduction The drying of a sessile droplet of colloidal suspension will leave a pattern of deposited particles on the substrate.

Since the patterns are mostly ring-like, the mechanism of such process is

known as the coffee ring effect.1

For some applications formation of such patterns is favorable,

as in biological patterning2-3 or colloidal self-assembly,4 while in other cases the coffee ring effects need to be suppressed or avoided, as in inkjet printing5 or spray coating.6-7

More

applications relied on the accuracy and reproducibility on the formation/suppression of the coffee rings, such as the development of microscale bio-separation,8-10 biomedical sensors11-12 and diagnose techniques.13-14

However, understanding the particle dynamics of this process is

difficult, since it is affected by at least two groups of factors.15 flow field of the evaporating droplet.

The first group comes from the

The second group comes from the inter-particle and the

particle-interface interactions. For the first group of factors, considerable experimental efforts have been focused on both the flow fields during evaporation, and the resulted deposition profiles of the particles.

These

can be experimentally accessed by particle imaging/tracking velocimetry (PTV/PIV),16 prolifometry17 and microscopic image analysis.18-19

One of the most significant factors of the

flow fields is the Marangoni effect induced by temperature or surface tension gradient.20 Other frequently studied and utilized factors of droplet evaporation include the wetting conditions,21 the pinning/unpinning of the contact line,22 the viscosity,23 and the evaporation rate.24

Variation of all these factors leads to different evolution of the flow fields inside the

droplet, and in turns affects the transport of the particles and the final patterns.

Physical

models are also developed to relate the deposition profile of the particles to the flow field of the 3


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droplet.20, 25-27

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More accurate predictions of the coffee ring were done by finite-element

method which simultaneously considered fluid dynamics, evaporation, heat and mass transfer together with either pinned or unpinned boundaries.28-29 For the second group of factors which comes from the colloidal particles, most experimental works relied on the final dried patterns to infer on the mechanism of the evaporation processes. Effects such as the shape of the particles,30 the inter-particle interactions,31-34 and the adsorption/desorption of the particles at the liquid-gas/solid interfaces19, 35 were reported. Direct analysis of the particle motions is rare. longer resulted exclusively from the flow field. suitable.

For such cases, the particles motions are no Therefore, the PTV/PIV methods are not

Moreover, under multiple factors, the behavior of the particles may be heterogeneous,

so the averaged behavior of all particles is insufficient to characterize the process of stain formation. In the present study, we used particle tracking technique to extract the trajectories of particles on the substrate of the sessile droplet.

Different salt concentrations were compared with the

intension to vary the strengths of particle-substrate interaction.

Several statistical quantities

were designed to separately characterized the different motions of the particles. the flow field and the particle-substrate interaction were separately quantified.

The effects of We found that

salts concentrations affect not only the adsorption of particles, but also on the non-adsorbed ones, which may be an evidence of the electrokineitc effect recently proposed by Das, et al. 36 methods proposed in the present study may be used in other time-dependent, multi-factor particles tracking problems, such as the activity inside living cells.

4


The

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Materials and Methods The aqueous suspensions of fluorescent polystyrene latex (ThermoFisher) with diameter 2a = 1.9 ± 0.04 µm were diluted to a weight fraction of 4.0 × 10-5.

Sodium chloride was added to the

suspensions to concentrations cNaCl = 0 mM, 25 mM, 50 mM, 100 mM and 250 mM. Cover glasses (Corning, size: 22 mm × 50 mm; thickness: 0.13 to 0.17mm) were used as the substrates for the sessile droplet.

The cover glasses were first dipped into “chromic acid”

cleaning mixture for 1 h, rinsed thoroughly with pure water, and dried in clean air at room temperature.

Then the cover glasses were used immediately for experiment.

1 µL of droplet was introduced to the cover glass using a contact angle measurement system (Dataphysics OCA20).

The diameter of the contact area was maintained to be 2R = 3.0 ± 0.1

mm among different experiments. 25°.

The resulted initial contact angles θ0 varied within 20° ~

The ambient temperature and the relative humidity were equilibrated by the air

conditioner at T = 25 °C ± 1 °C and H = 30 ± 1 %, respectively.

The total evaporation time tf

varied from 180 ~ 220 s. The drying process of the droplets was monitored under an inverted epifluorescence microscope (Nikon Eclipse Ti-S) with a 40× objective (N.A. = 0.6).

Videos were acquired by an sCMOS

(Andor Zyla) at a resolution of 640 × 540 pixels (corresponding to 409.6 × 345.6 µm with a resolution of 0.64 µm/pixel) frame rate of 15 fps.

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(A)

250 µm (B)

Obj.

Figure 1

(A) Size of the region-of-interest (ROI) of the microscopic observation (dash) related

to that of the droplet (solid) with the corresponding opening angle (dotted). (B) The related positions of the droplet, the objective, and the observed particles. Figure 1A shows the actual position of the region-of-interest (ROI) on a typical droplet.

In all

cases of present study, the contact lines were pinned throughout the drying process, and a ring of particles formed at the beginning of the microscopic observation, which served as an indication of the contact line.

We then maintained the location of the contact lines at a horizontal

position of ca. 64 µm from the left edge of the ROI among different observations, so that the ROI always covered the same portion of the sessile droplets of range 0.77R ~ R.

The observed

portion of droplet corresponds to an open angle of ca. 13.2°, as indicated by the dotted line in 6


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Figure 1A.

Due to the small open angle, the motion of the particles in the radial direction was

approximated by the change of their x coordinates for simplicity (with error less than 1-cos(13.2°/2) = 0.66%).

The focus plane was adjusted to observe the lowest layer of particles.

The depth-of-field (DOF) of microscopy is ca. 1.33 µm (shown by the dash line in Figure 1B), so that no more than one layer of particles was observed.

This height of the focus plane also

corresponds to 6.7 × 10-3h0, where h0 is the initial height of the droplet. The motions of the particles in focus were tracked using the algorithm of Crocker and Grier 37. Stages of evaporation were noted by the normalized evaporation time ̃ = /f , where t is the actual time of evaporation and tf is the total evaporation time. microscopic observation was regarded as t = 0.

The starting time of

The observation interval of ̃ was set to 0.05.

For each ̃, 100 frames of all tracks within the ROI were used for statistical analysis, which corresponding to Δ̃ ≈ 3.4 × 10-3 and regarded as the instant behavior at the corresponding ̃. These tracks were further analyzed by MATLAB. Results Since the temperature difference across the droplet during evaporation is typically very small (on a scale of less than 10-2 K),38 the effect of the temperature on the thermal motion of the particles can be regarded as isotropic inside the droplet. axisymmetric.

The flow field inside the droplet is considered

Since the flow field is axisymmetric, and the substrate interaction is isotropic

within the plane of focus, only the radial component of the recorded trajectories was analyzed in the present work.

And since only one layer of particles were observed, the radial component

of the motion of the particle can be analyzed by the model of one-dimensional random walk with 7


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a drift velocity v.

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Several statistical quantities will be used in the following texts to quantified

the heterogeneity of the particle mobility as well as the effects of the flow field and the substrate interaction.

Mathematical derivation of these quantities was provided in the supporting

information. It is intuitive to first look at the velocity v of the particles for any dependence on the salt concentrations.

To this end, the end-to-end velocity vee of a particle trajectory is defined as:

vee =

1 N ∑ ∆xi , N ∆t i =1

(1)

where N is the number of steps, ∆t is the time lag of one step, which is determined by the shutter time of the camera.

∆xi is the displacement of the i-th step.

(A) cNaCl = 0 mM

(B) cNaCl = 25 mM

1

1

0.5

0.5

0 0

0 0 39.5 0.5

18.7 1

22.4 0.5

10.6 1

0

8


0

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(C) cNaCl = 50 mM

(D) cNaCl = 100 mM

1

1

0.5

0.5

0 0

0 0 16.6 0.5

7.8 1

15.2 0.5

7.2 1

0

0

(E) cNaCl = 250 mM 1 0.5 0 0 8.4 0.5

4 1

Figure 2

0

Normalized distribution of the end-to-end velocity vee of the particles at different

reduced evaporation times ̃.

Salt concentrations: (A) 0 mM; (B) 25 mM; (C) 50 mM; (D)

100 mM; (E) 250 mM. Figure 2A-E show the normalized distribution of vee of the particles at different normalized evaporation times ̃.

In the salt-free droplet (cNaCl = 0 mM, Figure 2A), the distribution of vee

mostly lies in the smaller values at early stages of evaporation (e.g. ̃ < 0.5), and move to higher values as the evaporation proceeds, as indicated by the arrow.

For cNaCl = 25 mM (Figure 2B),

the fraction of particles with velocity close to zero increases with evaporation time, and becomes dominant at later stages of evaporation.

This suggests that the motions of the particles may be

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depressed with the addition of salt and a dominant population of the particles are nearly immobilized.

As the salt concentration further increases (Figure 2C~E), not only the portion

of small velocities remains dominant, but the span of velocity distribution also decreases as can be read from the axes.

The distributions of vee outside the smallest velocity are wide-spread

for the samples with added salt, which indicates the particle motions are heterogeneous in the presence of salt.

Figure 3

Observed trajectories within the ROI for the sample with cNaCl = 25 mM at evaporation times (A) ̃ = 0.05; and (B) ̃ = 0.85.

Figure 3 shows the observed trajectories inside the ROI for the droplet with cNaCl = 25 mM at evaporation time ̃ = 0.05 and 0.85.

For ̃ = 0.05 (Figure 3A), although most of the

trajectories show obvious drifting characteristics, a small number of them are apparently immobilized with trajectories appearing as dots (as indicated by red circles). (Figure 3B), the number of the dot-like trajectories increases. dot-like trajectories were not due to small number of steps.

For ̃ = 0.85

We have checked that the Therefore, these restricted

10


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trajectories are from particles immobilized by adsorption to the substrate, which contributes to the dominating distribution of nearly-zero vee as shown in Figure 2.

The results of Figure 3

further support that the particles can be sharply divided into the mobile group and the immobilized group. Figure 3 also shows that, as to the mobile group of the particles, their trajectories are more rippled at ̃ = 0.05 than at ̃ = 0.85.

This suggests that for the mobile particles, not only the

drift velocity but also the randomness of their motion should be characterized for fuller information.

This is done by defining the forward-moving factor F of a trajectory as the

probability of two neighboring positive displacements,

F = P ( sgn ( ∆xn+1 ) + sgn ( ∆xn ) = 2 ) , where n = 1, 2, …, N – 1.

(2)

F characterizes the directionality of a trajectory.

For large

enough number of steps, F = 0.25 if the displacements are purely random, and F = 1 if all steps are positively displaced.

F depends on the dimensionless drift velocity v∆t / σ (see supporting

information for details), i.e. it depends on the directionality rather than the absolute velocity.

(A) cNaCl = 0 mM

(B) cNaCl = 25 mM

1

1

0.5

0.5

0 0

0 0 1 0.5

1 0.5

1

0

0.25

1

11


0

0.25

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(C) cNaCl = 50 mM

(D) cNaCl = 100 mM

1

1

0.5

0.5

0 0

0 0 1

1

0.5

0.5 1

0

0.25

1

0

0.25

(F)

(E) cNaCl = 250 mM

cNaCl (mM): 1

0 25 50 100 250

1 0.8 0.5 0.6 0 0

0.4 1 0.5 1

Figure 4

0

0.25

0

0.2 0.4 0.6 0.8

(A) ~ (E) Normalized distribution of the forward-moving factor F for samples with

the indicated salt concentrations at different reduced evaporation times ̃. (F) Fraction of particles with F > 0.25, nF>0.25 as a function of ̃ for for samples with the indicated salt concentrations.

Figure 4 shows the normalized distribution of F as a function of the evaporation time ̃.

For

the salt-free sample (Figure 4A), F distributes evenly from 0 to 1 at early stages of evaporation, then the fraction of high F values increases sharply with evaporation time. motions become more directional as the evaporation proceeds. 12


This means the

For the cases of different

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added salt concentrations (Figure 4B~E), a fraction of F < 0.25 remains throughout the evaporation process, and the fraction of F > 0.25 decreases with increasing salt concentration. The case of F < 0.25 indicate some zig-zag nature in the trajectories.

This is not predicted by

the random walk model and should not happen without the influence of external forces.

In the

present study, the distribution of F < 0.25 is contributed by the immobilized particles (as will be later confirmed by Figure 5), whose trajectories are close to the resolution limit of the tracking algorithm and resulted in zig-zag shapes.

Figure 4F shows the number fraction nF>0.25 of

trajectories with F > 0.25, i.e. the mobile particles.

nF>0.25 of the salt-free sample remains

nearly 1 throughout the evaporation process, i.e. most of the particle motions are highly directional, even at early stages of evaporation where the velocities are small as shown in Figure 2A.

In contrast, nF>0.25 of samples with different salt concentrations is much lower than 1

which suggests a significant portion of particles with F < 0.25.

For the sample with cNaCl= 25

mM nF>0.25 decrease from nearly 1 to less than 0.4 with evaporation time.

With higher salt

concentrations (cNaCl =, 50, 100, and 250 mM), the initial values of nF>0.25 are lower than 1 and their decreases with evaporation time are less obvious.

The results of Figure 4B~F indicate

that, with the addition of salt, the fraction of the immobilized particles as well as the directionality of the mobile particles increases. The reason for the salt-dependence of the mobile particles, may be jointly affected by both the flow field and the substrate.

The flow field affects only the drifting velocity v, whereas the

substrate causes effective friction that affects both the drift velocity and the randomness of the motion, i.e. the variance of the displacement σ2 = Var[∆xi].

Without any friction from the

substrate, σ2 = 2D∆t where D = kBT / (6πηa) is the diffusion coefficient of a spherical particle of 13


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radius a in a medium of viscosity η; otherwise, σ2 < 2D∆t.

Utilizing this fact, the two effects,

the flow field and the substrate friction, can be resolved by considering at the same time F and the mean square radius of gyration Rg2 of a trajectory.

Rg2 =

The latter is defined as

1 N 2 ( xi − x ) ∑ N i =1

(3)

where xi is the coordinate of the i-th step and ̅ is the averaged coordinate among all steps of the trajectory.

Rg2 characterizes the typical range in which a particle has walked after N steps.

A dimensionless version of Rg2 is constructed as

R% g2 =

3N Rg2 . ( N − 1) D∆t

(4)

2

g characterizes the deviation of a particle trajectory from the theoretical prediction of the no-drift, frictionless Brownian motion case.

g > 1 means a trajectory is larger than

predicted average of a drift- and friction-free Brownian motion, and vice versa.

Considering

g together with F, there are 5 cases for the behavior of the particles: 1) For immobilized g ≪ 1. particles, F < 0.25 and

g = 1. 2) For frictionless, zero drift motion, F = 0.25 and

g < 1. 3) For frictional, zero drift motion, F = 0.25 and

4) For frictionless, drifting motion,

g ; a trajectory of such motion should have the there is a one-to-one relationship between F and g ) that lies on the theoretical F- g curve. value pair (F,

5) For frictional motion with

g ) that lies below non-zero drift velocity, a trajectory of such motion should have value pair (F, g curve. the theoretical F-

g curve is provided in the The calculation of the theoretical F-

14


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supporting information.

Therefore, the corresponding positions of all particle trajectories on a

g coordinates distinguish separately the effects of the flow field and the substrate friction F- forces.

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10 2 (A)

10 2

10 0

10 0

10 -2

10 -2

10 -4

10 -4 c NaCl = 0 mM

10 -6 0

0.25

0.5

0.75

(B)

c NaCl = 25 mM

10 -6 1

0

0.25

F (-) 10 2

10 0

10 0

10 -2

10 -2

10 -4

10 -4 c NaCl = 50 mM 0

0.25

0.5

0.75

1

F (-)

10 2 (C)

10 -6

0.5

0.75

(D)

c NaCl = 100 mM

10 -6 1

0

0.25

F (-)

0.5

0.75

1

F (-)

10 2 (E)

0.00 0.30 0.60 0.90

10 0 10 -2

0.05 0.35 0.65 0.95

0.10 0.40 0.70

0.15 0.45 0.75

0.20 0.50 0.80

0.25 0.55 0.85

10 -4 c NaCl = 250 mM

10 -6 0

0.25

0.5

0.75

1

F (-)

Figure 5

g plotted agains the forward-moving factor F for Reduced mean square radii

particle trajectories observed at all evaporation times. (A) ~ (E) samples with salt concentrations cNaCl = 0, 25, 50, 100, 250 mM, respectively. 16


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g diagram. Figure 5 shows the distribution of trajectories of all evaporation times on an F- g = 1 and the tracking resolution limit. Two horizontal dash lines were plotted for

A

g -F And a dash curve indicates the theoretical

vertical dash line was plotted for F = 0.25. relationship for frictionless Brownian motion.

For all samples, the F > 0.25 population of

g curve nicely, with only few ones lying below the prediction. particles follow the theoretical F- This means that almost all mobile particles are free from any substrate friction.

For the

samples with different concentrations of added salt (Figure 5B~E), a significant number g < 1 and F < 0.25. particles lie in the region of

g of these trajectories are In fact, the

close to the line of tracking limit, which is a strong indication that these particles are completely immobilized or adsorbed to the substrate. free sample (Figure 5A).

In contrast, this type of particles is rare for the salt

The judgement that most mobile particles (those with F > 0.25) are

free from substrate friction can be further checked by the variance of the displacements selected from particles with F > 0.25, which should all equal to 2D∆t. coefficient Dexp is shown in Figure 6.

The experimental diffusion

For all samples, the values of Dexp lie around the

predicted value of D = kBT / (6πηa) (black dash) independent of evaporation times and salt concentrations, which confirms that most mobile particles are free from substrate friction.

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1

0.5

cNaCl (mM): 0 25 50 100 250

0 0

Figure 6

0.2

0.4 0.6

0.8

Experimental determined diffusion coefficient Dexp (see text for definition) of the

mobile particles (those with F > 0.25) for samples with the indicated salt concentrations. We have confirmed that the particles can be divided into mobile and immobile ones by F > 0.25 and F < 0.25, respectively; and the mobile particles are free from substrate drag forces.

The

averaged velocity of the mobile part of the particles can be obtained by their mean square displacement MSD(n) which is defined for any n steps in the trajectory by 2

   ∑ ∆x j  ∑ i =1  j =i  . MSD ( n ) = N − n +1 N − n +1 i + n −1

18


(5)

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Figure 7

(A) Mean square displacements (MSD) of the particles with F > 0.25 at the indicated

evaporation times for sample with cNaCl = 25 mM. 2Dn∆t.

The black dash line represents MSD(n) =

The colored dash lines indicate the fitted curves corresponding experimental data

using eq. (6). (B) vMSD from the fitting as a function of evaporation time for samples with the indicated salt concentrations.

Figure 7A shows the typical results of MSD(n) of the mobile particles (F > 0.25) for sample with cNaCl = 25 mM.

The predicted MSD(n) for drift- and friction-free Brownian motion, i.e.

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MSD(n) = 2Dn∆t, were also plotted as black dash line for comparison, which indicates a slope of one as such a motion should have.

The MSDs of particles with F > 0.25 generally shows

larger values than dash line, with their slopes larger than one.

This reflects non-zero values of

v, since the mean of MSD scales with the square of n∆t: MSD ( n ) = v 2 ( n∆t ) + 2Dn∆t 2

(6)

Fitting the experimental results with eq. (6) gives an estimated averaged velocity vMSD of the mobile particles.

The colored dash lines in Figure 7A shows the fitted curves which indicates

good quality of fitting, and Figure 7B shows the dependence of vMSD on evaporation time for all the salt concentrations.

The reliability of the vMSD is double-checked by the velocity

auto-correlation function (see supporting information for the results).

Figure 7B shows that

the initial velocity for all samples are similarly small, which suggests that any salt concentration effect is not significant at the early stage of the evaporation.

As the evaporation continues, the

velocity of particles in the salt free droplet increases significantly, and suddenly drop near the end of evaporation.

The velocity for samples with cNaCl = 25 and 50 mM also increase with

evaporation time but to less extents, and also drop near the end of evaporation.

The velocity

for samples with higher salt concentrations remain small throughout the evaporation.

Note

that the results of vMSD are from selected particles with F > 0.25, which have been confirmed to be free from any substrate friction, i.e. they are only affected by the flow field.

Therefore,

their salt-concentration dependence indicates that the flow field is also affected by salt concentration.

20


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Discussions A particle in a drying sessile droplet near the substrate at any instance stage of evaporation suffers from not only the stochastic force due to the thermal motion and the inter-particle multi-body interactions, but also several sources of deterministic forces, as illustrated by Figure

8.

z

Fz = Fzdrag + FDLVO + Δ Fr = Frdrag

Fz

r

Figure 8

Fdrag

d

Scheme of the force analysis of the particle.

The Derjaguin–Landau–Verwey–Overbeek (DLVO) force FDLVO is the sum of the van der Waals interaction FvdW and the double-layer interaction Fdl, which are given by

FvdW = −

2 A132 a3 3d 2 ( 2a + d )

2

,

(7)

and 2  eψ s   eψ p  k T  Fdl = 16ε 0ε r a  B  κ exp ( −κ d ) tanh   tanh  . 4 4 k k T T  e   B   B 

21


(8)

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Page 22 of 38

In eq. (7), A132 is the Hamaker constant of polystyrene and silica glass interacting across water, which is calculated to be 0.404 × 1020 J by the combining relation;39-40 d is the surface-to-surface distance between the particle and the substrate.

In eq. (8), kB is the Boltzmann constant; T is

the temperature; ε0 and εr are the vacuum permittivity and the relative permittivity of water, respectively; e is the elementary charge; κ = ε 0ε r kBT

( 2n e ) 2

−1/2

s

is the Debye length which

depends on the electrolyte concentration ns; ψs and ψp are the surface potentials of the particle and the surface, respectively. ψs and ψp are estimated by considering the dissociation equilibrium of carboxyl groups and silanol groups on the surfaces of the particles (carboxyl-modified polystyrene) and the substrate (silica glass), respectively, from which the surface charge density σc can be expressed as41

σc =

103−pK NA Γe , nkw exp ( − ys ) + 103−pK NA

where Γ and pK are the total site density and the dissociation constant of carboxyl or silanol groups, respectively; nkw is the number density of dissociated water; NA is the Avogadro constant; and ys = eψ / (kBT) is the dimensionless surface potential.

The value of Γ = 8 nm-2 for the

silanol groups on glass surface is used,41 and the value of Γ for the carboxyl groups on polystyrene surface was given by provider of the latex.

The values of pK for silanol and

carboxyl groups are 7.5 and 4.7, respectively.42 The surface charge density/surface potential relationships for a highly-charged planar surfaces are given by43

22


(9)

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 σ%  ys = 2arcsinh  c  ,  2  where c = c / r B !" is the dimensionless surface charge density.

(10)

For highly charged

planar surface, the renormalized surface charge potential σc,r is given by43

σ% c,r =

4σ% c 2 + 4 + σ% c2

,

(11)

so that ys = c,r . For highly-charged spheres, the surface charge density/surface potential relationship is given by Ohshima 44 in the form of dimensionless surface potential:

  Q*2  ys = ln  *  6φ {ln (1/ φ ) + pQ } 

(12)

where Q* = c is the dimensionless particle charge; ϕ is the volume fraction of the particles; p is the ratio of the concentration of added counter-ions to the released ones. ψp and ψs are calculated by solving eqs. (9) ~ (12), and inserted to eq. (8) to calculate the double-layer force Fdl. The drag force on a particle is given by

Fdrag = 6πη a ( u − v ) , where u and v are the instantaneous flow and particle velocity, respectively. 23


(13)

The vector of

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flow velocity u = (ur, uz) is defined in the r-z coordinate system as shown in Figure 8.

u varies

according to the position and evaporation time of the drying sessile droplet, i.e. u = u($̃ , %̃ , ̃), where $̃ = $/ and %̃ = %/ℎ are the dimensionless radial and axial coordinates, respectively. In the present work we used the results by Hu and Larson 38. considered the effect of salt on the viscosity of water.45

During the calculation, we have

The addition of NaCl also increases

the surface tension of water by less than 3% at saturation,46 whereas typical surfactant decrease the surface tension by more than 50% at dilute concentration. the NaCl can create is thus limited.

The surface tension difference

On the other hand, the diffusion coefficient of NaCl (~

10-9 m2/s)47 is one order of magnitude higher than those of typical surfactants (e.g. SDS, ~ 10-10 m2/s).48

Therefore, the concentration gradient of NaCl near the water-air interface caused by

the evaporation flow is small. by NaCl can be neglected.

Considering these facts, the additional Marangoni flow caused Only the thermal Marangoni flow is considered in the calculation.

The calculated flow fields of four stages of evaporation are plotted in Figure 9A~D. circulation due to the Marangoni stress is observed.

The

At earlier stages of evaporation Figure

9A~C, the circulation is counterclockwise, while at later stages, Figure 9D the circulation becomes clockwise.

24


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Figure 9

(A)~(D) Velocity field predicted by Hu and Larson

38

at four stages of droplet

evaporation corresponding to contact angles θ = 23°, 18°, 14°, and 10°.

(E) The radial

component of the drag force calculated by the predicted flow field with the absolute value of the negative results plotted in red dash lines.

The time- and salt-dependent appearance of the immobilized particles can be explained by the analysis of the axial forces on the particles.

The total force acting on a particle in the axial

direction, Fz, is the sum of the axial component of drag force, Fzdrag, the DLVO force, FDLVO, and the gravity force, 4πa3∆ρg / 3, where ∆ρ is the density difference between the particles and the medium, and g is the gravitational acceleration.

Fz is dependent on the surface-to-surface

25


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distance d between the particle and the substrate. roughness of the latter. order of ~0.1 nm.49-51

Page 26 of 38

The closest distance dmin is limited by the

Typical roughness for commercial microscopic glass slides is in the In the present work the value of dmin = 4 Å was taken to be the

contacting distance between the particle and the substrate.

For a particle to be kept in focus,

its z position should be within the DOF of the microscopy, so dmax = DOF – a ≈ 0.38 µm.

Figure 10 shows the calculated Fz in the range of dmin < d < dmax at different evaporation times for samples of five salt concentrations.

The values of Fz are on the order of 1 nN, which is an

order of magnitude smaller than previous reported results.52-53

This may due to the high

surface charged density of particles and substrate in the present study.

The values of Fz is

negative close enough to the substrate (d < 1 nm), particles within this region are attracted and adsorbed by the substrate.

Above this attractive region, the particle-substrate interaction

becomes repulsive (Fz > 0).

At late stages of evaporation (̃ > 0.8), a sudden increase in the

repulsive force is observed at a large range of d for all samples, which is resulted from reversal of the flow field and hence the axial component of the drag force (see Figure 9).

For the

salt-free sample (Figure 10A), the repulsive region remains until d = dmax, which spans a wide region of over 100 nm.

In this repulsive region, the magnitude of Fz is also negligibly small.

Therefore, most particles observed in the salt-free droplet are neither adsorbed by the substrate nor repelled out of focus, and generally show frictionless motions as observed in Figure 5A. For samples with cNaCl = 25 ~ 100 mM (Figure 10B~D), however, a re-entrance to attractive region occur as d increases from the repulsive region at early stages of evaporation (̃ < 0.7), which is cut off by the merging repulsive region at later stages of evaporation. secondary attractive region is stabilized from adsorption.

Particles in this

However, the probability of leaping

over the repulsive barrier to the adsorption region may be also significant. 26


The occasionally

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adsorbed particles are also protected by the repulsive region and their number cumulates.

This

may be the reason of the coexistence of the mobile and immobile particles in the samples with added salts.

The width of the repulsive barrier decreases with increasing salt concentration,

and completely disappears for cNaCl = 250 mM.

This may explain why the fraction of

adsorbed particles increases with evaporation time for cNaCl = 25 ~ 100 mM, while remains large and unchanged for cNaCl = 250 mM.

27


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Figure 10

Page 28 of 38

Contour diagram of the axial component of total force Fz at different axial positions and evaporation times.

Although the salt concentrations significantly affect the evolution of Fz, the dependence of vMSD on the salt concentration, which is in the radial direction, is not resulted from the axial DLVO forces.

In the present study, the particle velocities are radially averaged among $̃ ≈0.77 ~ 1.

For comparison, the radial component of the radially averaged drag force Frdrag predicted using the flow fields by Hu and Larson 38 is plotted in Figure 9E for the case of v = 0.

The

magnitude of Frdrag first remains unchanged on the order of 0.1 pN at ̃ < 0.5, which is comparable with previous measurements.52

Afterwards it significantly increases to the order of

1 nN, then decreases again at the latest stage of evaporation

Compared with Figure 7B, the

trend of Frdrag with ̃ is very similar to that of vMSD for samples with cNaCl = 0, 25, and 50 mM. However, since the magnitude of Frdrag should be independent of the salt concentrations, the difference in vMSD among different salt concentrations are not resulted from the flow field by Hu and Larson 38.

This additional flow field effect is attributed to the electrokinetic interactions

proposed by Das, et al. 36.

For an evaporating droplet of electrolyte solution on a charged

substrate, the ions in the electrical double layer of the substrate undergo advection driven by the 28


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flow field of the droplet, causing a streaming potential that tends to slow down the rate of evaporation transport.

To maintain the constancy of the latter, an increase in the pressure

gradient occurs, resulting in an increased particle velocity.

In the case of present work, the

addition of salts may compress the electrical double layer, and reduce the electrokinetic enhancement on the flow field, resulting in lower particle velocities.

The difference in vMSD is

most significant in the latest stages of evaporation (̃ > 0.8), as shown in Figure 7B, this is consistent with the discussion of Das, et al. 36 that the electrokinetic effect is most prominent near the end of the evaporation. It was also reported that the charged polystyrene particles reveal an anomalous electrophoretic mobility in presence of salt, for which different explanations coexist.

One explanation is the

change in the surface charge by adsorption of co-ions,54 which is an electrostatic effect that will only affect the DLVO force between the particle and the substrate in the case of present study. The other explanation, the O’Brien and White model,55 is the relaxation of the ion cloud during electrophoresis, which does not affect the flow-driven motions of the particles.

Due to the

highly-charged nature, the particle surfaces adsorbed the counter-ions instead, which was considered in calculation the surface potential by eq. (12).

Conclusions The motions of the particles near the substrate in drying sessile droplets are studied by multi-particle tracking.

The use of end-to-end velocity, forwards moving factor, radius of

gyration, and mean square displacement could distinguish and quantified the adsorbed, frictional and frictionless particle motions.

In the salt-free droplet, nearly all the particles are

29


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non-adsorbed and undergo frictionless motions.

Page 30 of 38

With the addition of salt, significant fraction

of particles was adsorbed and appeared immobilized on the substrate.

The fraction of

immobilized particles increase with increasing initial salt concentration as well as the evaporation time.

This is explained by the changes in the repulsive and attractive force

regions above the substrate by the addition of salt.

The remaining mobile particles are

frictionless, but their motions are still affected by salt concentration.

For salt-free droplet, the

velocity of the mobile particles increase with evaporation time to a peak then decrease again. With the increase of salt concentration, the increase in the velocity is less obvious.

The results

of the present study suggested that effects of added salts were two-fold: 1) compressing the double-layer between the particles and the substrate leading to increased number of adsorbed particles, and 2) reducing the electrokinetic effect and hence the radial velocity of the mobile particles.

The present work shows how the factors other than the flow field of evaporating

droplet affect the particles motion before deposition.

The methods for analyzing directional

Brownian motions with friction proposed in the present work may be also helpful for other research in non-equilibrium colloidal physics.

Acknowledgements The financial support from the Natural Science Foundation of China (21427805 and 21574047) and the Fundamental Research Funds for the Central Universities (2015ZP005) are gratefully acknowledged.

Associated Content Supporting information is available free of charge via the Internet at http://pubs.acs.org/. 30


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Table of Contents graphic:

36


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Fz = Fzdrag + FDLVO + Fr = Frdrag

Δ

z

Fz

r

Fdrag

d

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Langmuir

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TOC graphic 86x64mm (300 x 300 DPI)


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Effect of Salt Concentration on the Motion of Particles near the

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