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Electroosmosis of Viscoelastic Fluids: The Role of Wall Depletion Layer Siddhartha Mukherjee, Sankha Shuvra Das, Jayabrata Dhar, Suman Chakraborty, and Sunando DasGupta Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02895 • Publication Date (Web): 25 Sep 2017 Downloaded from http://pubs.acs.org on September 27, 2017

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Electroosmosis of Viscoelastic Fluids: The Role of Wall Depletion Layer Siddhartha Mukherjee1, Sankha Shuvra Das2, Jayabrata Dhar2, Suman Chakraborty1,2, Sunando DasGupta1,3 1

Advanced Technology Development Centre, Indian Institute of Technology Kharagpur, Kharagpur, India-721302

2

Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, India-721302 3

Department of Chemical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, India-721302

ABSTRACT We investigate electroosmotic flow of two immiscible viscoelastic fluids in a parallel plate microchannel. Contrary to traditional analysis, the effect of depletion layer is incorporated near the walls, thereby capturing the complex coupling between the rheology and the electrokinetics. Towards ensuring realistic prediction, we show the dependence of electroosmotic flow rate on the solution pH and polymer concentration of the complex fluid. In order to assess our theoretical predictions, we have further performed experiments on electroosmosis using aqueous solution of Polyacrylamide (PAAm). Our analysis reveals that neglecting the existence of a depletion layer would result in grossly incorrect predictions of the electroosmotic transport of such fluids. These findings are likely to be of importance in understanding electroosmotically driven transport of complex fluids, including biological fluids, in confined microfluidic environment.

* E-mail address for correspondence: [email protected] ACS Paragon Plus Environment

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1. INTRODUCTION The transport of complex fluids has progressively become important in the engineering and the medical domain, owing to large number of applications in micro and nano-fluidics. 1–5 These also include biological fluids, bearing immense implications towards the development of lab-on-achip based devices for medical diagnostics.

6

The interplay between low Reynolds number

hydrodynamics, complex rheology and interfacial phenomena makes the transport processes of such complex fluids in microfluidic devices extremely intriguing. 7–10 Many of these devices use electric field as a driving force for the transport of ionic species through narrow confinements, under the broad purview of electroosmosis.

11–16

Such devices have applications in the bio-

chemical, bio-technological and pharmaceutical industries: DNA hybridization, 17 drug delivery, 18

mixing

19–24

and separation of chemical species

17

etc. Intrinsically, electroosmotic flows

(EOF) enjoy numerous advantages over traditional hydrodynamic transportation through micro and nano confinements.

25,26

These advantages appear to be more imperative for certain stress-

sensitive complex biofluids which would otherwise degrade under conventional transport process due to high shear generation in the bulk. As reported in previous studies, there occurs an existence of a depletion layer adjoining the fluid-substrate interface in case of transport of complex fluids. 27–31 Usually, the introduction of a non-adsorbing polymer in a solution phase results in the generation of a depletion layer formed due to the repulsive force between the polymer and the walls.

32–36

For higher polymer

concentration, the thickness of the depletion layer is proportional to the radius of gyration of the particles and it decreases with the increasing polymer concentration. The radius of gyration is equivalent to the length scale typically used in various micro and nanofluidic applications, which in turn, may bear immense consequences towards altering the functionalities of micro- and nanoscale devices. 37–39 Due to the presence of depletion layer, the flow domain is divided into two regimes of different viscosity: lower viscosity within the depletion layer and higher viscosity in the bulk. Previous studies have shown that using streaming potential phenomenon, 10 one can enhance the electrokinetic conversion efficiency by reducing the volume flow rate in the bulk fluid of higher viscosity.

40–42

This can simply be done without affecting the electrokinetics and by making the 2

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depletion layer thickness larger than Electrical Double Layer (EDL) thickness. Whenever EDL thickness is smaller than the depletion layer, the flow within the depletion layer can be assumed to be Newtonian (of constant viscosity) and thus electrokinetics within the EDL does not depend on the bulk rheology. However, in practice, the viscosity of the fluid within the depletion layer may change depending on the bulk polymer concentration.

34,43

Also, there can be a scenario

when the EDL thickness becomes larger than the depletion thickness, thereby penetrating into the bulk fluid. Accordingly, one may expect to observe the effect of bulk rheology within the EDL, thus ensuring complex coupling between complex constitutive behavior of the fluid and the electrokinetics in confined environments. Some theoretical studies illustrate the complex behavior of fluids for both single and multi-phase systems. However, there are issues which are yet to be addressed, for example, whether the microstructure of the complex fluids is comparable to the EDL thickness and how it influences the transport characteristics. Also, there is a lack of support from experimental studies.

44–46

Additionally, recent studies have been focused toward investigating flows of

complex superimposed fluids, since the higher concentration of ions near the solid-fluid interface ensures enhanced electroosmotic pumping.

1,25,47,48

Nevertheless, comprehensive studies are

lacking in various aspects, for example, the ionization mechanism of glass surfaces in deionized solutions shows that there can be variation in the charge density depending on the pH and ionic strength of the electrolyte.

49,50

Besides, there can be significant alteration in the electroosmotic

transport with varying polymer concentration, which in turn, also influences other parameters like viscosity, relaxation time and depletion layer thickness. Most of the previous works are directed towards describing the individual effects of the aforementioned parameters on the electrokinetic transport. Since these effects are interrelated and cannot to be varied individually, we need to consider the combined effect of all parameters to represent the actual experimental situation. To this end, we have attempted an analysis of electroosmosis of two immiscible superimposed viscoelastic fluids with consideration of induced depletion layers and EDL adjacent to the walls in a parallel plate microchannel. Instead of showing individual effects of parameters, we have presented their combined effects such that the results are practically feasible 3


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under experimental realms. To corroborate our theoretical predictions, we have performed an experimental investigation of electroosmosis using aqueous Polyacrylamide solution (PAAm) which exhibit viscoelastic properties. We have used fluorescence microscopy imaging technique for the measurement of the EOF driven velocity. Further, a validation with previous experimental data is also attempted. Also, some reported limiting cases have been established which theoretically match with the previous studies. We believe that the present concept towards analyzing realistic system involving complex rheology will be helpful towards designing microfluidic devices employing such fluids. 2. PROBLEM FORMULATION

Figure 1. Schematic representation of the electroosmosis of two immiscible viscoelastic fluids in a parallel plate microchannel.

We have considered steady, incompressible and fully developed flow of two immiscible viscoelastic fluids in a slit microchannel where the depletion layer effect is also incorporated near the walls (Figure 1). The channel surfaces are subjected to no-slip boundary condition and constant surface potentials. Also, the small value of surface potential ensures that the Debye4


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Hückel linearization can be used. The interfaces between the fluids are considered to be planar thus implying no deformation at the interfaces. POTENTIAL DISTRIBUTION We have assumed that the EDLs are non-overlapping and the ions are in equilibrium such that we can use the Poisson-Boltzmann equation to obtain the potential distribution. This equation, along with Debye-Hückel linearization, yields

d 2ψ = κ 2ψ d y2

(1) 2 n0 ( z e ) , no being the bulk ion ε kB T 2

where κ is the inverse of the EDL thickness expressed as κ =

concentration, z being the valence (assuming z : z symmetric electrolyte), e being the elementary charge, ε being the permittivity of the fluid, kB being the Boltzmann constant and T being the absolute temperature. While obtaining the potential distribution, we have chosen two different potentials at the surfaces (ψ 1 = ζ 1 at y = H1 , ψ 2 = ζ 2 at y = − H 2 ) , while symmetry of electric

 dψ1  dψ 2 at y 0  prevails at the interface between two viscoelastic = ε= displacement  ε1 2 dy  dy  fluids. It should be brought into notice that depletion layer overlap is not analogous to EDL overlap. The effective thickness of the depletion layer is of the order of few nanometers ~ O (10−8 ) m . Thus, for nano-confinement, although there may exist a possibility, at least

mathematically, of having overlapped depletion layer since their thickness is comparable to channel dimensions, physically such a scenario would be infeasible since there must always be a layer of viscoelastic fluid in between the channel walls that precludes the depletion layer overlap even for such nanometer-sized channels. Besides, there is no permittivity variation in a Newtonian fluid and its adjacent viscoelastic fluid. The governing equations can be written as

d 2ψ1 = κ12ψ 1 d y2 d 2ψ 2 = κ 2 2ψ 2 2 dy

    for 0 ≤ y ≤ − H 2  

for H1 ≤ y ≤ 0

(2)

5


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Using the boundary conditions as mentioned earlier, the solution of eq (2) is written in the following with the coefficients shown in Appendix A.

ψ 1 = ζ 1 A ( y )  

ψ 2 = ζ 1 B ( y ) 

(3)

VELOCITY DISTRIBUTION In presence of depletion layer, the flow domain is divided into four layers. The region close to the wall exhibits Newtonian fluid like behavior whereas away from the wall, nonNewtonian behavior can be seen. Although electroosmotic flow has numerous advantages over pressure-driven flow, one major concern is the Joule heating effect due to volumetric heat generation. This becomes more prominent at higher fluid conductivity and higher electric field such that the physical properties of the fluid like permittivity, density, conductivity and viscosity become temperature dependent thereby influencing the net throughput and the performance of the microfluidic devices.51 In the present study, the energy equation in absence of viscous dissipation takes the following form

∂T ∂x

ρ C p u= k

∂ 2T +σe E2 ∂y 2

(4)

∂T ∂ 2T where ρ C p u is the convective part, k 2 is the conduction part and σ e E 2 is the heat ∂y ∂x generation due to Joule heating with σ e being the electrical conductivity of the fluid. To assess the relative contribution of Joule heating , we have performed a scaling analysis by equating the ∂ 2T scale of the conduction term k 2 with that of the volumetric heat generation term σ e E 2 in ∂y the energy equation, resulting in a characteristic temperature difference ( ∆ T ) that scales as

∆T ~

σe E2H 2 k

(

H  O 10 −5

)

. Here, we have used the following range of parameters: E  O (105 ) V m ,

m, k  O (10−1 ) W m.K which yields ∆ T ~ 10 σ e . The value of σ e comes out to 6


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be ~ 10−2 S m even for high electrolyte concentration. This implies that the maximum ∆ T in this study can be ~ 10−1 K which is small enough to influence any electrothermal effect which occurs due to the temperature gradient. Thus, we have neglected the Joule heating effects in this study and focused on the hydrodynamic characteristics through the micro-conduit. Now, for steady, incompressible, fully developed and electroosmotically driven flow, the momentum equation is given by dτ xy dy

= − ρ e Ex

(5)

where ρe is the volumetric charge density, Ex is the applied electric field and τ is the stresstensor. For Newtonian fluid, the momentum equation simplifies to

µ

d2 u d 2ψ E = ε x d y2 d y2

(6)

For viscoelastic fluid, we have considered simplified Phan-Thien-Tanner model (sPTT) for the stress-tensor term which obeys the following constitutive relationship 52,53 ∇

f (τ kk )τ + λ τ = 2η D

(7) ∇

where λ is the relaxation time, D is the deformation tensor, η is the viscosity, τ is the upper convective derivative and f (τ kk ) is the stress-coefficient given by f (τ kk ) = 1 +

sλ

η

τ kk (for linear

PTT model), where s represents the extensibility of the fluid and τ kk is the trace of stress tensor . For fully developed flow, eq (7) reduces to

τ xx = 2

2 λ τ xy ) ( η

(8)

and the governing equation for a viscoelastic fluid becomes

7


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 2 d u 1  2 s λ2 dψ = +c  ε Ex 1 + 2 (τ xy )  τ xy where τ xy =  η dy η dy  

(9)

While obtaining the velocity distribution, we have used the no-slip boundary condition at the walls while at the interfaces, velocities and stresses are continuous where only the effect of viscous stress is considered. In electrohydrodynamics, Maxwell stress couples the electrostatics with hydrodynamics. For an electrically linear, incompressible dielectric medium, the Maxwell stress tensor reads as 54 1 = τ M ε E E − ε E2 I 2

(10)

where ε is the absolute dielectric constant of the form ε = ε 0ε r (where ε 0 is the permittivity of vacuum and ε r is the relative permittivity of the medium), and I is the identity matrix. From the stress expression, one can express the electric surface force on the interface in the form

{(

1 = τ M ⋅ n ε 0  ε1 E12n − ε 2 E22n − ε1 E12t − ε 2 E22t 2

) (

)} n + (ε

1

 E1n E1t − ε 2 E2n E2t ) t  

(11)

where Ei t and Ei n denotes the tangential and normal components of the electric field at the interface of the two fluids, which are functions of only the transverse coordinate (y). Here, n denotes the normal to the interface (which, for an un-deformed interface) is the transverse direction in the channel, whereas t denotes the tangential component (which, for an un-deformed interface, is the axial direction of the channel). Now, as no charge accumulation is assumed at the fluidic interface (it is considered to be perfectly dielectric in the present study), the normal component

of

the

displacement

field

becomes

continuous

across

the

interface55

ε1 E1n E1t = ε 2 E2n E2t (as considered while deriving the potential distribution across the viscoelastic interface). As a consequence, the tangential component of τ M ⋅n vanishes. The normal component of the Maxwell stress gets balanced by the jump in the pressure across the interface. 55 Thus, the Maxwell stress does not contribute to the total stress balance across the two viscoelastic-fluid interface. 8


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It must, however, be noted that another prerequisite for vanishing Maxwell stress (even for perfectly dielectric medium) is the condition of curvature-free interface as has been considered here. The interface is considered to be planar with absence of any deformations. Whether deformation takes place or not is governed by a parameter known as Capillary number Ca ( Ca ) which is defined as=

µ U c ε Eζ = where γ γ

µ is the viscosity, U c characteristic velocity,

γ surface tension, ε permittivity, E applied electric field and ζ is the surface potential. The

(

)

typical values of the parameters are as follows : ε ~ O 10−9 F m ,

( )

E ~ O 105

V m,

ζ ~ 10 to 100 mV and γ ~ O (10−2 ) N m . This implies that the value of Ca is very small, i.e. ~ O (10−4 ) to O (10−3 ) for which the assumption of no deformation remains valid. Hence, we

have considered the traditional simplification of the continuity of only viscous stresses at the interface. Now, the governing equations and the solution along with the coefficients can be found in Appendix B. LIMITING CASES Now, we can analyze some limiting cases on the basis of the present theoretical model. Electroosmosis of a Newtonian fluid

d= 0, H= H= H , ε= ε= ε , De = 0 In absence of depletion layer, we substitute d= 1 2 1 2 1 2 and simplify the momentum equation to obtain the flow field. The potential distribution subjected to equal zeta potentials at the surfaces yields

ψ=

cosh (κ y ) cosh (κ )

(12)

Using eq (12) in Navier-Stokes equation, we get the following velocity distribution cosh (κ y ) u = 1− uhs cosh (κ )

(13)

9


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Equation (13) is a well-known equation for purely electroosmotic flow of Newtonian fluids through parallel plate microchannel. Electroosmosis of a single viscoelastic fluid without depletion layer Similarly, the flow field for a single viscoelastic fluid without depletion layer can also be represented

by

substituting

d= d= 0, H= H= H , ε= ε= ε , s= s= s,η= η= η, 1 2 1 2 1 2 1 2 1 2

λ= λ= λ and solving the momentum equation (9) 1 2

{

}

u 2 s De 2 = uN + F1 ( H ) − F1 ( y ) uhs cosh 3 (κ )

(14)

where the coefficients F1 ( H ) and F1 ( y ) are given by 2 2 1  1  F1 ( H ) =  sinh 2 (κ ) cosh (κ ) − cosh (κ )  , F1 ( y ) =  sinh 2 (κ y ) cosh (κ y ) − cosh (κ y )  3 3 3  3 

Equation (14) is the same expression for purely electroosmotic flow as previously derived by Pinho et al. 3 with u N representing the Newtonian part described by eq (13). 3. EXPERIMENTAL PROCEDURE

10


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Figure 2. (a) Schematic of the experimental setup of electroosmosis. (b) Crosssectional view of the microchannel along with the flow visualization plane.

The representative diagram of the experimental setup is shown in Figure 2. This setup consists of a voltage source, a data acquisition system, a microchannel and a CCD camera. The data acquisition system consists of a computer which is connected with an Olympus microscope for recording of the experimental data. The solution is prepared by mixing and stirring of 1mM KCl solution with PAAm powder of molecular weight 5 M (HiMedia Leading BioSciences Company). To prepare the electrolyte solution, fresh deionized water (Millipore India Pvt. Ltd) is mixed with KCl salt (Merck Life Science Pvt. Ltd.). Three different concentrations of PAAm (0.01%, 0.05% and 0.1%) are used in the experiment to compare the electroosmostic velocity profiles with those predicted by the above mentioned theoretical model. A well established and standard protocol is followed for the fabrication of rectangular microchannel where photolithography followed by soft lithography process is used.

56

The

process starts with cleaning the glass slides using Piranha solution (H2O2 : H2SO4 = 2 : 1) 11


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followed by DI water and then purging it by pure nitrogen jet. Further, to remove the residual moisture from the substrate, baking is done at 95 °C by placing it in the oven for approximately one hour. Thereafter, the substrate is cooled to room temperature followed by plasma treatment (for ~ 1 min) to improve the adhesion property of the glass substrate. The plasma treated substrate is then spin coated with negative photoresist SU8-2050 (MicroChem Corp, Newton, USA) at 4000 rpm (35 - 40 µm film thickness) for 25 secs and a two stage baking is done at 65 °C for 3 mins followed by 95 °C for 6 mins. Thereafter, the substrate is exposed to UV (Hybralign 200 Mask aligner, OAI) at 100 mJ/cm2 energy density through the appropriate mask for 12 secs. Further, post-baking is done at 65 °C for 1 min which is followed by 95 °C for 5.5 mins. Finally it is developed in developer solution for ~ 4 mins and then placed in the oven at 95 °C for 2 - 3 hours. After obtaining the master mould, the PDMS based polymer and the curing agents (Sylgard184, Dow corning, USA) are mixed in a 10 : 1 ratio (w/w) and kept in the desiccator in order to remove the trapped air bubbles. Then, the master mould is filled by PDMS and is cured in the oven at 95 °C for 2 hours. After that, the PDMS layer containing the structure of the microchannel is peeled off and provision of the reservoir is made. The PDMS based microchannel is bonded to another glass substrate using plasma bonding (PDC 32G, Harrick Plasma Cleaner). Two cylindrical reservoirs (8 mm long x 5 mm diameter) are made in the inlet and outlet and filled with PAAm solution. To supply the electric field, platinum wires are immersed within the fluids ensuring a proper electrical connectivity. To avoid entrance and exit losses, the reservoirs are made large as compared to the microchannel dimensions. Before starting the measurement, the fluid level in the two reservoirs is balanced, which ensures the absence of any flow due to pressure head. For visualization of the electroosmotic velocity profile, we have used fluorescence microscopy imaging technique where 0.03 % (v/v) fluorescent particles (of mean diameter 1 μm) (FluoSpheres@carboxylate, Life Technologies pvt. ltd.) are added to the sample. An inverted microscope (Olympus) is used for observation and recording of the images in a CCD camera with a time interval of 50 ms. 4. RESULTS AND DISCUSSIONS

12


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Here we have developed the theoretical model for electroosmosis of viscoelastic fluids in a slit microchannel with depletion layer effect. This model involves numerous tunable parameters like ratios of viscosity, electrical conductivity and permittivity, relaxation time, extensibility; depending on which the volumetric flow rate through the channel can change significantly. For the sake of brevity, we only discuss some realistic situations by addressing the effects of polymer concentration, depletion layer thickness, pH and ionic strength of the electrolyte on the net electroosmotic transport. The rheological behavior of complex fluids depends strongly on the concentration and type of the solvent. Several theoretical and experimental studies depict that the viscosity of these fluids change drastically when there is a transition between different concentration regimes. These different regimes are known as dilute, semi-dilute unentangled and semi-dilute entangled regime which are discriminated by overlap concentration ( c* ) and entangled concentration ( ce ) where the transition from one regime to another takes place. Also, the quality of solvent is determined by a factor, termed as fractal polymer dimension (ν ) . Depending on whether good solvent or θ-solvent, it can vary between 0.5 to 0.6. Now, we introduce a dimensionless quantity specific viscosity (η sp ) which is defined as η= sp

η0 − 1 , where η0 and η s are solution viscosity ηs

and solvent viscosity. The dependence of the specific viscosities on the above parameters are given by 57 1 3ν −1 3 3ν −1 ηsp ,dilute ∝ c , η sp , semi-dilute ∝ c ( ) and ηsp , entangled ∝ c ( )

where c the concentration in the respective regimes. Also, the relaxation times ( λ ) of the polymer solution in the respective regimes are given by the following scaling laws 57,58

λ dilute ∝ c 0 , λ semi-dilute ∝ c(

2 −3ν ) ( 3ν −1)

and λentangled ∝ c

3(1−ν ) ( 3ν −1)

The introduction of non-adsorbing polymer in microchannel creates a depletion layer near the surface. To include the effect of depletion layer, we usually assume a position dependent viscosity profile while obtaining the electroosmotic velocity profile 34,35 13


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η ( x) =

ηb

(15)

η  1 +  b − 1  e − x λD  η0 

where ηb is the bulk solution viscosity and η0 is the viscosity at x = 0 with x representing the co-ordinate perpendicular to the surface and λD being the effective depletion layer thickness. Also, it depends on the bulk viscosity in the following way

 ηb + η 0    η0 

δ = λD ln 

(16)

where δ is the half-width of the depletion layer. 34 To support our theoretical model, we have compared our theoretical results with the experimental work of Huang et al..59 This comparison is shown in Figure 3 where the electroosmotically driven velocity (in mm/s) is plotted with varying electric field strength (in V/cm). Huang et al.

59

have performed experiments of electroosmosis using aqueous solution of

Polyethylene oxide (PEO) and also compared their experimental results with their proposed theoretical model which was based on the coupling between the Smoluchowski approach of Newtonian fluid with non-Newtonian power-law model. For comparison, we have considered two different concentrations of PEO solution, 0.3 wt % and 0.5 wt %; for which the corresponding viscosities are 0.005 Pa.s. and 0.01 Pa.s. respectively. Also, this concentrations of PEO fall in the semi-dilute unentangled regime for which relaxation time of the aqueous PEO is given by λ semi-dilute ∝ c

( 2−3ν ) (3ν −1)

. As reported in the literature, the value of the coefficient ν

for PEO solution is found to be ν = 0.55 which is in between the two limits of good solvent and θ-solvent. 57 For 0.5 wt % PEO solution, the relaxation time is λ  5 ms

60

and for 0.3 wt %, it is

taken as  3 ms , in agreement with the above scaling relationship. We have also incorporated the changes in the depletion layer thickness due to the variation in PEO concentration governed by eq (16). For example, we have chosen the depletion layer thickness for 0.5 wt % as

λD  4.7 nm ,

36

while it is λD  6.27 nm for 0.3 wt % (using Eq.(16)). While comparing these

results, we have presented our theoretical results at s = 0.01 and our predictions are well within 14


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the acceptable margin with that of Huang et al. Figure 3 also shows their theoretical predictions and interestingly, the electroosmotically driven flow can be better approximated when the depletion layer effect is considered. With depletion layer effect, our theoretical predictions are relatively closer for low concentration and this holds for higher electric field also.

Figure 3. Comparison between present theoretical study and the results of Huang et al. Dotted lines show the theoretical predictions of Huang et al. while solid lines represent the present theoretical results. For further strengthening our theoretical model, we have performed experiments using aqueous solution of PAAm where the average velocity is obtained with varying electric field strength and polymer concentration. For measurement of the viscosity, we have used commercially available Polyacrylamide (PAAm) of molecular weight 5 M (HiMedia Leading BioSciences Company). Three different concentrations are prepared, 0.01 wt %, 0.02 wt % and 0.05 wt %; by dissolving in deionized water and then stirring at a speed of 900 rpm for overnight. The viscosities of different PAAm solutions are measured using rheometer (Anton Paar MCR302). The viscosity variation with respect to different shear rates is shown in Figure 4. With increasing PAAm concentration, the characteristic Deborah number (De) and s ⋅ De 2 increases which is associated with the shear-thinning behavior of a viscoelastic fluid. Thus, shear-thinning effect becomes more prominent at higher PAAm concentration. Within the range of concentrations studied, η sp ∝ c and falls in the dilute regime. Transition can only occur at 15


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higher concentration (for > 0.1 wt% PAAm). In accordance with the Capillary break-up extensional rheometry measurement (CaBER)

61

and Zimm's theory,62 we have chosen the

relaxation time of the fluid as λ  1 ms . Here also, we have accommodated the change in the depletion layer thickness with varying PAAm concentration. For example, the depletion layer thickness decreases 1.8 times as we increase the PAAm concentration from 0.01 wt % to 0.05 wt % , which is taken using eq (16) .

Figure 4. Variation of viscosity for different concentration of aqueous PAAm solution.

In general, the flow velocity is increased with increased electric field strength but follows an inverse relationship with increasing PAAm concentration. Higher electric field means more energy will be transferred into the fluid leading towards the enhancement in the EOF driven velocity. On the contrary, increased PAAm concentration results in the significant increase in the viscosity (for example, viscosity increases up to ~ 4 times as we increase the PAAm concentration form 0.01 wt % to 0.05 wt %) thus leading to the reduction in the flow velocity. If we keep on increasing the PAAm concentration, situation may occur when the EOF driven flow is completely arrested due to extremely high apparent viscosity.

16


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Figure 5. Fluorescence images of electroosmotic flow of aqueous PAAm solution of 0.05 wt % concentration. Images show the displacement of particles at different time intervals for electric field strength 50 V cm-1. The displacement of fluorescent particles in electroosmotic flow of aqueous PAAm solution is depicted in Figure 5 which is processed to get the electroosmotic velocity profile. The comparisons between theoretical and experimental results are shown in Figure 6 where the inset shows the corresponding results for Newtonian fluid (1 mM KCl solution). It is clear from the figure that, our theoretical predictions are very close to that of experimental one for lower concentration. For Newtonian fluid, it is in excellent agreement with the experimental results and deviation starts to occur with increasing concentration of PAAm and becomes prominent at higher concentrations. For lower concentrations (0.01 wt% and 0.02 wt %), the maximum deviation is ~ 10% while it rises up to ~ 17% for higher concentration (0.05 wt %). The mismatch in the velocity profile in higher concentration is due to the higher value of characteristic Deborah number ( De = λ κ uhs ) for which the linear PTT model cannot describe completely the rheological behavior of PAAm solution. Although the microchannel is uniform throughout the length, there can be microscopic undulations or waviness in the channel affecting 17


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the charge distribution within the EDL. Another probable reason may be the assumption of constant zeta potential throughout the analysis which is often regarded as convenient since the complex rheological behavior of PAAm does not hinder the electroosmotic transport characteristics. Also, the deviation is more pronounced in higher electric field where the average velocity is not directly proportional to the electric field which may be due to the fact that the charge distribution is not uniform within the EDL.

Figure 6. Comparison between present theoretical and experimental study. Inset shows the comparison in case of Newtonian fluid. The theoretical dependence of the volume flow rate with concentration is predicted in Figure 7. Different concentration regimes are shown also which shows the transition from dilute to semi-dilute (occurs at c cref ~ 10 ) and from semi-dilute to entangled regimes (occurs at c cref ~ 45 ) where the overlap

( c *)

and entangled concentrations

( ce )

are also marked.

Theoretically, we can vary the parameters like concentration, depletion layer thickness, relaxation time individually to see their sole effect on the net throughput through the microchannel, but in practice, they are actually interrelated to each other. To see their combined effects, we incorporate these parameters via proper scaling relations in the respective regimes. Increasing polymer concentration significantly affects the pumping capacity thereby reducing the volume flow rate up to 50 times, evaluated at an ionic strength of 1 mM. This figure also shows 18


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the effect of concentration variation between the two viscoelastic fluids on the flow rate. As we increase the concentration ratio, depletion layer thickness gets reduced by following the prementioned scaling relation which leads to a slight increment in the flow rate, ~ 30 times (for c2 c3 ~ 4 ) lower with respect to the Newtonian fluid (Qref is the flow rate corresponds to the case

when the entire microchannel is filled with a Newtonian fluid). The corresponding results for a different ionic strength are shown in the inset. With increasing ionic strength (from 1mM to 10 mM), the reduction in the pumping capacity due to high polymer concentration can be arrested up to some extent, resulting in ~ 24 times lessening of the net throughput (for c2 c3 ~ 4 ).

Figure 7. Reduction in volumetric flow rate with increasing concentration ratio evaluated at ionic strength = 1 mM. c2 and c3 are the concentrations of the two Viscoelastic fluids while cref corresponds to the concentration of Newtonian fluid. Inset shows the corresponding results for higher ionic strength, i.e. 10 mM. The enhancement in volume flow rate for different pH values of electrolyte concentration is plotted in Figure 8. Considering the ionization of silica and silicate glass surfaces, the diffused layer potential as a function of charge density (σ ) is given by the following equation 49,63

= ψ d (σ )

1  −σ ln  β e  eΓ +σ

ln (10 ) σ  −  − ( p H − p K) βe C 

(17) 19


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where β −1 = k B T is the thermal energy, σ is the charge density given by σ =−e ΓSiO−

50

with Γ

representing the total site density defined as Γ = ΓSiO + ΓSiOH . At the silica-water interface, the −

value of dissociation constant pK is found to be 7.5 64 and C is the Stern layer capacity, related to the surface potential (ψ 0 ) and diffused layer potential (ψ d ) as C=

σ

(18)

ψ 0 −ψ d

The value of C shows little variation with the electrolyte concentration and its value can be chosen as 2.9 F/m2. 64 Also, the charge density depends on the distribution of the mobile charges. If it follows the Poisson-Boltzmann equation, then σ for a flat, isolated surface will satisfy the Grahame equation given by

σ (ψ d ) =

2ε l  β eψd sinh   β e   2

where l is the Debye screening length described by l =

  

(19)

β e2 n with n representing the ε

concentration of the monovalent small ions. If the curvature effect is also taken into account, then eq (19) can be rewritten as

σ (ψ d ) =

2ε l  β eψd sinh   β e   2

 2  β eψd  + tanh   la  4

  

(20)

where a is the radius of curvature such that l a ≥ 0.5 and the value of Γ is taken as Γ =8 nm −2 .65 We have solved eq (17) and eq (20) simultaneously to get the charge density and the diffused layer potential for varying pH values of the electrolyte solution ranging from 5 to 7. The effect of varying ionic strength is also incorporated by choosing three different concentrations; 1, 5 and 10 μMol/l. 49 With increasing pH of the electrolyte, there is a significant enhancement in the charge density thereby resulting in an increment in the diffused layer potential which in turn augments the flow rate considerably. The diffused layer potential (ψ d ) depends on the Debye screening

20


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β e2 n length ( l ) given by l = which means ψ d is proportional to n , the ionic strength of the ε electrolyte. Thus, increasing the ionic strength of the electrolytes enhances the flow a bit, but less significant as compared to the effect of pH.

Figure 8. Enhancement in volumetric flow rate with increasing pH of the electrolyte solution for three different values of ionic strength. Inset shows the variation of charge density with pH.

5. CONCLUSIONS We have developed the closed-form velocity distribution for electroosmostic flow of two immiscible viscoelastic fluids in a parallel plate microchannel by incorporating the depletion layer effect thereby employing the combined consequences of rheology and interfacial electrokinetics in narrow confinements. Additionally, we have compared the theoretical predictions with our experiments, which is done using different concentrations of aqueous Polyacrylamide solution (PAAm). The consideration of depletion layer existence approximates quite well with our experimental results as well as some previous works. The present study reveals that by properly tuning parameters like the concentration of fluid, pH and ionic strength of the electrolyte, we can control the volume flow rate through the microchannel which is

21


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extremely important in various microfluidic applications like drug delivery and species separation. ACKNOWLEDGEMENTS One of the authors, Siddhartha Mukherjee, gratefully acknowledges the help of Ms. Manikuntala Mukhopadhyay (Research scholar, Dept. of Chemical Engineering, IIT Kharagpur) regarding the post-processing of experimental data. All authors acknowledge the financial support provided by the Indian Institute of Technology Kharagpur, India [Sanction Letter no.: IIT/SRIC/ATDC/CEM/2013-14/118, dated 19.12.2013].

APPENDIX A The coefficients of eq (3) are given below

= a1 ε= where sinh (κ 2 H 2 ) cosh (κ1 H1 ) + a1 sinh (κ1 H1 ) cosh (κ 2 H 2 ) r κ r , a2 a3 a1 a4 a4 sinh (κ 2 H 2 ) + a1 ζ r sinh (κ1 H1 ) , a4 = cosh (κ 2 H 2 ) − ζ r cosh (κ1 H1 ) , a5 = , a6 = , a7 = , a3 = a2 a2 a2

ε2 κ2 ζ2 ,κr = ,ζ r = , A( y) = εr = a5 cosh (κ1 y ) + a6 sinh (κ1 y ) , B ( y ) = a5 cosh (κ 2 y ) + a7 sinh (κ 2 y ) ε1 κ1 ζ1 APPENDIX B The governing equations for the velocity distribution and the solution along with the coefficients are given in the following dψ1 + c1 dy

    2 d u2 1  2 s1 λ12 dψ1 = τ xy ,2 )  τ xy ,2 where τ xy ,2= ε1 Ex + c3 ; for H1 − d1 ≤ y ≤ 0 (  1 + 2 d y η1  dy η1    (21) 2 d u3 1  2 s2 λ2 2 dψ 2  τ xy ,3 )  τ xy ,3 where τ= ε 2 Ex = + c5 ; for 0 ≤ y ≤ − H 2 + d 2 ( xy ,3 1 + 2  d y η2  dy η2   d u4 dψ 2  = µ4 ε 2 Ex + c7 for − H 2 + d 2 ≤ y ≤ − H 2  dy dy  du dy

1 µ= ε1 Ex 1

for H1 ≤ y ≤ H1 − d1

22


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1 2 3 4 where µ1, η1, η2 and µ2 are viscosities of four fluids (µ corresponds to Newtonian fluid while η 5 6 corresponds to Viscoelastic fluid) while ε1 and ε2 are the permittivities of fluids. Now, 7 integrating eq (21) we get the following velocity distribution 8 9 10 u1  11 = − A ( y ) + c1 y + c2  12 uhs  13  A y ( ) c y u 2 14 = − + 3 + c4 + f1 ( y ) + c3 f 2 ( y ) + c 32 f3 ( y ) + c33 f 4 ( y )  15 µr1 µr1 uhs  16 (22)  ε r B ( y ) c5 y 17 u3 2 3 = − + + c6 + g1 ( y ) + c5 g 2 ( y ) + c5 g3 ( y ) + c 5 g 4 ( y )  18  µ µ uhs r2 r2 19  20 ε r B ( y ) c7 y  u4 = − + + c8 21  µr3 µr3 uhs 22  23 24 Now, the coefficients of eq (22) can be given as 25 26 27 ε E ζ η1 η2 µ2 λ2 s2 − 1 x 1 , De = µr1 = λ1 κ1 uhs , , µr2 = , µr3 = , sr = , λr = , uhs = 28 µ µ µ λ µ s 1 1 1 1 1 1 29 30  3 1  2  2 2 3 31  a5  3 sinh (κ1 y ) cosh (κ1 y ) − 3 cosh (κ1 y )  + a5 a6 sinh (κ1 y )    32  f1 ( y ) = −2 F  33   1 2   3 2 2 3 34  + a6  sinh (κ1 y ) cosh (κ1 y ) + sinh (κ1 y )  + a5 a6 cosh (κ1 y )  3 3    35 36 κ y κ y  6 F  2 1 1 37 f2 ( y ) = a5  sinh ( 2 κ1 y ) − 1  + a6 2  sinh ( 2 κ1 y ) + 1  + a5 a6 cosh 2 (κ1 y )   2  2  κ1   4 38 4  39 2 6 F A( y ) s1 De 2F 40 − f3 ( y ) = y, F = , f4 ( y ) = . 2 2 41 κ1 κ1 µr2 3 42 43  3 1  2 44  2 2 3  a5  3 sinh (κ 2 y ) cosh (κ 2 y ) − 3 cosh (κ 2 y )  + a5 a7 sinh (κ 2 y )  45    46 g1 ( y ) = −2 G ε r 3    47 1 2  3 2 2 3  + a7  sinh (κ 2 y ) cosh (κ 2 y ) + sinh (κ 2 y )  + a5 a7 cosh (κ 2 y )  48 3 3    49 2 50 6G εr  2 1 κ y κ y  1 g2 ( y ) a5  sinh ( 2 κ 2 y ) − 2  + a7 2  sinh ( 2 κ 2 y ) + 2  + a5 a7 cosh 2 (κ 2 y )  51=  2  2  κ2   4 4  52 53 6G εr B ( y ) s1 sr De 2 λr 2κ r 2 2G 54 − = = g3 ( y ) = g y y G , and . ( ) 4 κ 22 κ 22 µr2 3 55 56 57 58 23 59 60 ACS Paragon Plus Environment

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NOTE: While obtaining the numerical results, the functions f1 ( y ) , f 2 ( y ) , g1 ( y ) , g 2 ( y ) may blow up for very high values of κ since they involve hyperbolic functions, highly sensitive to changes. To avoid this numerical artifacts, this terms are needed to be identically zero for accurate prediction of the theoretical results. REFERENCES (1) (2)

(3)

(4) (5)

(6)

(7) (8)

(9) (10)

(11) (12) (13) (14) (15) (16)

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Electroosmosis of two immiscible viscoelastic fluids in a parallel plate microchannel.

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Comparison between theory and experiments – inset for Newtonian fluids.


Electroosmosis of Viscoelastic Fluids: Role of Wall ... - ACS Publications

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