Surface Tension Driven Filling in a Soft Microchannel: Role of

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Thermodynamics, Transport, and Fluid Mechanics

Surface Tension Driven Filling in a Soft Microchannel: Role of Streaming Potential srinivas gorthi, Harshad S Gaikwad, Pranab Kumar Mondal, and Gautam Biswas Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 26 Mar 2019 Downloaded from http://pubs.acs.org on April 1, 2019

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Industrial & Engineering Chemistry Research

Surface tension driven filling in a soft microchannel: Role of streaming potential Srinivas R. Gorthi1, Harshad Sanjay Gaikwad1, Pranab Kumar Mondal1, Gautam Biswas1* 1Department

of Mechanical Engineering, Indian Institute of Technology Guwahati, Assam, India – 781039.

_____________________________ *Email

address: [email protected]

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Abstract The filling characteristics of a Newtonian electrolyte through a microchannel with grafted polyelectrolyte layer on its inner walls have been investigated. A reduced order model has been deployed for the analysis. The streaming potential triggered by the interfacial electrochemistry, followed by an involved electrostatic interaction between polyelectrolyte macromolecules and the electrolytic ions alters the filling characteristics in the microchannel. An oscillatory behavior of the filling dynamics is observed as the interface moves in the capillary. It has been demonstrated that the capillary filling dynamics can be altered through selective and tunable input parameters used in the investigation. A qualitative study has been performed on the variation in the filling dynamics by invoking scaling-estimates of the reduced order model. The scales of various participating forces on a spatio-temporal map have been identified to demarcate various regimes of the filling dynamics.


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1. Introduction The phenomenon of capillarity and capillary flows play a significant role in numerous natural phenomena, in technological processes like detection and separation of chemical and biological samples

1–3,

in Industrial processes like lab-on-chip reactors4, chromatography5 and

design of efficient heat and mass transfer systems. Advancements in the field of microfabrication have enabled research community to explore capillary flows for biomedical applications like diagnostic testing and DNA manipulation. Lab-on-Chip (LOC) based microfluidic devices6,7 offer the unparalleled advantage of allowing different investigations to be performed concurrently and within a miniscule area. Even with these advantages, transportation of fluids and control over their flow in LOC based microfluidic devices/systems still remains a challenge owing to the length scales involved. Even in the backdrop of these scaling induced constraints, the requirement of propelling, mixing of analytes with reagents and furthermore, the need to restrain parameters at desired values, gave the impetus for research community to explore passive flow control methods in the past and even to date. Surface structuring6,8, chemical patterning of walls9, applied electric field in combination with chemical patterning10, transverse magnetic field using Lorentz force

11

and combined Electro-magneto-hydrodynamics (EMHD)1

for flow control, have been reported in literature. Studies into the effects of distribution of charges and dipoles in interfacial regions, the concept of Electrical Double Layer (EDL) and its associated effects 12, opened-up an avenue for use of electrokinetics as a popular choice for capillary transport 13. Some inherent advantages of electrokinetic mechanisms include absence of moving components and plug like velocity profiles that aid in minimizing biological species dispersion during capillary transport under axially invariant interfacial potential. An other electrokinetic phenomena that manifests due to electric potential difference between the ends of fluid column viz. streaming potential, is gaining attraction of researchers for controlling flow in narrow fluidic confinements. The feature of flow control in such devices is attributable

primarily to the electroviscous effects inherently

associated with streaming potential14,15 and realized through a reduction in forward flow rates when compared with the absence of EDL effects14,16. Viewed from a different perspective, electroviscous effects can be gainfully employed as effective flow control agent in capillary flows, in a rather judicious manner.



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Polymer coatings and surfaces are used in a variety of applications17 such as biosensors, tissue engineering scaffolds, drug delivery systems and in the field of microfluidics, to name a few. Polyelectrolytes elegantly combine the properties of electrolytes and polymer solutions. Besides possessing polymeric macromolecules specific to the desired application, the ready availability of ionizable groups makes them particularly suitable for electrokinetic mechanisms, although such a suitability is subject to the polyelectrolyte’s physico-chemical properties, electrical nature and the electric field applied. A layer of polylectrolyte brushes18–20 lined over rigid walls of microchannels could be used to manipulate flows through careful control over their molecular structure and chemical composition21 . Quite remarkably, ability of polyelectrolyte brush layers to alter the electrostatic field and consequently, the flow of ionic species under specific conditions

22

offers an excellent

avenue for passive control over microflows. In the light of the above aspects, there exists an avenue for passive control over transport and the filling dynamics in soft microchannels, of possible significance in biological and biomedical research. A polyelectrolyte soft layer between the channel wall and bulk electrolyte in a electrokinetic setup, can be expected to affect the associated electrohydrodynamics in a non-trivial way23. Donath and Voigt24 pioneered an attempt to model electrokinetic parameters viz. streaming potential and streaming current, for quantifying the properties of soft interfaces.

It may be mentioned here that several other

researchers have also studied similar problems and reported in earlier literature as well 17,25,26. As fundamental building blocks of all processes in living beings, proteins form an essential component of almost every biological sample. Specific biological samples may warrant transport over soft layers to prevent undesirable effects like degradation or cell lysing. Alternately, interaction of an electrolytic fluid sample with a specific polyelectrolyte may be required. Biomedical and pharmacological applications involving electrokinetic transport of fluids, have witnessed renewed research interest in the last two decades27. In a soft microchannel based streaming potential set up, the polyelectrolyte macromolecule-electrolytic ion interactions led electroviscous effects may allow a passive form of flow control. Furthermore, a microfluidic setup free from external power requirements and absence of moving components is highly desirable. To the best of authors’ knowledge, electroviscous effect modulated control over filling dynamics exploiting surface tension force (without assistance of any moving parts) in soft microchannels has not been attempted by research community till date. 4 ACS Paragon Plus Environment

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In the present study, capillary filling dynamics of a Newtonian fluid through a soft microchannel are analysed through a reduced order model. The flow initiates and progresses under the influence of surface tension force. Electroviscous effects that ensue due to generation of streaming potential, tend to restrain the meniscoidal advancement. The present study analyses the complex interplay of participating capillary forces and possible control over capillary dynamics through variation of select parameters. Mathematical Formulation The prime focus of the present study is to study the role of streaming potential over capillary filling dynamics through a soft microchannel. A microchannel in which a polyelectrolyte soft layer is sandwiched between the rigid, uncharged wall and bulk electrolyte flow, is termed as a soft microchannel. Accordingly, a microchannel of length L , height 2H and width W is chosen for the present study. The width W of channel is assumed to be much larger than that of the channel height i.e. W  2 H as shown in Fig. 1, which in essence, allows us to consider the flow through microchannel to be two dimensional. The coordinate system is chosen such that the origin is fixed on the center line at the entry to the channel as shown in Fig. 1.

Figure 1 (Color online) Schematic diagram of flow through soft microchannel in the present study. The channel is of Length L , height 2H and width W . We assume W  2 H and hence the flow can be considered to be two dimensional. The origin is to the left end of the channel. Electrolyte enters from left and progresses towards exit in the right. The small red color curves extending from channel wall into the channel upto distance d , are symbolic representation of polyelectrolyte brushes with polyelectrolyte macromolecules (cations) (shown in violet color) tethered onto them. The accumulation of ionic charges near the meniscus which causes the streaming potential, can be seen. 5 ACS Paragon Plus Environment


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Numerous wiggly polymer structures (shown in red color), a few of which are symbolically depicted in Fig. 1, are grafted to the surface of the rigid walls, constituting a brush18. Each brush constitutes a polymeric chain with numerous Polyelectrolyte (PE) macromolecules (shown in violet color and depicted larger in size compared to the bulk electrolyte ions for ease in identification) along the backbone of their structure. This type of brush layer in the channel is referred to as the polyelectrolyte layer28 (PEL) in the present work. In this analysis, we consider the mean length of PE brushes itself as the thickness ( d ) of the PEL (cf. Fig. 1). Accordingly, two distinct regions of the flow domain are identified in the present analysis: (i) the region described by [ H  y   H  d  ], which constitutes the PEL and (ii) flow domain comprising region excluding the PEL, known as Electrolyte layer (EL) and described by [  H  d   y  0 ]. Initially, the electrolyte enters the microchannel from the left end and advances through the microchannel solely under the influence of surface tension force. Hence, meniscoid advancement from left to right constitutes forward direction of flow in the present study. Flow of electrolyte through the PEL sets off electrostatic interactions between the PE macromolecules and the electrolytic ions, causing charges to advect downstream and accumulate at the meniscus as shown in the Fig. 1, creating a high charge region behind the meniscus. Charge imbalance in the flow field is the cause of a potential difference between the meniscus and the inlet. The generated EMF constitutes the conduction current, widely popular as the streaming potential. Electrostatics The rigid walls of the microchannel feature a polyelectrolyte soft layer grafted onto the inner face. Following those reported in literature29–32, we assume that the rigid walls of the microchannel are uncharged. Hence, the PEL itself performs the role of charged wall while at the same time, permits the electrolyte to flow through it. The soft PEL thus forms an intermediate, permeable media between the outer rigid wall and the bulk electrolyte flow, allowing for electrostatic interactions driven by the electrochemistry between the polyelectrolyte material and the bulk electrolyte. The competing forces in the flow domain are the surface tension force, the viscous resistance and the electroviscous force. The charged macromolecules bound to the PEL brushes 6 ACS Paragon Plus Environment

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interact with the ionic charges of the electrolyte solution, leading to alteration of the electrostatic field. Note that the governing equation describing the generation of electric field is given by Gauss’ Law5. The governing equations for EL and the PEL, respectively are as follows: For EL,

  2 e   r 0  2     ee  y 

(1a)

  2 p   r 0  2     ep  y 

(1b)

For PEL:

In Eqs. 1(a)-(b),  ee and  ep represent the space charge densities in EL and PEL respectively. The induced potentials in EL and PEL are given by  e and  p ,  r is relative permittivity,  0 permittivity of free space. The space charge densities in terms of number density of ions and valencies can be defined as:

 ee   ze  n  n 

(2a)

ep   ze  n  n   ZeN

(2b)

Where z and Z indicate valencies of the electrolyte ions and polyelectrolyte ions respectively, e

is the protonic charge, N is the number of polyelectrolyte ions and, n and n are the positive and negative ions in the electrolyte respectively for z : z symmetric electrolytes. The ionic concentration, following Boltzmann charge distribution theory, is given by:

n  n0 exp  mze e k BT  where n0 is the ionic concentration at neutral state, k B is the Boltzmann constant and T is the absolute temperature. Note that while formulating Eqs. 2(a) and 2(b), we assume that the electrostatic potential at the walls is less than 25 mV, which allows us to apply the DebyeHuckel approximation

sinh  ze

k BT   ze k BT



33,34.

Using this approximation for the

governing Poisson equations separately for EL and PEL, Eqs. 1(a) and 1(b) can now be represented as:  2 e   2 e 2 y

for  H  d   y  0

(3a)



 2 p

  2 p   p2

y 2

In above Eqs. (3a)   2 z 2e 2n0  r 0k BT

 p  zZe 2 N  r 0k BT

for H  y   H  d 

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(3b)

is the Debye-Huckel parameter and the term

in (3b) represents the equivalent Debye-Huckel parameter of PEL. It

may be noted that the above Debye-Huckel parameters are reciprocals of Debye-lengths for electrolyte and the polyelectrolyte layers respectively. Non-dimensionalization of Eqs. (3a) and (3b) using the reference parameters lref  H and  ref   k BT ze  yields:  2 e   2 e 2 y

 2 p y

2

  2 p   p2

for 1  d   y  0 for 1  y  1  d 

(4a)

(4b)

Further, to solve these Poisson-Boltzmann equations in both EL and PEL domains, we invoke the

following

 p y

p

0 e

 p y  e y

 y 1 d

0 y

    : Dirichlet condition at PEL EL interface    : Robin condition at PEL EL interface    : Symmetry condition along channel centerline  

conditions.

: Gauss Boundary condition at walls

y 1

y 1 d

boundary

y 1 d

 e y

y 1 d

(5)

The first condition is derived from Gauss’ law and does not permit flux through walls since the walls of the channels are uncharged. The second condition requires continuity in potential at the PEL/EL interface. The third condition mandates continuity of electric field at the PEL/EL interface. The last boundary condition is for electroneutrality along centerline of the channel. Using these above mentioned boundary conditions, the closed form expressions for the potential distribution in both EL and PEL can be obtained respectively as:

  p2  sinh  d  cosh  y  e   2     sinh  

for 1  d   y  0

(6a)


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   p2   sinh    d   p   2  1  cosh    y   sinh       

for 1  y  1  d 

(6b)

Hydrodynamics of flow through Soft capillary In this study, we consider the flow of a Newtonian fluid in the soft microchannel solely under the influence of capillary force. As the fluid front or meniscus advances through the microchannel, electrolytic ions advect downstream and accumulate near the meniscus as shown in Fig. 1. The forward momentum of flow through the soft microchannel, primarily attributable to the supportive capillary force encounters viscous resistance to the flow everywhere in the domain and an additional drag on the electrolyte flowing through PEL. An additional resistance is encountered in the form of electroviscous effects which stem from the generation of conduction current. The strength of electroviscous forces is proportional to the potential difference between the meniscus and the inlet. Given the forcing factors involved in the capillary flow under the present formulation, assuming steady, fully developed and unidirectional flow, we frame the equations governing the underlying electro-hydrodynamics as follows: For EL i.e. for  H  d   y  0 : 0

dp d 2ue  dx dy 2

(7a)

For PEL i.e. for H  y   H  d  :

d 2u p dp 0     2  c u p dx dy

(7b)

In Eqs. 7(a) and 7(b), ue and u p are the flow velocities in the EL and PEL respectively. The fluid dynamic viscosity of the fluid is represented by  . The PEL imposes an additional drag penalty on electrolyte flowing through it. This is modelled by the term c u p representing Darcy drag35 on the flow of electrolyte through the PEL where c stands for the drag parameter. The above governing equations must be solved subject to the following boundary conditions (Eqs. 7(a)-(b)):



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   up  ue y 1d : Dirichlet condition at PEL EL interface y 1 d   u p u   e : Robin condition at PEL EL interface y y 1d y y 1d   ue  0 : Symmetry condition along channel centerline  y y 0  up

y 1

0

: No slip condition at walls

(8)

In above boundary conditions, the first condition represents no slip condition at the walls of the channel. The second condition enforces velocity continuity at the PEL/EL interface. The third condition ensures continuity in the velocity gradient across the PEL/EL interface. The last condition characterizes the symmetrical condition of flow along the center of the channel. Using above boundary conditions, velocity profiles in the EL and PEL can be obtained as: For EL i.e. for  H  d   y  0 :

A   d  H  c sinh  Ad   1 dp y 2 dp 1  2  ue   2 A  A  d  H  c  2   dx 2 dx 2 Ac  cosh  Ad  

(9a)

For PEL i.e. for H  y   H  d  : up  

dp 1 dp 1  A cosh  A  d  H  y     d  H  c sinh  A  H  y       dx c dx Ac  cosh  Ad  

(9b)

where A is defined as A  c  and has units of m 1 . Dimensionless expressions of above velocity profiles are obtained using the following reference parameters:

uref

dp H 2  dx 2

 xref   tref

  , 

tref 

H   3

S

and

A  AH .

The

resulting

dimensionless velocity profiles ue and u p are: For EL i.e. for 1  d   y  0 :

 2 1   d  1 A sinh  Ad   2 ue  uref  2   d  1  y 2  2  A2 cosh  Ad   A 

(10a)

For PEL i.e. for 1  y  1  d  : 10 ACS Paragon Plus Environment

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up 





2uref  cosh A  d  1  y    d  1 A sinh  A 1  y    1    A2  cosh  Ad   

(10b)

By integrating above equations in their respective domains i.e. EL and PEL respectively, we obtain the dimensionless expression of average velocity  u  .

uavg

3  1  2  d  1   d  1 tanh  Ad   2   2uref  2  2  d   A2  d  1  1  3 A3  A   cosh  Ad    





(11)

Capillary Filling Dynamics The capillary filling dynamics are governed by force balance in the soft microchannel. Below, we write the governing equation of the capillary filling dynamics, d  2 xuavg H   FST  FV  FElec dt

(12)

In above Eq. (12),  is density of the liquid, FST is surface tension force, FV is viscous force and FElec is the electrical body force, modelled to account for a pseudo force generated due to the streaming potential (U S ) . Note that it opposes the flow driven by the capillary force or surface tension force FST , given as

FST  2 S cos  

(13)

Now, we find the expression of viscous force below. The shear stress at the walls is given as,

 xy

y  H

  xy

yH



2 uref H

(14a)

From which, the viscous force at the walls can be calculated as, FV  2 xy x 

4 x uref H

(14b)

The electrical body force FElec , stemming from generation of streaming potential (U S ) in the channel can be calculated as, US Q dy H x

FElec  x 

H

(15)

Where Q represents the charge density, being ee for EL and ep for PEL respectively. As the streaming potential is dependent on linear advancement of the meniscus along the microchannel, 11 ACS Paragon Plus Environment


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the streaming field in above equation is given by U S x  . During generation of streaming potential, counterions from the electrolyte accumulate at the meniscus. The charge density of the counterions thus accumulated at the meniscus is represented by qa . From Gauss’ law we have, US qa qa    0 r Ac  0 r  2 H  x

Here, using Eq. (1) and the above relation, Eq. (15) can be rewritten as, FElec 

H 2 xqa  H  d ee dy   ep dy    d 0 H   0 r  2 H  

(16)

We now recall the equation governing filling in the capillary following a reduced order analysis. Accordingly, we replace the average velocity uavg in Eq. (12) with a simple, first order differential representing the velocity of the advancing liquid-air interface, dx dt . Using Eqs. (13), (14) and (16), reconfigured Eq. (12) takes the form: H d dx  4 x 2 xqa  H d  ee dy    ep dy  uref   2  Hx   2 S cos     H d   0 r  2 H   0 dt  dt  H

(17)

Below, we non-dimensionalise Eq. (17) using the following reference parameters: lref  H ,

tref 

H   3

S

,

  S

H  ,

 p   pH ,

 e   k BT  0 r   H 2e  ref

and

qaref   S eH 2   k BT  . Using the above non dimensionalization scheme, Eq. (17) can be written

as: d dt

 1  dx  2  x dt   cos   2 xuref   2 xqa d  p S

(18)

Charge Balance Downstream advection of electrolyte’s ionic charges and their accumulation at the meniscus, alters the local charge density. The charge density at the meniscus is calculated from a charge balance analogy. Under equilibrium conditions, the streaming current and the conduction currents balance each other. In tune with this aspect, we assume that the total current in the channel ( I C  I S ) counterbalances the temporal changes in the charge density at the meniscus until an equilibrium is achieved. IC  I S 

dqa dt

(19) 12

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The conduction current in the channel (streaming potential) can be expressed using the Ohm’s law as: IC 

US RC

Here, RC , the resistance to conduction current, is given as x   AC  where  is the mean conductivity of the electrolyte and AC   2 H  is cross sectional area. Therefore, the expression for conduction current becomes:

IC 

U S  AC x

(20)

We now proceed to find the expression for mean conductivity of the electrolyte from the conductivity of the electrolyte given by the expression,

   i   zi evi ni i

(21)

i

where  is local conductivity of electrolyte and vi is the electrical mobility of the ionic species. It should be noted that polyelectrolyte ions are strongly tethered to the PE brushes and are immobile and as such, we do not consider them while forming Eq. (21). Assuming Boltzmann distribution of ions, Eq. (21) can be rewritten as,

  n0  zi evi exp   zi 

(22)

i

The average or mean conductivity of the electrolyte across the channel section can then be calculated as,



1 2H



H

H

 dy 

H 1 n0e   zi vi exp   zi dy  H 2H i

(23)

Using Debye-Huckel approximation, Eq. (23) can be written as,



H 1 n0e   zi vi 1  zi dy H 2H i

(24)

For monovalent electrolytes,



 e v  v



k BT v   v 

  n0e  v  v   1 



H

H



 dy  

(25)

In order to simplify the above expression, we use molar conductivity for monovalent electrolyte which is given as, 13 ACS Paragon Plus Environment


Page 14 of 46

 m   evi N A i

Hence, the final expression of the mean conductivity of the electrolyte becomes,

  n0

m   e v  v 1  N A  k BT v   v 



H

H



 dy  

(26)

For many binary electrolyte solutions, the mobility of the positive and negative ions is similar. Therefore, Eq. (26) now reduces to,

  n0

m NA

(27)

Using the modified expression of the mean conductivity of electrolyte given by Eq. (27), the expression for conduction current given by Eq. (20) becomes, IC 

U S n0  m AC xN A

(28)

An expression for streaming current can now be obtained as, H

I S   u Q dy H

(29)

Where the charge density of ions Q is replaced by  ee in the bulk electrolyte layer (EL) and

 ep in the polyelectrolyte layer (PEL). Substituting Eq. (28) and Eq. (29) in Eq. (19), the final expression for the current neutrality condition (equilibrium condition) can be obtained as,



H

H

u e dy 

U S n0  m AC dqa  xN A dt

(30)

Using the reference quantities and dimensionless parameters mentioned in the preceding section, Eq. (30) can be obtained in dimensionless form as, 1 dqa  B1   u  e dy   qa B2  1  dt

(31)

Where the dimensionless parameters B1 and B2 are as follows:

B1 

 0 r k B2T 2 H and B2  2  Se H  0 r 

It may be noted that for ease in presentation, we drop the over bar signs from the dimensionless parameters in further discussions.


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Results and discussion In this section, we show the variation in capillary filling characteristics with different geometrical and physical parameters such as PEL thickness ( d ) , Debye-Huckel parameter of the EDL ( ) , height of the channel ( h ) , Darcy drag in soft PEL. Before proceeding to elaborate the results on hand, we discuss here the selection of parameters that are intended to affect a variation in capillary filling characteristics. Selection of parameters for the study Debye-Huckel parameter In electrokinetically driven nanofluidic flows, for symmetric electrolyte such as KCl, the Debye length    1   of the EDL varies from 9.6nm to 304nm 36 in response to a change in the concentration from 10-3 to 10-6 M. Hence, we adopt Debye lengths of the order of 100nm for investigations in the present study. Thickness of PEL We select the value of dimensionless PEL thickness  d  be within the range 0.1 to 0.3 reported in literature earlier37. It is important to mention here that while selecting the PEL thickness, the inherent shrinking and swelling effects of the PEL due to the physicochemical interaction between the brushes within the PEL are neglected. Physical parameters 8 10 In the present study we assume the value of drag parameter to vary from 10 to 10

following literature, to account for the drag in the PEL. This assumed range of drag parameter is

consistent with the value reported in previous studies35, which effectively allows us to focus on the underlying hydrodynamics and electro kinetics through PEL grafted microchannels. We consider that the walls are completely wetted by the electrolyte. Accordingly, we here assume the contact angle of the liquid with the walls of the channel to be zero. The basic physical properties of the flowing electrolyte are assumed to be similar to that of water owing to the fact that most of the biological samples are aqueous based and since water is a widely used solvent for electrolytic solutions. The surface tension and dynamic viscosity of the liquid is assumed to be  S  0.072 N m and   1Pa .s . For electrolytic solution, the electrical permittivity of 15 ACS Paragon Plus Environment


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free space is 7.08  1010 F m while relative permittivity is 78.7. The molar conductivity of the electrolyte

m 

is

selected as 1.0  105 S m . Relevant discussions about the physical

significance of above parameters can be found in literature23,28,38–40. Model benchmarking Here, we attempt to benchmark capillary penetration rate obtained with the mathematical model employed in the present study with that reported in Bandopadhyay et. al.41. Although the cited work relates to a viscoelastic (PTT fluid) model, the authors chose to benchmark their model against the findings, on the premise that viscoelastic fluids characterized by very low relaxation times (of the order of 10-4 sec), tend to exhibit Newtonian fluid rheology. Accordingly, we select the a case of a viscoelastic fluid with relaxation time of 10-4 sec. We consider a horizontal channel of half height  H   100  m , and thickness of the PEL as 1nm, which can be considered to be very thin, as compared to the height of the channel. We strive to explain through Fig. 2, the capillary filling response obtained in the present study and plot the capillary penetration dynamics against the results of the benchmarked model41, for similar channel dimensions and matching relaxation time of the fluid.

Figure 2 (color online): Plot illustrating capillary penetration distance as a function of time. Temporal variation of the capillary penetration distance obtained from the present formulation exhibits a good match with that reported by Bandopadhyay et al.41, for PTT fluid in open horizontal channel with half-height of 100𝜇m and relaxation time 10-4 sec (behavior tends to that of a Newtonian fluid) with results of our study. 16 ACS Paragon Plus Environment

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From Fig. 2, it may be observed that the capillary penetration distance as a function of time under the present formulation and values of parameters mentioned, exhibits a good match with that reported in the literature41. A fairly accurate match between the present and reported results indeed vouches for the accuracy of the present model. The streaming potential: parametric dependencies The streaming potential alters the electrostatic field in the soft microchannel in a rather non-trivial manner.

There exists a trove of literature dealing with streaming potential in

microflows 12,34,36,42 and the profound influence of EDL on the flow behavior of fluids. Reduced flow rates due to electroviscous effects, is a characteristic of electrokinetic set ups with streaming potential feature. The quantitative treatment of electroviscous effects is theoretically dealt as an effect of apparent increase in the fluid viscosity while it is quantified as the magnitude of conduction current (also known as streaming potential). While exploring yet another avenue to manipulate, or to control the filling dynamics in soft microchannel under the present formulation, we remind that the role of streaming potential is the focus of the present study. Accordingly we identify pertinent parameters on which the magnitude of streaming potential under the present formulation, depends. We analyse the identified parameters sequentially and in a systematic manner. The present study analyses the (i) effect of varying the thickness ( d ) of the polyelectrolyte layer, (ii) effect of variation in the drag parameter ( c ) of the PEL, (iii) effect of varying the height of the microchannel ( h ) and (iv) effect of variation in the thickness of the EDL, parameterized by reciprocal of Debye-Huckel parameter ( 1 ) . It is to be mentioned here that in all subsequent discussions about electroviscous effects, we assume that the actual viscosity of the fluid is unaffected by the presence of EDL i.e. the microscopic structure of the fluid is assumed to be unaltered by local fields. Under the above assumption, viscosity of the fluid in EDL is the same as the bulk viscosity. Surface tension force being the principal forcing factor behind the fluid flow, is primarily opposed by the viscous resistance and the streaming potential under the present formulation. Superimposed on this force scenario, is the dynamics resulting from the electrostatic and ionic interactions between the polyelectrolyte soft layer, the bulk electrolyte and presence of the EDL. 17 ACS Paragon Plus Environment


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The resulting complex interaction is likely to bring some interesting outcomes into light, especially in the wake of the solutions to Eq. (19). In a systematic and sequential manner we now attempt to study the influence of parameters identified, on the streaming potential and the resulting capillary filling dynamics. Variation in the thickness  d  of the PEL Under the present formulation, polyelectrolyte layers comprising of numerous brushes with specific properties and modelled for a designed level of electrostatic interaction with the ions of bulk electrolyte, forms the soft layer. The soft layer forms an intermediate, permeable media between the outer rigid wall and the bulk electrolyte flow. As such, it performs the role of a charged wall while at the same time, acts as a permeable media for the electrolyte to flow through it, allowing for electrostatic interactions. Under the present formulation, electrolyte in the region away from the longitudinal axis of the channel, permeates into and out of the PEL. However, owing to its physical constitution, the PEL offers an additional resistance to the flow through it, over and above the viscous resistance that is encountered by the fluid everywhere else in the channel. This additional resistance in the PEL is taken into account in the present study through the drag parameter c the effect of which, is discussed separately. The additional drag is varies as thickness of the PEL. There are various parameters influence the nature and structure of polyelectrolyte soft layers18,23. To a varying degree, these parameters determine the extent and vigor of electrostatic interactions between the charged macromolecules on PE brushes and the bulk electrolytic ions. As such, there are some tunable parameters viz. grafting methodology, brush grafting density, optimum brush length, the physico-chemical properties of the brushes and the working fluid18, the electrical properties of brushes to name a few, which can be tailored suitably for specific applications. Presence of a polyelectrolyte soft layer between the rigid wall and the bulk electrolyte alters the electrostatic field in the soft microchannel. Precisely, the nontrivial alteration of this electrostatic field influences the underlying hydrodynamics through the soft microchannel. The overall effect then, is the cumulative effect of extensive physico-chemical and electrostatic interactions between charges in the PEL and the bulk electrolyte ions. Hence, choice of material and the electrochemical nature of the soft PEL can be said to have a deterministic influence on 18 ACS Paragon Plus Environment

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both the extent and vigor of electrostatic interactions between the PEL and electrolytic ions. In addition, physical attributes like thickness of the PEL and drag parameter are also likely to influence the overall hydrodynamics of the problem under consideration. As tunable parameters that can employed for passive control over capillary filling dynamics, study of influence of physical attributes of the PEL on electrohydrodynamics of a soft microchannel may be useful for applications. Accordingly, we first consider a change in PEL thickness and study the effects on capillary filling dynamics through the soft microchannel. Capillary filling is defined as the distance penetrated by the fluid along the centerline of the soft microchannel. We choose to study the variation in capillary filling, for three different values of PEL thickness d =500nm, 1000nm and 1500nm. A discussion on variation in magnitude and extent of electrostatic potential with thickness of the PEL would help in deeper understanding of the contribution of various factors behind the interesting oscillatory tendency observed in Fig. 3(a)-(b) and the overall capillary filling dynamics. These fluctuations precede the instant of attainment of equilibrium in the channel and thereafter, a steady flow rate prevails through the microchannel. Before proceeding further, it is worthwhile to examine the inset of Fig. 3(a) wherein the non-dimensional electrostatic potential

 

that develops in the soft channel is plotted against normalized inward distance in a direction

perpendicular to the rigid wall, for various PEL thickness values considered (as mentioned earlier, the overbar symbols over parameters have been dropped). It is to be mentioned that under the present formulation, the mean length of these brushes is adopted as the thickness  d  of PEL.



Page 20 of 46

Figure 3 (color online): Fig. 3(a): Temporal variation in the electrical body force  FElec  for different values of PEL thickness d (= 500nm, 1000nm and 1500nm) chosen. Inset Fig. 3(a): Variation in the electrostatic potential ( ) with a change in the value of PEL thickness d from 500nm through 1500nm. The legend for inset is same as for main figure and note that both  FElec  and t are dimensionless. Fig. 3(b): Capillary filling length (dimensionless) plotted against dimensionless time t for the PEL thickness 1 cases considered. The other parameters chosen for the study: H  30  m ,  p  500 nm , 

1

 400 nm ,

5 c  109 Ns m 4 and   10 S m .

Referring to the inset of Fig. 3(a), the normalized distance y  1 corresponds to location of the rigid wall and, decreasing values of ( y ) indicate a greater normal distance inwards. In essence, it is the inward normal distance upto where the effect of electrostatic potential can be felt. The potential   on the ordinate axis represents the potential at the EL  e  or the potential at the PEL  p  , depending on location of y respectively. Away from the wall, in a normal direction towards the central region of channel, electroneutrality condition12,34 is assumed to exist. Referring to inset Fig. 3(a), it may be observed that, as the thickness of the PEL increases from 500 nm through 1500 nm, the magnitude of electrostatic potential is also progressively higher. From the same inset figure, it may be observed that the inward normal distance y varies with PEL thickness. This means, that the extent of electrostatic effect felt due to the presence of PEL increases progressively with its thickness, being maximum when d  1500 nm . In other words, the region where electroneutrality condition exists, is farther away from the rigid wall when compared to y in the case of d  500 nm . The effects of both the above observations 20 ACS Paragon Plus Environment

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realized from the plot in inset of Fig. 3(a) will be explained in connection with the overall electrohydrodynamics in succeeding paragraphs of this section. We now analyze the overall alteration in capillary filling dynamics in response to a variation in the PEL thickness. The temporal variation in electrical body force FElec in the fluid is mathematically expressed in Eq. (16). As can be examined from the equation, the magnitude of

FElec is proportional to linear dimension x , the distance upto which the meniscus physically advances through the microchannel, and the charge density qa of the ions accumulated at the meniscus. The electrical body force FElec represents the body forces purely due to electrokinetic attributes i.e. conduction current. However, it is to be mentioned that it is a component of the total body force. The surface tension force FST and the viscous resistance FV are the two other components of the total body force. As per the present formulation, the surface tension force is the principal forcing factor that causes flow through the channel i.e. advancement of meniscus, while viscous resistance and electrical body force oppose the flow. Hence, it is mentioned that the overall capillary filling dynamics as can be observed from Fig. 3(b) is dependent on the resultant total body force given by the RHS of Eq. (12) and not FElec alone. However, it is mentioned here that, we have presented plot of FElec in Fig. 3(a) since the focus of the present work is to study the role of streaming potential. Fluid enters the channel initially (at t  0 ), solely under the influence of surface tension forces overcoming the viscous resistance. In this process, fluid in the region away from the longitudinal axis of the channel is free to permeate into and out of the PEL. Electrostatic interaction ensues between the macromolecules on brushes and the electrolyte’s ionic charges flowing through the PEL. As the meniscus continues its advancement downstream, the ionic charges of the electrolyte advect through the PEL and the EL and reach the meniscus. Downstream accumulation of the ionic charges in close proximity behind the meniscus (cf. Fig. 1) under the influence of electrostatic interactions, constitutes the streaming current I S . As can be observed from Eq.(15), the electrical body force FElec increases due to enhanced downstream advection of ions from the incremental extent of PEL covered by the meniscus. The potential difference between the entry of the microchannel and the instantaneous location of meniscus is widely known as streaming potential. The current due to this potential, known as conduction 21 ACS Paragon Plus Environment


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current I C , is in competition with streaming current I S as long as the rate of charge buildup

 dq

a

dt  (cf. Eq. (19)) at the meniscus is non-zero. The system settles when the streaming

current and conduction current become in magnitude when charge build up rate at the meniscus is zero. To assist in our analysis, we resort to Fig. 3(a) that depicts an interesting phenomenon remaining unrevealed in the open literature till date. From the figure, oscillations can be observed in initial stages of the plot of FElec for all three values of PEL thickness considered. The initial-oscillatory regime is marked by an intense competition between participating forces, resulting in a fluctuating magnitude of electrical body force FElec as can be seen from Fig. 3(a). Corresponding undulations in capillary filling are observable from respective plots in Fig. 3(b) as well. After oscillations, the filling follows establishment of an equilibrium between the participating forces wherein the rate of filling is relatively steady, under the present formulation. It needs to be mentioned here that in tune with the focus of the present work, we do not intend determining the spatio-temporal instant of equilibrium in this study. The undulations in capillary filling during the oscillatory regime are in fact a fallout of the intense competition between various forces viz. the driving surface tension force FST , resistive viscous resistance FV and resistive electroviscous force FElec . The oscillation, albeit a damped one, results in an advanceretreat or lead-lag motion of the meniscus through the microchannel. These oscillations exist for a brief period before reaching an equilibrium, determined by the solution of the governing equation (Eq. (12)). Post equilibrium, a steady rate of flow continues for the remaining length of the microchannel at a rate determined by equilibrium state variables. The intricate aspects of variation in forces, establishment of equilibrium and capillary filling are elaborated in the succeeding paragraphs. Fluid initially enters the channel solely under the influence of surface tension force. As the meniscus continues its advancement in the microchannel overcoming the viscous resistance, ions from the electrolyte in the PEL get advected downstream under the influence of both, the electrostatic interaction with macromolecules and the surface tension force. Ions from the bulk i.e. the EL although much lesser in number, also advect downstream under the influence of 22 ACS Paragon Plus Environment

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surface tension force. These ionic charges accumulate behind the meniscus, building up charge density in the proximity. The streaming current I S of ions contributes to this accelerated buildup of electrical body force FElec , which can be seen in common for all the three cases of variation in PEL thickness considered, from Fig. 3(a). However, the rate of increment of

FElec and its

maximum value varies in all the three cases. It can also be seen that the slope and magnitude are highest for d  1500nm and corresponding values are lowest for d  500nm . This is due to the fact that for the same linear advancement of meniscus in the channel, a larger proportion of electrolyte can flow through a thicker PEL and consequently contribute a greater number of ions for downstream accumulation, than a thin PEL. As a result, the rate of increase in buildup of

FElec is highest in case of d  1500nm as against rate of increase in the case of d  500nm which can be observed from Fig. 3(a). The maximum magnitude of FElec when d  1500nm and successively lower maximum magnitudes for d  1000nm and for d  500nm is also attributable to the same reason. The presence of EDL alters the electrostatics of the region in a non-trivial way. A thicker PEL affects the magnitude of potential and the extent of EDL, as explained in the initial paragraphs of this section. As can be observed from inset of Fig. 3(a), the potential is proportional to the PEL thickness, being highest for d  1500nm and lowest when d  500nm . Also, the inward normal distance upto which the effect of potential can be felt, follows the same trend as the magnitude of potential. The overall effect of a greater magnitude of the electrostatic potential as well as the extent of EDL is that, a greater number of ionic charges can electromigrate and contribute for streaming potential. Thus, a sharp increase in FElec corresponding to thicker PEL as can be observed from main plots in Fig. 3(a) as against thin PEL. Referring to Eq. (12), buildup of FElec implies increasing streaming potential. An increasing streaming potential offers resistance to the advancement of the meniscus as it causes reverse electromigration. The meniscus begins to halt, owing to the severe opposition it encounters. The viscous resistance, a minor component of total body force during advancement of meniscus, appears in the scenario as if it is a major component, especially in the wake of reduced total body force. Consequently, the total body force is the lowest when FElec is greatest 23 ACS Paragon Plus Environment


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(cf. RHS of Eq. (12)). In the background of a reducing total body force scenario, the meniscus experiences retardation. We now address the events that follow in the case of PEL thicknesses corresponding to

d  1500nm and d = 1000nm after FElec reaches the maximum value in the initial buildup regime, and then correlate it with the subsequent capillary filling dynamics. Unable to advance any further in the background of a strong electroviscous force due to steep increase in FElec , the meniscus experiences retardation. The positive slope corresponding to velocity plots of

d  1500nm and d = 1000nm tend to turn negative under severe opposition to advancement and meniscus starts receding, after negotiating through a local inflection point as observed from Fig. 3(b). As FElec is dependent on linear advancement x and the charge density behind the meniscus, a decreasing trend is observed in FElec from Fig. 3(a). The decrease in electroviscous force continues until it is comparable to surface tension force. From that instant, the meniscus resumes its forward motion and the events leading to buildup of FElec repeat themselves. At this point, the rate of charge build up at the meniscus is zero and streaming current is equal in magnitude to the streaming potential

I

S

  I C  . Beyond this equilibrium instant, the filling (advancement of

meniscus) in the microchannel continues at a steady rate determined by the equilibrium as observable from Fig. 3(b). This oscillatory or lead-lag phenomenon is in actuality, an effort by participating forces (identified in Eq. (12)), to achieve equilibrium and reach steady state capillary filling. The interesting fact being that the lead-lag motion of meniscus exhibits a successively reducing amplitude (decrease in magnitude of FElec ) until the competing forces attain a force equilibrium. This trend indicates the involvement of forces within, which enforce a settlement. The forces involved, and how damping takes place is explained as follows: From entry and throughout the meniscoid advancement, a constant magnitude surface tension force FST encounters viscous resistance FV and electroviscous force FElec everywhere in the flow domain. There is an additional Darcy drag on the portion of electrolyte flowing through the PEL proportional to the linear extent covered by the advancing meniscus. On the other hand, during receding motion of the meniscus, the Darcy drag penalty is not incurred. The electrical body force FElec itself weakens owing to not accrual of contribution from ionic charges released from the extent retreated. Under such circumstances, the linear extent upto which FElec can push 24 ACS Paragon Plus Environment

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the meniscus backwards, by countering the surface tension force and viscous resistance, is severely restricted. This effect manifests itself as a decadence in amplitude (decrease in magnitude of FElec ) in successive cycles, in the physical advance-retreat motion through the channel. Once the forces attain an equilibrium, the flow is positively forward and steady. It is to be mentioned here that, there are two significant aspects of advancing and receding motion of the meniscus. First, the viscous resistance penalty borne by the driving surface tension force FST during advancement, is the burden of electrical body force FElec during receding motion of meniscus. Second, there is no flow through the PEL during meniscoid receding motion and hence Darcy drag is inapplicable. It may be observed from Fig. 3(a) that, the rate of increase in FElec corresponding to

d  500nm is slower when compared to rates of increase when d  1500nm or d = 1000nm . Also, the streaming potential FElec in Fig. 3(a) and the corresponding plot for capillary filling in Fig. 3(b) displays tendency to attain steady state earlier when d  500nm when compared to

d  1500nm or d = 1000nm . This can be attributed to the fact that the linear extent which the meniscus must cover along the length of soft microchannel, for FElec to increase to a given value, will be greater for a thin PEL, as compared to the corresponding extent when the PEL is thick. Another interesting observation that can be made from Fig. 3(b) is that, the driving surface tension force FST is successfully able to resist the opposing forces upto a longer distance into the microchannel

when

d  500nm

as

compared

to

corresponding

distance

when

d  1500nm and d = 1000nm . The cumulative effect being that at any given instant, the capillary filling is greater when PEL thickness is low. The rate of filling beyond the equilibrium state is also marginally higher when PEL thickness is low, as in the case of d  500nm . Hence, it can be inferred that a careful control of PEL thickness can be gainfully employed to alter the capillary filling dynamics through a soft capillary. Variation in the drag parameter of PEL Under the present formulation, the soft PEL is permeable in nature. The portion of bulk electrolyte away from the longitudinal axis of microchannel, flows through the soft PEL layer. The fluid flowing through the soft PEL encounters an additional drag force, known as Darcy drag, on the movement of electrolytic ions, over and above the viscous drag encountered 25 ACS Paragon Plus Environment


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everywhere in the flow domain. As previously mentioned, the present formulation accounts for the drag in PEL through the term c u p appearing in the equation of motion Eq. 7(b) where c is the drag parameter.

Figure 4 (color online): Plots showing the temporal variation in streaming potential and capillary filling dynamics with variation in the drag parameter of the PEL c (= 108, 109 and 1010). Fig. 4(a): Variation in the streaming potential U S  with a change in selected values of drag parameter. Fig. 4(b): Temporal

variation in filling length for the drag parameter values chosen. The other parameters chosen while 1 plotting are: H  30  m ,  p  500 nm ,

 1  400 nm , d  1000 nm and   105 S m .

We now attempt to correlate the variation in magnitude of drag parameter with the corresponding effects on the overall capillary filling dynamics in the soft microchannel. The dimensionless additional drag parameter  c  adopted for the soft brush matrix of PEL is in conformance to values reported in literature 35, and is generally of the order of 1010 . In an effort to clearly bring out the effects of variation in the drag parameter on the capillary dynamics of the system under the present formulation, we select three different values of dimensionless drag parameter viz. 108, 109 and 1010. Figure 4(b) depicts the temporal variation of streaming potential U S in the channel for three chosen values of the drag parameter. It may be observed that in a manner similar to variation of FElec in the preceding section, the streaming potential displays an initial oscillatory tendency here, in the case of variation in drag parameter also. The effect of an increase in drag 26 ACS Paragon Plus Environment

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parameter on hydrodynamics of ionic charges, is rather straightforward. Increase in drag parameter characterizes increase in drag force through the PEL, which affects the advection of charges through it. The higher the drag parameter, the higher drag experienced by ionic charges in advecting towards the meniscus. In other words, for a given flow and electrostatic field, the ionic charges in PEL characterized by a low drag parameter advect with relative ease and contribute to a healthier accumulation behind the meniscus as compared to a PEL with higher drag parameter. Consequently there is a sharp increase in U S , as can be seen when the drag parameter is lowest i.e. c  108 . The lower resistance to advancement of fluid front manifests itself as improved filling characteristics through the soft microchannel, as can be confirmed from Fig. 4(b). The additional penalty incurred by ionic charges at higher value of drag parameter, i.e. when c  1010 clearly shows both, a lower maximum magnitude and also a delayed increase in US .

It may be noted that due to the fact that ionic charges in a PEL characterized by c  1010 , the force balance or equilibrium is established in the early stages of the flow and the initial

oscillations on the variation of streaming potential are not prominent. On the other hand, the relative ease in ionic charge migration through a PEL with lower drag parameter, sets off the typical competition among participating forces, leading to oscillations as can be observed from Fig. 4(a) and corresponding lead-lag motion in menisci before achieving a steady rate of filling through the microchannel. Thus it can be inferred that a passive control over the capillary filling can be exercised through a careful choice of material of the PEL as characterized by its drag parameter. Variation in the channel height ( H ) Under the present formulation, the flow progresses solely under the driving influence of surface tension force. We now attempt to study the capillary filling dynamics through the soft microchannel corresponding to a variation in height H of the channel, keeping the Debye-Huckel parameter  1  unaltered. Accordingly, we choose three different heights of the microchannel viz. H = 10𝜇m, 30𝜇m and 50𝜇m. It may be appreciated that variation in height of the channel simply means that the length scale of the study is being altered. Figure 5(a) illustrates temporal variation in the generated streaming potential U S  corresponding to each 27 ACS Paragon Plus Environment


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value of height chosen, while Fig. 5(b) depicts penetration of the bulk electrolyte into the soft microchannel (capillary filling) as a function of time. Before proceeding further, it is worthwhile to examine the inset Fig. 5(a) wherein, the non- dimensional electrostatic potential  

Figure 5 (color online): Plots showing spatio-temporal variation in streaming potential and evolution of capillary filling dynamics, with a variation in height H of the channel. Fig. 5(a): Temporal variation of the streaming potential U s for different heights of the channel as considered for the study. The inset depicts the electrostatic potential in the soft microchannel against dimensionless distance in the inward direction, normal to the rigid wall. Fig. 5 (b): shows the variation in the capillary filling length with time (non-dimensional) t . The inset figure shows a view of the selection marked A of the original plot. The 1 1 other parameters considered while plotting the above figures are:   400 nm ,  p  500 nm ,

d  1000 nm , c  109 Ns m 4 and   10 4 S m . developed in the soft channel is plotted against inward normalized distance perpendicular to the rigid wall ( y ) for the different channel heights chosen. It may be noted in this connection that the normalized distance y  1 corresponds to location of the rigid wall and decreasing values of

( y ) indicate larger normal distances into the channel. The potential   on the ordinate axis represents the potential at the EL  e  or the potential at the PEL  p  , depending on location of y respectively. As can be observed from inset Fig. 5(a), the magnitude of electrostatic potential   is similar in all the three cases viz. H  10  m , H  30  m and H  50  m . This is because for a 28 ACS Paragon Plus Environment

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given electrolyte and absolute temperature condition, the magnitude of electrostatic potential   given as  k BT ze  , remains unaltered. However, it may also be observed that the location of inward normal distance from the rigid wall y , where the electrostatic potential   reduces to zero, differs with change in length scale, being maximum when H  10  m and successively decreasing as H increases. This in effect, means that the inward normal distance upto where the effect of electrostatic potential can be felt, decreases as the height of the channel is increased. The scale effect is the reason behind the variation. The extent of EDL represented by value of Debye length   although remains unaltered in magnitude, assumes significance and comparable with the channel length scale when H is smaller. In the background of the scale effect on electrostatic potential   explained above, we now discuss the generation of electrostatic potential and capillary filling dynamics with variation in channel height H , through Figs. 5(a)-(b). The undulation appearing in the filling characteristics through the channel of H  10  m as illustrated in Fig. 5(a), is a manifestation of the competition among the forces resulting in an equilibrium condition. The magnitude of streaming potential towards the end of the equilibrium state is higher (Fig. 5(a)) which is also due to the relative scale effect, leading to a reduced filling rate through the capillary verified through Fig. 5(b). It is worth to add here that such variation in capillary filling rate depicts the effect of scaling in a variety of microfluidic lab-on-a-chip devices such as for PE grafted nano-particle drug delivery, flow control valves, sensors and biomoieties using pH-responsive PEL grafted microchannel28. Thus, it can be inferred that the filling characteristics in a microchannel are controllable by suitable invocation of scaling effect i.e. by adopting suitable height H of the soft microchannel. Variation in the Debye-length In the endeavor to explore various avenues to exercise passive control over the capillary filling dynamics through the soft microchannel, variation in Debye length was one of the influencing factors identified. Accordingly, we now investigate the effect of variation in EDL thickness represented by the inverse of Debye Huckel parameter ( 1 ) , on the filling dynamics in the soft channel. Accordingly, we consider three different values of Debye-length,  1 = 100nm, 29 ACS Paragon Plus Environment


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250nm and 400nm. The extent of EDL where the potential due to the charged surface (the soft PEL layer, under the present formulation) is felt, is characterized by Debye length. It is to be mentioned here that unlike in a conventional electrokinetic set up, the EDL under the present formulation forms under the influence of the charges in soft PEL. This layer performs the role of charged wall, except for the fact that it is permeable and allows electrolyte to flow through it. The temporal variation in streaming potential U S  in response to a variation in EDL

Figure 6 (color online): Plots showing temporal variation in the streaming potential and capillary filling 1 characteristics for three different values of Debye-length (  = 100nm, 250nm and 400nm). Fig. 6(a) depicts the variation in the streaming potential U S  with dimensionless time, for the different Debyelengths considered. Inset Fig. 6(a) represents plots electrostatic potential

 

variation against

dimensionless distance inwardly normal to the rigid wall. Fig. 6(b) depicts the variation in the filling length  x  against non-dimensional time  t  . The other parameters for the analysis are: H  30  m ,

 p1  500 nm , d  1000 nm , c  109 Ns m 4 and   105 S m . thickness as characterized by Debye length variation, is depicted at Fig. 6(a). The effects of varying EDL thickness on the associated capillary filling dynamics, are depicted in Fig. 6(b). The electrostatic potential of an EDL dictates the extent upto which the effect of charge held within, is felt. In other words, the Debye length depends on the potential within an EDL. The inset of Fig. 6(a) depicts the potential   plotted for various values of EDL thicknesses considered in the present analysis. As can be observed from inset of Fig. 6(a), the electrostatic 30 ACS Paragon Plus Environment

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1 potential corresponding to an EDL characterized by smaller thickness   100 nm is lower

when compared to an EDL with higher value of  1 . The charge density within the EDL is dependent on the Debye length. Charge density in a thin EDL as characterized by a low value of

 1 , is lower when compared to the charge density of a thick EDL. Under the present formulation, charge density influences the extent of electrostatic interaction between PEL macromolecules and electrolytic ions. Referring to Fig. 6(a), the effect of variation in EDL thickness from 100nm to 400nm on the magnitude of streaming potential U S is found to be marginal. The streaming potential under the present formulation, shows no appreciable variation with a change in EDL thickness. This confirms the findings by Chanda et. al.23 that in soft microchannels, development of streaming potential was observed to be virtually independent of EDL thickness. The effect of reduced U S being a weak electroviscous force that opposes the driving surface tension force. When compared to an EDL with higher value of  1 , the meniscus in the case of a thin EDL (lower value of  1 ) is able to advance further into the microchannel, before the electroviscous force increases to such a strength so as to halt its advance. Though the effect being investigated in the present section is a variation in EDL thickness, the underlying mechanism behind the variation in magnitude of U S and corresponding undulation in the capillary filling dynamics observed from Fig. 6(b) is similar to that elaborated previously, while discussing variation in the thickness of PEL. From Fig. 6(b), it may be observed that for the three values of  1 considered, there is no appreciable difference in the variation of U S . From Fig. 6(b),t he filling rates in all the three cases of variation in  1 also do not show an appreciable difference. Damping rate of oscillations and its dependence on the physical parameters In present study, the filling length oscillates (the meniscus moves back and forth) due to the instantaneous competition among different body forces that exert influence on the filling dynamics viz. surface tension, viscous force at the walls and the electroviscous effect due to streaming potential. The magnitude of these forces in the underlying phenomenon highly depends on the strength of the physical parameters considered in the analysis such as Debye31 ACS Paragon Plus Environment


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Hückel parameter of electrolyte, thickness of the polyelectrolyte layer (PEL), height of the channel and drag parameter of the PEL. With a change in the aforementioned parameters, we observe in Figs. 3-6 that a significant variation is obtained either in electrostatics or the hydrodynamics of the fluid, which ultimately affects the magnitude of streaming potential being generated in the process of filling through the soft microchannel. As a result, this gives rise to the modulations in the magnitude of oscillations or the damping rate of the oscillations. Therefore, we observe in Figs. 3-6 that for different physical parameters of the system and for their different magnitudes, the dampening occurs at different instants of filling, primarily attributed to the variation in the electroviscous effect with the chosen set of physical parameters. To be precise, higher the streaming potential, lower will be the dampening rate (cf. Figs. 3-6). Therefore, with the extent of this understanding, it can be said that either an increment in the PEL thickness and Debye length of the electrolyte or a decrement in the drag parameter and the height of the channel, leads to a reduction in the damping rate of the oscillations, which can be verified in Figs. 3-6. Scaling analysis In addition to an analysis of the capillary filling dynamics in response to variations of parameters of the system, it is also important to understand the regimes of qualitative variations in the filling dynamics due to the presence of electroviscous effect and the polyelectrolyte layer at the walls of the channel. We resort to scaling estimates in the reduced order model wherein, we compare the scales of different participating body forces and demarcate the significant filling regimes. Referring to Fig. 7, it may be observed that the logarithmic plot depicting capillary filling in the soft microchannel can be explained through two distinct regimes. Regime-A, depicts a linear regime  x ~ t  in which the surface tension and inertia forces are active participating members. Regime-B is non-linear, and the widely popular Modified Washburn regime

x ~ t 

which

compares surface tension force with combined effect of viscous force and electroviscous effects. We elaborate these regimes in the following paragraphs:


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Figure 7 (color online): Scaling analysis of the present mathematical model. The dotted line represents the result obtained from the reduced order model whereas the distinct segments of the straight line represent the results obtained from the scaling analysis. Here, we obtain two distinct regimes A: inertia surface tension regime and B: surface tension and combined electroviscous effect and viscous force regime. The other parameters considered while plotting above results are: H  100  m ,  S  0.072 N m , 1  p  500 nm ,   400 nm , d  1000 nm and   10 S m .

1

4

Regime-A: inertia ~ surface tension In the incipient stages of filling marked as regime-A in Fig. 7, the surface tension takes a lead role in building a liquid column in the soft microchannel as there are no other body forces active at the commencement of filling. Here, we compare the surface tension force with the inertia of the liquid column and obtain the qualitative result of early stage of filling dynamics. To mention, there is no recognizable contribution of electroviscous effect as well as the viscous force at this stage. This is because in regime-A, owing to the small liquid column in the microchannel, the downstream accumulated ionic charges are unable to generate a streaming potential of a strength, sufficient enough to oppose the surface tension driven flow. From Fig. 7, it can be seen that the above observations qualitatively match with the analytical result of Eq. (18) for solely surface tension driven flow. In presence of electroviscous effects, a different type of proportionality for underlying filling dynamics can be observed, which is discussed in the following section. Regime-B: surface tension ~ (sum of electroviscous effects and viscous force) 33 ACS Paragon Plus Environment


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From Fig. 7, it can be seen that beyond regime-A the viscous force as well as the electroviscous effect starts to dominate and oppose the flow, implying a reduction in the filling rate as compared to regime-A. Therefore beyond regime-A, the trend of filling dynamics starts to deviate from linear characteristics, as seen in Fig. 7. In regime-B, the magnitude of filling rate leads to generation of streaming potential in the soft microchannel, which introduces electroviscous effects into the underlying filling dynamics. Note that an increment in the filling rate (i.e. flow velocity) enhances the convection of counter-ions and hence leads to the generation of streaming potential and associated electroviscous effect. Thus, in present regime (regime-B), we resort to the summation of electroviscous effect and the viscous force at the walls and equate this sum with the surface tension force to obtain the qualitative nature of the filling dynamics in the soft microchannel. While performing the scaling estimates for regime-B, we also take into account the instantaneous accumulated charge density at the meniscus  qa  , which depends on the filling length of liquid column  x  . By comparing the scales of the expressions of the streaming current and the conduction current mentioned in the RHS of Eq. (31), we get:

qa ~ x t since qa ~ uavg and uavg ~ x t . Using this scale, the scale of electroviscous effect can be obtained as : x 2 t . Notably, the scale of electroviscous effect exactly matches with the viscous force considered in this analysis. Now, comparing this scale with the scale of surface tension force, we get final scale for regime-B as: x ~ t . As this scale is reminiscent of Washburn dynamics43 for simple surface tension driven flow, we term this streaming potential modulated filling dynamics as the modified Washburn regime. The actual scale for the modified Washburn regime as observed in the present study is follows: x:

t 12

 2  A3 B1d p2 2     A1B2  A1 S 

(34)

where 3  1  2  d  1   d  1 tanh  Ad   2  A1  2  2  2  d   A2  d  1  1  3 A3  A   cosh  Ad    



A3 



 C1  C2  C4  C5  C7  C10   2 p 2 A3  A     5  A    34 ACS Paragon Plus Environment

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C1  4 3 A3  d  1 cosh  d  sech  Ad 



C2  2 A3 cosh 2  d  A2  1  d    coth    A2   2   2 sech  Ad  



C3  A4  A4d  2 2 A2d  2 4



C4  2 A C3   1  d   2  A     2  A    sech  Ad   A4 sinh 2  d  



C5  2 A3  A2   2   2 sech  Ad   sinh  2 d  C6  A4  1  d  cosh   2 d   2

  A

2

  2  sinh    A2 sinh    d 



C7  2 2 cos ech   C6 tanh  Ad  C8  3  A    A     cosh  2 d   A2   2   2 sech  Ad  



C9    sech  Ad   3  4  1  d   sinh  d    2 A  1  d   A sinh  2 d    tanh  Ad  



C10  A3 coth   C8  C9  In Eq. (34), the first term in the denominator represents the viscous force at the walls and the second term represents the electroviscous effect. Conclusion In present study, we propose a mathematical model for streaming potential modulated capillary filling dynamics of a Newtonian fluid in the polyelectrolyte grafted microchannels. To track the liquid meniscus, we invoke the reduced order model and use the conservation of ionic species to estimate the instantaneous charge on the meniscus. Through this investigation, we shed a light on the temporal evolution of the filling dynamics and time dependent generation of streaming potential and the electroviscous effect at various locations in the soft microchannel. The intriguing and central observation of this analytical study is the presence of oscillatory trend of filling length at the genesis of the filling phenomenon, which is a consequent of the intricate interplay between the surface tension force and the filling rate dependent electroviscous effect. In regard to this aspect, we identify several physical and geometrical parameters (such as height of the channel, Debye length of the electrolyte, thickness of the PEL and drag parameter of PEL) that not only tune the characteristics of the filling fluid but manage to dampen the oscillatory behavior of the filling phenomenon. It is conjectured from the analysis that the dampening of the oscillations starts once the current neutrality condition is satisfied in the channel. In addition, the 35 ACS Paragon Plus Environment


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intense electroviscous effect obtained for more compact microchannels (smaller height), higher Debye length and the PEL thickness as well as lower Darcy drag, resist the surface tension driven flow in soft microchannel and reduce the filling rate of the fluid in profound manner. Furthermore, we perform a scaling analysis and successfully show that the scale obtained for modified Washburn regime has a close match with the analytical result of reduced order model, implying that the streaming potential modulated filling phenomenon in soft microchannel follows the standard Washburn dynamics, i.e. x ~ t . Albeit as limiting cases, we strongly believe that the inferences gained from this analytical work would provide inputs for some pertinent modifications and design of autonomous capillary filling systems and Lab-on-a-chip assays. We would like to mention here that the significant extensions to the present basic analysis include aspects such as imposition of axial temperature gradient in the flow field44, consideration of effect of steric interactions on the overall filling dynamics45,46, accounting for variation in the conductivity47 within PEL and EL and modeling the problem considering viscoelasticity48 of the complex fluids into account etc., to name a few. ACKNOWLEDGMENT The authors acknowledge the financial support provided by the SERB (DST), India, through Project No. ECR/ 2016/000702/ ES. References (1)

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Figure 1 (Color online) Schematic diagram of flow through soft microchannel in the present study. The channel is of Length L , height 2H and width W . We assume W  2H and hence the flow can be considered to be two dimensional. The origin is to the left end of the channel. Electrolyte enters from left and progresses towards exit in the right. The small red color curves extending from channel wall into the channel upto distance d , are symbolic representation of polyelectrolyte brushes with polyelectrolyte macromolecules (cations) (shown in violet color) tethered onto them. The accumulation of ionic charges near the meniscus which causes the streaming potential, can be seen.


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Figure 2 (color online): Plot illustrating capillary penetration distance as a function of time. Temporal variation of the capillary penetration distance obtained from the present formulation exhibits a good match with that reported by Bandopadhyay et al.41, for PTT fluid in open horizontal channel with halfheight of 100𝜇m and relaxation time 10-4 sec (behavior tends to that of a Newtonian fluid) with results of our study.



Figure 3 (color online): Fig. 3(a): Temporal variation in the electrical body force

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F  Elec

for different

values of PEL thickness d (= 500nm, 1000nm and 1500nm) chosen. Inset Fig. 3(a): Variation in the electrostatic potential ( ) with a change in the value of PEL thickness d from 500nm through 1500nm. The legend for inset is same as for main figure and note that both  FElec  and t are dimensionless. Fig. 3(b): Capillary filling length (dimensionless) plotted against dimensionless time t for the PEL thickness cases 1 considered. The other parameters chosen for the study: H  30 m ,  p  500nm , 

1

 400 nm ,

5 c  109 Ns m4 and   10 S m .


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Figure 4 (color online): Plots showing the temporal variation in streaming potential and capillary filling dynamics with variation in the drag parameter of the PEL c (= 108, 109 and 1010). Fig. 4(a): Variation in the streaming potential U S  with a change in selected values of drag parameter. Fig. 4(b): Temporal

variation in filling length for the drag parameter values chosen. The other parameters chosen while plotting 1 are: H  30 m ,  p  500nm ,

 1  400nm , d  1000nm and   105 S m .



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Figure 5 (color online): Plots showing spatio-temporal variation in streaming potential and evolution of capillary filling dynamics, with a variation in height H of the channel. Fig. 5(a): Temporal variation of the streaming potential U s for different heights of the channel as considered for the study. The inset depicts the electrostatic potential in the soft microchannel against dimensionless distance in the inward direction, normal to the rigid wall. Fig. 5 (b): shows the variation in the capillary filling length with time (nondimensional) t . The inset figure shows a view of the selection marked A of the original plot. The other 1 1 parameters considered while plotting the above figures are:   400 nm ,  p  500nm , d  1000nm , c  109 Ns m4 and   10 S m . 4


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Figure 6 (color online): Plots showing temporal variation in the streaming potential and capillary filling 1 characteristics for three different values of Debye-length (  = 100nm, 250nm and 400nm). Fig. 6(a) depicts the variation in the streaming potential U S  with dimensionless time, for the different Debyelengths considered. Inset Fig. 6(a) represents plots electrostatic potential

 

variation against

dimensionless distance inwardly normal to the rigid wall. Fig. 6(b) depicts the variation in the filling length  x  against non-dimensional time  t  . The other parameters for the analysis are: H  30m ,

 p1  500nm , d  1000nm , c  109 Ns m4 and   105 S m .



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Figure 7 (color online): Scaling analysis of the present mathematical model. The dotted line represents the result obtained from the reduced order model whereas the distinct segments of the straight line represent the results obtained from the scaling analysis. Here, we obtain two distinct regimes A: inertia surface tension regime and B: surface tension and combined electroviscous effect and viscous force regime. The other 1 parameters considered while plotting above results are: H  100 m ,  S  0.072 N m ,  p  500 nm ,

  400 nm , d  1000 nm and   104 S m . 1


Surface Tension Driven Filling in a Soft Microchannel: Role of

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