Zeta Potential and Slip Coefficient Measurements of Hydrophobic

Article pubs.acs.org/IECR

Zeta Potential and Slip Coefficient Measurements of Hydrophobic Polymer Surfaces Exploiting a Microchannel Hung Mok Park* Department of Chemical and Biomolecular Engineering, Sogang University, Seoul, South Korea ABSTRACT: The surface charge and wettability of polymer surfaces are important properties that affect various physicochemical properties such as adhesion, hydrophilicity, and swelling. The zeta potential gives information of the ions distribution at the electric double layer (EDL), and the Navier slip coefficient is also an important indicator of the wettability of surfaces. In the present investigation, a technique is devised to measure the zeta potential and the slip coefficient of polymer surfaces nondestructively exploiting electrokinetic flows in a microchannel, which is formed by combining a probe cell and a macroscopic polymer surface under consideration. By measuring zeta potential and volumetric flow rate of electroosmotic flow at various bulk ionic concentration, one can estimate the zeta potential and slip coefficient accurately even under significant experimental errors in the measurement of streaming potential and volumetric flow rate. The method devised in the present investigation may be employed to estimate zeta potential and slip coefficient of various polymeric surfaces, which are important characterizing various physicochemical properties of polymers.

1. INTRODUCTION When polymer surfaces are in contact with aqueous electrolyte solutions, the acidic or basic groups at the surface dissociates and this induces the preferential adsorption of cations or anions to the surface in competition with water adsorption. Typical dissociable groups are carboxylate, sulfate, sulfonate, phosphate, and amino groups. These groups cause the polymer surface charged and influence various surface properties of polymers, such as adhesion,1 hydrophilicity,2 swelling,3 graft layer configuration4 and many other physicochemical properties. Sometimes, such charged groups are introduced intentionally using chemical or physical methods such as substitution reaction, graft polymerization, and plasma treatment.5 The measurement of the zeta potential has provided a good means for studying the structural and chemical charge distribution at the surface which affects the various surface properties of the polymers.6 In this regard, electrokinetic measurements are more useful for a porous substrate, fiber, and film in aqueous media than conventional X-ray diffraction (XRD), infrared spectroscopy (IR), and contact angle measurement. Among various electrokinetic methods for the measurement of zeta potential, streaming potential, and volumetric flow rate of electro-osmotic flow have been widely adopted especially for macroscopic polymer surfaces.7 To measure streaming potential of macroscopic surfaces, they are usually cut to fit a cylindrical cell. A method is desirable that allows measurement of zeta potential of macroscopic polymer surfaces nondestructively. Recently, a device called the asymmetric clamping cell is developed by Anton Paar (Graz, Austria) and is currently marketed by Brookhaven Instruments Corporation.8 This device exploits electrokinetic flows through the microchannels formed by the probe cell and the macroscopic surface under consideration (cf. Figure 1). The streaming potential of electrokinetic flow and the volumetric flow rate of electroosmotic flow through these microchannels are affected by the zeta potential of both the probe cell and the macroscopic polymer surface. Therefore, the zeta potential of the macroscopic © 2012 American Chemical Society

Figure 1. (a) Asymmetric champing cell composed of probe cell and a macroscopic surface before construction; (b) microchannels formed after construction; (c) coordinate system for numerical simulation.

surface can be determined by measuring either the streaming potential or the volumetric flow rate of electroosmotic flow if the zeta potential of the probe cell is predetermined. Usually the probe cell are made of polymeric materials such as poly Received: Revised: Accepted: Published: 6731

January 16, 2012 March 16, 2012 April 12, 2012 April 12, 2012 dx.doi.org/10.1021/ie300143f | Ind. Eng. Chem. Res. 2012, 51, 6731−6744

Industrial & Engineering Chemistry Research

Article

the polymer surface under consideration. However, as explained in Park,13 the measurement of −∂ϕstr/∂z and Qeof are not independent as long as ζ and b are concerned, since both −∂ϕstr/∂z and Qeof depend on ζ and b in a special functional form under a given ionic strength. Namely, we can infer one measurement value from the measurement of the other. It is imperative to separate the effect of slip coefficient from that of zeta potential on −∂ϕstr/∂z and Qeof before determining b and ζ of a hydrophobic polymer surface accurately employing the asymmetric clamping cell. In the present investigation, we have devised a method where the effects of ζ and b on −∂ϕstr/∂z and Qeof appear separately. It shall be shown in the sequel that this can be realized by varying the bulk ionic concentration in the electrokinetic flows of the asymmetric clamping cell. This method is found to allow accurate determination of slip coefficient b and zeta potential ζ of various macroscopic polymer surfaces employing the asymmetric clamping cell.

methylmethacrylate (PMMA) or PDMS (poly dimethylsiloxane) because a variety of fabrication techniques are available for polymeric material microchannels. In practice, aqueous electrolyte solutions with concentration in the range of 1 mM to 100 mM are employed in the microchannel of 1000 μm wide and 100 μm deep. Then, the thickness of the electric double layer (EDL) is so thin that semianalytic expressions for the streaming potential and volumetric flow rate can be found.9 For an accurate prediction of zeta potential using the asymmetric clamping cell, it is important to adopt a probe cell made of materials having low zeta potentials.10 A surface treatment may be adopted to reduce the zeta potential of the probe cell. Although many polymeric materials with appropriate surface treatment have low zeta potentials, they are sometimes hydrophobic11 and the Navier velocity slip12 must be taken care of in the analysis of electrokinetic flows in the microchannels. Frequently, synthetic hydrophobic polymers have been modified by grafting with various kinds of hydrophilic monomers to improve their surface performance such as wettability, adhesion, lubrication, and biocompatibility. When measuring zeta potential of these modified polymer surfaces to study the structural and chemical charges of ionic graft chains, it is important to take into consideration the Navier velocity slip at the surface. The liquid slip velocity vb at the liquid−solid interface is usually represented using the slip length.12,13

vb = b

∂vb ∂η

2. STREAMING POTENTIAL OF ELECTROKINETIC FLOW WITH VELOCITY SLIP IN THE ASYMMETRIC CLAMPING CELL It is assumed that a 1:1 electrolyte flows in the microchannel of asymmetric clamping cell (Figure 1b,c). The governing equations in dimensionless form may be written as8−13

(1)

Here η is the normal distance from the solid surface. The slip length or the slip coefficient b depends on contact angle, wall roughness, and other physicochemical properties of the solid surface.12 Electrokinetic flows in the microchannels are driven by the electroosmotic force and the pressure force. The pressuredriven flow vp has a parabolic velocity profile, while the electroosmotic flow ve has a very sharp velocity gradient near the wall.8 It can be proven that, under the condition of the same volumetric flow rate through a slit microchannel,13 ⎛ ∂v e ⎞ ⎛ ∂v p ⎞ ⎛ h ⎞ ⎜ ⎟/⎜ ⎟=⎜ ⎟ ⎝ ∂η ⎠ ⎝ ∂η ⎠ ⎝ 3κ ⎠

∂p ∂ϕ 1 ⎛ ∂2 ∂2 ⎞ ⎜ 2 + 2 ⎟v z = − 2δ sinh(αψ ) Re ⎝ ∂x ∂z ∂z ∂y ⎠

(3)

⎛ ∂2 ∂2 ⎞ ⎜ 2 + 2 ⎟ψ = β sinh(αψ ) ∂y ⎠ ⎝ ∂x

(4)

where z indicates the flow direction. The relevant boundary conditions are

at the walls of probe cell ∂v z , ∂η

v z = ba

ψ = ζa

at the wall of macroscopic substrate

(2)

where h is the half depth of the slit channel and κ is the Debye length representing the thickness of electric double layer. This ratio is very large since the Debye length is less than one thousandth of the channel half depth. Therefore, a very small slip length b can have appreciable effect on the volumetric flow rate of electroosmotic flow and streaming potential. This also implies that it is more advantageous to employ electroosmotic flows than pressure driven flow in the experimental determination of slip length b, since the former is more sensitive to b and shall reduce the effects of experimental error. Since, the small velocity slip present in the hydrophobic polymeric surfaces affects the estimated value of zeta potential of the macroscopic polymer surfaces when it is determined by measuring the streaming potential or the volumetric flow rate of electroosmotic flows in the microchannel of asymmetric clamping cell, it is very important to take care of the Navier velocity slip when measuring zeta potential of polymer surfaces. Since both the streaming potential −∂ϕstr/∂z and volumetric flow rate of electroosmotic flow Qeof are affected by slip coefficient b and zeta potential ζ, it may be possible to estimate b and ζ using measurements of both −∂ϕstr/∂z and Qeof. The magnitude of slip coefficient b indicates the hydrophobicity of

v z = bt

∂v z , ∂η

ψ = ζt

(5)

where η is the normal distance from the wall, ζa is the dimensionless zeta potential of probe cell, ζt is that of macroscopic substrate, ba is the dimensionless slip coefficient of probe cell, and bt is that of substrate. In the above equations, the dimensionless variables and dimensionless groups are defined using the dimensional variables indicated with superscript asterisk as follows. x=

x* , L

ψ* , ζ0

v* , U

v=

t=

t* , L/U

ϕ=

ϕ* , ζ0

α=

ezζ0 , kbT

β=

L2 2n0ez , ε0εζ0

Re = 6732

ψ=

ρLU , μ

ζ=

ζ* , ζ0

p=

p* ρf U 2

ε0εkbT

κ=

2n0ρ2 z 2 δ=

b=

b* L

zen0ζ0 ρf U 2

,

, ,

(6)

dx.doi.org/10.1021/ie300143f | Ind. Eng. Chem. Res. 2012, 51, 6731−6744


Article

Here, ψ* is the induced electric potential, P* is the pressure field, φ* is the external potential imposed on the system, ζ* is the zeta potential at the wall, b* is the slip coefficient, n0 is the bulk ionic concentration, e is the elementary charge, kB is the Boltzmann constant, T is temperature, ζo is the reference zeta potential, L is the width of the microchannel, U is the characteristic velocity, ε0 is the permittivity of vacuum, ε is the dielectric constant, κ is the dimensionless Debye length given by 1/(αβ)1/2 and Re is the Reynolds number. For typical microchannels with ζ0 = 0.1 V, L = 10−4 m, n0 = 10−3 mol L−1 and U = 8 × 10−4 ms−1, the dimensionless parameters α, β, and Re have values approximately, α = 3−4, β = 106∼107, Re = 0.05−0.1, and the Debye length κ is less than ten thousandth of the channel width. Since eq 3 is linear, we may represent vz as follows. ⎛ ∂ϕ ⎞ p p ⎛ ∂p ⎞ + bav(1) v z = (v(0) )⎜ − ⎟Re + v e⎜ − ⎟Re ⎝ ∂z ⎠ ⎝ ∂z ⎠

other two directions. Under this condition, eq 4 is solved analytically as follows,8 ⎡ − 2 ⎢1 + e ψ (η) = ln⎢ α ⎢ 1 − e− ⎣ i

∂x

+

p ∂ 2v(0) 2

∂y

2 e

= −1.0;

p ∂ 2v(1) 2

∂x

p ∂ 2v(1) 2

+

∂y

f ≡ ve −

∂ 2f ∂x

∂η

,

p v(1)

= 0,

p bt ∂v(0) = a , b ∂η

∞

=

∑ m=0

∂ 2f

+

= 0;

∂y 2

=0 (13)

∂f 2δ a − (ζ + bag a) ∂η β

at the wall of substrate f = bt

(8)

∂f 2δ t − (ζ + bt g t ) ∂η β

(14)

where v e = ba

∂v e ∂η

4 g = α i

e

t ∂v

v =b

∂η

(9)

⎡ ⎢ (1 − ( −1)m )(1 − ( −1)n ) sin(mπx) sin ∑⎢ ⎡ nπ 2 ⎤ ⎢ n=1 (mπ )(nπ )⎢(mπ )2 + a ⎥/4 ⎢⎣ ⎣ ⎦

( )

(15)

2δ a 2δ a (ζ + bag a) + [(ζ + bag a) − (ζ t + bt g t )]v β β 2δ + ψ β (16)

where v satisfies the following equation with the relevant boundary conditions:

⎤ ⎥ ⎥⎦

(i = a or t )

ve = −

∂ 2v ∂ 2v + 2 =0 2 ∂x ∂y

( naπy ) ⎥⎥

∞

( α4 ζ i) 2 α 1 − {tanh( 4 ζ i)} αβ tanh

which is the negative of gradient of the induced potential ψ at the wall. Since the order of magnitude of ζ is O(1), b is O(10−4), and g is O(104), the first terms in the right-hand sides of eq 14 may be safely ignored. Then, ve is found from eqs 12−14 as follows.

e

where η is normal distance from the wall. The first two equations for the vp(0) and vp(1) fields can be solved analytically for a rectangular microchannel. If the macroscopic surfaces are flat or the curvature is not significant at the scale of the microchannel, the cross section of the microchannel may be regarded as rectangular. It shall be shown that it is only vp(0) that enters in the final expression for the streaming potential. For a rectangular microchannel, 0 ≤ x ≤ 1 and 0 ≤ y ≤ a, vp(0) is given by the following expression. p v(0)

2

f = ba

at the wall of substrate v0p

(12)

at the walls of probe cell

at the walls of probe cell p ∂v(0)

2δ ψ β

The governing equation and boundary conditions for f are found from eqs 4, 8, and 9 as follows:

(7)

∂v ∂v = 2δ sinh(αψ ) + 2 ∂x ∂y 2

p = v(1)

(i = a or t ) (11)

2 e

v0p = 0,

tanh

where the superscripts a and t denote probe cell and macroscopic substrate, respectively. Exploiting eq 11, we can solve the third of eq 8 for ve as follows. Define an auxiliary variable f such that

where we have assumed a regular perturbation solution for the pressure driven velocity field vp, keeping only the first order term since the slip coefficient b is usually very small. Then vp(0), vp(1) and ve fields satisfy the following equations with relevant boundary conditions. p ∂ 2v(0) 2

( α4 ζ i) ⎤⎥ ⎥ α αβη tanh( 4 ζ i) ⎥⎦ αβη

(10)

v=0

at the walls of probe cell

For an annular shaped cross-section, as depicted in Figure 1b, which occurs for a macroscopic polymer surface whose curvature is significant at the scale of microchannel, vp(0) can be found by solving numerically the first equation of eq 8 in the transformed rectangular domain as depicted in Figure 1c. Contrary to the original set of equations, eqs 3−4, which requires tremendous grid numbers to resolve the electric double layer, numerical solution of vp(0) field needs only a small number of grids. On the other hand, when the EDL is very thin, ψ varies only in the normal direction to the wall and constant in

v=1

at the wall of substrate

(17)

For a rectangular microchannel, 0 ≤ x ≤ 1 and 0 ≤ y ≤ a, v is found analytically as ∞

v=

∑ m=1

2(1 − (− 1)m ) sin(mπx)(emπy − emπ(2a − y)) mπ(1 − e 2mπa)

(18)

For an annular shaped microchannel as depicted in Figure 1b, v can be found by solving numerically eq 17, employing only a small grid numbers in the transformed rectangular domain. 6733



Article

pt is that formed by the substrate, the bracket ⟨vp(0)⟩a means the peripheral average of vp(0) on the probe boundary of the cross section, ⟨vp(0)⟩awall means the value of ⟨vp(0)⟩a at the wall and ⟨vp(0)⟩t is the peripheral average of vp(0) on the substrate boundary. The integrals in the third line of eq 21 may be evaluated exploiting the fact that ∂ψ/∂η decreases from wall value to zero very rapidly while ⟨∂vp(0)/∂η⟩ varies slowly within the EDL. Thus,

The dimensionless current through a long straight microchannel is given by8−13

∫Ω δv z(n+ − n−) dΩ

I=

−

α ScRe

⎛

⎞

∫Ω δ(n+ + n−)⎜⎝ ∂∂ϕz ⎟⎠ dΩ

(19)

Here, Sc is the Schmidt number given by the ratio of kinematic viscosity to diffusivity of ions, n+ the concentration of cations, n− the concentration of anions, Ω the cross section of the microchannel and ∂ϕ/∂z is the streaming potential or externally imposed electric field. The Schmidt number is about 400 for K+ and Cl− ions. In the sequel, we take Sc = 400 and Re = 0.1. Assuming the equilibrium Boltzmann distribution for ions, n± = e∓αψ, eq 19 may be rewritten using eqs 7 and 16 as I = [− 2Reδ

⎛

∫

∫

1 β

=

pa 1 a a a (p ζ g + pt ζ tg t ) − β β pt − β

1 = β

p ∂ψ v(0)

∫∂Ω

∂2 ⎞ ⎟ψ d Ω ∂y 2 ⎠

1 dΓ − ∂η β

p p ⎛ ∂v(0) ∂v(0) ∂ψ ∂ψ ⎞ ⎜⎜ ⎟⎟d Ω + Ω ⎝ ∂x ∂x ∂y ∂y ⎠

−

1 a p β

1 − pt β

∫η=0 S

∫η=0

p ∂v(0)

a

∂η p ∂v(0)

∂η

t

+ pt

⎛ ∂ψ ⎞a ⎜ ⎟ dη ⎝ ∂η ⎠

−

⎛ ∂ψ ⎞t ⎜ ⎟ dη ⎝ ∂η ⎠

∂p ∂z

2

+

∫0

∫0

∂2 ⎞ ⎟ψ d Ω ∂y 2 ⎠ +

∂ψ ∂ψ dΩ ∂y ∂y

η⎧ ⎪⎛ ∂ψ

⎞a ⎫ ⎨⎜ ⎟ ⎬ dη ⎩⎝ ∂η ⎠ ⎭ 2

⎪

⎪

⎪

2 ⎞t ⎫ ⎪ ⎨ ⎬ ⎜ ⎟ ⎪ dη ⎪ ⎩⎝ ∂η ⎠ ⎭

⎧ ⎩

⎧

p ∂v(0)

(23)

∂η

a

(ζ a + bag a) wall

⎡ ⎫⎤ ⎪ ⎛ ∂p ⎞ 2αδ (ζ t + bt g t )⎬⎥⎥⎜− ⎟ + ⎢ A ⎢ ScRe ⎪ ⎝ ∂z ⎠ ⎣ ⎭⎦ wall

t

p ∂v(0)

∂η

⎧ ⎛ 4Reδ 2 ⎪ a a a 2 ⎨−p b (g ) − pt bt (g t )2 + ⎜⎜pa 2 ⎪ β ⎩ ⎝

+ pt

(21)

1 ⎨pa αβ

∂ϕ str − ∂z A 2

(22)

wall

∂v ∂η

a

ζa wall

⎫⎤ ⎞ ⎪ ⎛ ∂ϕ ⎞ ∂v ζ t ⎟⎟((ζ a + bag a) − (ζ t + bt g t ))⎬⎥⎜− ⎟ ⎪⎥⎝ ⎠ ∂η wall ⎠ ⎭⎦ ∂z t

Equation 4 is used for the first equality of eq 21 and integration by parts has been done for the second equality. In the above equation, pa is the microchannel perimeter formed by the probe cell,

( ) = Sc(Re) (− ) 1+

ζa

η⎧ ⎪⎛ ∂ψ

⎡ ⎧ 2Reδ ⎪ a ⎨−p I = ⎢⎢ β ⎪ ⎩ ⎣

∫

1 1 p a p t = pa ⟨v(0) ⟩wall g a + pt ⟨v(0) ⟩wall g t β β 1

a

Here, A is the cross sectional area of the microchannel. The integrals, ∫ 0η {(∂ψ/∂η)a}2 dη and ∫ 0η {(∂ψ/∂η)t}2 dη, are negligible since the integrands are nonzero only near η = 0. Exploiting the fact that b ≈ O(10−4) and g ≈ O(10−4), we may rewrite eq 20 as follows.

∫ v(0)p sinh(αψ ) dΩ +

∂η

∫dΩ ψ ∂∂ψη dΓ − 1β ∫ ∂∂ψx ∂∂ψx

=

Each term in eq 20 may be evaluated exploiting the fact that ψ varies only in the normal direction η within the thin electric double layer and vanishes outside the EDL. For example,

2

wall

∫ ψ sinh(αψ ) dΩ = 1β ∫ ψ ⎜⎝ ∂∂x 2

∫

⎛

∫

∂η

p ∂v(0)

⎛ ∂ψ ⎞a ⎜ ⎟ dη = − η = 0 ⎝ ∂η ⎠ 1

∫ cosh(αψ )dΩ ≈ A

∫

∫ v(0)p⎜⎝ ∂∂x 2

a

p ∂v(0)

Other terms in eq 20 can be evaluated in a similar manner. Further examples are

∫Ω v(0)p sinh(αψ ) dΩ − 2Reδba∫Ω v(1)p

1 β

⎛ ∂ψ ⎞a ⎜ ⎟ dη ⎝ ∂η ⎠

∂η

=

⎛ ∂p ⎞ ⎡ 2αδ × sinh(αψ ) dΩ]⎜ − ⎟ + ⎢ cosh(αψ ) dΩ ⎝ ∂z ⎠ ⎣ ScRe 2δ + 2Reδ (ζ a + bag a) sinh(αψ ) dΩ β 2δ − 2Reδ {(ζ a + bag a) − (ζ t + bt g t )} v sinh(αψ ) dΩ Ω β ⎤⎛ ∂ϕ ⎞ 2δ ψ sinh(αψ ) dΩ⎥⎜ − ⎟ − 2Reδ ⎦⎝ ∂z ⎠ β (20)

=

a

p ∂v(0)

1

∫η=0

2ScRe 2 δ ⎨pa ba(g a)2 A αβ 2 ⎩

p ∂v(0)

∂η

a

(24)

Streaming potential is found by setting I = 0 and (−∂ϕ/∂z) = (−∂ϕ/∂z)str in eq 24. Equation 24 may be rewritten for a later convenience as (ζ a + bag a) + pt

wall

⎛ + pt bt (g t )2 + ⎜pa ⎝

∂v ∂η

a wall

p ∂v(0)

∂η

ζ a + pt

⎫ (ζ t + bt g t )⎬ ⎭ wall t ⎫ ⎞ ∂v ζ t ⎟((ζ t + bt g t ) − (ζ a + bag a))⎬ ∂η ⎭ wall ⎠ t

(25)

6734



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where detailed expressions for ⟨∂vp(0)/∂η⟩awall, ⟨∂vp(0)/∂η⟩twall, ⟨∂v/ a t ∂η⟩wall and ⟨∂v/∂η⟩wall are given in the Appendix for a rectangular microchannel of aspect ratio a. On the other hand, the volumetric flow rate Q under an external pressure gradient and an electric field (−∂ϕ/∂z) which may be streaming potential (−∂ϕ/∂z)str or externally imposed one (−∂ϕext/∂z) is expressed, using eqs 7 and 16, as Q = [Re

employment of Qeof instead of Qp shall reduce the experimental errors in the determination of slip length bt. Since we measure (−∂ϕ/∂z)str and Qeof at various bulk ionic concentrations n0 to remove the degeneracy in the dependence of (−∂ϕ/∂z)str and Qeof on ζt and bt, we may take either (−∂ϕ/∂z)str or Qeof at various n0 to determine both ζt and bt. For a later convenience, we may rewrite the third equation of eq 27 as follows:

⎛ ∂p ⎞

Q eof

∫Ω v(0)p dΩ + Reba ∫Ω v(1)p dΩ]⎜⎝− ∂z ⎟⎠

⎡ 2δ + ⎢Re {(−ζ a − bag a)A + (ζ a + bag a − ζ t − bt g t ) ⎣ β + pa

S

∫η=0

ψ a dη + pt

⎤⎛ ∂ϕ ⎞ ψ t dt }⎥⎜ − ⎟ ⎦⎝ ∂z ⎠ η=0

∫

∂ϕ ext ( −Re)2 − ∂z

( )

∫Ω v dΩ

δ [A(ζ a + bag a) β

=

+ (ζ t + bt g t − ζ a − bag a)

S

∫Ω v dΩ]

(26)

(28)

A detailed expression for ∫ Ωv dΩ is given in the Appendix for a rectangular microchannel. In the above equation, the integrals

where v is given by eq 18 for a rectangular microchannel, 0 ≤ x ≤ 1 and 0 ≤ y ≤ a, and obtained by solving eq 17 numerically for annular shaped microchannels. Exploiting eqs 6 and 15, we pick out terms in eqs 25 and 28 that depend on the bulk ionic concentration n0 as follows:

pi ∫

l

η= 0

ψ i dη are negligible compared with other terms. Then

we rewrite eq 26 as Q = Q p + Q eof

αβ = c1̂ n0̂ δ = c3̂ β

⎛ ∂p ⎞

∫Ω v(0)p dΩ + ba ∫Ω v(1)p dΩ]⎜⎝− ∂z ⎟⎠

Q p = Re[

Q eof = −Re ×

ĉ β = 4 n0̂ α c3̂

2δ [A(ζ a + bag a) + (ζ t + bt g t − ζ a − bag a) β ⎛

⎞ext

∫Ω v dΩ]⎜⎝− ∂∂ϕz ⎟⎠

2c4̂ n0̂ αζ i /2 i i β αζ i /2 − e−αζ /2) = − e−αζ /2) (e (e α c3̂ (i = a or t )

gi = 2

(27)

Here v is given by eq 18, vp(1) depends on bt as expressed in eq 9 and Qeof is the volumetric flow rate of electroosmotic flow. In principle, the two equations, eqs 25 and 27, allow us to estimate the slip coefficient bt and the zeta potential ζt of a polymer surface under consideration. To reduce the experimental errors, it is important to make the sensitivity of Q with respect to b as large as possible. However, sensitivity of Qp with respect to bt is so small that it is very difficult to estimate both bt and ζt using measured values of (−∂ϕ/∂z)str and Qp.9 On the other hand, the sensitivity of Qeof with respect to bt is largely due to the presence of gt, which is of the order of 104, under an externally imposed electric field (−∂ϕ/∂z)ext. However, we find from eqs 25 and 27 that (−∂ϕ/∂z)str and Qeof depend on ζt and bt in a special functional form, that is, (ζt + btgt). Therefore, measurements of (−∂ϕ/∂z)str and Qeof are not independent; that is, from measurement of one quantity we obtain the other quantity. Thus, it is not possible to determine ζt and bt from measuring (−∂ϕ/∂z)str and Qeof simultaneously. In the present work, we overcome this difficulty by introducing a method where the streaming potential and the volumetric flow rate of electroosmotic flow are measured under various bulk ionic concentration. As we shall demonstrate in the next section, the variation of bulk ionic concentration removes the degeneracy in the dependence of (−∂ϕ/∂z)str and Qeof on ζt and bt.

(29) 3

where the bulk ionic concentration n0 (molecules/m ) is replaced with n̂0 (mol/L) to ensure numerical stability and accuracy since n0 is very big numbers. The relation between n0 and n̂0 is n0̂ =

n0 (6.022 × 1023)(1000)

(mol/L)

(30)

Other parameters in eq 29 are given by ⎡ e 2z 2L2 ⎤⎡ 2 ⎤ n0 c1̂ = ⎢ ⎥ ⎥⎢ ⎣ kBT ⎦⎣ ε0ε ⎦ n0̂ c3̂ =

c4̂ =

ε0εζ02 2ρf L2U 2 ζ0 2

ρf U L

ε0εkBT 2

n0 n0̂

(31)

The zeta potentials, ζa and ζt, also change as the bulk ionic concentration varies. Bruin et al.14 showed that increasing the buffer ionic strength or concentration lowered the zeta potential when the capillary temperature is controlled. Van Orman15 also showed that zeta potential decreased as the ionic strength of the buffer increased. Usually, plots of zeta potential versus the logarithm of ionic strength are approximately linear at a given pH for organic or inorganic buffers.16,17 Thus, we represent the dimensionless zeta potential as

3. ESTIMATION OF THE SLIP COEFFICIENT BT AND THE ZETA POTENTIAL ζT OF A MACROSCOPIC SURFACE Since Qp is rather insensitive to the variation of bt, we choose Qeof in addition to (−∂ϕ/∂z)str for the simultaneous estimation of ζt and bt. As mentioned previously, the sensitivity of Qeof with respect to bt is large since gt is a large quantity. Thus,

ζ i = [Ai + Bi ln n0̂ + C i(ln n0̂ )2 ]/ζ0 6735

(i = a or t )

(32)



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Figure 2. (a) Variation of streaming potential with respect to the slip coefficient of substrate bt when ζ t = −1.0; (b) variation of streaming potential with respect to ζ t when bt = 10−4 and aspect ratio a = 0.5; (c) variation of streaming potential when bt = 10 −4 and a = 0.1.

where C i is a small number representing a slight deviation from the linear relationship between ζ and lnn̂0, and ζ0 is the reference zeta potential taken as 0.1 V. Therefore, the unknown parameters to be determined are At, Bt, Ct, and bt, the first three parameters determining the zeta potential of the macroscopic surface ζt. The estimation of At, Bt, Ct, and bt using measured data of (−∂ϕ/∂z)str and Qeof at various n̂0 becomes a nonlinear curve fitting problem which can be solved using the Levenberg−Marquardt method.18 We may rewrite eq 25 symbolically as y(ϕi) = f (x(i); a)

and eq 28 as y(Qi) = g (x(i); a)

(34)

where the parameter vector a denotes (At, Bt, Ct,bt)T, subscript (i) means the ith concentration of ions tried in the experiment and χ(i) = ln n̂0(i). For a given value of bulk ionic concentration eof or equivalently x(i), we measure (−∂ϕ/∂z)str (i) and Q(i) under the pressure gradient −∂p/∂z and the external electric field (∂ϕ/∂z)ext, which are equivalent to yϕ(i) and yQ(i) in eqs 33 and 34. The nonlinear curve fitting problem under consideration is to find the parameter vector a ≡ (At, Bt, Ct, bt) that makes eqs 33 and 34 valid as closely as possible. We may employ

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Figure 3. (a)Variation of Qeof vs bt when ζt = −1.0; (b) variation of Qeof vs ζt when bt = 10−4 and a = 0.5; (c) variation of Qeof vs ζt when bt = 10−4 and a = 0.1; (d) variation of Qp vs bt.

either yϕ(i) or yQ(i), or employ both of them in the determination of a. The following χ2 merit function is posed to apply the Levenberg−Marquardt method to the present estimation problem when both yϕ(i) and yQ(i) are employed.18 ⎡ y ϕ − f (x ; a ) ⎤ 2 (i) (i) ⎥ + χ 2 (a ) ≡ ∑ ⎢ ⎢ ⎥ σ i ( ) i=1 ⎣ ⎦ N

The above equation may be rewritten as M

∑ αk S(aS,min − aS,cur) = βk S=1

⎡ yQ − g (x ; a) ⎤2 (i) ⎥ ∑ ⎢⎢ (i) ⎥ σ ( i ) ⎦ i=1 ⎣

(37)

N

with 1 2 2 (∇ χ )k S 2 1 βk ≡ − (∇χ 2 )k 2

αk S ≡

(35)

where σ(i) is the measurement error of the i th data point, presumed to be known. If we use measurements of yQ(i) only, the first term in the right-hand side of eq 35 is deleted. A similar statement is valid when we exploit measurements of only yϕ(i). When a becomes the exact value, amin, χ2 is at a minimum and ▽χ2|amin = 0. If acur is the current parameter vector, ∇χ 2 |a min = 0 = ∇χ 2 |acur + ∇2 χ 2 ·(a min − acur)

(38)

Here, (∇2 χ 2 )k S is the k Sth component of the matrix ▽ 2χ2 and (▽χ 2) k is the kth component of the vector (▽χ 2). Differentiating χ 2 with respect to ak and aS , which are the kth and Sth components of the parameter vector a, yields

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Figure 4. Estimation of ζt and bt exploiting measurements of (−∂ϕ/∂z)str and/or Qeof at various bulk ionic concentration (a) variation of yϕ and yQ with respect to ln n0 when the aspect ratio is 0.8 and ba is 10−5; (b) estimated ζt and bt when measurement error is 0%; (c) when measurement error is 0.5%; (d) when measurement error is 1%. 2 2 (∇ χ )k S . In the steepest descent method, eq 37 is approximated as

λαkk(ak ,min − ak , cur ) = βk

The procedure for the Levenberg−Marquardt method is summarized as follows: 1. compute χ2|acur 2. set λ = 0.001 3. solve eq 41 to find amin and evaluate χ2|amin 4. if χ2|amin ≥ χ2|acur, increase λ by a factor of 10 and go back to step 3 5. if χ2|amin < χ2|acur, decrease λ by a factor of 10, and update acur ← amin. Go back to step 3 6. stop iteration on the first or second occasion that χ2 decreases by a negligible amount

(39)

In the Levenberg−Marquardt method, we approximate α matrix as α′ jj = αjj (1 + λ) α′ jk = αjk

(j ≠ k )

(40)

and solve the following equation to update the parameter vector a.

4. RESULTS The accuracy of the formulas for the streaming potential and the volumetric flow rate of electroosmotic flow, eqs 25 and 28, is corroborated by comparing the results of these formulas with

M

∑ αkl′ (aS,min − aS,cur) = βk S=1

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Figure 5. Effects of aspect ratio of microchannel a on the accuracy of estimation (a) a = 0.1 or a = 0.8 with ba = 10−5; (b) estimation error of ζt and bt when measurement error is 0%; (c) when measurement error is 1%; (d) when measurement error is 3%.

to be noted that the stream function versus the ζt curve has a maximum when the aspect ratio a is reduced to 0.1 (cf. Figure 2c). As shown in Figure 2a−c, the sensitivity of the streaming potential with respect to either bt or ζt is so large that the streaming potential may well be employed in the determination of bt or ζt under nontrivial experimental errors. Figure 3a shows the variation of Qeof/[(−∂ϕ/∂z)ext(−Re)2] with respect to bt when a = 0.5, ζt = −1.0, (α, β) = (3.89, 5 × 106) or (3.89, 107), ba = 10−5, and ζa = −0.5. It is revealed that eq 28 yields very accurate results as compared with the exact solutions. As expected, the volumetric flow rate of electroosmotic flow Qeof increases as the slip coefficient of the macroscopic surface bt increases. It is also shown that Qeof increases as β or the bulk ionic concentration increases. Figure 3 panels b and c show variation of Qeof as ζt changes when bt = 10−4 and a = 0.5 (Figure 3b) or a = 0.1 (Figure 3c). It is revealed that Qeof increases with respect to both ζt and β. As plotted in Figure 3a−c, the

those of exact governing equations, eqs 4, 8, and 9. Exact values of the streaming potential are found by solving eqs 4, 8, and 9 for vz defined in eq 7, substituting it into eq 20 and setting I = 0. Exact values of volumetric flow rate are found by integrating ve found from eq 8 over the cross-section of the microchannel. Figure 2a shows [(−∂ϕ/∂z)str A]/[Sc(Re)2(−∂p/∂z)] against bt for a rectangular microchannel of aspect ratio a = 0.5 when ζt = −1.0, (α, β) = (3.89, 5 × 106) or (3.89, 107), ba = 10−5 and ζa = −0.5. It is shown that (−∂ϕ/∂z)str increases almost linearly as bt increases. The results from the analytic formula, eq 25, coincide with the exact results. Figure 2b plots [(−∂ϕ/∂z)str A]/ [Sc(Re)2(−∂ϕ/∂z)] versus ζt under the same condition as in Figure 2a except bt is fixed to be 10−4, and Figure 2c plots the streaming potential against ζt when bt = 10−4 and a = 0.1. In all cases, eq 25 yields very accurate predictions of streaming potential. It is also revealed that the streaming potential decreases as the bulk ionic concentration no or β increases. It is 6739



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Figure 6. Effects of the slip coefficient of probe cell ba on the accuracy of estimation (a) ba = 10−5 or ba = 10−3 with a = 0.8; (a) estimation error of ζt and bt when measurement error is 0%; (b) when measurement error is 1%; (c) when measurement error is 3%.

scopic surface are, ζt = ((−0.1163) + (2.145 × 10−3)(ln n̂0) + (8.979 × 10−4)(ln n̂0)2)/0.1 and bt = 10−4, respectively, while the zeta potential and slip coefficient of the probe cell are, ζa = ((−2.39 × 10−2) + (7.52 × 10−3)(ln n̂0) + (7.50 × 10−4)(ln n̂0)2)/0.1 and ba = 10−5. The microchannel has a rectangular cross-section with an aspect ratio a = 0.8. Simulated measurements have been adopted, which are constructed by adding Gaussian distributed random errors to the exact values found by solving the exact governing equations, eqs 4, 8, and 9. Figure 4a plots graphs of yϕ(i) versus x(i)(≡ln n̂0(i)) and yQ(i) versus x(i) for 14 equally distributed values of x(i) between −11.5 and −5. Figure 4 panels b−d show the results of estimation when the measurement errors of (−∂ϕ/∂z)str and Qeof are 0.0%, 0.5%, and 1%, respectively. In the estimation, we employ only yϕ(i) or only yQ(i) or both yϕ(i) and yQ(i) in the nonlinear curve fitting procedure (cf. eq 35). It is worth noting that the magnitude of ζa is smaller than that of ζt in this experiment so that yϕ(i) and yQ(i)

sensitivity of Qeof with respect to either bt or ζt is also so large that Qeof may also be employed in the estimation of bt and ζt. On the other hand, Figure 3d shows that Qp, found by solving the first two equations of eq 8, is so insensitive to bt that it cannot be employed in the determination of bt due to the inherent experimental errors. Previously, we mentioned that the simultaneous measurements of Qeof and (−∂ϕ/∂z)str cannot determine bt and ζt, since they depend on bt and ζt in the same functional form, (ζt + btgt), as revealed in eqs 25 and 28, at a constant bulk ionic concentration. However, if we change the bulk ionic concentration, the degeneracy in the dependence of (−∂ϕ/∂z)str and Qeof on bt and ζt is removed as explained in the previous sections. Next, we present the results of estimation of bt and ζt exploiting measurements of (−∂ϕ/∂z)str and/or Qeof at various bulk ionic concentration n0. Figure 4 shows the results when the exact value of zeta potential and slip coefficient of the macro6740



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Figure 7. Effects of the magnitude of zeta potential of probe cell on the accuracy of estimation (a) two cases with ba = 10−5 and ba = 5 × 10−4 are considered (b) the magnitude of zeta potential of probe cell |ζa| is larger than that of |ζt| (measurement error is 0%) (c) when measurement error is 1% (d) when measurement error is 3%.

of yϕ(i) and yQ(i) with respect to x(i)(≡ln n̂0(i)) are reduced as a decreases from 0.8 to 0.1, the accuracy of estimation is slightly improved as shown in Figure 5d, since the sensitivity of yϕ(i) and yQ(i) with respect to bt and ζt increases as a decreases. Figure 6 shows the effect of slip coefficient of the probe cell ba on the accuracy of estimation. We consider the case where the slip coefficient of macroscopic surface bt, which is to be estimated, is 10−4 and ba is either 10−5 or 10−3. It is found that as ba increases from 10−5 to 10−3, which shall diminish the effect of bt on (−∂ϕ/∂z)str and Qeof, the estimation error increases. This trend is revealed clearly for the case where the measurement error is 3% (Figure 6d). Figure 6 panels c and d show that the estimation error, especially that for bt, deteriorates easily when ba is much larger than bt. Next consideration is the effect of both the zeta potential of probe cell ζa and ba on the accuracy of estimation of ζt and bt. As depicted in Figure 7b, the

are more influenced by ζt than by ζa. It is shown that the estimation errors increase as the measurement errors increase. It is also shown that it is better to adopt both yϕ(i) and yQ(i) in the estimation procedure to obtain an accurate estimation of ζt and bt. If we have to employ only one among yϕ(i) and yQ(i), a better estimation is obtained when yQ(i) is adopted. From now on, we adopt both yϕ(i) and yQ(i) in the estimation procedure. Figure 4 shows that the estimation error of ζt is about the same magnitude as the measurement error when both yϕ and yQ are employed. These results demonstrate that the method developed in section 3 yields reasonably accurate estimates of ζt and bt even though there are significant measurement errors. The aspect ratio of microchannel, a, determines the ratio of surface area to cross-sectional area of the microchannel. Figure 5 shows the effect of a on the accuracy of estimation when the slip coefficient of probe cell ba is 10−5. Although the variations 6741



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Figure 8. A nonrectangular microchannel for a macroscopic surface with a large curvature (a) a nonrectangular microchannel with a = 0.2, A = 0.7, pa = 3.9, pt = 3.5 (b) estimation error of ζt and bt when there is no measurement error (c) when measurement error is 1% (d) when measurement error is 3%.

magnitude of ζa is larger than that of ζt in this experiment. Therefore, the zeta potential of probe cell affects (−∂ϕ/∂z)str and Qeof much more significantly than in the previous cases we have considered. This is expected to induce a larger estimation error. For the slip coefficient of substrate ba, we consider two values, 10−5 and 5 × 10−4. As ba increases, the influence of the surface state of probe cell on (−∂ϕ/∂z)str and Qeof increases, resulting in deterioration of the accuracy of estimation as displayed in Figure 7c,d. Therefore, it is desirable to manufacture a probe cell whose magnitudes of zeta potential and slip coefficient are smaller than those of the macroscopic polymer surface to ensure an accurate estimation of ζt and bt. Finally we consider a nonrectangular microchannel. This case occurs when the curvature of the macroscopic polymer surface is very large as in the case of thin cylindrical surfaces. For a nonrectangular microchannel depicted in Figure 8a, where a ≈ 0.2, A ≈ 0.7, pa = 3.9, and pt = 3.5, we solve the first equation

of eq 8 and eq 17 for vp(0) and v numerically employing a coarse grid set to find ⟨(∂vp(0))/(∂η)⟩ awall, ⟨(∂vp(0))/(∂η)⟩ twall, ⟨(∂v)/(∂η)⟩ awall and ⟨(∂v)/(∂η)⟩ twall and ∫ Ωv dΩ, which are needed in the formulas for (−∂ϕ/∂z) str and Qeof. Figure 8 shows the results of estimation. Figure 8 demonstrates that the present method of estimating ζ t and bt works quite well even for curved polymer surfaces. The estimation errors of (ζ t, bt) are (0%, 0%), (0.43%, 0.2%), and (1.45%, 0.37%) when the measurement error are 0%, 1% and 3%, respectively.

5. CONCLUSION The zeta potential measurement is a valuable tool to characterize surface properties of polymers such as adhesion, hydrophilicity, swelling, graft layer configuration, biocompatibility, and composite compatibility. Electrokinetic measurements are the most convenient method to measure zeta potential of 6742



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macroscopic polymer surfaces. However, the zeta potential measurement of polymer surfaces is never trivial, especially when they are hydrophobic. Hydrophobicity induces velocity slip at the liquid−solid interface, which affects electrokinetic phenomena such as streaming potential and electroosmotic flows. It is imperative to estimate accurate value of the slip coefficient before finding accurate value of zeta potential of a given polymer surface. In the present investigation, we analyze electrokinetic flows in the microchannel of the asymmetric clamping cell, which allows determination of the zeta potential of macroscopic polymer surfaces nondestructively, and devise a method that permits determination of the zeta potential and slip coefficient simultaneously. From a fluid mechanical analysis, it is found that both the streaming potential (−∂ϕ/∂z)str and the volumetric flow rate of electroosmotic flow Qeof depend on zeta potential ζ and slip coefficient b in the same functional form, ζ+bg, where g is the gradient of induced potential at the wall. However, measurements of (−∂ϕ/∂z)str and Qeof at various bulk ionic concentration remove this degeneracy and allow accurate determination of both ζ and b of polymer surfaces. We have investigated the effects of the aspect ratio of microchannel, slip coefficient, and zeta potential of probe cell on the accuracy of estimation of ζ and b of macroscopic polymer surfaces. It is found that as long as the zeta potential and slip coefficient of the probe cell are not dominant in the electrokinetic flows of the microchannel, the present method allows accurate determination of slip coefficient and zeta potential of macroscopic polymer surfaces even though there are nontrivial experimental errors in the measurements of streaming potential and volumetric flow rate. The present technique may also serve as a simple and accurate method of determining the Navier slip coefficient of macroscopic surfaces, which is much cheaper than the traditional methods such as the microparticle velocimetry.

■

where wm = mπ , (0) = A mn

Cm =

■

a

∂η

1 = 1 + 2a

wall

∞

∞

∑∑ m=1 n=1

t

p ∂v(0)

∂η

∂v ∂η

∞

=

m=1 n=1

wall a

wall

∞

(0) ∑ ∑ A mn

1 = 1 + 2a

■

ACKNOWLEDGMENTS This work was supported by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea Government Ministry of Knowledge and Economy (No. 20114010203090).

■

∂v ∂η

t

wall

m

m=1

∫

(A3)

∑ Cm(1 − (−1)m )(1 + e2w a) m

m=1 ∞

v dΩ =

(A2)

∞

=

∑ m=1

2(1 − ( −1)m )2 (2e wma − e 2wma − 1) 3 2wma wm(1 − e )

REFERENCES

(1) Stana-Kleinschek, K.; Strnad, S.; Ribitsch, V. Surface characterization and adsorption abilities of cellulose fibers. Polym. Eng. Sci. 1999, 39, 1412. (2) Möckel, D.; Staude, E.; Dal-Cin, M.; Darcovich, K.; Guiver, M. Tangential flow streaming potential measurements: Hydrodynamic cell characterization and zeta potentials of carboxylated polysulfone membranes. J. Membr. Sci. 1998, 145, 211. (3) Bismarck, A.; Kumru, M. E.; Springer, J. Characterization of several polymer surfaces by streaming potential and wetting measurements: Some reflection on acid−base interactions. J. Colloid Interface Sci. 1999, 217, 377. (4) Jacobasch, H.-J. Surface phenomena at polymers. Makromol. Chem. Macromol. Symp. 1993, 75, 99. (5) Uchida, E.; Uyama, Y.; Ikada, Y. Zeta potential of polycation layers grafted onto film surface. Langmuir 1994, 10, 1193. (6) Jacobasch, H. -J. Characterization of the adhesion properties of polymers by electrokinetic measurement. Angew. Makromol. Chem. 1984, 128, 47. (7) Van wagenen, R. A.; Andrade, J. D. Flat plate streaming potential investigations: Hydrodynamics and electrokinetic equivalency. J. Colloid Interface Sci. 1980, 76, 305. (8) Masliyah, J. H.; Bhattacharjee, S. Electrokinetic and Colloid Transport Phenomena; Wiley-Interscience: New York, 2006; pp 653−659. (9) Park, H. M.; Sohn, H. S. Measurement of zeta potential of macroscopic surfaces with Naver velocity slip exploiting electrokinetic flows in a microchannel. Int. J. Heat Mass Transfer 2011, 54, 3466. (10) Walker, S. L.; Bhattacharjee, S.; Hoek, E. M. V.; Elimelech, M. A novel asymmetric clamping cell for measuring streaming potential of flat surfaces. Langmuir 2002, 18, 2193. (11) Kirby, B. J.; Hasselbrink, E. F., Jr. Zeta potential of microfluidic substrates. 2. Data for polymers. Electrophoresis 2004, 25, 203. (12) Neto, C.; Evans, D. R.; Bonaccurso, E.; Butt, H. J.; Craig, V. S. J. Boundary slip in Newtonian liquids: A review of experimental studies. Rep. Prog. Phys. 2005, 68, 2859. (13) Park, H. M. A method to determine zeta potential and Navier slip coefficient. J. Colloid Interface Sci. 2010, 347, 132. (14) Bruin, G. J. M.; Chang, J. P.; Kuhlmay, R. H.; Zegers, K.; Kraak, J. C.; Poppe, H. Capillary zone electrophoretic separations of proteins in polyethylene glycol-modified capillaries. J. Chromatogr. 1989, 471, 429−436. (15) Van Orman, B. B.; Liversidge, G. G.; McIntire, G. L.; Olefirowicz, T. M.; Ewing, A. G. Effects of buffer composition on electroosmotic flow in capillary electrophoresis. J. Microcol. 1990, 2, 176−180.

∑ Cm{2(2ew a − e2w a − 1)

+ 2((− 1)m − 1)e wma}

AUTHOR INFORMATION

The authors declare no competing financial interest.

∞ m

(A6)

Notes

(A1)

ζn (1 − ( −1)m ) wm

2(1 − ( −1)m ) wm(1 − e 2wma)

*Tel: +82 2 705 8482. E-mail: [email protected].

(0) A mn [wm 2(1 − ( −1)n ) wmζn

+ ζn 2( −1)n (( −1)m − 1)]

(1 − ( −1)m )(1 − ( −1)n ) , wmζna(wm2 + ζn2)/4

Corresponding Author

APPENDIX p ∂v(0)

ζn = nπ /a ,

(A4)

(A5) 6743



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(16) Baker, D. R. Capillary Electrophoresis; John Wiley & Sons, Inc.: New York, 1995. (17) Kirby, B. J.; Hasselbrink, E. F., Jr. Zeta potential of microfluidc substrates. 1. Theory, experimental technique, and effects on separations. Electrophoresis 2004, 25, 187. (18) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numenical Recipes; Cambridge University Press: Cambridge, UK, 1986.

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Zeta Potential and Slip Coefficient Measurements of Hydrophobic

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