The Streaming Current and Effective Zeta Potential for Particle

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Streaming Current and Effective ζ‑Potential for Particle-Covered Surfaces with Random Particle Distributions Maria L. Ekiel-Jeżewska,† Zbigniew Adamczyk,‡ and Jerzy Blawzdziewicz*,§,∥ †

Institute of Fundamental Technological Research, Polish Academy of Sciences, ul. Pawińskiego 5B, 02-106 Warsaw, Poland Jerzy Haber Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Niezapominajek 8, 30-239 Krakow, Poland § Department of Mechanical Engineering, Texas Tech University, Box 41021, Lubbock, Texas 79409, United States ∥ Department of Physics and Astronomy, Texas Tech University, Box 41051, Lubbock, Texas 79409, United States Downloaded via IOWA STATE UNIV on February 7, 2019 at 04:56:43 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

ABSTRACT: A detailed theoretical and experimental study is presented concerning the streaming current and the derivative effective ζ-potential for a planar surface covered by a monolayer of adsorbed particles. Precise simulation results are obtained for the equilibrium and random-sequential-adsorption (RSA) distributions of monodisperse spherical particles interacting via the excluded-volume potential. The streaming current is calculated in the thin-double-layer regime for all physically accessible particle area fractions. The results are expressed as a linear combination of the interface and particle contributions DI and DP weighted by the interface and particle ζpotentials ζI and ζP. We find that in the area-fraction regime where both particle distributions exist, the equilibrium and RSA results for the streaming current are nearly indistinguishable. Our numerical data show that DI exponentially decays to zero when the particle area fraction θ is increased, whereas DP exponentially tends to a linear behavior. The results are described (with the accuracy better than 1.5% of the maximal value) by the exponential and linear plus exponential approximations, with only one fitting parameter. The numerical and theoretical predictions are in agreement with experimental data obtained for a wide range of ζ-potentials of the interface and the particles. Results obtained for a rough surface with spherical asperities indicate that the roughness can reduce the effective ζ-potential (as evaluated from the streaming current) by more than 25%; this prediction is also confirmed by experiments.

I. INTRODUCTION Particle deposition on solid surfaces, leading to monolayer formation of a controlled structure, has potential applications in the production of nano- and microstructured materials such as narrow-band optical filters, optical switches, photonic bandgap materials, waveguides, and other electro-optical and magneto-optical devices. Noble metal (e.g., silver) nanoparticle monolayers are widely used as biocidal (antibacterial) coatings, in catalysis, and in analytical chemistry as sensors and nanostructured surfaces in surface-enhanced Raman spectroscopy.1 On the other hand, in other processes such as membrane filtration, flotation (slime coating formation), and production of microelectronic or optical devices (wafers, disks, chips, photovoltaic modules), particle deposition processes are undesirable. In addition to this extensive range of applications, quantitative studies of particle deposition interpreted in terms of adequate theoretical models furnish essential information about interactions between particles and interfaces and between adsorbed and moving particles. This is a crucial issue for colloid science, biophysics, medicine, soil chemistry, etc.2 Moreover, results of particle deposition experiments can be exploited as reliable reference data for analyzing protein © XXXX American Chemical Society

adsorption phenomena involved in thrombosis, angiogenesis, blood clotting, wound healing, tumor growth, artificial organ failure, plaque formation, fouling of contact lenses and heat exchangers, ultrafiltration and membrane filtration units, etc.3 However, despite an extensive research effort, the understanding of nanoparticle deposition phenomena is still incomplete. This is caused by the lack of direct experimental techniques to provide reliable information about the electric charge of interfaces and adsorbed particles under various physicochemical conditions. One of the few exceptions is a class of electrokinetic techniques based on measurements of the streaming current or streaming potential. The streaming current results from the convective flux of ions in the thin-double-layer region adjacent to the interfaces.4−7 The flux is produced by a macroscopic flow of the fluid, usually driven by the hydrostatic pressure gradient for channel and capillary flows, or by motion of interfaces. As a result of charge separation caused by the streaming current, an electrostatic potential difference appears along the interface, referred to as the streaming potential, which can be directly Received: October 16, 2018 Revised: January 7, 2019 Published: January 17, 2019 A

DOI: 10.1021/acs.jpcc.8b10068 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C measured using reversible electrodes. In this way, the electrokinetic charge of surfaces can be reliably determined. Adsorbed species attached to an interface, for example, polymers, proteins, and nano- and microparticles, modify the streaming current/potential in two ways. First, the particles damp the local fluid flow near the interface, reducing convection of the charge in its double layer. Second, since the particles are charged, the double layers adjacent to their surfaces provide an additional contribution to the ion flux,2,3,8 which depends on the particle ζ-potential. Because these effects are dependent on particle coverage (referred to later on as area fraction), the streaming-current/potential measurements enable unique, in situ measurement of particle and protein adsorption/desorption phenomena.9−12 Recent investigations provided some important theoretical and numerical tools for interpretation of experimental measurements of streaming current (or potential).2,8 In these studies, exact expression has been developed describing the streaming-current contributions in terms of hydrodynamic multipolar Stokes flows induced on the particles, and accurate numerical calculations were performed for the particle area fractions up to θ = 0.5,8 assuming equilibrium distribution of the adsorbed particles. Numerical results were interpolated in terms of rational functions of the particle coverage alone. However, these results are limited to moderate area fractions of spherical particles and the equilibrium structure of the particle monolayer. In experiments, the formation of the particle monolayer often occurs in an irreversible sequential adsorption process and does not involve equilibration of the microstructure via desorption and readsorption or mobility of the adsorbed particle along the surface. The applicability of our prior results to systems such as irreversibly adsorbed protein monolayers or equilibrium monolayers at high area fractions has not been yet justified. Two main goals of our present study are (i) to evaluate the streaming current for equilibrium systems for the broad range of coverage up to 0.85, including the regime where the phase transition from a disordered to ordered phase occurs,13−15 and (ii) to evaluate the streaming current for a random-sequentialadsorption (RSA) particle distribution, which is a common distribution in many experimental systems. The results allow us to assess the influence of the particle distribution on the streaming current and to meet the third important goal of this work, that is, to (iii) construct simple analytical approximations that are valid in the entire range of area fractions. We demonstrate that the theoretical results and interpolation expressions agree with experimental data. Additionally, the results allow one to infer the particle area fraction from measurements of the streaming current or assess the effective ζ-potential of rough surfaces.

Figure 1. Geometry of the control region for evaluation of the streaming current produced by an imposed shear flow v0 over a particle-covered interface.

(parallel to the interface). Since simple shear flows appear in the vicinity of most solid/liquid interfaces exposed to laminar ambient flows (e.g., for channel flows in capillaries, ducts, and packed-bed columns2), theoretical and numerical results for the streaming current derived for simple shear (1) are of a broad utility. As a result of a different surface charge (acquired either by ionization of surface groups or by irreversible adsorption of ionic species from the solution), the interface and the particles usually have different ζ-potentials, ζI and ζP, respectively. Accordingly, the charge density ρe in the diffuse part of the electric double layer is different near the interface and the particles. In the presence of ambient flow (1), the macroscopic motion of the electrolyte produces a convective flux of the charge ρe. This flux, that is, the streaming current Is, is evaluated by calculating the amount of charge flowing per unit time through a control surface ΔSc that extends from the charged interface into the bulk solution and is perpendicular to the flow2,8,16 Is =

c

(2)

Here, v is the local fluid velocity and dS is the surface element; the system geometry is illustrated in Figure 1. Using the Poisson equation that relates the charge distribution to the electrostatic potential ψ in the electric double layer7,17 ε∇2 ψ = −ρe

(3)

(where ε is the dielectric permittivity of the electrolyte) and averaging over the channel length L, the streaming current (2) can be expressed in the form of the volume integral ε (∇2 ψ )v·ex̂ dV Is = − L Vc (4)

∫

where Vc = L × ΔSc is the control volume in which the streaming current is calculated. According to eq 4, evaluation of the streaming current in systems where the double-layer thickness κ−1 (the Debye screening length17) is comparable to the particle diameter d requires the knowledge of the electrostatic potential field ψ and the hydrodynamic velocity field v in the entire region occupied by the adsorbed particles. However, the problem simplifies considerably in the thin-double-layer limit

II. THEORETICAL ANALYSIS II.I. Streaming Current in the Thin-Double-Layer Limit. To gain quantitative insights into the effect of adsorbed particles on the streaming current, we consider electrokinetic charge convection near a planar solid substrate covered with a monolayer of spherical particles (see Figure 1). This interface− particle system is subject to the ambient fluid flow v0 = γ0̇ z ex̂

∫ΔS ρe v·ex̂ dS

(1)

(κd)−1 ≪ 1

(5)

because only the ζ-potentials ζI and ζP and the flow field in the immediate vicinity of the solid surfaces are needed. The thin-

where γ0̇ is the shear rate, z is the coordinate perpendicular to the interface, and ex̂ is the unit vector in the flow direction B


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Figure 2. Mechanism of generation of the streaming current in an interface−particle system subject to the external shear flow v0. The flow transports the ions in the thin double layers near the negatively charged interface and the positively charged particle surface.

current driven by the flow next to the particle surfaces. Both terms depend on the particle area fraction θ = NSg/SI (where 1 Sg = 4 πd 2 is the particle area projected onto the interface SI) and on the particle distribution. We note that the streaming current (7) linearly depends on the ζ-potentials of the interface and of the particles and on the local shear rate (which is proportional to the shear rate of the imposed flow, γ0̇ ). Since the surface charge and the charge distribution in the diffuse layer depend on the composition of the ionic solution, ζ-potentials ζI and ζP can be controlled by changing parameters such as salt concentration and pH. II.III. Expressions for the Streaming Current in Terms of Hydrodynamic Forces. Using relation (6) for the streaming current of the bare interface, relation (7) can be rewritten in a concise form

double-layer condition (5), assumed herein, is satisfied for a variety of systems that involve particles larger than 10 nm and solvent of ionic strength above 10−3 M.17 II.II. Interface and Particle Contributions to the Streaming Current. In the case of a particle-free planar interface, eq 4 can be explicitly integrated. Namely, combining (4) with (1) and integrating by parts (assuming also that the electric potential vanishes far from the interface), we find4,16 I0 = −εγ0̇ lζI

(6)

where l is the width of the control region. A theoretical evaluation of the streaming current is more complicated for particle-covered surfaces. As for a bare interface, we assume that the streaming current is driven by an imposed shear flow (1); therefore, the flow field v tends to the unperturbed flow v0 far from the interface. Near the interface, however, the flow v is perturbed by the presence of the particles, resulting in augmented charge convection. Moreover, there are two contributions to the streaming current, which stem from charge convection in the electric double layers near the interface SI and near the surface of the particles SP (see Figure 2). Thus, for example, if the ζ-potential of the particles, ζP, has the opposite sign to the sign of the ζ-potential of the interface ζI (which is a usual situation illustrated in Figure 2), the particle and interface streaming currents counteract. Moreover, a layer of particles adsorbed at an interface produces a hydrodynamic screening effect that reduces the flow near the interface and lowers the corresponding streaming-current contribution. In what follows, we analyze these effects. To obtain convenient expressions for the streaming current in terms of the interface and particle contributions, eq 4 is integrated by parts twice using the Gauss divergence theorem, and only the leading contribution in the dimensionless doublelayer thickness parameter (κd)−1 is retained. This procedure yields the following expression for the streaming current Is in the presence of the particles8,16,18 ÄÅ ÉÑ ε ÅÅÅ ÑÑ eẑ ·∇v(r) ·ex̂ dS + ζP n(r) ·∇v(r) ·ex̂ dS ÑÑÑ Is = − ÅÅζI Å ÑÑ L ÅÅÇ SI SP ÑÖ

∫

Is ζ = D I (θ ) + P D P (θ ) I0 ζI

(8)

where DI =

1 γ0̇ SI

∫S eẑ ·∇v(r)·ex̂ dS

1 γ0̇ SI

∫S

(9)

I

and DP =

n(r) ·∇v(r) ·ex̂ dS

(10)

P

are the interface and particle contributions to the streaming current, respectively. Integrals (9) and (10) can be expressed in terms of hydrodynamic forces acting on the adsorbed particles. The relevant forces are the total hydrodynamic force FPx =

∫S

n(r) ·σ ·ex̂ dS

(11)

P

and the viscous component of the hydrodynamic force FP′ x =

∫

∫S

n(r) ·σ ′·ex̂ dS P

(12)

where

(7)

Here, SI = lL is the area of the bare interface, SP = ∑Nk=1 SPk is the combined surface of all particles, N is the number of particles attached to SI, and n(r) is the unit vector normal to SP at the position r and pointing into the fluid. The first of the integrals on the right side of eq 7 represents the streaming current produced by the fluid flow v near the planar interface, and the second integral is the streaming

σ = σ ′ − pI ̂

(13)

is the stress tensor and σ ′ = η[∇v + (∇v)T ]

(14)

is its viscous component, p is the pressure, I ̂ is the unit tensor, and the superscript T denotes the matrix transposition. For C


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(21), which contributes to the interface streaming current DI, but not to the particle streaming current DP. In particular, as discussed in Section IV.IV, this distinct role of the pressure force has important consequences for the streaming current produced by the imposed flow next to a rough surface. Expression (23) was discussed in our previous study,8 and expressions (9) and (10) were evaluated using the HYDROMULTIPOLE expansion method. Relation (24) complements these results by providing a unifying interpretation of expressions for the streaming current in terms of hydrodynamic forces.

isotropic particle distributions, the streaming current depends only on the component of these forces in the flow direction x. To obtain the relation for DI in terms of the particle force (11), we first note that 1 DI = FIx ηγ0̇ SI (15) where FIx =

∫S eẑ ·σ ·ex̂ dS I

(16)

is the total hydrodynamic force acting on the interface. Equations 9 and 15 are equivalent because the pressure term and the transpose velocity gradient do not contribute to the integral (16). Thus, applying the force balance FIx + FPx − ηγ0̇ SI = 0

III. NUMERICAL AND EXPERIMENTAL METHODS III.I. Numerical Approach. III.I.I. Equilibrium and Random-Sequential-Adsorption Distributions. In our previous publication,8 the coefficients DI and DP were calculated for the equilibrium distribution of spherical adsorbed particles in a limited range of moderate area fractions. In the following section, we expand these results by providing new simulation data in the entire range of accessible area fractions for two experimentally accessible random particle distributions: the equilibrium and random-sequential-adsorption (RSA) distributions. As in our previous work, the particles interact via a hard-sphere excluded-volume potential. In experimental systems, a particle distribution similar to the RSA distribution is obtained in an irreversible adsorption process from a dilute suspension. The equilibrium distribution can be achieved if the adsorbed monolayer undergoes a relaxation process, for example, due to particle adsorption and desorption or surface diffusion. Thus, the RSA and equilibrium distributions are relevant for electrokinetic investigations of the particle deposition processes. In our numerical simulations, the equilibrium distribution of an adsorbed monolayer of spheres was generated using a compression/equilibration technique, where a low-density system is compressed to the desired area fraction θ in a sequence of compression steps, each followed by the Brownian-dynamics equilibration of the particle configuration. The RSA distribution was obtained by randomly placing spherical particles at the interface and rejecting adsorption steps leading to particle overlap. The simulations were performed for N = 100 and 256 particles in a doubly periodic system with a square unit cell. The above numbers of particles in the periodic cell are much smaller than in experiments where one typically uses a single particle-covered surface with a large number of particles arranged into a fixed configuration. To compensate for this difference, in numerical simulations we performed additional ensemble averaging over different initial configurations of particles. In this way, we obtained statistically converged results in the simulations performed in a small system. Experimental systems involve a large number of particles, and therefore the measurement results are self-averaging. III.I.II. Calculation of the Streaming Current. The streaming current (8) was calculated by solving the Stokes equations using the multipolar expansion method,19,20 as described in our earlier studies.2,8 Accordingly, the normalized hydrodynamic forces in eqs 23 and 24, FPx̅ and FPx ̅′ (the total and viscous contributions, respectively), are expressed as ensemble averages of certain combinations of multipolar friction coefficients.8 The contributing friction coefficients are evaluated using the Cartesian representation algorithm for

(17)

on the control volume Vc yields DI = 1 −

FPx ηγ0̇ SI

(18)

(see Appendix A for more details). In contrast to relation (16) for the interface force, the pressure term in eq 13 contributes to the total particle force (11). However, this pressure contribution is not present in the integral (10) defining the particle component of the streaming current. Thus, taking into account eq A11, the expression for DP is obtained in terms of the viscous part of the hydrodynamic force DP =

FP′ x ηγ0̇ SI

(19)

and not in terms of the total force. The total force and its viscous part differ by the force QP associated with the hydrodynamic pressure FPx = FP′ x + Q P

(20)

where QP = −

∫S

pn(r) ·ex̂ dS P

(21)

It is convenient to re-express the above relations in terms of the dimensionless force contributions FP̅ x =

QP FPx F′ , FP̅′x = Px , Q̅ P = ηγ0̇ SI ηγ0̇ SI ηγ0̇ SI

(22)

In the dimensionless formulation, the interface and particle contributions to the streaming current are DI = 1 − FP̅ x

(23)

DP = FP̅′x

(24)

where the normalized forces FP̅ x and FP̅′x differ by the pressure contribution FP̅ x = FP̅′x + Q̅ P

(25)

Equations 23−25 show that there is a unique relation between the streaming-current contributions resulting from charge convection in the double layers adjacent to the interface and particles and the hydrodynamic forces acting on the particle layer. Especially noteworthy is the role of the pressure force D


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Figure 3. Examples of the equilibrium and RSA particle distributions at different area fractions. The equilibrium distribution at θ = 0.8 has a hexagonal crystalline structure.

Stokes flow in a parallel wall channel with periodic boundary conditions21−23 and extrapolated to a one-wall periodic system using a limiting procedure of shifting the second, virtual wall to infinity.8 This procedure allows us to utilize doubly periodic algorithms developed for two-wall systems to obtain accurate results for a one-wall system with fluid extending to infinity. We estimate that the accuracy of our numerical results for the coefficients DI and DP is 0.002. Accuracy has been estimated by changing the multipole truncation order L0 (in the same way as described in ref 8), the distance H between the wall and virtual second wall, the number of particles N in the periodic cell, and the number of trial configurations in the statistical ensemble. In this paper, we present the results extrapolated to large values of L0 with the use of data from numerical computations obtained for regular particle lattices, described elsewhere. III.II. Streaming-Potential Measurements. In experiments discussed in this work, molecularly smooth and electrostatically homogeneous mica sheets were used as the substrate. Polystyrene particles of diameter d equal to 520 nm and dispersity index of ca. 5% (hereafter referred to as A500) were used as the model colloid suspension from which particle monolayers were deposited. Experimental results are also shown for monolayers of 1130 and 870 nm polystyrene particles of different surface properties (particles referred to as L +39 and L −70) and for 1520 nm melamine particles (referred to as M25). The physicochemical properties of the particles are described in more detail in refs 9 and 24. The screening-length parameter (κd)−1 in all experiments was much smaller than unity, which corresponds to the thin electric double-layer regime considered in our theoretical analysis. The ζ-potential of the particles, ζP, was regulated by varying the composition, concentration, and pH of the background electrolyte. For example, the A500 particles in 10−2 M NaCl solution were positively charged for pH below 10.5 and negatively charged otherwise. The change of sign of ζP for the L +39 and M25 particles in phosphate buffer (PB) occurred at pH = 7. The ζ-potential ζI of the negatively charged mica substrate was varied within wide limits by controlled adsorption of either Mg 2+ ions or cationic surfactant dodecyltrimethyl-ammonium chloride (DTACl). Investigating a variety of systems with different interface and particle ζpotentials (including the important cases where ζI ≈ 0 or ζP ≈

0) allowed us to discern the interface and particle contributions to the streaming current. The primary quantity measured in the experiments was the electric potential difference Es (streaming potential) rather than the streaming current because this ensures a much higher precision of the results. The streaming potential Es is connected with the streaming current Is by the ohmic dependence6,25−27 Es = IsR e

(26)

where Re is the overall electric resistance of the cell (mainly governed by the specific conductivity of the electrolyte). In the general case, Re can be expressed as Re =

L L = ΔScKe ΔSc(Keb + Kes/dc)

(27)

where ΔSc is the capillary or channel cross-section area, Ke is the overall electric conductivity of the cell, Keb is the specific conductivity due to the bulk electrolyte, Kes is the surface conductivity, and dc is the characteristic cell dimension in the direction perpendicular to the ambient flow vector. By measuring the slope of the potential Es vs the hydrostatic pressure difference ΔP (governing the flow in the experimental cell), one can evaluate the effective ζ-potential of a particlecovered substrate (37) using the formula9,18 ζ (θ ) =

η ΔEs(θ ) εKe Δ(ΔP)

(28)

which results from (26) and (27) and the relation between the pressure drop and shear rate at the walls of the cell. It should be mentioned that eq 28, first derived by Smoluchowski,4,5 has a universal significance, indicating that the effective ζ-potential derived from streaming-potential measurements does not depend on the cell geometry. The experimental procedure of determining the ζ-potential of particle-covered mica consisted of three stages: (i) the reference ζ-potential of bare mica in the background electrolyte was determined; (ii) particle monolayer of controlled coverage was deposited under the diffusioncontrolled conditions; and (iii) the cell was flushed with pure electrolyte and the streaming potential was measured as a function of time. E


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Table 1. Numerically Evaluated Contributions to the Streaming Currents AI, AP, DI, and DP for Particle-Covered Interfaces with the Equilibrium and RSA Distributions of the Adsorbed Particles equilibrium

RSA

θ

AI

AP

DI

DP

AI

AP

DI

DP

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.52 0.55 0.60 0.65 0.70 0.75 0.80 0.85

10.204 7.830 6.276 5.155 4.312 3.676 3.192 2.795 2.477 2.215 1.999

6.510 5.044 4.081 3.389 2.873 2.480 2.181 1.938 1.740 1.579 1.446

1.000 0.608 0.372 0.227 0.138 0.081 0.042 0.022 0.009 0.003 0.001

0.000 0.252 0.408 0.508 0.575 0.620 0.654 0.678 0.696 0.710 0.723

10.204 7.838 6.262 5.155 4.326 3.694 3.201 2.803 2.481 2.217 2.000 1.923

6.510 5.047 4.073 3.391 2.883 2.496 2.192 1.948 1.750 1.585 1.447 1.398

1.000 0.608 0.374 0.227 0.135 0.076 0.040 0.019 0.007 0.002 0.000 0.000

0.000 0.252 0.407 0.509 0.577 0.624 0.658 0.682 0.700 0.713 0.724 0.727

1.819 1.668 1.539 1.429 1.334 1.251 1.177

1.331 1.236 1.155 1.085 1.027 0.974 0.927

−0.0006 −0.0006 −0.0005 −0.0005 −0.0006 −0.0006 −0.0004

0.732 0.742 0.750 0.760 0.771 0.779 0.788

Afterward, the coverage of particles was determined under wet conditions by a direct optical microscope enumeration procedure.9 For particle area fractions exceeding θ = 0.2, atomic force microscopy (AFM) working in the semicontact mode was applied for imaging deposited particles. Typically, the particles were counted over 10−20 equal-sized areas randomly selected over the mica substrate. The total number of deposited particles was above 1000, which ensures a relative precision of these measurements below 2%. From the particle count, the average particle surface concentration, ⟨n ⟩ = ⟨N⟩/ SI, and the area fraction, θ = Sg⟨n⟩, were evaluated (where ⟨...⟩ denotes the average over the selected areas). The maximal area fraction achieved was approximately θ = 0.52, which is close to the limiting value of 0.547 that can be attained after infinite time of adsorption in irreversible particle deposition processes governed by the RSA mechanism.17,28 It was confirmed that particle deposition was irreversible over the time of the streaming-potential measurements under the prevailing flow conditions. No detectable detachment of particles from mica surface was observed, not even in the case where their ζ-potential was converted to negative values.9

Figure 4. Interface (Dα = DI) and particle (Dα = DP) contributions to the streaming current (8). The simulation results for the equilibrium particle distribution (blue triangles) and RSA distribution (red circles) are shown along with the exponential and linear plus exponential approximations (32) and (33) (solid lines).

AI = (1 − DI)/θ and AP = DP /θ

(29)

An examination of the numerical results indicates that for a given area fraction θ, the streaming currents for the equilibrium and RSA particle distributions are nearly the same. In the whole range of area fractions, the difference between the equilibrium and RSA results for DI and DP is significantly below 0.01. Thus, for random distributions of particles interacting via an excluded-volume potential, measurements of the streaming current can be used to estimate the area fraction of a particle monolayer without considering the details of a specific distribution. IV.II. Simple Analytical Approximations. In practical applications,1−3,9−12,16 it is useful to have accurate analytical approximations for the streaming current. Such simple and accurate theoretical expressions for the coefficients DI and DP are developed below and in Appendix B. The analysis is performed for the equilibrium distribution, but since the

IV. RESULTS AND DISCUSSION IV.I. Simulation Results. Examples of the equilibrium and RSA particle distributions are shown in Figure 3. The equilibrium and RSA distributions are fairly similar up to the area fraction of approximately θ = 0.54, where jamming of the RSA distribution occurs. The equilibrium distribution is disordered till the area fraction of approximately θ = 0.7, above which the system undergoes phase transition first to hexatic and then hexagonally ordered states.14,15 The streaming-current contributions DI and DP for particlecovered surfaces with equilibrium and RSA distributions are listed in Table 1 and depicted in Figure 4. For compatibility with the streaming-current analysis carried out in previous publications8,16 in Table 1, we also provide data for F


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The Journal of Physical Chemistry C equilibrium and RSA numerical data are nearly the same, the results are applicable to both cases. The starting point of our analysis is the virial expansion of the streaming-current contributions DI = 1 − c I1θ + ...,

DP = c P1θ + ...

(30)

where the virial coefficients c I1 = 10.2037 and c P1 = 6.5097

(31)

Figure 5. Schematic of the screening mechanism. In the screened regions between particles, the flow is reduced (practically to zero at sufficiently high area fractions). The shear rate in the active regions (exposed to the flow) approximately equals the applied shear rate γ̇0 .

8,16

have been evaluated in our earlier studies. For our present purpose, the leading-order expressions are sufficient; see Appendix B for a discussion of the higher-order terms. Note that the corresponding first virial coefficients (31) are equal to θ = 0 values of AI and AP listed in the first row of Table 1. In the case of the interface contribution to the streaming current, the results in Table 1 and Figure 4 indicate that DI rapidly tends to zero at high area fractions. The analysis shown in Appendix B indicates that this decay is exponential and is well described by a one-mode exponential approximation DI = e−c I1θ

surfaces and a linear increase of the exposed area when θ (and therefore the number of particles N) is increased. The exponential and linear plus exponential expressions (32) and (33) improve significantly over the rational-function approximations developed in ref 8, which are limited to a narrower range of area fractions θ ≤ 0.5. Relations (32) and (33) are simple, accurate, and valid for all area fractions up to close packing. The exponential approximation for the interface contribution (32) was earlier proposed in refs 16 and 18 based on an analysis of experimental results and is verified here with high accuracy. Let us stress the importance of the linear term in the behavior (33) of the particle contribution to the streaming current DP; the approximations previously considered do not have such a term and are much less accurate than the linear plus exponential approximation proposed here. We note that approximation (32) is free of fitting parameters and (33) has only one fitting parameter. Exponential form (32) can be interpreted as the first-cumulant formula based on the viral expansion of the function DI. An analysis of higher-order virial and cumulant approximations is presented in Appendix B. IV.III. Evaluation of the Effective ζ-Potential of a Particle-Covered Interface. Based on the analogy to relation (6) between the streaming current and ζ-potential of a uniformly charged smooth surface, streaming-current measurements for nonuniform surfaces are often interpreted in terms of the effective ζ-potential ζ defined by the relation I ζ=− s εγ0̇ l (37)

(32)

where the decay rate cI1 is determined by matching (32) to the viral expansion, eqs 30 and 31. The analysis of the particle contribution is more subtle. In this case, we observe that DP exponentially approaches a linear behavior at large area fractions, rather than a constant value (see Figure 4 and Appendix B), and we find that DP is well approximated by the combination of the linear and exponential terms DP = aθ + b(1 − e−c I1θ )

(33)

where

a = 0.202

(34)

is a fitting parameter, and b = (c P1 − a)/c I1 ≈ 0.6182

(35)

as determined by matching to the low-density virial expansion. We find that the accuracies of the exponential approximation (32) and linear plus exponential approximation (33) are better than 1.5 × 10−2 for both the equilibrium and RSA distributions. We note that the rate of exponential approach to the limiting behavior at large area fractions is the same for the interface and particle contributions (32) and (33). This implies that DI and DP satisfy the following approximate relation DP(θ ) = aθ + b[1 − DI(θ )]

The above equation is often used to characterize electrokinetic properties of particle-covered, nonuniformly charged, or rough surfaces. In the present section, we analyze the effective ζ-potential for interfaces covered by a monolayer of randomly distributed adsorbed particles. Combining eq 37 with eq 8, we obtain the following expression

(36)

As discussed in Appendix C, relation (36) is satisfied with accuracy better than 1% of the maximal value. Approximate relations (32) and (33) not only provide convenient fitting formulas for interpretation of experimental results but also shed light on physical interpretation of the behavior of the streaming-current contributions. First, the exponential decay captured by eq 32 stems from the hydrodynamic screening of the flow near the interface by immobile adsorbed particles (see the schematic shown in Figure 5). Second, relation (33) suggests that the exponential approach of DP to the linear behavior results from hydrodynamic screening of the unexposed parts of the particle

ζ ζ = D I (θ ) + P D P (θ ) ζI ζI

(38)

for the effective ζ-potential in terms of ζ-potentials of the interface and the particles. In Figure 6, our accurate approximations (32) and (33) are used to determine the effective ζ-potential (38) (normalized by the ζ-potential of a bare interface) as a function of the area fraction θ for different values of the ratio ζP/ζI. According to the results shown in Figure 6, the effective ζpotential is the most sensitive to the area fraction in the regime G


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Figure 6. Normalized effective ζ-potential (38) of a particle-covered interface evaluated from the exponential and linear plus exponential approximations (32) and (33). The values of ζP/ζI as labeled.

Figure 7. Normalized effective ζ-potential for a rough surface with spherical asperities modeled by a particle monolayer. Simulation results for random (blue triangles) and RSA (red circles) particle distributions are shown along with the linear plus exponential approximation (41) (solid line).

θ ≤ 0.2 and ζP/ζI ≤ 0. This rapid variation results from the exponential contribution in eqs 32 and 33; at larger area fractions, ζ/ζI crosses over to a linear behavior associated with the corresponding behavior of particle contribution DP. For ζP/ζI ≈ 1/b ≈ 1.6, the dependence of ζ on the area fraction is nearly linear for all area fractions because of the cancellation of the exponential terms. We note that for 0 < ζP/ζI < 1.57, the function ζ(θ) is nonmonotonic; thus, not always can the area fraction θ be uniquely determined from measurements of the effective ζpotential ζ. Also, at larger area fractions, the function ζ(θ) is relatively insensitive to θ. However, our results also predict that the effective ζ-potential is sensitive to the particle coverage at low area fractions, especially in the case of the opposite signs of the interface and particle ζ-potentials (which is the usual case under practically met situations). Thus, techniques based on streaming-current measurements are well suited for assessing the particle coverage in the regime where alternative in situ techniques are not available. IV.IV. Effective ζ-Potential of a Rough Surface. An important consequence of our analysis can be derived for a system where ζ-potentials of the interface and adsorbed particles are the same, ζP/ζI = 1. In this case, eq 38 gives the effective ζ-potential of a rough surface with spherical asperities corresponding to the monolayer of the adsorbed particles. The effective ζ-potential can be represented either as a sum of the interface and particle contributions ζ = D I (θ ) + D P (θ ) ζI

rough surface can be smaller than the microscale ζ-potential ζI = ζP by more than 25%. For a surface with randomly distributed monodisperse spherical roughness asperities, an approximately 25% reduction is observed in a wide range of area fractions. This effect can be explained by considering characteristic features of the flow in the system of adsorbed particles, as illustrated in Figure 5. Namely, the local shear rate γ ̇ in the valleys between particles is reduced due to the hydrodynamic screening, whereas γ ̇ next to the exposed portions of the particle surfaces remains close to the value γ0̇ corresponding to the shear rate of the ambient flow. Since only the exposed parts of the particles strongly contribute to the convective flux of the charge, the total streaming current is reduced. The same conclusion can be derived from the analysis of the pressure force contribution in eq 40. Since the pressure is increased on the upstream and decreased on the downstream surface of each particle, the pressure force (21) is positive, resulting in a reduced streaming current (40), consistent with the analysis based on the examination of the flow field. The above considerations are based on general hydrodynamic arguments and are valid for particles (asperities) of arbitrary shape. We thus conclude that ζ < ζI

independent of the specific microstructure of the roughness. Relations (40) and (42) are applicable to systems where the electric double-layer thickness is much smaller than the size of roughness asperities, κ−1 ≪ d. For small roughness asperities, that is, for d ≪ κ−1, the standard interpretation of the effective ζ-potential (37) as the microscopic ζ-potential of the surface is valid because the roughness only slightly perturbs the flow field in the double-layer region, that is, in the region where the charge convection occurs. However, in electrolytes of ionic strength above 10−3 M, the typical double-layer thickness is less than 10 nm; thus, the effect of roughness on the effective ζ-potential may be significant for roughness asperities of size above 10 nm. IV.V. Comparison with Experimental Data. The above theoretical results will now be compared with the experimental data obtained in refs 9 and 24 using the streaming-potential method. A typical example of the particle distribution at the

(39)

or, equivalently, in terms of the pressure force ζ = 1 − Q̅ P(θ ) ζI

(40)

and this relation is a consequence of eqs 23−25. Figure 7 presents simulation results for the normalized effective ζ-potential ζ/ζI for the rough-surface case ζP/ζI = 1 along with the analytical approximation ζ = aθ + b + (1 − b)e−c I1θ ζI

(42)

(41)

that follows from eqs 32 and 33. Approximation (41) is also represented in Figure 6, where it is shown by the long-dashed line. The results indicate that the effective ζ-potential of a H


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The Journal of Physical Chemistry C mica interface, derived from AFM imaging at the area fraction θ = 0.2, is depicted in Figure 8.

Figure 10. Normalized effective ζ-potential ζ/ζI for a monolayer of particles with ζP = 0 adsorbed on a negatively charged mica substrate. (Note that ζ/ζI = DI in this case). Experimental data24 for L +39 particles (solid triangles) and M25 particles (open triangles) in PB buffer at pH = 7 are shown along with the exponential approximation (32) (solid line). Figure 8. AFM image of the A500 particles adsorbed at mica at area fraction θ = 0.2 (courtesy of Dr Małgorzata Nattich-Rak).

The experimental results for the normalized effective ζpotential ζ/ζI for systems with several values of ζP/ζI are presented in Figures 9 and 10. Figure 11 shows the effective ζ-

Figure 11. Normalized effective ζ-potential ζ/ζP for a monolayer of positively charged particles on a mica substrate with significantly reduced ζ-potential ζI. (Note that ζ/ζP ≈ DP in this case.) Experimental results (symbols) are shown along with the linear plus exponential approximation defined by eqs 32, 33, and 43. The results are plotted for A500 particles in the MgCl2 electrolyte,9 ζI/ζP = −0.16 (golden solid diamonds and golden solid line) and for L −70 particles in DTACl solution,24 ζI/ζP ≈ 0 (black open circles and black dashed line). The electrolyte concentrations, pH, and ζ-potentials of the interface and particles are listed in Table 2.

Figure 9. Normalized effective ζ-potential ζ/ζI for a monolayer of A500 particles adsorbed on a mica substrate in NaCl electrolyte as the background solvent. Experimental data9 (symbols) are shown along with the linear plus exponential approximation defined by eqs 32, 33, and 38 (solid lines). Values of ζP/ζI as indicated; the electrolyte concentration, pH, and ζ-potentials of the surface and particles are listed in Table 2.

positive and there is no particle−interface electrostatic repulsion) and then flushing the system with a buffer solution regulating pH to a value within the range from 10 to 12 to obtain negative charge of the A500 particles. In all cases shown, the experimentally determined normalized effective ζ-potential ζ/ζI is in agreement with the theoretical results represented by the linear plus exponential approximation defined by eqs 32, 33, and 38. The theory quantitatively agrees with the experimental data (within the experimental error) and correctly describes qualitative features such as the rapid variation of ζ/ζI at low area fractions (especially for ζP/ζI < 0), followed by a region of much slower variation. To provide further evidence for good accuracy of approximations (32) and (33), we show results for two systems where the effective ζ-potential ζ is dominated by the

potential in a different normalization, ζ/ζP, for the case where |ζP/ζI| ≪ 1. The experimental parameters for the measurements depicted in these figures are listed in Table 2. Error bars shown in Figures 9 and 11 represent two standard deviations. Figure 9 shows results for A500 particles in the NaCl electrolyte at different pH and electrolyte concentrations. Representative results are shown for the usual case of particles having the opposite charge to that of the substrate, ζP/ζI < 0, and for a less typical system with ζP/ζI > 0, that is, where both the particles and the interface are negatively charged. The latter case is experimentally realized by depositing particles to the desired coverage at pH = 5.5 (where particle charge is I


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The Journal of Physical Chemistry C Table 2. System Parameters for Experiments Depicted in Figures 9−11 particles

electrolyte

concentration (M)

pH

ζI (mV)

ζP (mV)

ζP/ζI

figure

reference

A500 A500 A500 A500 L+39 M25 L−70 A500

NaCl NaCl NaCl NaCl PB PB DTACl MgCl2

10−3 10−2 10−2 10−2 3.3 × 10−4 3.3 × 10−4 1.2 × 10−4 9 × 10−3

5.5 5.5 10.2 11.8 7.0 7.0 6.0 5.5

−85 −70 −80 −80 −120 −120 0 −14

53 72 −20 −70 0 0 45 85

−0.62 −1.0 0.25 0.87 0 0 ∞ −6.1

9 9 9 9 10 10 11 11

9 9 9 9 24 24 24 9

interface and particle contributions DI and DP, respectively. These results are depicted in Figures 10 and 11. The experimental data presented in Figure 10 were obtained for latex particles in a phosphate buffer at pH 7, where the particle ζ-potential approximately vanishes; thus, ζI/ζP ≈ 0. In this case, there only appears the interface contribution to the streaming potential in this system. The experimental results clearly show an exponential decay of ζ/ζI, consistent with approximation (32). The observed decay is strong evidence for the underlying hydrodynamic screening mechanism illustrated in Figure 5. Figure 11 depicts results for two experimental systems where |ζI| ≪ |ζP|. The effective ζ-potential is dominated here by the particle contribution to the streaming current; therefore, the results are normalized by the particle ζ-potential ζ ζ = I D I (θ ) + D P (θ ) ζP ζP

initially decreases with the increasing roughness and then saturates. For θ above 0.3, the effective ζ-potential ζ assumes values significantly below the ζ-potentials of both the interface and particles, confirming our conclusion that roughness can reduce the effective ζ-potential determined from the streaming-potential or streaming-current measurements.

V. CONCLUSIONS This paper presents a new analysis of the streaming current produced by external shear flow near a charged interface covered by a monolayer of spherical particles. The thin-doublelayer regime is assumed, and this condition is satisfied in many practical applications that involve particles suspended in moderate and large ionic strength electrolytes. Since the substrate and the particles have their separate electric double layers, the streaming current of a particle-covered interface includes an additive interface and particle contributions DI and DP . An important new result of the present study is a set of precise numerical-simulation data for the equilibrium and RSA distributions of the adsorbed particles. The calculations have been performed for all accessible area fractions of the particle monolayer, including the high-density regime in the equilibrium systemthe domain where the monolayer undergoes a phase transition to a hexagonally ordered state. Previously, there were no simulation results for the RSA distribution, and the data for the equilibrium distribution were available only for small and moderate area fractions. We find that at a given area fraction θ, the values of the streaming current Is for the equilibrium and RSA particle distributions are nearly identical. Moreover, the phase transition that occurs at θ ≈ 0.7 for the equilibrium monolayer is not reflected in the function Is(θ), which shows that the streaming current is insensitive to the details of the monolayer microstructure. We thus conclude that for random particle distributions based on excluded-volume interactions, similar to the equilibrium and RSA distributions considered here, the particle coverage θ is the most important parameter determining the streaming current, whereas the microstructure of the particle monolayer plays a secondary role. This conclusion has an essential practical significance for measurements of nanoparticle and protein coverage under in situ conditions where the monolayer random structure is not known. Our numerical results for the interface contribution to the streaming current reveal that the function DI(θ) decays rapidly to zero when the area fraction θ is increased. We find that this decay can be accurately described by the exponential function with the decay-rate coefficient obtained from the low-density virial expansion. For the particle contribution, we have determined that at high area fractions, the quantity DP linearly

(43)

and the ratio ζI/ζP becomes a small perturbation parameter. In the experimental system represented by the solid diamonds in Figure 11, the small ratio ζI/ζP = −0.16 was achieved by reducing the ζ-potential of the negatively charged mica substrate to a small absolute value of |ζI| = 14 mV using the controlled adsorption of Mg2+; moreover, the particle ζpotential was increased to ζP = 85 mV by an appropriate choice of pH. In the second system (open circles), the ζpotential of the mica surface was reduced practically to zero using the adsorption of cationic surfactant DTACl to neutralize the surface charge; we thus have ζI/ζP ≈ 0, and there is only the particle contribution to the streaming current. The experimental data are plotted in Figure 11 along with the theoretical result obtained using the exponential and linear plus exponential approximations (32) and (33) for the interface and particle contributions to the normalized effective ζ-potential (43). The theoretical curves indicate that the difference in the effective ζ-potential ζ/ζP for the ratios ζI/ζP = 0 and −0.16 is on the order of ζI/ζP. This difference occurs only in the small area-fraction regime θ ≲ 0.2; for larger area fractions, the difference is negligible because of the exponential decay of the interface streaming-current contribution. The experimental data are consistent with our theoretical calculations within the scatter of experimental data, which further supports our analysis. One of our key results of general importance can be derived from the set of experimental data shown in Figure 9 for the case where the ratio ζP/ζI = 0.87 is almost equal to one. Since the ζ-potentials of the mica substrate and the particles are nearly the same, this system approximates a rough surface. The degree of surface roughness can be identified with the area fraction of spheres θ. As can be seen in Figure 9, the normalized effective ζ-potential for such a rough surface J



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depends on θ; the approach to this linear behavior is exponential, with the same exponent as the one obtained for DI. These results can be qualitatively interpreted in terms of hydrodynamic screening of the flow near the interface to which the particles are attached and near the portions of the particle surfaces that are not exposed to the ambient flow. Our analytical expressions for the streaming-current components can also be viewed as first-order cumulant approximations; higher-order cumulant expressions have also been discussed. We note that exponential approximation (32) for DI was previously proposed by Adamczyk, Warszyński, and Zembala16,18 based on the analysis of experimental data. So far, however, there has been no corresponding accurate approximation for DP. Here, we demonstrate that eq 32 for DI is satisfied with high accuracy for equilibrium and RSA particle distributions, and we provide a complementary relation (33) for DP. Due to their simplicity and precision, approximations (32) and (33) provide a useful tool for interpretation of experimental data. By inverting relations (32) and (33), one can determine in situ the particle coverage from streaming current or streamingpotential measurements. This procedure can be carried out for arbitrary particle size under the thin electric double-layer conditions. The streaming current/potential method for measuring particle coverage is especially sensitive in the lower coverage range, that is, in the regime where optical techniques such as ellipsometry or reflectometry become imprecise. Thus, the streaming current/potential method is an essential electrokinetic tool for investigating particle deposition as it complements other techniques in the regime where they become inadequate. It is expected that numerical and analytical results obtained in this paper for model systems of monodisperse spherical particles can be exploited as useful reference data for the analysis of macromolecule adsorption, especially protein molecules. Some of our conclusions also apply to more general microstructures. In particular, using the adsorbed particles as a model of roughness asperities, we have shown that the effective ζ-potential of a rough surface, evaluated from the streaming current, may be significantly smaller than the microscale ζ-potential of the material of which the surface is composed. Thus, our analysis has important consequences for interpretation of results of electrokinetic measurements of ζpotential for microstructured surfaces. The qualitative explanation of our results in terms of hydrodynamic screening helps better understand the influence of the microscale particle distribution on the streaming current. Since the behavior of the streaming current strongly depends on the screening effect, we expect that the conclusion that Is is insensitive to the particle distribution may have its limitations: the screening occurs in small gaps between the particles; thus, we anticipate that significant microstructural effects can occur for those particle distributions that leave relatively large particle-free areas exposed to the flow. Such exposed areas may occur, for example, for polycrystalline microstructures experimentally investigated in ref 29 or in systems where particle aggregates form before deposition. An analysis of these problems will be a subject of a forthcoming study.

APPENDIX A: DERIVATION OF EXPRESSIONS RELATING THE STREAMING CURRENT TO HYDRODYNAMIC FORCES In this appendix, we derive eqs 15, 18, and 19, which relate the interface and particle streaming-current contributions (9) and (10) to the hydrodynamic forces (11), (12), and (16). Derivation of Equation 15

To derive eq 15, we compare the integrals in the streamingcurrent and force relations (9) and (16). The pressure component of the stress tensor (13) does not contribute to the integral (16) because vectors ex̂ and eẑ are orthogonal; therefore, to show that the integrals on the right-hand sides of eqs 15 and 16 are equivalent, we need to demonstrate that the transpose velocity gradient contribution also vanishes

∫S eẑ ·(∇v)T ·ex̂ dS = 0

(A1)

I

To derive eq A1, we note that the no-slip boundary condition at the interface SI (which is in the plane z = 0) implies ∂vy ∂vx = 0 for z = 0 = ∂y ∂x

(A2)

Combined with the flow incompressibility condition ∇·v = 0

(A3)

relation (A2) yields ∂vz = 0 for z = 0 ∂z

(A4)

Equation A4 and the no-slip boundary condition indicate that the flow near SI has the form of a local shear flow tangent to the interface v = γż t + O(z 2)

(A5)

where γ ̇ = γ(̇ x , y) is the local shear rate, and t = tx ex̂ + t y eŷ is a unit vector parallel to SI. Relation (A1) is obtained from (A5) by performing the derivative and setting z = 0. Derivation of Equation 18

To derive eq 18, we need to show the force balance (17), which links the surface force Fix in eq 15 to the particle force FPx. The required relation (17) is obtained by considering the balance of hydrodynamic forces in a control volume Vc = L × l × H that occupies the cuboidal region 0 ≤ x ≤ L, 0 ≤ y ≤ l, 0≤z≤H

(A6)

where the streaming current is measured through the control area ΔSc = l × H. The lower boundary of the control volume, z = 0, coincides with the position of the interface SI. The flow field at the upper boundary SH in the plane z = H ≫ d equals to the applied flow (1). We assume the periodic boundary conditions for the velocity and pressure fields in the flow direction x and vorticity direction y. The force balance (17) is obtained by considering the hydrodynamic stress field in the fluid region Vf = Vc − VP

(A7)

where VP is the volume occupied by the particles. Integrating the stress balance K


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The Journal of Physical Chemistry C (A8)

∇·σ = 0

coefficients cIk and cPk, k = 1, 2, 3, were evaluated in our earlier study8 for the equilibrium distribution of adsorbed particles; their values are listed in Table 3. The expansions (B1) and (B2) are shown in Figures 12 and 13 along with our numerical results to illustrate the key features of the functions DI and DP and their expansions.

(valid in the creeping-flow regime assumed here) over the volume Vf and applying the Gauss divergence theorem yields

∫∂V n·σ dS = 0

(A9)

f

where ∂Vf is the boundary of the volume Vf. Taking into account the periodic boundary conditions in the x and y directions, the integral (A9) can be reduced to

∫S n·σ dS + ∫S I

n · σ dS +

∫S

n · σ dS = 0

H

P

Table 3. Virial Coefficients cIk and cPk for the Interface and Particle Contributions DI and DP to the Streaming Current8 and the Corresponding Cumulant Coefficients gIk for DI Calculated Using Equations B3, B4a and B4b a

(A10)

The force balance (17) is obtained from eq A10 by using the far-field form (1) of the velocity field on the upper boundary SH and employing the force definitions (11) and (16). a

Derivation of Equation 19

first

second

third

DI

cI1 = 10.2037

DP

cP1 = 6.5097

cI2 = −59.43 gI2 = −0.723 cP2 = −36.82

cI3 = 292 gI3 = 3.9 cP3 = 181

Results for the equilibrium distribution of adsorbed particles.

To derive the force relation for the particle contribution to the streaming current, eq 19, we need to show that the integrals on the right side of the streaming-current expression (10) and force expression (12) are equivalent. Taking into account relation (14) for the viscous stress, we thus need to demonstrate that the transpose velocity gradient (∇v)T does not contribute to the viscous force (12), that is,

∫S

n ·(∇v)T ·ex̂ dS = 0

(A11)

P

The above equation can be derived by relating it to the surface integral (A1). To this end, the identity

∇·(∇v)T = 0

(A12)

which follows from the flow incompressibility condition (A3), is integrated over the fluid region Vf, and the Gauss divergence theorem is applied. Using the periodic boundary conditions in the x and y directions and the far-field form (1) of the flow field near the interface SH, the resulting surface integral

∫∂V n·(∇v)T dS = 0

Figure 12. Analysis of the interface contribution to the streaming current. Simulation results for the equilibrium (blue triangles) and RSA (red circles) particle distributions are shown along with the virial expansion (B1) (main panel) and cumulant approximation (B3) (inset). Virial and cumulant approximations of the first order (solid lines), second order (dashed), and third order (dotted).

(A13)

f

can be reduced to

∫S eẑ ·(∇v)T dS + ∫S I

n ·(∇v)T dS = 0 P

(A14)

Equation A11 is obtained by projecting (A14) onto the x direction and using (A1).

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APPENDIX B: CUMULANT AND LINEAR PLUS CUMULANT APPROXIMATIONS Here, we develop the cumulant and linear plus cumulant approximations for the interface and particle streaming-current contributions DI and DP. The leading-order approximations in this family yield the convenient and precise approximate expressions (32) and (33), shown in Figure 4, and used in Sections IV.II−IV.V to interpret our numerical and experimental data. Our analysis combines the low-density virial expansions 2

3

DI = 1 − c I1θ − c I2θ − c I3θ + ...

(B1)

DP = c P1θ + c P2θ 2 + c P3θ 3 + ...

(B2)

Figure 13. Analysis of the particle contribution to the streaming current. Simulation results for the equilibrium (blue triangles) and RSA (red circles) particle distributions are shown along with the virial expansion (B1) (main panel) and linear plus cumulant approximation based on eqs B10 and B3 (inset). Virial and cumulant approximations of the first order (solid lines), second order (dashed), and third order (dotted). The dash-dotted line in the main panel shows the limiting linear behavior of DP at large values of θ.

with pertinent observations regarding the behavior of the functions DI(θ) and DP(θ) at higher area fractions. The virial L


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The Journal of Physical Chemistry C Analysis of the Interface Contribution DI

we choose here an alternative approach, which utilizes a very accurate numerical observation that the functions D̃ P and DI are nearly identical

Our analysis of the function DI is based on the observation that the interface streaming-current contribution exponentially tends to zero when the area fraction θ is increased. According to the inset in Figure 12, at moderate area fractions, the exponent of this decay is constant, and at higher area fractions, its absolute value gradually increases. This behavior of the function DI suggests using approximations based on the cumulant expansion ln DI = −c I1θ(1 + gI2θ + gI3θ 2) + ...

D̃ P = DI

(B9)

as depicted in Figure 14 and discussed in Appendix C.

(B3)

with the coefficients gI2 = c I2/c I1 + c I1/2

(B4a)

gI3 = c I3/c I1 + c I2 + c I12/3

(B4b)

determined from the condition that (B3) is consistent (to the order O(θ3)) with the virial expansion (B1) (see Table 3 for the explicit values). The cumulant expansion (B3) truncated at the first order gives eq 32, and the third-order truncation yields 2

DI = e−c I1θ(1 + gI2θ + gI3θ )

Figure 14. Comparison between the exponential component D̃ P of the particle streaming-current contribution and the interface streaming-current contribution DI for the equilibrium particle distribution. Simulation results for DI (blue triangles up) and D̃ P (green triangles down). The inset shows the ratio (C1) to provide an estimate for the accuracy of relation (B9). The fitting parameter a is given by eq 34.

(B5)

These approximations given by eqs 32 and B5 are represented in the inset of Figure 12 by the solid line and dotted line, respectively. The results shown in Figures 12 and 4 reveal that already the first cumulant approximation (32) provides very accurate results. The increasing relative error seen in the inset of Figure 12 for θ ≳ 0.4 occurs only in the regime where DI has already decayed nearly to zero; thus, this deviation (corrected in the third order) is not important in applications of eq 32 to analyze experimental data. The second cumulant approximation is not shown in Figure 12 because as a result of the negative sign of the second cumulant coefficient, gI2 < 0, the corresponding approximation decays too slowly at high θ. We find that the second-order cumulant approximation is accurate only in the range of small area fractions, in contrast to the first and third cumulant approximations, which are accurate for all values of θ.

Combining eqs (B7) and (B9) yields DP(θ ) = aθ + b[1 − DI(θ )]

Relation (B10) involves only one independent fitting parameter a (see eq 34 for its value); the other parameter b is evaluated from eq 35 to ensure that (B10) gives the correct value of the first virial coefficient cP1. By combining eq B10 with the first- or third-order cumulant approximation for the function DI (eq 32 or (B5)), we obtain accurate first and third cumulant approximations for the particle contribution to the streaming current. A comparison of these approximations with numerical results is shown in the inset of Figure 13 and in Figure 4. We emphasize that these cumulant approximations have only one fitting parameter a; the other parameters have been obtained from the exact lowarea-fraction expansion. The leading-order approximation (33) is accurate and sufficient for the practical purpose of analyzing experimental data.

Analysis of the Interface Contribution DP

Unlike DI, the particle streaming-current contribution DP does not decay to a constant value but, as shown in Figure 13, tends to a linear behavior

D P ≈ aθ + b

(B6)

for large θ (where a and b are constant parameters discussed below). Our numerical results indicate that the approach to this linear form (B6) is exponential. The exponential approach is illustrated in the inset of Figure 13, in which we plot the results for the function D̃ P(θ ) = −b−1(DP − aθ − b)

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APPENDIX C: NUMERICAL EVIDENCE FOR APPROXIMATE IDENTITY (B9) Here, we discuss the evidence for validity of the accurate proportionality relation (B9) between the interface streaming current DI and the exponential component of the particle streaming current D̃ P. The evidence is based on our numerical results and the analysis of the virial expansions. The direct numerical evidence is presented in Figure 14. The simulation results plotted in the main panel show that the quantities DI and D̃ P coincide within the graph resolution. The accuracy of relation (B9) can be estimated from the results depicted in the inset for the ratio

(B7)

that is, DP from which the linear behavior (B6) has been subtracted and which has been normalized to give D̃ P(0) = 1

(B10)

(B8)

The observed exponential decay of D̃ P suggests using a cumulant expansion for this function. Such cumulant approximations involve two fitting parameters: the slope a and the constant term b of the high-density linear behavior (B6). To reduce the number of independent fitting parameters, M


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The Journal of Physical Chemistry C R=

1 − D̃ P 1 − DI

part by the Statutory Activity Fund of the Jerzy Haber Institute of Catalysis and Surface Chemistry PAS. J.B. was supported by NSF Grant CBET 1603627. He would also like to acknowledge the hospitality of IPPT PAN and financial support from the National Science Center under grant 2014/15/B/ST8/ 04359 during his summer visits. M.L.E.-J. was supported in part by the National Science Center under grant 2014/15/B/ ST8/04359.

(C1)

The results indicate that (C2) R≈1 with the accuracy better than 0.01, demonstrating that relation (B9) is satisfied within 1% of the maximal value D̃ P = DI = 1. Further evidence is obtained by comparing expansion (B1) to the virial expansion of the function D̃ P 2

3

D̃ P = 1 − c P1 ̃ θ − c P2 ̃ θ − c P3 ̃ θ + ...

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(1) Oćwieja, M.; Adamczyk, Z.; Morga, M.; Kubiak, K. Silver particle monolayers−Formation, stability, applications. Adv. Colloid Interface Sci. 2015, 222, 530−563. (2) Adamczyk, Z.; Sadlej, K.; Wajnryb, E.; Nattich, M.; EkielJeżewska, M.; Blawzdziewicz, J. Streaming potential studies of colloid, polyelectrolyte and protein deposition. Adv. Colloid Interface Sci. 2010, 153, 1−29. (3) Adamczyk, Z.; Nattich, M.; Wasilewska, M.; Zaucha, M. Colloid particle and protein deposition−Electrokinetic studies. Adv. Colloid Interface Sci. 2011, 168, 3−28. (4) Smoluchowski, M. Contribution a la theorie de l’endosome electrique et de quelques phenomenes correlatifs. Bull. Akad. Sci. Cracovie 1903, 1, 182−199. (5) Smoluchowski, M. In Marian Smoluchowski, Selected Scienitfic Works; Cichocki, B., Ed.; Wydawnictwa Uniwersytetu Warszawskiego, 2017; pp 59−75. (6) Lyklema, J. Fundamentals of Interface and Colloid Science, Solid− Liquid Interfaces; Academic, 1995; Vol. II. (7) Ohshima, H.; Furusawa, K. Electrical Phenomena at Interfaces: Fundamentals, Measurements, and Applications; Marcel Dekker, 1998. (8) Sadlej, K.; Wajnryb, E.; Bławzdziewicz, J.; Ekiel-Jeżewska, M.; Adamczyk, Z. Streaming current and streaming potential for particle covered surfaces: virial expansion and simulations. J. Chem. Phys. 2009, 130, No. 144706. (9) Adamczyk, Z.; Zaucha, M.; Zembala, M. ζ-Potential of Mica Covered by Colloid Particles: A Streaming Potential Study. Langmuir 2010, 26, 9368−9377. (10) Wasilewska, M.; Adamczyk, Z. Fibrinogen Adsorption on Mica Studied by AFM and in Situ Streaming Potential Measurements. Langmuir 2011, 27, 686−696. (11) Da̧bkowska, M.; Adamczyk, Z. Human Serum Albumin Monolayers on Mica: Electrokinetic Characteristics. Langmuir 2012, 28, 15663−15673. (12) Da̧bkowska, M.; Adamczyk, Z. Ionic strength effect in HSA adsorption on mica determined by streaming potential measurements. J. Colloid Interface Sci. 2012, 366, 105−113. (13) Henderson, D., Davison, S. G. In Physical Chemistry, An Advanced Treatise, Vol. II, E 361; Eyring, H., Ed.; Academic Press: NY, 1967. (14) Bernard, E. P.; Krauth, W. Two-Step Melting in Two Dimensions: First-Order Liquid-Hexatic Transition. Phys. Rev. Lett. 2011, 107, No. 155704. (15) Engel, M.; Anderson, J. A.; Glotzer, S. C.; Isobe, M.; Bernard, E. P.; Krauth, W. Hard-disk equation of state: First-order liquidhexatic transition in two dimensions with three simulation methods. Phys. Rev. E 2013, 87, No. 042134. (16) Adamczyk, Z.; Warszyński, P.; Zembala, M. Influence of adsorbed particles on streaming potential of mica. Bull. Pol. Acad. Sci., Chem. 1999, 47, 239−258. (17) Adamczyk, Z. Particles at Interfaces: Interactions, Deposition, Structure; Interface Science and Technology V. 20, Academic Press, 2017. (18) Zembala, M.; Adamczyk, Z. Measurements of streaming potential for mica covered by colloid particles. Langmuir 2000, 16, 1593−1601. (19) Cichocki, B.; Felderhof, B. U.; Hinsen, K.; Wajnryb, E.; Bławzdziewicz, J. Friction and mobility of many spheres in Stokes flow. J. Chem. Phys. 1994, 100, 3780−3790.

(C3)

where c P1 ̃ = (c P1 − a)/b

(C4a)

c P2 ̃ = c P2/b

(C4b)

c P3 ̃ = c P3/b

(C4c)

are the virial coefficients determined from eqs (B2) and (B7). The results listed in Tables 3 and 4 show a good agreement Table 4. Virial and Cumulant Coefficients c P̃ k and gP̃ k for the Exponential Component D̃ P of the Particle StreamingCurrent Contribution Evaluated from Expressions (C4a−C4c) and (C6a and C6b)a D̃ P

first

second

third

c P1 ̃ = c I1

c P2 ̃ = − 59.56

c P3 ̃ = 293

gP2 ̃ = − 0.735

gP3 ̃ = 3.8

a

Results for the equilibrium distribution of adsorbed particles.

between the coefficients cIk and c P̃ k for k = 1, 2, 3 (i.e., for all values of k for which explicit results are available). This agreement carries over to the cumulant expansion ln D̃ P = −c P1 ̃ θ(1 + gP2 ̃ θ + gP3 ̃ θ 2) + ...

(C5)

with the coefficients gP2 ̃ = c P2 ̃ /c P1 ̃ + c P1 ̃ /2

(C6a)

gP3 ̃ = c P3 ̃ /c P1 ̃ + c P2 ̃ + c P1 ̃ 2 /3

(C6b)

The results, listed in Table 4, agree well (approximately within three significant digits for cPk ̃ and two significant digits for gP̃ k ) with the corresponding results for the virial and cumulant coefficients for DI (see Table 3).

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Zbigniew Adamczyk: 0000-0002-8358-3656 Jerzy Blawzdziewicz: 0000-0002-2446-3654 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This paper is dedicated to the memory of our friend and collaborator Eligiusz Wajnryb, who passed away in 2016. He shared with us numerical codes, and we benefited from discussions with him. This work was financially supported in N


Article

The Journal of Physical Chemistry C (20) Cichocki, B.; Ekiel-Jezewska, M.; Wajnryb, E. Lubrication corrections for three-particle contribution to short-time self-diffusion coefficients in colloidal dispersions. J. Chem. Phys. 1999, 111, 3265− 3273. (21) Bhattacharya, S.; Bławzdziewicz, J.; Wajnryb, E. Many-particle hydrodynamic interactions in parallel-wall geometry: Cartesianrepresentation method. Phys. A 2005, 356, 294−340. (22) Bhattacharya, S.; Bławzdziewicz, J.; Wajnryb, E. Hydrodynamic interactions of spherical particles in suspensions confined between two planar walls. J. Fluid Mech. 2005, 541, 263−292. (23) Bhattacharya, S.; Bławzdziewicz, J.; Wajnryb, E. Far-field approximation for hydrodynamic interactions in parallel-wall geometry. J. Comput. Phys. 2006, 212, 718−738. (24) Zembala, M.; Adamczyk, Z.; Warszyński, P. Streaming potential of mica covered by latex particles. Colloids Surf., A 2003, 222, 329− 339. (25) Delgado, A. V.; Gonzalez-Caballero, F.; Hunter, R. J.; Koopal, L. K.; Lyklema, J. Measurement and interpretation of electrokinetic phenomena. J. Colloid Interface Sci. 2007, 309, 194−224. (26) Werner, C.; Korber, H.; Zimmermann, R.; Dukhin, S.; Jacobasch, H. Extended electrokinetic characterization of flat solid surfaces. J. Colloid Interface Sci. 1998, 208, 329−346. (27) Lyklema, J. Electrokinetics after Smoluchowski. Colloids Surf., A 2003, 222, 5−14. (28) Hinrichsen, E.; Feder, J.; Jossang, T. Geometry of random sequential adsorption. J. Stat. Phys. 1986, 44, 793−827. (29) Suzuki, Y.; Hattori, Y.; Nozawa, J.; Uda, S.; Toyotama, A.; Yamanaka, J. Adsorption, Desorption, Surface Diffusion, Lattice Defect Formation, and Kink Incorporation Processes of Particles on Growth Interfaces of Colloidal Crystals with Attractive Interactions. Crystals 2016, 6, No. 80.

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