Hydrodynamic versus surface interaction impacts of roughness in

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Hydrodynamic versus surface interaction impacts of roughness in closing the gap between favorable and unfavorable colloid transport conditions Anna Rasmuson, Kurt VanNess, Cesar Ron, and William P. Johnson Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/acs.est.8b06162 • Publication Date (Web): 14 Feb 2019 Downloaded from http://pubs.acs.org on February 14, 2019

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Revision for Environmental Science & Technology

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Hydrodynamic versus surface interaction impacts

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of roughness in closing the gap between

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favorable and unfavorable colloid transport conditions

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Anna Rasmuson1, Kurt VanNess1, Cesar A. Ron1, William P. Johnson1,*

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Department of Geology and Geophysics, University of Utah,

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Salt Lake City, Utah 84112, United States

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Corresponding author. Email: [email protected]; Tel: (801)585-5033; Fax: (801)5817065.Engineering, University of Utah, Salt Lake City, Utah 84112, United States *

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Abstract

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Recent experiments revealed that roughness decreases the gap in colloid attachment between

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favorable (repulsion absent) and unfavorable (repulsion present) conditions through a

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combination of hydrodynamic slip and surface interactions with asperities. Hydrodynamic slip

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was calibrated to experimentally-observed tangential colloid velocities, demonstrating that slip

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length was equal to maximum asperity relief, thereby providing a functional relationship

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between slip and roughness metrics. Incorporation of the slip length in mechanistic particle

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trajectory simulations yielded the observed modest decrease in attachment over rough

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surfaces under favorable conditions, with the observed decreased attachment being due to

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reduced colloid delivery rather than decreased attraction. Cumulative interactions with

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multiple asperities acting within the zone of colloid-surface interaction were unable to produce

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the observed dramatic increased attachment and decreased reversibility with increased

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roughness under unfavorable conditions, necessitating inclusion of nanoscale attractive

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heterogeneity that was inferred to have codeveloped with roughness. Simulated attachment

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matched experimental observations when the spatial frequency of larger heterodomains

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(nanoscale zones of attraction) increased disproportionately relative to smaller heterodomains

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as roughness increased. Whereas attachment was insensitive to asperity properties, including

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the number of interaction per asperity and asperity height, colloid detachment simulations

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were highly sensitive to these parameters. These cumulative findings reveal that hydrodynamic

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slip moderately decreases colloid bulk delivery, nanoscale heterogeneity dramatically enhances

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colloid attachment, and multiple interactions among asperities decrease detachment from

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rough surfaces. 3 ACS Paragon Plus Environment

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Introduction

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Predicting colloid transport in porous media underlies applications ranging from targeted

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delivery of engineered nanoparticles (e.g., Zhang1) to understanding the underlying causes of

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waterborne disease outbreaks (e.g., Worthington and Smart2). Natural surfaces exhibit some

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degree of roughness, as has been demonstrated for various mineral surfaces,3,4 as well as for

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biological,5,6 and non-biological7,8 colloids. Whereas roughness has been recognized to have a

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significant impact on colloid transport for several decades, until recently, reports regarding the

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impacts of roughness on colloid retention in porous media were seemingly contradictory,

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demonstrating both increased3,9,10- and decreased11,12 retention with increasing roughness.

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By delineating roughness impacts under favorable (repulsion absent) and unfavorable

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(repulsion present) conditions, Rasmuson et al.8 demonstrated for carboxylate-modified

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polystyrene microspheres (CML) on smooth glass that roughness closes the gap between

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favorable and unfavorable collector efficiencies () (Figure 1, symbols). Observed values of 

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for smooth surfaces under favorable (Figure 1, blue triangles) versus unfavorable (Figure 1, red

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triangles) conditions differed by one to three orders of magnitude. Roughness reduced this gap

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from two directions; decreasing  under favorable conditions (Figure 1, blue circles and

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squares), and increasing  under unfavorable conditions (Figure 1, red circles and squares).

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Whereas AFM pulloff forces demonstrate that roughness can increase13 or decrease14 adhesion,

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in our transport experiments, roughness consistently increased adhesion compared to the

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smooth surface, as evidenced by decreased reversibility of attachment8 (Figure 1, panel d,

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dashed versus solid lines). That roughness closed the gap both from above (decreased  under



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favorable conditions) and from below (increased  under unfavorable conditions), reflects a

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previously unknown combination of hydrodynamic and surface interactions that we herein

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explore.

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Because surface interactions scale directly with radius of surface curvature,16 asperities reduce

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interactions relative to equivalent smooth surfaces. 17-22 However, cumulative interaction of

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multiple asperities across the zone of colloid-surface interaction (ZOI) can enhance surface

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interactions,7 depending on asperity height (radius of curvature) and the number of asperities

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interacting within the ZOI, as we explore below. The notion of surface coverage by asperities is

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well utilized in previous theoretical literature, wherein roughness is conceptualized as

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occupying only a fraction of the surface. 17-22However, this notion deserves further

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consideration (as we develop below), since surfaces are often entirely rough, being

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characterized by surface maxima and minima lying within a continuum of heights, with

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contiguous boundaries between asperities.8

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Roughness on a chemically homogenous (fully repulsive or attractive) surface reduces the

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surface interaction (via reduced radius of curvature) 17-22 regardless of whether the interaction

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is repulsive or attractive. However, reduced radius of curvature cannot reverse the interaction

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from net repulsive to net attractive, or vice versa, as extensively demonstrated in the

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literature15,17-20 and calculated herein. This is a critical consideration because reduced radius of

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curvature therefore cannot explain the well-reported increased retention of colloids with

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increased roughness under unfavorable conditions,3,9,8,10 nor can it explain the observed

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increased irreversibility of adhesion with roughness wherein colloids arrest with multiple


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interactions per asperity and/or multiple points of contact, such as attachment in

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concavities.8,15

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Colloid attachment on smooth surfaces under unfavorable conditions is explained by nanoscale

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surface heterogeneity (e.g., in charge, van der Waals, or other properties) wherein net

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repulsion is reversed to net attraction when a nanoscale heterodomain occupies a critical

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fraction of the ZOI. 21,24-26 It seems improbable to expect that nanoscale surface heterogeneity

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would be absent on rough surfaces, given its importance on smooth surfaces. It is therefore

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expected that nanoscale heterogeneity would be present, or even enhanced, on rough

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surfaces, where defects may create charge imbalance.27 Existing techniques (e.g., via X-ray

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photoelectron spectroscopy, AFM force volume, or other techniques) lack sufficient spatial

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resolution to reliably characterize nanoscale surface heterogeneity in -potential, Hamaker

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constant, and Lewis acid-base forces.3,28-30 While multiple forms of nanoscale heterogeneity

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may exist, we lack an analytical basis from which to distinguish them, and so for the sake of

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parsimony we attribute all heterogeneity to charge, while admitting that other forms may

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contribute.

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The observed increased attachment on rough surfaces under unfavorable conditions therefore

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requires consideration of nanoscale heterogeneity on rough surfaces. This approach was

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implicitly adopted by Bradford et al.,15 who placed attractive heterodomains at the tops of

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pillars intended to represent asperities on rough surfaces. Whereas Pazmino et al.26 found that

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Pareto distributed nanoscale heterogeneity provided good fits to attachment on smooth

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surfaces, there is no reason to expect that the distribution would remain equivalent on rough

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The topological impact of roughness on colloid-surface interaction has been investigated in

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terms of colloid interaction with concavities, both experimentally30,32 and theoretically.15 The

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as-yet unconsidered possibility of multiple ZOIs and multiple points of contact is a

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generalization of interactions with concavities, and depends on colloid:asperity size ratio

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(Figure 2, panel c), as well as the alignment of asperities, which can range from directly

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opposed (the equivalent of simple cubic packing) (Figure 2, panel a) to complementary (the

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equivalent of dense cubic packing) (Figure 2, panel b). Contact mechanics theory33,34 yields a

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contact radius for the smooth surface that is approximately 10% of the colloid radius for CML

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interacting with silica across water35 (Figure 2, panel c red bars). For smooth surfaces, the

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arresting torque lever arm is equal to the contact radius, whereas for rough surfaces with low

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colloid:asperity size ratios, the arresting torque lever arm may increase as multiple points of

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contact are established (Figure 2, panel c yellow bars).

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In contrast to the above-described surface interactions, hydrodynamic impacts of roughness

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are well described in existing literature. For smooth surfaces, the planes of colloid contact and

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fluid no-slip are coincident.26,36 However, on rough surfaces, fluid no-slip and colloid contact

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conditions occur across a continuum of heights bounded by the asperity maxima (Maxt), and

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minima (-Maxt) (Figure 4). Flow over rough surfaces can be represented by setting the effective

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no-slip boundary at some distance (slip length, b) below the effective contact surface, which is

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the effective plane within the asperity distribution where colloids come into contact with the

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collector.37-39 A finite tangential velocity (vslip) exists at the effective contact surface which lies

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within the range of asperity heights above the effective fluid no-slip boundary37-39 (Figure 4).

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Slip lengths exceeding asperity relief are possible because the no-slip boundary may be 7 ACS Paragon Plus Environment

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subsumed beneath asperity minima39-41 For simulation of colloid attachment, the absolute

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location of the effective contact surface is far less important than b, which dictates the value of

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vslip.

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Whereas slip is well recognized over rough surfaces, a relationship between b and roughness

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metrics (i.e. RMS, maximum asperity height, asperity spacing etc.) has not been established.

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Furthermore, existing literature does not address the extent to which slip impacts colloid

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delivery to, versus colloid immobilization on, the effective contact surface.

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The objective of this paper is to determine the relative impacts of the above-described

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hydrodynamic versus surface interaction impacts of roughness on colloid attachment and

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detachment. To achieve this objective, hydrodynamic impacts (i.e., the relationship between b

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and roughness metrics) were determined by comparison of particle trajectory simulations

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incorporating b and experimentally-observed tangential colloid velocities. Particle trajectory

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simulations incorporated surface interaction impacts of roughness via hemispherical

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asperities,17,19 and discrete representative nanoscale heterogeneity (DRNH)4,26,41 for comparison

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to experimentally-observed values of . The effects of roughness on adhesion were explored

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via consideration of multiple interactions per asperity. These hydrodynamic and surface

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interaction impacts of roughness were delineated by considering colloid attachment and

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detachment under both favorable conditions and unfavorable conditions.

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Methods

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Experiments



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Colloid velocities, collector efficiencies and detachment results from CML transport

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experiments on smooth and roughened silica8 were used to calibrate slip length, asperity

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coordination number, and heterodomain surface coverage as a function of roughness.

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Experiments examined colloid attachment and detachment for three colloid sizes (0.25, 1.0,

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and 2.0 m) (Molecular Probes Inc., Eugene, OR) in an impinging jet flow cell for three levels of

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collector roughness (silica) and two fluid velocities (1.7E-3 ms-1 and 5.94E-3 ms-1). Colloid

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attachment was quantified as the collector efficiency (η = number attached/number injected).

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Colloid detachment was examined in response to a factor of 90x increase in flow following

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loading, and was quantified as the percent remaining after flow perturbation. Unfavorable

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conditions and favorable conditions were obtained with solution chemistries of pH 8, 6 mM

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NaCl, and pH 2, 50 mM NaCl, respectively. Collector and CML electrophoretic mobility were

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measured in suspensions using a -potential analyzer (Mobiu, Wyatt Technology Corp., Santa

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Barbara, CA) and were -80.0 mV, -74.9 mV, -91.0 mV, -80.5 mV, for soda-lime glass, 0.2 m, 1.0

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m, and 2.0 m CML, respectively under unfavorable conditions, and -10.0 mV,-2.3 mV, -5.1

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mV, -5.4 mV, for soda-lime glass, 0.2 m, 1.0 m, and 2.0 m CML, respectively under favorable

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conditions.

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Collector and CML roughness were measured with an atomic force microscope (AFM) (model

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N9451A Agilent Technologies; Santa Clara, CA). Glass surface RMS roughness was 1 ± 0.7 nm

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(untreated), 38 ± 11 nm (NaOH-treated), and 550 ± 209 (HF-treated) (Supporting Information,

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Table SI-1). Colloid RMS roughness was 13 ± 7 nm (1.0 m), 10± 7 nm (2.0 m), 13 ± 6 nm (4.4

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m), and 27 ± 9 (6.8 m).


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Tangential colloid velocities (ut) were tracked in the near-surface using constant streaming

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images and were averaged for each colloid size for a minimum of 10 attached colloids. A

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student’s t-test was conducted to compare ut as a function of solution chemistry, surface

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roughness, and colloid diameter (Supporting Information Table SI-2).

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Further details regarding these experiments (e.g., microsphere suspensions, surface roughness

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measurements, and experimental setup) are provided in the Supporting Information and

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Rasmuson et al.8

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Colloid Trajectory Model

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A Lagrangian colloid trajectory model developed for impinging jet systems26 was used to

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simulate colloid trajectories. The model includes a full force-torque balance to determine

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whether the arresting torque exceeds the driving torque when the colloid is in contact with the

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surface. A more detailed description of the force and torque balance including fluid drag,

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hydrodynamic retardation, gravity, diffusion, XDLVO forces, and virtual mass is described in

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previous publications.25,26,426 Measured colloid and collector ζ potentials were input to simulate

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favorable and unfavorable conditions.

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Unfavorable conditions were simulated with DRNH in the form of oppositely charged

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heterodomains. Whereas surfaces are comprised by a continuum of heterodomain sizes with

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varied spatial frequencies, we represented this complexity with DRNH, wherein a given colloid

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size is arrested by heterodomains of a given size and larger.26,35,43 The colloid sizes examined

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ranged an order of magnitude (200 nm to 2000 nm) across which dramatic differences in

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diffusion, VDW interactions, ZOI, settling and fluid drag exist, constraining the size and spatial 10 ACS Paragon Plus Environment


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frequency of the DRNH.43 DRNH was also constrained by colloid reversibility in response to flow

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and ionic strength perturbations, wherein detachment occurred if adhesion to a given

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heterodomain generated insufficient arresting torque relative to the driving torque.35 Colloid

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detachment in response to a factor of 90x flow perturbation was simulated using final

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attachment and contact area parameters from attachment simulations.35,44 Contact mechanics

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parameters for colloid adhesion were determined via experiments examining detachment in

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response to flow and ionic strength perturbations, described in detail in VanNess et al.35

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Surface Interactions

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A conceptual description of surface interactions with asperities is provided below. Detailed

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calculation procedures are provided in the Supporting Information. Asperities were nominally

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represented as hemispheres on an otherwise smooth surface. The asperity height was set to

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the RMS roughness of the colloid, as smaller asperities are superimposed on larger asperities8

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(Supporting Information, Table SI-1) and have the greatest impact on the magnitude of surface

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interactions in the domain of the ZOI (Supporting Information Figure SI-1). 18,22,45, Furthermore,

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we restricted asperity size range to avoid generation of additional impinging surfaces that

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change the nature of the continuum flow field. A power-law relationship between colloid

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asperity height (hasp) and colloid radius (ap) was determined from AFM measurements of CML

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RMS roughness using four asperity8 (Supporting Information, Figure SI-2):

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(1)

ℎ𝑎𝑠𝑝 = 0.97𝑎𝑝0.34


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where hasp and ap are in units of nm. Asperities were placed contiguously such that surface

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coverage by asperities (hemispheres) was equal to the jamming limit (J=0.79) (Supporting

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Information, Figure SI-3).

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Asperity interactions considered three potential scenarios: 1) roughness on a single surface

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(colloid or collector); 2) roughness on both surface (colloid and collector); and 3) roughness

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insignificant on both surfaces (i.e. RMS roughness ≤ 5 nm).18 In this paper, interactions with the

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≤ 1 nm RMS rough surface were calculated using Scenario 1 because an RMS roughness

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exceeding 5 nm was present on the colloids (Supporting Information Table SI-1). Interactions

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with the 38 nm and 550 nm RMS rough surfaces were calculated using Scenario 2.

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For Scenarios 1 and 2, asperity-collector interactions were represented using a sphere-plate

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and sphere-sphere geometry, respectively, with asperity heights assumed equal on opposing

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surfaces in Scenario 2 to maintain strictly normal interactions. Lateral interactions were

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considered by multiplying interactions between individual asperities by the number of adjacent

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asperities (Nco), which may range between 1-4 for directly opposed and close-packed

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hemispheres, respectively (Figure 2, panels a and b). These interactions were normal between

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asperities but lateral relative to the equivalent smooth surface. .

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Surface interactions between asperities and offset equivalent smooth surfaces were calculated

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as the sum of electric double layer (EDL),46 van der Waals (VDW),47,48 Lewis acid base (LAB),49

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Born,50 and steric16 forces. Interactions between the offset smooth surfaces were calculated

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using sphere-plate interactions, with the sphere radius set to ap. The offset equivalent smooth

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surface is defined by average asperity minima, with the separation distance (H’) between offset



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smooth surfaces being defined by asperity coordination (Figure 2, panels a and b). Because real

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surfaces display a range of asperity heights, an average value of H’ (Figure 2, panel b) was used

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to calculate the offset equivalent smooth surface separation distance when roughness was

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present on both surfaces (Scenario 2). Furthermore, net surface interactions between offset

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equivalent smooth surfaces were similar for the range of H’ corresponding to opposite versus

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complimentary hemispheres (Supporting Information, Figure SI-4).

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Interactions between asperities were applied to VDW, EDL, and LAB forces as these interactions

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are functions of the radius of curvature. Asperity interactions were calculated as a function of

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separation distance between asperities (H). Interactions between asperities were multiplied by

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the number of asperities within the area of the ZOI (AZOI)51,52 and were multiplied by the

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jamming limit. Because asperities were contiguous and roughly equal to their height

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(Supporting Information, Table SI-1), roughness smaller than the colloid was evenly distributed

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throughout the ZOI. For larger roughness, an enhanced lever arm (rlever) is expected when the

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colloid experiences multiple contact points (each with its own ZOI), which occurs when the

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asperity size approaches the colloid size (Figure 2, panel c), and increases for decreasing

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colloid:asperity ratios, as described in the Supporting Information.

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Since VDW interactions are volume-based, the total VDW interaction was calculated by

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combining interactions with spherical asperities (sphere-sphere expressions using the radius of

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asperities) and interactions with the offset equivalent smooth surface (sphere-plate expressions

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using the radius of the colloid)(Supporting Information, Figure 2). While portions of the smooth

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surface are over accounted for by including the rear-hemisphere of the asperity in the van der

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Waals interaction, the interaction with the offset smooth surface including the rearward 13 ACS Paragon Plus Environment

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hemisphere was negligible relative to the contribution from the forward hemispheres7

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(Supporting Information, Figure SI-5). Therefore, the rear hemisphere was included in the total

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VDW interaction.

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As EDL interactions are surface-based, only the fraction of the offset equivalent smooth surface

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not occupied by asperities (smooth = 1-J) was included to account for EDL interactions between

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equivalent smooth surfaces (Supporting Information, Figure SI-3). LAB interactions are short-

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ranged (i.e. < 10 nm), therefore interactions between offset equivalent smooth surfaces were

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negligible. Spherical asperity interactions for EDL and LAB were added to the total interaction.

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There are no existing hemispherical expressions for EDL and LAB interactions, and the

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corresponding solution to the nonlinear Poisson-Boltzmann equation would be difficult if not

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impossible to derive.7 Although the EDL interaction between small asperities is less accurate for

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separation distances that approximate the Debye length (~4.3 nm), sphere-sphere interactions

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serve as a good approximation.7 Because these interactions are surface based, the interaction

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with the rear hemisphere can be assumed to be negligible relative to the forward hemisphere.16

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As LAB forces are shorter ranged than EDL and VDW interactions, AZOI for LAB interactions was

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calculated using the acid-base decay length (). Note that the inverse Debye length () serves

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as a decay length for calculating AZOI for EDL interactions.46

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Born and steric forces were calculated using sphere-plate interactions as a function of H. These

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interactions were not modified to account for interactions with asperities as they interactions

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are not a function of the radius of curvature. Short ranging repulsive steric interactions can

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originate from multiple sources at multiple scales, including structured water at interfaces,



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molecular-scale roughness emanating from polymeric structures or mineral defects, and nano-

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to micro-scale topographical features.16

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In the absence of steric interactions (i.e., Born repulsion in vacuum), colloid-collector

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separation distances for attached colloids do not exceed 0.2 nm,16 and corresponding force and

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torque simulations predict zero detachment in response to flow and ionic strength

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perturbations.35,44,53 In contrast, detachment from surfaces is routinely observed,26,35,53 thereby

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demanding additional repulsive interactions, which can be attributed variously to

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roughness,15,23 steric,35,44 or modified Born53 interactions. The net impact of these modifications

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is to push the attached colloid away from the collector surface, thereby decreasing attraction

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(e.g., VDW and possibly LAB), and rendering the colloid prone to detachment. Determination of

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steric parameters is described in VanNess et al.35

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Implicit Slip Length (b) Calibrated from Colloid Velocities

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In order to represent increased tangential fluid velocities (vt) over rough surfaces, the analytical

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flow field developed for the impinging jet system was modified to set an effective no-slip

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surface a distance b below the effective contact surface38,39 (Figure 4):

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v*t(z) = vt(z +b)

(2)

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Where z is the distance from the mean plane of the rough surface (here defined as the

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arithmetic mean taken from an arbitrary datum) to the center of the colloid, b is the slip layer

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length, and v*t is the modified tangential fluid velocity at distance z (Figure 4).


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For the normal component of flow (vn),the no-slip boundary condition was set at the effective

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contact plane54,55 (Figure 4). This decision was supported by COMSOL-simulated vn profiles that

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decayed to zero equivalently over the rough and smooth surfaces (Supporting Information,

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Figure SI-6 left panel). In contrast, vt profiles decayed to a finite tangential velocity over the

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rough but not the smooth surface (Supporting Information, Figure SI-6, right panel). Simulations

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using COMSOL are described in the Supporting Information.

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In order to reflect the lessened hydrodynamic resistance that takes place between approaching

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rough surfaces37.38,56-58the separation distance was shifted below the contact surface. The

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modified hydrodynamic functions are: fi(H) = fi(H+b)

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(3)

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Where fi are the original hydrodynamic correction functions59,60 with i equal to 1-4, with

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maximum values of fi occurring at H=0, and b is the slip length. In the modified version, the

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maximum values occur at H=-b, resulting in higher normal and tangential drag forces and higher

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tangential and normal fluid velocities than would be expected near a smooth surface.

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The slip length was calibrated to simulated tangential colloid velocities (ut), averaged from a

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minimum of 10 trajectories. Average ut was calculated for attached colloids simulated under

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unfavorable conditions after they had traveled a radial distance 10 times the injection radius

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(i.e. R>50 m) to ensure trajectories were predominantly tangential to the surface and had

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traveled a significant distance before attachment.

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Results



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Surface Interactions with Asperities

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That roughness cannot reverse an interaction from repulsive to attractive was true for all

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colloid sizes (Supporting Information, Figure SI-7), and is exemplified by 2.0 m colloids (Figure

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5). Over a favorable (repulsion absent) surface, the primary minimum depth decreased and

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shifted outward with increasing asperity size up to 20 nm diameter, and this trend reversed

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with further increased asperity size (Figure 5, panel a). The maximum asperity size in

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simulations was 20 nm since smaller asperities are superimposed on larger asperities.18,22 Over

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an unfavorable surface (repulsion present) (Figure 5 panels b and c), the energy barrier (Figure

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5, panel b), and the depth of the primary minimum (Figure 5, panel c), each expanded as

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asperity size increased to 4 nm, and then contracted with further increased asperity size. When

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the number of asperities within the ZOI was small (i.e. for asperities > 10 nm), the magnitude of

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the interaction was weakened, however when the number of asperities within the ZOI was

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large (i.e. for asperities ≤ 10 nm) the interaction was enhanced (Figure 5, panels b and c). This

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result is analogous to enhanced interactions calculated for concavities,15,20 which are another

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expression of interactions with multiple asperities.

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Colloid-surface interaction differed with the favorable surface versus heterodomains as

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roughness increased. For interaction with the favorable surface, primary minimum depth

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decreased with asperity size, whereas for interaction with heterodomains, primary minimum

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depth increased with asperity size up to 4 nm, then decreased with further increased asperity

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size. This contrast reflected lesser opposite colloid and collector -potentials for the low

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pH/high IS conditions that created favorable interaction versus strong opposite colloid and

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heterodomain -potentials under unfavorable conditions. 17 ACS Paragon Plus Environment

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Asperities of all sizes yielded weaker interactions when present on only one surface versus

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when present on both surfaces (Supporting Information, Figure SI-7). This was because multiple

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interactions per asperity can occur when roughness is present on both surfaces (Figure 2,

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panels a and b).

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Whereas the magnitudes of repulsion and attraction (barrier and minimum) were altered by

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roughness, they were not reversed (Figure 5). Similar trends were observed for all colloid sizes

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investigated (Supporting Information, Figure SI-7). To explain the observed increased

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attachment and decreased reversibility on rough surfaces under unfavorable conditions,

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nanoscale zones of attraction (heterogeneity) were required, as explored below.

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Hydrodynamic Interactions via Experimentally-Observed Tangential Colloid Velocities

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Experimentally-observed tangential colloid velocities (ut) increased modestly with roughness,

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by less than a factor of two under both favorable and unfavorable conditions for all colloid sizes

342

(p 0.05). The large standard deviation in ut for this colloid size (Figure 3,

346

panel a) reflects greater diffusion, also demonstrated by their relatively small Peclet numbers

347

(Supporting Information Table SI-3).

348

Explicit simulations of fluid flow around AFM-determined asperities in an impinging jet

349

geometry with the same dimensions as the experimental setup were simulated via CFD

350

(COMSOL Multiphysics® v. 5.1) for the 550 nm RMS surface (Supporting Information, Figure SI18 ACS Paragon Plus Environment


351

10). The resulting CFD-solved flow field was used in colloid trajectory simulations (also in

352

COMSOL, and lacking surface interactions) to determine whether the largest (and sparsely

353

located) asperities impacted near-surface colloid trajectories, as described in detail in the

354

Supporting Information. The maximum asperities for the 550 nm RMS surface (4.5 m),

355

influenced colloid trajectories upgradient and downgradient of these features. Relative to a

356

smooth surface, the mean colloid-surface separation distance and mean tangential velocity

357

increased by factors of 4.4 and 2.4 respectively (Figure 3, panel d) analogous to an atmospheric

358

boundary layer over mountainous terrain.61 CFD-simulated ut values (flow field with explicit

359

asperities) matched experimentally-observed values (Figure 3, panel c), indicating that spatially

360

sparse large asperities drove the experimentally-observed increase in colloid tangential

361

velocities with increased roughness.

362

Slip Layer (b) as a Function of Roughness

363

To avoid computational intensity of explicit simulation of flow around asperities and allow

364

colloid trajectory simulations over larger regions of rough surfaces, our goal was implicit

365

representation of asperities by determining a relationship between roughness and effective slip

366

length (b).38,39 Among a number of tested relationships for b ranging from 10 nm to 10 m

367

(equal to colloid RMS roughness and twice the relief of the roughest surface, respectively)

368

(Supporting Information Figure SI-11), the best fit to experimentally-observed ut values in

369

trajectory simulations was obtained by setting b equal to the relief between asperity maxima

370

and minima (2Max) considering asperities on both the colloid and collector (Figure 3, panel a).

371

The use of maximum relief as the relevant roughness parameter corroborates the explicit

372

simulations described above wherein the large sparse asperities determined the near-surface 19 ACS Paragon Plus Environment

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flow field (Figure 3, panels c and d), as well as previous studies.58 Slip lengths (b) equaled 110

374

nm, 1 m, and 4.5 m, for RMS roughness values ≤ 1 nm, 38 nm, and 550 nm, respectively.

375

Notably, this treatment of b was not dependent on colloid size, yet yielded experimentally-

376

observed ut values for each colloid size and for each roughness investigated (Figure 3, panel a).

377

Although the < 1nm surface possessed relatively insignificant roughness (2Max = 3.5 nm),

378

colloid roughness (2Maxt= 110 nm) was sufficient to increase ut relative to the smooth surface

379

(Supporting Information Figure SI-11, left panel black ovals). Simulated ut values using b = 2Max

380

slightly underestimated mean experimental ut values for the 0.25 m colloids, however they

381

were well within one standard deviation of the experimental mean, the standard deviation

382

being large due to higher diffusion for this colloid size (Figure 3, panel a, blue diamonds). These

383

colloids had average separation distances greater than the largest asperities (Figure 3, panel b,

384

blue lines) due to their enhanced diffusion.

385

Because fluid drag scales with colloid size, larger colloids would be expected to experience

386

higher ut in the absence of other considerations. However, our observations and simulations

387

demonstrated that relatively low diffusion and relatively strong secondary minimum interaction

388

held the larger (2 m) colloids closer to the surface relative to smaller colloids (Figure 3, panel

389

b, purple lines), resulting in lower ut values as they translated along the surface (Figure 3, panel

390

a, purple line and symbols).

391

Favorable Condition Attachment Predicted with Hydrodynamic Slip and Surface Interactions

392

Incorporating the above-determined slip (b = 2Max) and surface interaction (cumulative

393

asperity interactions across the ZOI) relationships into mechanistic trajectory simulations under 20 ACS Paragon Plus Environment


394

favorable conditions yielded a good match between predicted and experimentally-observed

395

collector efficiencies () (Figure 1 solid, panel a stippled and dashed blue lines). Simulated 

396

under favorable conditions was reduced by up to 0.2%, 24% and 70%, for the ≤ 1 nm RMS, 38

397

nm RMS, and 550 nm RMS roughness surfaces, respectively, relative to simulations

398

corresponding to a perfectly smooth surface (Figure 1, panel a blue lines). Under the factor-of-3

399

increased fluid velocity condition (5.94E-3 ms-1), simulated  under favorable conditions

400

bracketed experimental values (Figure 1, panel b blue lines).

401

Surface interactions with asperities were simulated with aasp set to the RMS of the colloid and

402

the number of interactions per asperity set to 2.5 (average between opposed and

403

complimentary spheres). Simulated attachment was equivalent when surface interactions with

404

asperities were eliminated (aasp = 0) (Supporting Information, Figure SI-12), indicating that

405

reduced colloid attachment was driven by decreased colloid delivery (via vn  0 at asperity

406

maxima and vt  0 at asperity minima), rather than via decreased arresting torque due to

407

reduced colloid-collector interaction (Figure 5, panel a).

408

Attachment under Unfavorable Conditions Predicted with Hydrodynamic Slip and Surface

409

Interactions

410

Values of  for the ≤ 1 nm RMS roughness surface were well predicted (Figure 1, panel a, solid

411

red line) by inclusion of a near-power law distribution of 60 nm, 120, and 170 nm

412

heterodomains at spatial frequencies of 6719, 575 and 456, heterodomains/mm2,

413

respectively26,35,43 (Figure 1, panel c solid red line). However, incorporating asperity interactions

414

with the DRNH determined for the smooth surface surfaces predicted  values that were far 21 ACS Paragon Plus Environment

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415

below those observed for rough surfaces in experiments (Supporting Information, Figure SI-13).

416

This was because radius of curvature effects alone cannot reverse unfavorable surface

417

interactions (Figure 5, panel b), necessitating increased spatial frequency of heterodomains to

418

facilitate attachment with increased roughness. In contrast to attachment, detachment was

419

highly sensitive to parameters describing surface interactions with asperities under both

420

favorable and unfavorable conditions (i.e. with aasp and Nco set to RMScolloid and 2.5,

421

respectively), as described below.

422

Increasing spatial frequency of heterodomains as a function of increased roughness in

423

simulations (Figure 1, panel c) yielded predicted values of  that reasonably matched those

424

observed on the rough surfaces (Figure 1, panels a and b stippled and dashed red lines), with

425

greatly improved prediction relative to other mechanistic models.4,25,26,57

426

Unfavorable simulations that incorporated DRNH but without the above-described slip

427

relationship (i.e., b = 0), predicted attachment similar to the smooth favorable surface, far

428

greater than those observed on rough surfaces (Supporting Information Figure SI-14, dash dot

429

lines). The same slip length was used under favorable versus unfavorable conditions because

430

favorable and unfavorable attachment converged with increasing roughness (Figure 1, panels a

431

and b, blue and red stippled and dashed lines), suggesting that slip was equivalent under

432

favorable and unfavorable conditions, as expected from the fact that EDL interactions extend at

433

most several tens of nm from the collector surface whereas slip lengths extend up to several

434

m. For 2.0 m colloids, simulated attachment was actually greater for 38 nm relative to 550

435

nm RMS roughness, due to the greater slip length associated with the 550 nm RMS roughness.

436

Under the factor of three higher fluid velocity, the gap between unfavorable and favorable  22 ACS Paragon Plus Environment


437

values did not close in simulations, reflecting experimental observations (Figure 1, panel b), and

438

indicates an enhanced impact of slip under higher fluid velocities.

439

Influence of Roughness on Reversibility of Attachment

440

Experimentally-observed colloid detachment in response to 90x flow perturbations was

441

greatest from smooth surfaces (Figure 1, panel d, triangles), with little to no detachment from

442

rough surfaces (Figure 1, panel d, open circles and squares), indicating roughness increased

443

adhesion under both favorable and unfavorable conditions. Simulated reversibility yielded

444

qualitative agreement with experimental reversibility for the ≤ 1 nm RMS surface (Figure 1,

445

panel d solid blue and red lines), as well as for rough surfaces (Figure 1, panel d, dashed and

446

dotted lines) for which attachment was irreversible under both favorable and unfavorable

447

conditions.

448

Irreversibile attachment on the rough surfaces was attributable to increased adhesion as the

449

number of interactions per asperity increased (Figure 1, panel d, open blue line). Attachment

450

ranged from partially reversible to irreversible when the number of interactions per asperity

451

ranged from 1 to ≥ 2.5, respectively for both favorable (Supporting Information, Figure SI-15,

452

panel d), and unfavorable conditions (Supporting Information, Figure SI-16, compare compound

453

versus simple red lines, respectively, for both solid and dashed). Additionally, simulated

454

detachment was sensitive to asperity height, wherein attachment ranged from irreversible to

455

fully reversible for aasp ranging from ≤ 12 nm to ≥ 20 nm, respectively (Supporting Information,

456

Figure SI-15, panels a and b).


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457

Adhesion under unfavorable conditions was increased not only by interactions among multiple

458

asperities, but also by increased opposite charge of heterodomains. For example, adhesion was

459

modestly increased (Supporting Information, Figure SI-16, compare solid versus dashed red

460

lines, respectively, for both simple and compound) when heterodomain -potential increased

461

from +10 to +80 mV, yielding stronger attraction (Figure 5, panels a and c). However, this

462

condition was still more reversible than equivalent interactions with multiple asperities

463

(Supporting Information, Figure SI-16, compare compound versus simple lines, respectively, for

464

both solid and dashed), indicating that irreversibility arose predominantly from interactions

465

among multiple asperities, and secondarily from charge heterogeneity. Note however, that on

466

the unfavorable smooth (i.e., ≤ 1 nm RMS) surface, sensitivity to heterodomain charge was

467

demonstrated by improved fit of simulated reversibility in response to decreased het (Figure 1,

468

panel d, red open line).

469 470

Discussion

471

Whereas others have demonstrated that roughness generates slip as described above, no

472

existing studies have incorporated the effects of slip in trajectory simulations. Only one

473

previous study has incorporated slip into trajectory simulations for favorable conditions,57 and

474

we have extended this effort to: a) develop a functional relationship between roughness

475

metrics and slip length that bypasses computationally intensive explicit representation of

476

asperities; b) extended the corresponding trajectory simulations to unfavorable conditions and

477

showed that radius of curvature effects cannot explain experimental attachment or



478

detachment: and c) incorporated interactions among multiple asperities and surface charge

479

heterogeneity to explain observed attachment and reversibility of attachment.

480

The primary impact of hydrodynamic slip was a moderate reduction in colloid delivery to the

481

effective contact plane (and moderately reduced attachment). Our results clarify that reduced

482

colloid attachment to rough surfaces under favorable conditions was driven by decreased

483

colloid delivery (i.e., decreased bulk transfer) rather than decreased adhesive torque (Figure 5,

484

panel a), as was previously claimed in the literature.31,62 Our findings also contrast against those

485

of Saiers and Ryan63 who placed hemispherical asperities into a Happel sphere-in-cell collector

486

flow field, and found that colloid mass transfer to the rough surface increased. However, their

487

approach did not alter the near-surface flow field, thereby introducing new surfaces on which

488

flow impinged, a characteristic we purposely avoided, as described above.

489

By considering both attachment and detachment, this study resolved that asperity height and

490

the number of interactions per asperity negligibly affected colloid attachment, but sensitively

491

impacted colloid detachment. In corollary, charge heterogeneity had a relatively minor impact

492

on colloid detachment from, but a major impact on colloid attachment to, rough surfaces under

493

unfavorable conditions. .

494

Experimentally-observed values of  increased with roughness disproportionately for larger

495

colloids (Figure 1, red circles and squares), necessitating a disproportionate increase in large

496

versus small heterodomains (Figure 1, panel c, stippled and dashed red lines). The extent to

497

which this actually reflects a larger nanoscale zones of attraction versus nanoscale zones of

498

reduced fluid drag (e.g., Scheurman et al.64) is unknown and worthy of further investigation. For


Page 26 of 43

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499

simplicity, we considered nanoscale heterogeneity to encompass increased delivery to net

500

attractive zones including potential nanoscale regions of low fluid drag.

501

Notably our simulations did not require an enhanced arresting torque lever arm (rlever) to

502

explain experimental observations, although we expect enhanced rlever for colloid:asperity ratios

503

below 10 (Figure 2, panel c) which may also contribute to decreased reversibility. For example,

504

simulated reversibility of 2.0 m colloids decreased when rlever exceeded the contact radius, and

505

was irreversible when rlever approximated ap (Supporting Information, Figure SI-15, panel c).

506

The combined experimental results and simulations indicate that the impacts of roughness and

507

nanoscale heterogeneity are inextricably linked. We acknowledge that the DRNH incorporated

508

here reflects acid-etched silica, and may or may not reflect environmental surfaces roughened

509

by natural processes; however, we believe that our finding that nanoscale heterogeneity is

510

inseparable from roughness is general, and is relevant to natural surfaces. To help others

511

explore this and related phenomena, we provide our executable codes at:

512

http://www.wpjohnsongroup.utah.edu.

513 514

Notes

515

The authors declare no competing financial interest.

516

Acknowledgements

517

We would like to thank three anonymous reviewers for their suggestions. This article was

518

developed under support to A. R. from the STAR Fellowship Assistance Agreement no. FP26 ACS Paragon Plus Environment


519

91780501-0 awarded by the U.S. Environmental Protection Agency (EPA). It has not been

520

formally reviewed by EPA. The views expressed in this publication are solely those of A. R. and

521

EPA does not endorse any products or commercial services mentioned in this article. Support

522

for W.P.J., K.V, and C.A.R., was provided by the National Science Foundation Hydrologic Science

523

Program (1547533). Any opinions, findings, and conclusions or recommendations expressed in

524

this material are those of the authors and do not necessarily reflect the views of the National

525

Science Foundation.

526

Supporting Information Available

527

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

528 529

Supporting Information: 4 tables, 23 figures, descriptions of Experimental Methods, Surface Interactions with Asperities, and CFD Simulated Colloid Trajectories: Explicit Roughness.

530


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Figure Captions

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Figure 1: Experimental (symbols) and simulated (lines) collector efficiencies () for favorable (blue lines and symbols) and unfavorable (red lines and symbols) conditions under 1.7E-3 m/s (panel a) and 5.94E-3 m/s (panel b) fluid velocity conditions. Three surfaces were examined with RMS roughness values of ≤1 (blue and red triangles), 38 nm (blue and red circles) and 550 nm (blue and red squares). Detachment experiments and simulations were in response to a factor of 90 flow (panel d). Simulated  values and % remaining after flow perturbation are shown with solid, stippled, and dashed lines for the ≤1 nm RMS, 38 nm RMS, and 550 nm RMS surfaces, respectively, with the slip length set to 2Max. Nanoscale asperities were included in simulations under both favorable and unfavorable conditions. Under unfavorable conditions 60 nm, 120 nm, and 170 nm heterodomains were used to capture attachment for 0.25 m, 1.0 m, and 2.0 m diameter colloids, respectively (panel c). The number of heterodomains/mm2 for the ≤1 nm RMS, 38 nm RMS, and 550 nm RMS surfaces are shown with solid, stippled, and dashed red lines, respectively. Modified from Rasmuson et al.6

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Figure 2: Schematic showing geometries for opposed (simple cubic packed, Nco = 1) spheres (panel a) and complimentary (dense packed, Nco = 4) spheres (panel b), and contact radius versus lever arm as a function of colloid:asperity ratio (panel c). Colloids are shown in green and asperities in blue. Black text and lines in panels a and b show calculations for the separation distance (H’) between the equivalent smooth surfaces located at asperity minima. Approximate contact radius is shown as red bars (moved from contact point for comparison to rlever) as a function of asperity height (hasp). Lever arm (rlever) is shown as yellow bars for the explicit rough surface. Colloid asperity ratios shown in white text. Contact radii for the CML glass system were approximately 10% of the colloid radius. Inset shows 0.1:1 colloid:asperity ratio.

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Figure 3: Lagrangian (Implicit Roughness) and COMSOL (Explicit Roughness) simulated ut (panels a and b), , and average simulated separation distance (panels c and d). Experimental ut values for 0.25 m, 1.0 m, and 2.0 m diameter colloids under unfavorable conditions are shown with blue diamonds, red squares, and purple triangles, respectively. Lagrangian simulated values with b set to 110 nm, 1.0 m, and 4.5 m, are shown with blue, red, and purple lines, respectively. COMSOL simulated values for 1.0 m colloids and streamlines are shown with red and green lines, respectively. Error bars represent standard deviation across ten or more trajectories. Note that results are staggered slightly to distinguish ut values corresponding to different colloid sizes. The explicit rough surface is shown by the jagged gray line.

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Figure 4: Schematic of parameters modified in the impinging jet model to account for slip, with the explicit rough surface shown with jagged grey line. The continuum expressions for vt are a function of z + b, with b = 0 for smooth surfaces. For rough surfaces, the no-slip condition for vn occurs at the contact plane and the no-slip condition for vt occurs at distance b below the effective contact plane. The slip velocity (vslip) is the finite tangential fluid velocity at the contact plane. 36 ACS Paragon Plus Environment


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Figure 5: Interaction energy profiles for 2 m colloids under favorable (left), unfavorable (center) and over an oppositely-charged heterodomain occupying 0.55 fraction of the ZOI (right). Interactions for a smooth surface (i.e. no asperities present) are shown with black line and interactions with 4 nm, 10 nm, and 20 nm asperities are shown with stippled red, dash-dot green, and dashed blue lines, respectively. Inset in central and right panels shows favorable interaction with smaller y-axis range to show primary minimum. Unfavorable conditions (pH 8, 6 mM) were calculated using colloid = -80.5 mV and collector = -80.0 mV, favorable conditions (ph2, 50 mM) were calculated using colloid = -5.1 mV and collector = 10.0 mV and heterodomain conditions were calculated using colloid = 80.5 mV and collector = -80.0 mV.

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Figure 5


Hydrodynamic versus surface interaction impacts of roughness in

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