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Environmental Processes

Why variant colloid transport behaviors emerge among identical individuals in porous media when colloid-surface repulsion exists William P. Johnson, Anna Rasmuson, Eddy Pazmino, and Markus Hilpert Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/acs.est.8b00811 • Publication Date (Web): 11 Jun 2018 Downloaded from http://pubs.acs.org on June 11, 2018

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Environmental Science & Technology

Draft manuscript for ES&T, 2018-06-02

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Why variant colloid transport behaviors emerge among identical individuals in

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porous media when colloid-surface repulsion exists

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Johnson W.P.*,1, Rasmuson A.1, Pazmiño E.2, Hilpert M.3

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Department of Geology & Geophysics, University of Utah, Salt Lake City, UT, USA

Department of Extractive Metallurgy, Escuela Politecnica Nacional, Quito, Ecuador 3

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Mailman School of Public Health, Columbia University, New York City, NY, USA

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Abstract

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We herein demonstrate the cause of well-observed variant transport behaviors for apparently

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identical colloids in porous media under conditions of colloid-collector repulsion (unfavorable

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attachment conditions). We demonstrate that variant colloid transport behavior under

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Corresponding Author: [email protected]; (801) 664-8289 ACS Paragon Plus Environment


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unfavorable conditions can be explained by inherently variable colloid residence times prior to

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arrest on grains (collectors). We demonstrate that the residence time distributions derived

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from particle trajectory simulations incorporating representative nanoscale heterogeneity

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provide quantitative prediction of colloid transport under unfavorable conditions. We

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quantitatively predict hyper-exponential retention profiles in glass beads from representative

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nanoscale heterogeneity determined for glass, and we qualitatively predict non-monotonic

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retention profiles in quartz sand from an estimated representative nanoscale heterogeneity for

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quartz. We also demonstrate that the transition from hyper-exponential to non-monotonic

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profiles among glass beads versus quartz sand under otherwise equivalent conditions is

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primarily driven by greater grain angularity and consequent greater length and number of grain

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to grain contacts in quartz sand relative to glass beads. That continuum-scale transport

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behaviors emerge from upscaling of simulated pore-scale colloid residence times corroborates

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the utility of representative nanoscale heterogeneity for quantitative prediction of colloid

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transport under unfavorable conditions.

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Introduction

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The profound impacts of bulk repulsion that confound prediction of colloid transport

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The ability to predict nano- and micro-particle (herein referred to as colloid) transport distances

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in porous media has relevance in applications ranging from groundwater resource protection

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from pathogens to in-situ groundwater contaminant remediation, as well as optimization of

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natural granular filters for water treatment.1-4 Under environmentally-relevant conditions, the

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distribution of retained colloids with distance from source (herein referred to as colloid 2 ACS Paragon Plus Environment

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retention profiles) deviate strongly from those predicted by colloid filtration theory (CFT) as

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described below, making colloid transport distances impossible to predict, even in cases where

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the spatio-temporal relationship between pathogen occurrence in sewage and underlying

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drinking water wells was demonstrated.5

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Both physical (e.g., advection, diffusion, and settling) and physicochemical (e.g., close-range van

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der Waals and electric double layer) interactions govern colloid transport in porous media, 6-8

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the latter being significant within colloid-collector (grain) separation distances of approximately

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200 nm, referred to herein as the near-surface fluid domain. Colloid filtration theory idealizes

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porous media as collections of forward flow stagnation zones (FFSZs) where flow impinges on,

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and diverges around, a grain. The FFSZ is defined by an axis and limiting radius within which

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fluid streamlines allow colloid delivery to the near-surface domain.6-8 The limiting radius

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expands with increased combined colloid diffusion and settling, such that depending upon their

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location in the up-gradient bulk fluid, colloids may enter the near-surface fluid domain of a

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grain somewhere between the FFSZ and rear-flow stagnation zone (RFSZ) where near surface

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flow converges and returns to bulk fluid.6,9,10

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Opposite versus like-charged colloid and collector surfaces generate electric double layer

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attraction versus repulsion, the latter condition yields a repulsive barrier at intermediate

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colloid-collector separation distances (Figure 2, panel a) that makes like-charged conditions

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unfavorable to colloid attachment (Figure 2, panel a). Under favorable conditions (lacking net

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repulsion) experimentally-observed colloid retention profiles are log-linear (in agreement with

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CFT). In contrast under unfavorable conditions (thought to be predominant in environmental

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contexts), retention profiles are either hyper-exponential or non-monotonic.11-14 They can

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transform from hyper-exponential to non-monotonic with increased ionic strength (IS).14

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Moreover, non-monontonic profiles may translate down-gradient with increased elution.14

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Understanding the cause(s) of these observed deviations from CFT is critical for predicting

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colloid transport distances under environmental conditions.

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Hyper-exponential deviation from log-linear retention profiles was first reported for bacteria,15

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and interpreted to reflect a distribution of “stickiness” (i.e., likelihood of attachment upon

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interception) among the cell population, with stickier cells retained up-gradient of less-sticky

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cells. Subsequently, hyper-exponential deviation was reported even for non-biological colloids

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such as carboxylate-modified polystyrene latex microspheres11,12 (CML), indicating that either a

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range of “stickiness” existed even among these apparently identical non-biological colloids, or

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alternatively, that the observed deviation reflects a process more fundamental than

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heterogeneity among the colloid population. Tufenkji and Elimelech16 posited that fast versus

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slow attaching colloids correspond to those colloids associated with secondary versus primary

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energy minima, respectively. However, the fact that colloids associated solely with secondary

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minima are not arrested10 refutes attribution of fast attachment to secondary minimum

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interaction. Bradford et al.17 posited that the fast-attaching subpopulation corresponded to

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those colloids transported into “dead-end” pores. However, hyper-exponential retention

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profiles are observed in well-sorted porous media lacking dead-end pores.11-12 Over the

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subsequent decade to present, hyper-exponential and non-monotonic deviation were reported

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to dependably emerge under bulk repulsion conditions, for biological, non-biological, micro-

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and nano-sized colloids.18,19 4 ACS Paragon Plus Environment

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In mechanistic particle trajectory simulations underlying CFT, near-surface colloids will not

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arrest when energy barriers exceed approximately 20 kT, and environmentally-relevant

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colloids/surfaces yield much higher energy barriers (hundreds to thousands of kT).20-22 To

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predict colloid immobilization on unfavorable surfaces, the notion of nanoscale attractive zones

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(herein called heterodomains) was introduced.21,22 The zone of interaction (ZOI) over which

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surface charge repulsion and van der Waals attraction act is a fraction of the colloid projected

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area because the colloid surface is convex to the collector (Figure 1), and colloid-collector

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interactions decrease rapidly with increased separation distance. The radius of the ZOI (RZOI)

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increases with colloid size and increases with decreased solution IS (Figure 2).21,23 The net

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colloid-surface interaction under bulk repulsive conditions can be attractive or repulsive

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depending on whether the ZOI lies over a heterodomain, as well as on heterodomain size

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relative to the ZOI.21,23 Furthermore, decreased IS may flip the interaction from net attractive

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to net repulsive as the ZOI increases in size relative to the heterodomain (Figure 2).24

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Representative nanoscale heterogeneities (heterodomain size and surface coverage) for silica

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and other minerals were determined via simulation of impinging jet experiments varying colloid

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size, fluid velocity, pH and IS, 21,22 yielding a representative nanoscale heterogeneity for silica

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(glass slides) that was Pareto-distributed (4:1 prevalence of 120 and 240 nm heterodomains)

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with 0.04% surface coverage by heterodomains.21

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The profound influence of nanoscale repulsion on colloid retention profiles described above is

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also obvious in pore scale transport experiments and simulations, where upon entering the



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near surface fluid domain colloids may attach rapidly or slowly corresponding to favorable

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versus unfavorable conditions, respectively.10,25 In addition to attachment, outcomes of near

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surface colloids include diffusion-driven re-entrainment to bulk fluid, and persistence in the

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near surface domain with eventual translation to the rear flow stagnation zone (RFSZ), where

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translation directly to the near surface domain of a down-gradient collector or expulsion to bulk

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fluid may occur.10 Pore scale observations show that hydrodynamic impacts of roughness are

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modest relative to colloid-collector impacts of roughness27. This, combined with the fact that

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these impacts were maximized rapidly as RMS roughness increased to several tens of nm,27

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supports findings reported by others34 that representative nanoscale heterogeneity simulates

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well the impacts of both charge and roughness heterogeneity.

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We show below that the link between bulk repulsive conditions and variant transport behaviors

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among identical colloids is the distribution of colloid residence times prior to arrest on the

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collector. The distribution is obtained from mechanistic pore-scale particle trajectory

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simulations with representative nanoscale heterogeneity that is responsible for colloid

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attachment under bulk repulsion conditions. We show that pore-scale trajectory simulations in

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a Happel sphere-in-cell collector with representative nanoscale heterogeneity quantitatively

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predict hyper-exponential retention profiles in glass beads at the column scale under bulk

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repulsion conditions. We show that the same mechanistic approach using an estimated

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representative nanoscale heterogeneity for quartz qualitatively predicts non-monotonic

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retention profiles observed in quartz sand under bulk repulsion conditions. While determining

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a quantitative representative heterogeneity for quartz sand lies beyond the scope of this paper,

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we demonstrate that non-monotonic retention profiles reflect increased accumulation and 6 ACS Paragon Plus Environment

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down-gradient translation of near surface (non-arrested) colloids, as driven by greater grain

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angularity, and greater number and length of grain to grain contacts per grain in quartz sand.

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Materials and Methods

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Experimental results examined here are from Li et al.11,13 which concerned transport of 1 µm

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carboxylate-modified polystyrene latex microspheres (CML) in uniform 510 µm diameter glass

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beads and quartz sand under favorable and unfavorable conditions with multiple IS. Additional

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experiments were herein run under equivalent conditions to Li et al.11 except that prior to

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experiments the glass beads were cleaned via the simpler SC-1 method26 rather than the more

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laborious acid cleaning method of Li et al.11 These new experiments examined the steady state

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shape of the hyper-exponential retention profiles by varying elution times following consistent

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injection times.

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Roughness of glass beads and quartz sand was measured with an atomic force microscope

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(AFM) (model N9451A Agilent Technologies; Santa Clara, CA) following SC-1 cleaning using

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contact mode in air with silicon nitride probes (type DNP-S10; Bruker Nano, Inc.) with a nominal

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spring constant of 0.12 N/m. Roughness was evaluated using SPIP software (Image Metrology;

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Hørsholm Denmark) and was characterized according to the root-mean-square height of

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asperiities. Further details are provided in Rasmuson et al.27

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Colloid-surface interactions included van der Waals, electric double layer, and non-DLVO

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interactions (steric and Lewis acid-base) as described in Pazmiño et al.21,24 Surface parameters

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for CML, glass and quartz from Li et al.,11,13 Tong et al.,26 and Pazmiño et al.,21,24 are provided in



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the Supporting Information (Table SI-1) for those parameters that varied with collector surface

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and IS. All experiments and parameters correspond to pH 6.72. Solutions were pH-buffered by

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addition of 0.46 g/L MOPS and 0.04 g/L NaOH. NaCl was added to achieve desired IS.

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Additional surface parameters held constant for all simulations are provided in the Supporting

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Information (Table SI-2). Colloid-collector interaction profiles as a function of separation

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distance are provided in the Supporting Information (Figure SI-1a-1e).

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Pore-scale trajectory simulations based on Lagrangian solution of forces and torques were

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performed as described in Pazmiño et al.21 Forces included colloid-surface interaction, fluid

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drag, diffusion, settling, virtual mass, and lift, where virtual mass is the inertia from

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displacement of fluid during acceleration or deceleration. The trajectory simulations were

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performed in a Happel sphere-in-cell collector.6,9 The representative nanoscale heterogeneity

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used in the simulations were Pareto-distributed 120 and 240 nm heterodomains (determined

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for glass slides)21, and at 1.0E-5 areal surface coverage for glass beads, and 2.0E-5 surface

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coverage for quartz sand, as explained further below. The heterodomain surface coverage for

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glass beads was determined by matching η determined from C/Co for the 0.02 M unfavorable

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condition, and the surface coverage was increased for quartz sand based on its greater

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roughness, as described below. These heterodomain surface coverages are more than an order

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of magnitude less than those determined for glass slides in an impinging jet21, possibly

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reflecting differences in materials as well as flow field (collector) geometry, as will be explored

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in future work.


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As described in the Results section below, two populations emerge from the pore scale

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mechanistic trajectory simulations. Among colloids that enter the near surface fluid domain in

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the Happel sphere-in-cell collector exists a fast-attaching population (α1) with pre-arrest

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residence times in the range of those attained under favorable conditions, and a slow-attaching

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population (α2) with pre-arrest residence times greater than those attained under favorable

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conditions. Near surface colloids may re-entrain to bulk solution at locations other than RFSZs

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(αreentrain), or may persist in the near surface fluid domain to the RFSZ (αRFSZ), where a

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subpopulation of RFSZ colloids may be translated directly to the near surface of down-gradient

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collectors (αtrans) or be expelled to bulk fluid (1-αtrans). Because near surface colloids must

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undergo one of the above options, these sub-fractions of the collector efficiency (η) total to

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unity:

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1 = + + + + 1 −

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This treatment is equivalent to that in Hilpert and Johnson28 wherein the first two terms (for

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arrest) on the right hand side are denoted as probabilities of attachment from bulk fluid and

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from near surface (ppa = α1 and pna = α2 respectively). The third-to-last and last terms on the

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right hand side of equation (1) are lumped into a probability of return from near surface to bulk

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pore fluid (pnp = αreentrain). The second-to-last term on the right hand side of equation (1) is

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referred to as a probability of direct transfer in near surface to down-gradient grains (pnn =

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αRFSZαtrans). In the tradition of preceding publications in the chemical literature we use “α” to

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denote these probabilities.


(1)


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Upscaling these pore-scale parameters to attachment rate constants for the two attaching sub-

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populations requires a decision regarding the influence of transport on these subpopulations.

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We herein posit that α1 (fast-attaching near surface colloids) is equal to the fraction of bulk

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fluid colloids (α1C0) that rapidly find arresting heterodomains, and this sub-population will not

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be replaced by colloids from the remainder of the colloid population (1-α1)C0 during transport

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through subsequent collectors. The remainder of the bulk population (1-α1)C0 is considered

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subject only to the possibility of slow attachment (with extended transport in the near surface

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fluid domain). The power of this hypothesis to explain experimental observations is explored

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below. We further posit that the α1C0 sub-fraction of the population arises from starting

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positions that place them on-trajectory proximal to heterodomains upon entry to the near

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surface. Previous work strongly suggests this possibility,29 where simulated values of η

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predicted from force/torque balance were found to decrease with transport distance in a series

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of contiguous collectors. We observe similar outcomes (unpublished), with factors of

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approximately 4, 10 and 15 decrease in η predicted for 0.5, 1.0, and 6.0 µm CML, respectively,

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between the first and second collectors in a series of dense cubic packed collectors.30 Such

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findings indicate that under favorable conditions the subset of colloids existing at starting

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points with trajectories likely to intercept the surface will rapidly deplete (within a few

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collectors) during transport, and so is not evident in retention profiles, which have segment

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lengths in the range of cm (typically tens of collectors). Under unfavorable conditions, this

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depletion should occur over larger transport distances since attachment requires both

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interception of the surface as well as the on-trajectory proximity of heterodomains. We will


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examine this possibility, and the influence of diffusion and settling, in future work involving unit

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cells connected in series.

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The continuum-scale rate constant (kf) for the fast-attaching subpopulations of colloids (α1) can

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be obtained by the upscaling strategy as detailed in Johnson and Hilpert:31

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= −

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where Nc/L is the number of collectors per unit length of transport, v is the bulk pore water

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velocity, η is the collector efficiency or fraction of introduced colloids that intercept the near

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surface fluid domain. Expressions for η and Nc/L are provided in Supporting Information (Table

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SI-3).

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This traditional upscaling strategy can be adopted to predict the rate constant (kf2) for

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attachment of remaining near surface colloids not subject to fast attachment:

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= −

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where v2 is the near surface fluid velocity. The values for kf and η were obtained from the

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observed steady state breakthrough plateau13, C/C0 (=10-4.0), under favorable conditions.

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The product of Nc/L (number of collectors per unit length of transport) and v or v2 (the

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characteristic velocities) in equations (2) and (3), respectively, are inverse residence times per

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collector over which the probability of retention per collector (Ln term) is applied (Figure 3).

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While the characteristic residence time for fast attachment of colloids from bulk fluid

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corresponds to the bulk fluid velocity (v), the characteristic residence time for slow attachment

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of near surface colloids corresponds to the near surface velocity (v2).

1 −

(2)

1 − α + α α !


(3)


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The expression to predict colloid retention profiles from attachment rate constants in the

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absence of explicit consideration of the near-surface fluid domain is:

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"# = $%& '(& exp ,−

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where: S(x) is the number of colloids retained per segment of sediment, V is the volume of the

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segment, t0 is the duration of colloid introduction, θ is the volumetric water content, C0 is the

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influent colloid concentration, kf* is a placeholder for the attachment rate for the two sub-

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populations (kf and kf2), v is the average bulk fluid velocity, and x is the midpoint of each

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segment.

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Non-monotonic retention profiles require explicit treatment of the near surface fluid domain,

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within which the slow-attaching sub-fraction of the bulk population (1-α1) must undergo at

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least one of the outcomes described above. The rate constant for net colloid loss from bulk

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fluid (kns) (excluding fast attachment, α1) is developed from the parameters described above

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using the characteristic velocity v (rather than v2) because this rate constant drives net loss from

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bulk fluid rather than from the near surface:

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= − 1 − α + α α !

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The explicit representation of the near surface fluid domain can be performed in a two-layer

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continuum model. For this purpose, we adapted the continuum Lagrangian model of Li et al.11

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to include two attachment rate constants. We validated the adapted two-layer continuum

-.∗ 0

# 1

(4)

(5)


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model by reproducing the hyper-exponential profiles of of Li et al.11 using the rate constants

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calculated from the pore scale simulations (Supporting Information, Figure SI-2).

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In the two-layer continuum model, the rate constant for slow attachment is not kf2, since the

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slow attaching colloids are explicit in the near-surface fluid domain, and are subject to

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translation to down-gradient collectors prior to arrest. We introduce k*f2 as the rate constant

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for attachment of near surface colloids that undergo advection in the near surface, which under

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steady state conditions is:

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∗ = −1 −

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where ac is the radius of the collector.

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Results and Discussion

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Roughness and Near Surface Velocities

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The root mean square (RMS) roughness of the glass beads and quartz sand determined from

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atomic force microscopy averaging 5 randomly selected locations scanned over a 5 µm x 5 µm

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area was 0.55 ± 0.32 nm and 0.82 ± 0.43 nm, respectively, with both values being similar to that

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for SC1-cleaned glass slides36 (0.57 ± 0.13 nm). These values were equivalent for larger scan

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sizes (20 µm x 20 µm) except for quartz sand for which RMS increased to 12.9 ± 5.9 nm. CML

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near-surface velocities under favorable conditions on silica increased only modestly (~50%) in

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response to more than an order of magnitude increased roughness (root mean square ranging

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< 1 nm to 38 nm),27 indicating that among the two impacts of roughness on colloid attachment,

03

(6)

4



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hydrodynamic interactions were dominated by colloid-collector interactions. Based on these

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results, we expect quartz sand and glass beads to have similar near-surface fluid velocities and

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distinct heterodomain surface coverages (greater for quartz sand).

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The average pore water velocity, v = 0.1667 m/hr, was obtained from flow rate, porosity (0.375)

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and cross-sectional area of the columns. The near-surface fluid velocity, v2, was indicated by

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pore-scale simulated near-surface colloid velocities for αRFSZ colloids, and ranged from

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approximately 0.001 to 0.003 m/hr, inversely dependent on IS (Table 1). The highest simulated

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pore-scale near-surface colloid velocity corresponding to the lowest IS (0.003 m/hr) was found

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to produce a down-gradient location for the maximum in the simulated non-monotonic

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retention profiles that matched those observed in experiments, as described below. Therefore,

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v2 = 0.003 m/day was considered representative of the near surface velocities for glass beads

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and quartz sand.

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The link between bulk repulsion and variant transport behaviors

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The goal of our pore scale trajectory simulations with representative nanoscale

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heterogeneity21,22,24 has been not only to capture colloid attachment quantitatively, but also to

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capture near surface transport dynamics. The pore scale simulations yield the fraction of near

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surface colloids that rapidly arrest (α1), slowly arrest (α2), return to bulk fluid (αreentrain), and

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persist in the near surface without arrest and translate to the RFSZ (αRFSZ). Also provided are

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the fractions of RFSZ colloids that translate directly to the near surface of a down-gradient

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collector (αRFSZαtrans) or are expelled to bulk fluid (αRFSZ(1-αtrans)).


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The impact of bulk repulsion on colloid residence times prior to arrest is demonstrated in Figure

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1, which shows simulated pore scale colloid trajectories with colloids colored according to their

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distance from the collector surface (bulk fluid domain = yellow, near surface fluid domain =

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green). Bulk attractive conditions (Figure 1, panel a, blue profile) yield colloid trajectories

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dominated by bulk fluid transport, where colloid trajectories are in bulk fluid (yellow) until

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attachment (Figure 1, panel b, bottom). In contrast, bulk repulsion (Figure 1, panel a, red

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profile) yields pore scale colloid trajectories with long residence in the near-surface fluid

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domain (green) (Figure 1, panel b, top).

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The means and variances of simulated pore-scale residence times prior to arrest on glass beads

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show were significantly lower under bulk attraction (blue) versus bulk repulsion conditions (red)

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(Figure 1, panel c). This is particularly dramatic for residence times in the near surface fluid

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domain (Figure 1, panel c, top), which ranged < 10 s versus 10 < t < 900 s for bulk attraction

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versus bulk repulsion conditions, respectively. The combined (total) residence times in the bulk

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and near surface fluid domains (Figure 1, panel c, bottom) also show the influence of repulsion,

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with total residence times ranging 15 to 76 s versus 114 to 865 s for bulk attraction versus bulk

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repulsion conditions, respectively. In the pore-scale trajectory simulations, two subpopulations

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emerged under bulk repulsion conditions: a) those with total residence times prior to arrest

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corresponding to those under bulk attraction (favorable) conditions (α1); and b) those with

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greater total residence times prior to arrest, corresponding to bulk repulsive conditions (α2)

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(Figure 1, panel c, top).



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The residence time distributions from simulated trajectories for 20 mM (Figure 1, panel c) and 6

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mM (Figure 3, panel b) IS demonstrate that the histograms of pre-arrest residence times shift

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with IS under the experimental conditions of Li et al.11 Different values of fast-attaching (α1)

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versus slow-attaching (α2) subpopulations, and different attachment rate constants (kf and kf2,

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respectively) emerged for the different IS (Table 1). All parameters (α1, α2, kf and kf2) increased

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with IS (secondary minimum depth) except αreentrain and the near surface colloid velocity (u2)

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which were inversely dependent on IS (secondary minimum depth). Note however, that the

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repulsive barriers to attachment remained thousands of kT strong regardless of IS (Supporting

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Information, Figure SI-1a-1e) consistent with the insensitivity of bulk surface zeta potentials to

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IS.20 The sensitivity of attachment to IS instead arises from the direct relationship between IS

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and the radius of the zone of colloid-collector interaction (RZOI),23 and the consequent inverse

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relationship with IS of the fraction of the ZOI occupied by attractive heterodomains.21-24

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Substituting the rate constants (Table 1) into equation (4) above and superimposing the values

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of S(x) for the α1 and 1-α1 subpopulations produced an excellent match to the experimentally-

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observed colloid retention profiles (Figure 3, panel c). The hyper-exponential retention profiles

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are explained by the two subpopulations identified in the residence time distribution as follows:

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a) the near-inlet portion of the retention profile reflects the fast-attaching fraction that has pre-

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arrest residence times corresponding to favorable conditions. This fraction shows the log-linear

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decrease with transport distance that is characteristic of favorable conditions (red dashed red

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line Figure 3, panel c); b) the distal portion of the retention profile reflects the slow-attaching

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fraction with pre-arrest residence times greater than under favorable conditions (red dotted

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line Figure 3, panel c). 16 ACS Paragon Plus Environment

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The significance of the match between predicted and experimentally-observed hyper-

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exponential retention profiles (Figure 3) is three-fold:

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a) The match strongly indicates that the observed variant behavior of identical colloids

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under bulk repulsion conditions is driven by their variant residence times prior to arrest,

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and so can be expected to emerge from any colloid population (even with identical

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individuals) under bulk repulsion conditions;

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b) The match suggests that representative nanoscale heterogeneity though not literal, is

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meaningful in terms of the resulting residence time distributions of colloids prior to

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arrest. Given the complexity of surface charge and roughness contributions to

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nanoscale heterogeneity, we cannot expect that the representation would be literal;

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however, the results indicate that the essential pore-scale transport behaviors emerge

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from this representation of nanoscale heterogeneity, since experimentally-observed

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complex continuum-scale transport behaviors emerge during up-scaling of pore-scale

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transport.

337

c) The results indicate that the starting positions of fast-attaching fraction of colloids

338

dictate their likelihood of rapidly finding arresting heterodomains upon entry to the

339

porous media, and that these positions are not rapidly replenished by the remaining

340

population during transport from pore to pore.

341

The simulations above used a value for v2 of 0.003 m/hr, and a value for αtrans of 0.25. The

342

calculated value of kf2 is directly related to each of these parameters (equation 3). The value of

343

v2 reflects the near surface colloid velocity at low IS (Table 1) as described above, and the value



344

of αtrans was set to achieve a good match to data (Figure 3, panel c). A one-in-four likelihood of

345

αRFSZ colloids being directly (or nearly so) translated to the near surface of a down-gradient

346

colloid seems reasonable based on movies of direct observation micromodel transport

347

(Supporting Information, SI-mp4 file). However, an equally good match to the retention

348

profiles can be obtained with v2 = 0.005 m/hr, and a value for αtrans of 0.0 (Supporting

349

Information, SI-3). The results for v2 = 0.003 m/hr, and αtrans = 0.0 are also shown for reference

350

(Supporting Information, SI-3). While these sensitivities will be further examined in future

351

work, they do not impact the points we make herein.

352

Tufenkji and Elimelech16 demonstrated mathematically that two independent deposition rates

353

within a single colloid population could produce hyper-exponential retention profiles. They

354

inferred slow deposition as corresponding to attachment in primary minima at heterogeneities,

355

and fast deposition as corresponding to arrest in secondary minima. In response to their paper,

356

we commented43 on the absence of mechanistic simulation of attachment in primary minima at

357

heterogeneities versus arrest in secondary minima, and their corresponding rates. Our work

358

here provides these mechanistic relationships, and furthermore shows that contrary to the

359

inference of Tufenkji and Elimelech,16 the fast attaching colloids are not those that persist in

360

secondary minimum association with the collector. Rather, persistence in secondary minimum

361

association with the collector reflects lack of attachment, and fast-attaching colloids do not

362

persist in secondary minimum association. Notably, in the two-layer continuum transport

363

model described below, the colloids that exit the column are solely those in the 1-α1 fraction.


Page 18 of 35

Page 19 of 35


364

The shape of the hyper-exponential profile is expected to be steady state based on equations 2-

365

4; i.e., the hyper-exponential profile shape does not reflect transient down-gradient translation

366

of near surface colloids away from the inlet. To confirm this expectation, we repeated the

367

experiments of Li et al.11 in SC1-cleaned glass beads with variable elution times (1, 6 and 19

368

pore volumes) following one pore volume of injection. The experimental results show identical

369

retention profiles regardless of elution time and therefore support this expectation (Supporting

370

Information, Figure SI-4).

371

It would seem reasonable that shorter versus longer pre-arrest residence times under bulk

372

repulsion conditions in pore scale simulations represent those colloids that arrested proximal

373

versus distal to the FFSZ of the collector, respectively. However, final locations of rapidly-

374

arrested colloids show the opposite (Figure 4). While the initial locations of both rapidly- and

375

slowly-arrested colloids corresponded to the FFSZ (Figure 4a), the final locations (Figure 4b) of

376

rapidly-attaching colloids corresponded to regions where near-surface tangential fluid velocities

377

are higher (outside the FFSZ), indicating that these shorter pre-arrest residence times

378

corresponded to entry into the near surface fluid domain proximal to on-trajectory

379

heterodomains in regions where greater tangential near surface fluid velocities drove rapid

380

encounter with on-trajectory heterodomains. That slowly-arrested colloids also attached in this

381

region (Figure 4b) indicates that on-trajectory heterodomains were not proximal upon entry to

382

near surface for these colloids.

383

Transition to non-monotonic retention profiles



384

Li et al.13 observed non-monotonic retention profiles in quartz sand (uniform 510 µm) despite

385

conditions otherwise equivalent to Li et al.11 in glass beads (1 µm CML colloids, 4 m/day

386

average pore water velocity, 3 PV injection, 7 PV elution, etc.) (Figure 5 panels a & b). In

387

contrast to the steady-state shape of the hyper-exponential retention profiles described above,

388

non-monotonic retention profiles can translate down-gradient with increased elution.14 This

389

transience demonstrates translation of near surface colloids down-gradient from grain to grain,

390

as considered in Johnson and Hilpert31, and recently simulated using a binomial model in

391

Hilpert and Johnson.28 Translation of colloids to down-gradient collectors requires increased

392

persistence of colloids in the near surface fluid domain (without arrest) (αRFSZ), which depends

393

on spatial coverage by heterodomains, depth of secondary minimum, and colloid diffusion. The

394

fate of near surface colloids upon reaching the RFSZ also determines their likelihood of

395

translation to down-gradient collectors, since they may be translated directly to the near

396

surface of down-gradient collectors (αRFSZαtrans) or be expelled to bulk fluid (αRFSZ(1-αtrans)).28,31

397

In contrast to glass beads, we can posit that the dual pore-grain network in quartz sand

398

facilitated greater translation of near surface colloids directly (or nearly so) to the near surface

399

of down-gradient collectors. This possibility is corroborated by previously reported x-ray

400

microtomography images and corresponding skeletonization results show that the number of

401

grain-to-grain contacts per grain, and the length of grain-to-grain contacts, are greater in

402

uniform quartz sand relative to uniform glass beads.32 This work also showed that under

403

favorable conditions, colloid deposition in grain to grain contacts was approximately a factor of

404

two higher for quartz sand relative to equivalent sized glass beads.32,33 Under unfavorable

405

conditions, colloid retention overall (although predominantly in grain to grain contacts), and 20 ACS Paragon Plus Environment

Page 20 of 35

Page 21 of 35


406

specifically in grain-to-grain contacts, were factors of 30 and 50 greater for quartz sand relative

407

to glass beads.33 The possibility that αRFSZ colloids are less likely to translate directly to down-

408

gradient collectors in glass beads relative to quartz sand is also corroborated in micromodel

409

experiments (Supporting Information, SI-avi files) under conditions equivalent to Li et al.11,13

410

with fluid velocities ranging from 6 to 30 m/day. In these experiments, longer grain-to-grain

411

contacts were apparent in quartz sand relative to glass beads, yielding longer near-surface

412

colloid residence times in quartz sand relative to glass beads. During elution with colloid-free

413

solution, greater translation of up-gradient near surface colloids was observed in quartz sand

414

relative to glass beads (Supporting Information, SI-avi files). Notably, the greater grain-to-grain

415

translation of near-surface colloids in quartz sand versus glass beads is not expected to be

416

driven by greater roughness in quartz sand. Pore scale observations27 demonstrate that

417

roughness increases colloid attachment and decreases colloid detachment (decreases

418

reversibility), likely through establishment of multiple points of colloid-collector contact.27

419

While optimization of the representative nanoscale heterogeneity for quartz is beyond the

420

scope of this paper, we can assume a representative heterogeneity for quartz to demonstrate

421

the transition to non-monotonic retention profiles. Based on the greater surface roughness of

422

quartz relative to glass (shown above), which impacts colloid-surface interaction predominantly

423

relative to colloid near-surface velocity,27 and which has already been demonstrated to be

424

representable via heterodomains34, we estimated a factor of two higher heterodomain surface

425

coverage on quartz sand (2E-5) relative to glass beads (both with Pareto-distributed 120/240

426

nm radii). The quartz-water-CML system also has a factor of two higher combined Hamaker

427

constant (1.96E-20 J) than the glass-water-CML system.35 These parameters were used in pore 21 ACS Paragon Plus Environment


428

scale simulations of CML transport on quartz sand (conducted at 3 mM and 20 mM IS) to

429

provide residence time parameters that were then upscaled to rate constants (Equations 5 & 6,

430

Table 1) implemented in the two-layer Lagrangian continuum model adapted from Li et al.11 in

431

order to predict the column-scale observations of Li et al.13

432

The predicted breakthrough-elution curves (Figure 5, panel a) and retention profiles (Figure 5,

433

panel b) demonstrated the experimentally-observed transition to non-monotonic retention

434

profiles. The imperfect agreement with experiments is expected because an assumed

435

representative nanoscale heterogeneity was used; however, the simulations indicate that the

436

transition from hyper-exponential to non-monontonic retention profiles is partly governed by

437

the intersection of the flow field with grain packing structure, since αtrans was 0.25 in glass

438

beads versus 0.5 in quartz sand (Table 1). The values of αRFSZ and its near corollary αreentrain,

439

show similarly strong dependence on IS in glass beads and quartz sand (Table 1). The simulated

440

first-segment retention did not increase as significantly with increased IS as was observed in

441

quartz sand experiments (Figure 5, panel b), indicating that the representative heterogeneity

442

determined for glass does not reflect quartz. The extent to which this sensitivity is due to

443

charge21,22 versus roughness27 heterogeneity will be examined in future work in development of

444

representative nanoscale heterogeneity for quartz.

445

Fast and slow attachment rates that characterize hyper-exponential profiles are also significant

446

in non-monotonic profiles, as demonstrated by the significant α1 and α2 values for quartz sand

447

(compare Table 1), and the fact that the near-source (column inlet) arrested colloid

448

concentrations were greater in the non-monotonic than in the hyper-exponential profiles


Page 22 of 35

Page 23 of 35


449

(Figures 3c versus 5d). In contrast to the hyper-exponential profiles on glass beads, which

450

lacked non-arrested near surface colloids, the non-monotonic profiles in quartz sand were

451

dominated by non-arrested near-surface colloids (Figure 5, panel d). In short, the non-

452

monotonic profile reflects relative abundance of retained non-arrested colloids rather than

453

reduction in attached colloids. Increased simulated elution time would translate the peak of

454

the simulated non-monotonic profile down-gradient, as observed in experiments14 and

455

simulations via a binomial model.28

456

Future refinement of the representative heterogeneity on quartz will allow determination of

457

whether it is possible for the non-monotonic profile shape to be steady-state as is the case for

458

hyper-exponential profiles on glass beads (Supporting Information, SI-4). In this work, we

459

developed simple dual rate constants (Table 1) from the distributions of residence times prior

460

to arrest. We expect that future work using distributions of rate constants developed from

461

distributions of residence times prior to arrest (e.g. Figures 1c, 3a, and 5c) will demonstrate

462

improved fits to experimentally-observed extended tailing in breakthrough-elution curves

463

(Figure 5a) and shapes of retention profiles (Figure 3b, and Figure 5b).

464

Tong et al.36 examined transport of three CML sizes in glass bead columns in series under the

465

same conditions as Li et al.11, and observed that the hyper-exponential character of retention

466

profiles was diminished in down-gradient relative to up-gradient columns as well as parallel

467

columns with reduced Co. They concluded that a faster-attaching fraction of the colloid

468

population was removed during transport in the up-gradient column; however, no compelling

469

characteristic was identified to distinguish influent versus effluent colloid populations. The



470

reduced slope and reduced magnitude of the down-gradient retention profiles are qualitatively

471

consistent with reduced flow via diversion for sampling between columns. Since flow was

472

equivalent in up-gradient and parallel columns (1.7 mL/min), the contribution of reduced flow

473

to the down-gradient profile shape is, unfortunately, uncertain.

474

Significance

475

Our pore-scale simulations indicate that nanoscale heterogeneity that is responsible for colloid

476

attachment under bulk repulsion conditions produces variant near-surface colloid residence

477

times prior to their arrest on porous media grains, even for identical colloids. The variant pore-

478

scale residence times up-scale to continuum-scale rate constants from which emerge hyper-

479

exponential and non-monotonic retention profiles, as well as extended tailing, all of which are

480

ubiquitously observed under unfavorable conditions. The results indicate that hyper-

481

exponential retention profiles reflect fast- and slow-attaching colloids (among an identical

482

population) with negligible accumulation of non-arrested near-surface colloids. In contrast, the

483

results indicate that non-monotonic retention profiles reflect dominance of non-arrested near-

484

surface colloids. These different transport behaviors are in part determined by colloid

485

persistence in the near surface without arrest, as well as by whether RFSZ colloids are expelled

486

to bulk fluid versus translated directly (or nearly so) to the near surface of down-gradient

487

grains. The latter outcome is facilitated by longer grain-to-grain contacts associated with

488

angular non-spherical grains. The result suggests that near-surface colloid accumulation

489

without arrest is more significant in natural media with angular oblate grains relative to model

490

media composed of spheres.


Page 24 of 35

Page 25 of 35


491

It is not hyperbole to state that for the first time we are in a position to predict colloid transport

492

distances under unfavorable (bulk repulsion) conditions. We can now seek to understand the

493

significance of attachment versus near surface accumulation without arrest at the field scale.

494

More generally, we are in a position now to assess these impacts of nanoscale repulsion

495

relative to other impacts that arise at the field scale such as preferential flow paths,37 flow

496

transience,38 bedforms39 and other contributors to field scale colloid transport. This

497

understanding may allow determining the role of nanoscale repulsion on important field scale

498

processes described in the introduction, as well as other phenomena such as the coincidence of

499

disease outbreaks with heavy rainfall.40-42

500 501

Acknowledgements

502

The authors thank Dr. Tim Scheibe of Pacific Northwest National Laboratory for providing the

503

continuum scale Lagrangian transport model used in Li et al.16 We greatly appreciate the

504

helpful input of four anonymous reviewers. This article is based upon work supported by the

505

National Science Foundation Hydrologic Science Program (1547533 and 1721660). Any

506

opinions, findings, and conclusions or recommendations expressed in this material are those of

507

the authors and do not necessarily reflect the views of the National Science Foundation.

508 509

Supporting Information Available

510

Supporting information is available including 3 tables, 4 figures, and links to a movie showing

511

contrasting CML transport in glass beads versus quartz sand. This information is available free

512

of charge via the Internet at http://pubs.acs.org.



513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554

References Cited 1. Harvey, R. W.; Garabedian, S. P. Use of colloid filtration theory in modeling movement of bacteria through a contaminated sandy aquifer. Environ. Sci. Technol. 1991, 25, 178- 185. 2. Kersting, A. B.; Efurd, D. W.; Finnegan, D. L.; Rokop, D. J.; Smith, D. K.; Thompson, J. L. Migration of plutonium in groundwater at the Nevada Test Site. Nature 1999, 397, 56-59. 3. Cushing, R. S.; Lawler, D. F. Depth filtration: fundamental investigation through threedimensional trajectory analysis. Environ. Sci. Technol. 1998, 32, 3793-3801. 4. Tufenkji, N.; Ryan, J. N.; Elimelech, M. The promise of bank filtration. Environ. Sci. Technol. 2002, 36(21); 422A-428A. 5. Bradbury, K.R; Borchardt, M. A.; Gotkowitz M.; Spencer, S.; Zhu, J.; Hunt, R.J. Source and Transport of Human Enteric Viruses in Deep Municipal Water Supply Wells. Environ. Sci. Technol. 2013, 47, 4096−4103. dx.doi.org/10.1021/es400509b. 6. Rajagopalan, R.; Tien, C.; Trajectory analysis of deep-bed filtration with the sphere-in-cell porous media model. AIChE J. 1976, 22(3), 523-533. 7. Elimelech, M.; O'Melia, C. R.; Kinetics of deposition of colloidal particles in porous media. Environ. Sci. Technol. 1990, 24(10), 1528-1536. 8. Elimelech, M. Particle deposition on ideal collectors from dilute flowing suspensions – mathematical formulation, numerical solution, and simulations. Sep. Technol. 1994, 4(4), 186-212. 9. Happel, J. Viscous flow in multiparticle systems – slow motion of fluids relative to beds of spherical particles. AICHE J. 1958, 4(2), 197-201. 10. Johnson, W. P.; Pazmiño, E.; Ma, H. L. Direct observations of colloid retention in granular media in the presence of energy barriers, and implications for inferred mechanisms from indirect observations. Water Research 2010, 44(4), 1158-1169. 11. Li, X. Q.; Scheibe, T. D.; Johnson, W. P. Apparent Decreases in Colloid Deposition Rate Coefficients with Distance of Transport under Unfavorable Deposition Conditions: A General Phenomenon. Environ. Sci. Technol. 2004, 38(21), 5616-5625. 12. Tufenkji, N.; Elimelech, M. Deviation from the classical colloid filtration theory in the presence of repulsive DLVO interactions. Langmuir 2004, 20(25), 10818-10828. 13. Li, X.; Johnson, W.P. Non-Monotonic Variations in Deposition Rate Coefficients of Microspheres in Porous Media under Unfavorable Deposition Conditions. Environ. Sci. Technol. 2005, 39, 1658-1665. 14. Tong, M.; Li, X.; Brow, C. N.; Johnson, W. P. Detachment-Influenced Transport of an Adhesion-Deficient Bacterial Strain within Water-Reactive Porous Media. Environ. Sci. Technol. 2005, 39(8), 2500-2508. 15. Albinger, O.; Biesemeyer, B. K.; Arnold, R. G.; Logan, B. E. Effect of bacterial heterogeneity on adhesion to uniform collectors by monoclonal populations. FEMS Microb. Lett. 1994 124, 321-326 pp. 16. Tufenkji, N.; Elimelech, M. Breakdown of colloid filtration theory: Role of the secondary energy minimum and surface charge heterogeneities. Langmuir 2005, 21(3), 841-852.


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17. Bradford, S. A.; Yates; S. R. Bettahar; M. Simunek, J. Physical factors affecting the transport and fate of colloids in saturated porous media. Wat. Resour. Res. 2002, 38(12), 1327. 18. Molnar, I. L.; Johnson, W.P; Gerhard, J.I.; O’Carroll, D.M. Predicting colloid transport through porous media: A critical review. Wat. Resour. Res. 2015, 51, 6804–6845. DOI: 10.1002/2015WR017318. 19. Wang, D.; Ge, L. ; He, J.; Zhang, W.; Jaisi, D. P.; Zhou D. Hyperexponential and nonmonotonic retention of polyvinylpyrrolidone-coated silver nanoparticles in an Ultisol. J. Cont. Hydrol. 2014, 164(0), 35-48. 20. Elimelech, M.; Nagai, M.; Ko, C.-H.; Ryan, J. N. Relative Insignificance of Mineral Grain Zeta Potential to Colloid Transport in Geochemically Heterogeneous Porous Media. Environ. Sci. Technol. 2000, 34(11), 2143-2148. 21. Pazmiño, E.; Trauscht, J.; Dame, B.; Johnson, W.P. Power-law size-distributed heterogeneity explains colloid retention on soda-lime glass in the presence of energy barriers. Langmuir 2014, 30(19), 5412–5421, http://dx.doi.org/10.1021/la501006p . 22. Trauscht, J.; Pazmiño, E. ; Johnson, W.P. Prediction of Nanoparticle and Colloid Attachment on Unfavorable Mineral Surfaces Using Representative Discrete Heterogeneity. Langmuir 2015, 31 (34), 9366–9378, DOI: 10.1021/acs.langmuir.5b02369. 23. Duffadar, R.; Kalasin, S.; Davis, J. M.; Santore, M. M. The impact of nanoscale chemical features on micron-scale adhesion: Crossover from heterogeneity-dominated to mean-field behavior. J. Coll. Interface Sci. 2009, 337(2), 396-407. 24. Pazmiño, E.; Trauscht, J.; Johnson, W.P. Release of Colloids from Primary Minimum Contact Under Unfavorable Conditions by Perturbations in Ionic Strength and Flow Rate. Environ. Sci. Technol. 2014, 48(16), 9227–9235, DOI: http://dx.doi.org/10.1021/es502503y. 25. Ma, H.; Pazmiño, E.; Johnson, W. P. Surface Heterogeneity on Hemispheres-in-Cell Model Yields All Experimentally-Observed Non-Straining Colloid Retention Mechanisms in Porous Media in the Presence of Energy Barriers. Langmuir 2011, 27(24), 14982-14994. 26. Tong, M.; Johnson, W.P. Excess colloid retention in porous media as a function of colloid size, fluid velocity, and grain angularity. Environ. Sci. Technol. 2006, 40(24), 7725-7731, DOI: 10.1021/es061201r. 27. Rasmuson, A.; Pazmiño, E.; Assemi, S.; Johnson, W.P. The Contribution of Nanoscale Roughness to Heterogeneity: Closing the Gap between Unfavorable and Favorable Colloid Attachment Conditions. Environ. Sci. Technol. 2017, 51 (4), pp 2151–2160, DOI 10.1021/acs.est.6b05911. 28. Hilpert, M.A.; Johnson, W.P. A binomial modeling approach for upscaling colloid transport under unfavorable attachment conditions: Emergent prediction of non-monotonic retention profiles. Wat. Resour. Res. 2018, 54, 46-60. https://doi.org/10.1002/2017WR021454. 29. Messina, F.; Tosco, T.; Sethi, R. On the failure of upscaling the single-collector efficiency to the transport of colloids in an array of collectors. Wat. Resour. Res. 2016, 52(7): 5492-5505. 30. Johnson, W.P.; Li, X.; Yal, G. Colloid Retention in Porous Media: Mechanistic Confirmation of Wedging and Retention in Zones of Flow Stagnation. Environ. Sci. Technol. 2007, 41,12791287, doi: 10.1021/es061301x. 31. Johnson, W.P; Hilpert, M. Upscaling Colloid Transport and Retention under Unfavorable Conditions Wat. Resour. Res. 2013, 49, 1–14, doi: http://dx.doi.org/10.1002/wrcr.20433. 27 ACS Paragon Plus Environment


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32. Li, X.; Lin, C.L.; Miller, J. D.; Johnson, W.P. Pore-scale Observation of Microsphere Deposition at Grain-Grain Contacts over Assemblage-scale Porous Media Domains Using X-ray Microtomography. Environ. Sci. Technol. 2006, 40 (12), 3762-3768. 33. Li, X.; Lin, C.L.; Miller, J. D.; Johnson, W.P. Role of grain to grain contacts on profiles of retained colloids in porous media in the presence of an energy barrier to deposition. Environ. Sci. Technol. 2006, 40 (12), 3769-3774. 34. Bendersky, M.; Davis, J. M., DLVO interaction of colloidal particles with topographically and chemically heterogeneous surfaces. . J. Colloid Interface Sci. 2011, 353 (1), 87-97. 35. Chinju, H.; Kuno, Y.; Nagasaki, S.; Tanaka S Deposition Behavior of Polystyrene Latex Particles on Solid Surfaces during Migration through an Artificial Fracture in a Granite Rock Sample. J. Nuclear Sci. Technol. 2001, 38(6), 439–443. 36. Tong, M.; Johnson, W.P. Colloid Population Heterogeneity Drives Hyper-Exponential Deviation from Classic Filtration Theory. Environ. Sci. Technol. 2007, 41(2), 493-499, DOI: 10.1021/es061202j. 37. McKay, L.D.; Cherry, J.A.; Bales, R.C.; Yahya, M.T.; Gerba, C.P. A field example of bacteriophage as tracers of fracture flow. Environ. Sci. Technol. 1993, 27(6), 1075-1079. 38. Zhuang, J.; Tyner, J. S.; Perfect, E. Colloid transport and remobilization in porous media during infiltration and drainage. J. Hydrology 2009, 377, 112-119. 39. Cardenas, M.B.; Wilson, J.L.; Zlotnik, V.A. Impact of Heterogeneity, Bed Forms, and Stream Curvature on Subchannel Hyporheic Exchange. Wat. Resour. Res. 2004, 40, W08307, doi:10.1029/2004WR003008. 40. Curriero, F. C.; Patz, J. A.; Rose, J. B.; Lele, S. The association between extreme precipitation and waterborne disease outbreaks in the United States. Am. J. Public Health 2001, 91 (8), 1194 − 1199. 41. Auld, H.; MacIver, D.; Klaassen, J. Heavy rainfall and waterborne disease outbreaks: The Walkerton example. J. Toxicol. Environ. Health, Part A 2004, 67 (20-22), 1879 − 1887. 42. Dorner, S. M.; Anderson, W. B.; Slawson, R. M.; Kouwen, N.; Huck, P. M. Hydrologic modeling of pathogen fate and transport. Environ. Sci. Technol. 2006, 40 (15), 4746-4753. 43. Johnson, W.P.; Li, X. Comment on: Tufenkji and Elimelech Breakdown of colloid filtration theory: role of the secondary energy minimum and surface charge heterogeneities, Langmuir 2005, 21, 841-852, Langmuir 2005, 21, 10895-10895.


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631

Table 1. Kinetic rate constants (kf and kf2) for fast- versus slow-attaching fractions of near-

632

surface colloids (α1 and 1-α1), respectively, from pore-scale trajectory simulations under

633

conditions corresponding to the experiments of Li et al.11. The average bulk and near surface

634

fluid velocities (v and v2) used to represent the experiments of Li et al.11 were 4 and 0.072

635

m/day (0.167 and 0.003 m/hr), respectively. The inverse residence times of the near surface

636

population that traveled to the RFSZ (αRFSZ); i.e., did not arrest or re-entrain, were multiplied

637

the maximum translation distance (πac) to determine the corresponding near-surface colloid

638

velocity (u2). Pore scale parameters from trajectory simulations for the CML-water-quartz

639

system under the conditions of Li et al.13 and rate constants for implementation in the two-

640

layer continuum model.

Glass beads η = 0.0181 7.677, k f (1/hr) = IS (M) = 0.006 0.02 α1 = 0.0033 0.010 α2 = 0.0091 0.065

641

k f2 (1/hr) = 0.0022 αreentrain = 0.961 αRFSZ = 0.0267 α trans = 0.25 k ns (1/hr) = 0.1447 u 2 (m/hr) = 0.0030

0.0166 0.0230 0.9016 0.25 0.643 0.0012

Quartz sand η = 0.0181 k f (1/hr) = 7.677 IS (M) = 0.003 0.006 0.02 α1 = 0.0053 0.0051 0.0169 α2 = 0.0132 0.0257 0.1460 k * f2 (1/hr) = 0.0497 αreentrain = 0.9314 αRFSZ = 0.0501 αtrans = 0.5000 k ns (1/hr) = 0.3271 u 2 (m/hr) = 0.0018

0.0975 0.0026 0.9670 0.5000 4.3730 0.00092

642


0.5910 0.0056 0.8315 0.5000 4.8290 0.00075


643 644

Figure 1. Example colloid-surface interaction profiles (panel a) representing interaction under bulk

645

attractive (blue) versus bulk repulsion (orange and red) conditions, the latter for greater and lesser IS,

646

respectively. Simulated colloid trajectories (panel b) under bulk attraction (bottom) versus bulk

647

repulsion (top) conditions with colloid-surface separation distance represented for bulk fluid and near

648

surface fluid domains by yellow and green coloring, respectively. Histograms of colloid residence times

649

prior to arrest (panel c) with near surface residence times (top) and total (bulk plus near surface)

650

residence times (bottom) shown for bulk attraction (favorable, blue) and bulk repulsion (unfavorable,

651

red) conditions. Parameters corresponding to the pore scale results corresponded to Li et al.11,13 for 1

652

µm colloids, 510 µm glass beads, 4 m/day average pore water velocity, pH 6.72, 20 mM NaCl. The

653

representative heterodomain sizes correspond to Pazmiño et al.21 (Pareto-distributed 120 & 240 nm

654

heterodomains) and fractional surface coverage by heterodomains = 1E-5.


Page 30 of 35

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655 656 657 658 659 660 661 662 663


Figure 2. Schematic representation of colloid-collector interaction force profiles as a function of colloidcollector separation distance (H) for 1.1 µm colloids at three IS (e.g., 0.006, 0.01 and 0.02 M) shown in red, yellow and blue series respectively. The projected colored disk represents the ZOI, and the inner green disk represents a heterodomain. The ZOI color corresponds to the force profile, with the greatest net repulsion corresponding to the largest ZOI (red) and lowest IS (0.006 M) and the lowest ZOI coverage by heterodomain. Net repulsion was absent for the highest IS (0.05 M, blue) because the heterodomain occupied a sufficient fraction of the ZOI to eliminate net repulsion.



Page 32 of 35

664

0.1

Favorable Unfavorable

Log (# microspheres)

Frequency

1

0.01

0.001

0.0001 (a)

Near Surface Residence Time (s)

10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5

1

Fav 0.02 M 0.006 M

0

0.05

Frequency

(c)

0.1

0.1 Distance (m)

0.15

0.2

Fraction attaching fast (α1):

0.01

Fraction attaching slow (1-α1 ): 0.001 (b)

Total Residence Time (s)

1/residence time (1/τ)

probability

665

666

Figure 3. Residence time distributions (left panel) from pore scale trajectory simulations under the

667

conditions of Li et al.11 at 6 mM NaCl (also corresponds to Figure 1 excepting 20 mM condition).

668

Predicted retention profiles (right panel) using attachment rate coefficients predicted from residence

669

time distribution as implemented in the expression for S(x) (retained # of colloids).

670


Page 33 of 35


671 672

673 674 675 676 677 678 679 680 681

Figure 4. Final and initial x-y locations (meters) of injected particles according to pore-scale trajectories simulated in a Happel sphere-in-cell collector under the conditions of Li et al.11 (CML on glass bead at 20 mM, 4 m/day). Z-direction (direction of flow) is into the page. Low near-surface tangential velocities are associated with FFSZ and RFSZ, the axes of which occur at the origin of the x-y plane. Non-attaching colloids occupy a subset of the x-y radius of the Happel grain (2.55E-4 m) due to imposition of a limiting radius (2.5E-4 m) for colloid injection to facilitate simulations, since colloids injected beyond the limiting radius will not enter the near surface domain. Values of η from simulations account for non-simulated colloids outside the limiting radius.



-1

3 mM 20 mM 3 mM Fav

1 Frequency

1 mM 6 mM Fav 20 mM

0

-2

Log (C/Co)

Page 34 of 35

Fav

Unfav

0.1

0.01

-3

0.001

-4

(c) -5

Total Residence Time (s)

-6 2


10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5

4 6 Pore Volumes

8

1 mM 6 mM Fav 20 mM

0

0.05

(b)

0.1 Distance (m)

10

3 mM 20 mM 3 mM Fav

0.15


0

(a)

0.2

10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 0 (d)

3 mM Arrested + Near Surface Arrested Near Surface

0.05

0.1 Distance (m)

0.15

0.2

682

683

Figure 5. Breakthrough-elution (a) and retention profiles (b) experimental data (symbols) for the data of

684

Li et al.13. In panels (a) and (b) are included two layer simulations (solid and dashed lines for 0.006 and

685

0.003 M IS, respectively). Histograms of total (bulk plus near surface) colloid residence times prior to

686

arrest (c). Contributions of arrested (dashed line) and near surface colloids (dotted line) to 0.006 M

687

retention profile (d).

688


Page 35 of 35


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