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Rotation and Retention Dynamics of Rod-Shaped Colloids with Surface Charge Heterogeneity in Sphere-in-Cell Porous Media Model ke li, and Huilian Ma Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b00748 • Publication Date (Web): 29 Mar 2019 Downloaded from http://pubs.acs.org on March 29, 2019

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Langmuir

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Rotation and Retention Dynamics of Rod-Shaped Colloids with Surface

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Charge Heterogeneity in Sphere-in-Cell Porous Media Model

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Ke Li, Huilian Ma

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Department of Geology and Geophysics

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University of Utah, Salt Lake City, UT 84112

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Abstract

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Colloid surface charge heterogeneity was incorporated into a three-dimensional trajectory model, which

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simulated particle translation and rotation via a force/torque analysis, to study the transport and retention

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dynamics of rod-shaped colloids over a wide size range in porous media under unfavorable conditions

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(energy barriers to deposition exist). Our previous study1 for rod transport under favorable conditions

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(lacking energy barriers) demonstrated that particle rotation due to the coupled effect of flow

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hydrodynamics and Brownian rotation governed rod transport and retention. In this work, we showed that

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the shape of a colloid affected both transport process and colloid-collector interactions, but shape alone

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could not make rods to overcome energy barriers of over tens of kT for attachment under unfavorable

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conditions. The location of colloid surface heterogeneity did not affect transport, but predominantly

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affected colloid-surface interactions by influencing the likelihood of heterogeneity patches facing the

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collector due to particle rotation. For surface heterogeneity located on the end(s) of a colloid, rods

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displayed enhanced retention compared with spheres; for surface heterogeneity located on the middle

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band, rods showed less retention compared with spheres. It was more effective to arrest a traveling rod

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when surface heterogeneity was located on the end relative to the side, because the tumbling motion

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greatly increased the likelihood of the end to intercept collector surfaces, and also because a rod would 

Corresponding author. E-mail: huilian. [email protected]; Tel: (801)585-5976; Fax: (801)581-7065.

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experience less repulsion with an end-on orientation relative to the collector surface compared to a side-

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on orientation due to curvature effect. The influences of particle aspect ratio on retention strongly

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depended upon the location of colloid surface heterogeneity. Our findings demonstrated that rods had

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distinct rotation and retention behaviors from spheres under conditions typically encountered in the

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environment; thus, particle rotation should be considered when studying the transport process of non-

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spherical colloids or spherical particles with inhomogeneous surface properties.

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Introduction

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Groundwater is one of the most important resources on earth. An enormous amount of groundwater is

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needed for irrigation to grow food. It also supplies drinking water for more than half of the U.S.

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population.2 Studying transport and deposition of colloids in saturated porous media is essential in

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protecting and utilizing groundwater, for example, in the process to track contaminant fate and to produce

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drinking water.3-4 Natural colloids in groundwater are often non-spherical in shape (e.g., rod-shaped

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bacteria,5-7 plate-like clay particles8-9) and possess varied degrees of surface heterogeneities. General

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filtration theory and many researches concerning colloid transport usually simplify natural colloids as

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spheres with effective radii and evenly charged surfaces.10-13 Mounting evidences14-22 have demonstrated

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that these approaches are inadequate in describing colloid transport and retention behaviors.

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For the past several decades, many studies have contributed to understanding the influences of particle

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shape on particle transport and adhesion, mainly in biomedical and aerodynamic fields.23-33 For instance,

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rod-shaped particles within micron size range were investigated for targeted drug delivery due to their

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better propensity to migrate towards blood vessel walls compared to spheres of equivalent volume.25-30, 34

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Zhang et al. reported that particle aspect ratio played an important role in leading to deposition of rod-


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shaped aerosols onto channel walls.35 Among such studies, some considered the influences of fluid

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hydrodynamic interactions induced by non-spherical shape,25-28, 31 whereas others investigated the

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translational and rotational dynamics of non-spherical particles in the Brownian regime.29, 36-38 However,

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very few studies took into account of the combined influences of flow hydrodynamics and Brownian

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rotation for colloids of non-spherical shape. Consequently, a comprehensive understanding of the effects

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of particle shape on transport and deposition over a wide range of particle sizes (e.g., from nanometers to

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tens of microns) is still lacking. In addition, many studies were carried out in parallel plate flow

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chambers.29-32 Often times, those studies did not consider colloidal interactions,23, 25-34 which might be

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justified by high ionic strength conditions associated with blood or lack of surface charges in air flow

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environments. As a result, it is questionable that findings from such researches can directly translate to

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colloidal filtration field involving porous media, where both colloids and collecting medium surfaces are

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usually negatively charged. As of now, in colloid filtration field, only a few experimental studies15, 39-40

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investigated the influences of particle shape on transport and retention. These experimental studies

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demonstrated that rod-shaped particles generally exhibited higher retention compared to spheres of equal

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volume.15, 39-40 However, the underlying mechanisms for the enhanced retention of rod-shaped particles

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were not well understood. Further, inconsistent findings regarding the role of rod aspect ratio on retention

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had been reported.15, 39-40 For example, Wang et al.39 observed that the slender, longer rod-shaped

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bacterial cells attached to the micro-pore filter walls less than the corresponding shorter ones; whereas

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Weiss et al.40 and Salerno et al.15 reported the opposite retention results. Also, the range of rod-shaped

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particle size and aspect ratio examined from those experiments was limited.15, 39-40 In a very recent study,

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we presented a three-dimensional particle transport model to investigate the effect of particle shape on

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transport and compared the retention behaviors of rods versus spheres of equivalent volume in porous

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media under favorable conditions (i.e., no energy barrier to retention).1 While particle rotation may be

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irrelevant to spheres of uniform properties, it was critical for the transport and retention of rod-shaped

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colloids.1 In shear flow, inertial rod-shaped particles underwent periodical tumbling motions when

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external forces and Brownian motions were not considered (see Video S1 rendered from our simulated 3 ACS Paragon Plus Environment

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trajectory in the parallel plate geometry in the Supporting Information), as predicted by Jeffery in 1922.41

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The positions of rods also oscillated periodically in response to flow hydrodynamics,1, 42-45 which caused

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rods to drift away from flow streamlines (Video S1 in the Supporting Information). Additionally, rotation

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due to diffusion changed rod orientation constantly and greatly altered the trajectories of rod-shaped

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particles under hydrodynamic effects. We also found that the rotation dynamics of rods varied with

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particle size as well as fluid flow velocity. Our previous study on rod rotation and transport under

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favorable conditions1 laid a foundation to continue the exploration of rod retention under unfavorable

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conditions (i.e., repulsive energy barriers exist).

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Predicting the attachment of colloids onto collector surfaces in the presence of an energy barrier remains

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as a challenge. The conventional mean-field zeta-potential approach to represent repulsive colloid-surface

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interactions yielded zero direct colloid attachment in porous media under unfavorable conditions;46-47

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whereas significant colloid attachments have been observed from column experiments packed with clean

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glass beads or quartz sand.21, 48-49 Many researchers incorporated the influences of surface heterogeneity,

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such as roughness,50-55 chemical heterogeneities,21-22, 56-59 onto collector surfaces and greatly improved the

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retention prediction of colloids (especially for spheres with uniform surface properties) relative to

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experimental observations under unfavorable conditions. However, very few studies have investigated the

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role of colloid surface heterogeneity on transport and retention.60-64 Until recently, Shave et al.60

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experimented on nanoscale functionalized microspheres flowing in a shear flow and found that the

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capture rate of particles onto chamber walls depended strongly on particle rotation, which controlled the

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likelihood of a functionalized group on particle surface facing the collector surface.

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For non-spherical colloids (e.g., rod-shaped ones), rotation becomes important not only in transporting

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particles to the vicinity of a collector surface,1 but also in properly orienting them to regulate colloid-


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surface interaction.6, 65 Since colloidal interactions scale with particle size, a rod may experience less

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repulsion with an end-on orientation toward the collector surface due to the small radius of curvature of

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its end. For instance, many rod-shaped bacteria have adhesins (ligands or pili) attached at one end of their

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bodies;66-70 take Thiothrix nivea as an example, the fimbriated end of Thiothrix nivea usually initiated

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contact with surrounding surfaces.67 Similarly, E. coli bacteria (rod-shaped) were observed to adhere to

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polystyrene particles by one end over 90% of the times.66 The tumbling motion of a rod-shaped particle

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may render its end to intercept with surrounding surfaces more frequently than the middle body sections.1

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However, under completely unfavorable conditions, our simulations indicated that shape alone (even with

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an end-on orientation for a rod) could not make the particle to overcome energy barriers above certain

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height (e.g, over tens of kT) for attachment based on mean-field surface potential approach. This called

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for the need to consider colloidal surface heterogeneity (here, we will incorporate surface charge

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heterogeneity as a first attempt) to better understand bacterial adhesion. Because of particle rotation, the

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location of charge heterogeneity over colloid surface becomes important.

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In this work, we will extend our particle trajectory model1 previously developed for rod-shaped (prolate

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spheroid) particle transport to incorporate colloid surface charge heterogeneity. This model predicts both

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the translational and rotational motions of an ellipsoidal particle based on an analysis of all the forces and

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torques acting on it. From this trajectory model analysis, the retention probability, attachment distribution

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and orientation of heterogeneous rod particles will be derived and compared to those of heterogeneous

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spheres of equivalent volume. Simulations will be performed primarily in the well-known Happel sphere-

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in-cell porous media model (Figure 1). No heterogeneity will be considered on the Happel model

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collector surfaces. To clearly demonstrate the orientation of retained colloids onto the collector surfaces,

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we will also present corresponding simulation results performed in the parallel plate geometry, due to its

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flat collector surfaces and simple flow field, under similar conditions as those in the Happel model.

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However, we stress that in this work, it is not our focus to quantitatively compare colloidal retention from 5 ACS Paragon Plus Environment

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these two geometrical models, despite that this study examines colloidal transport and retention at the

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near-surface laminar flow regimes in both models. The specific questions we will address include: i) what

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is the minimal colloid surface charge heterogeneity coverage required for particle capture? ii) how does

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the location of charge heterogeneity over colloid surface affect the retention of rod-shaped particles? iii)

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How does particle shape (i.e., aspect ratio) affect transport and retention of colloids with surface charge

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heterogeneity?

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Numerical Experimental Section

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Particle Transport Model. A Lagrangian trajectory approach was adapted in this model to simulate the

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time evolution of the three-dimensional (3D) translation and rotation of rod-shaped particles during

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transport in shear flow. The translation and rotation of a rigid particle can be described by the following

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equations of motion:

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{

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{

𝑑𝐮

𝑚 𝑑𝑡 = ∑𝐅

(𝟏)

𝑑𝐱 𝑑𝑡

(𝟐)

=𝐮

𝑑(𝐈 ∙ 𝛚) 𝑑𝑡 = 𝑑𝛀 𝑑𝑡 = 𝛚

∑𝐓

(𝟑) (𝟒)

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where t is time, m is the mass of the particle, 𝐈 is the moment of inertia of the particle; x and u are the linear

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position and velocity vectors of the particle, respectively;  and 𝛚 are the respective angular position and

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velocity vectors of the particle; and ∑𝐅 and ∑𝐓 represent the total forces and torques acting on the particle,

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respectively. Integrating all the forces acting on a particle yields particle linear velocity; likewise, integrating

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all the torques acting on a particle yields particle rotational velocity.1, 14, 49, 71-73 Among the forces a particle

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experienced, the following deterministic forces were accounted for in this research: virtual mass force, gravity,

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hydrodynamic fluid drag, shear lift, colloidal forces (van der Waals, electrostatic double layer) and non-

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colloidal (steric) forces. In this work, Brownian motions (including translation and rotation) were modeled


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separately as a stochastic process for each degree of freedom using Einstein’s equations.74 The torques acting

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on the particle are derived from the forces that do not pass through the particle center of mass, e.g., colloidal

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forces and hydrodynamic fluid drag. Detailed descriptions on how to solve the equations of motion, account

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for all the forces and torques, and incorporate Brownian translation and rotation can be found in our previous

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work.1 In this work, simulations were primarily performed in the Happel model (Figure 1), which represents

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porous media as a collection of spherical collectors each enclosed in a concentric spherical fluid shell. The

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flow field within the Happel model can be described analytically,75-76 and a representation of the laminar flow

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field in the Happel model was also provided in Figure S1 in the Supporting Information.

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Because of dilute particle concentration and small particle sizes (in the range of nanometers to microns)

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investigated, the fluid field disturbance due to the presence of particles was neglected, meaning that the

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velocity at the particle center was represented by the fluid velocity at that location in the absence of the

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particle. Likewise, particle-particle interactions or aggregations were ignored as a result of the assumption of

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dilute particle concentration. In this study, we used the universal hydrodynamic functions (𝑓1,𝑓2,𝑓3,𝑓4) that

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were based on the equivalent hydrodynamic Stokes sphere assumption to approximate the wall effects of

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ellipsoid particles (please refer to our previous work and other references for details).1, 13-14, 71

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Coordinate Systems. To represent the 3D rotation and translation of ellipsoidal particles, the following

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coordinate systems are needed (Figure 2): (1) an inertial frame with its origin often located on the

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collector, 𝐱 = [x,y,z]; (2) a particle frame with its origin at the center-of-mass of the particle and its axes

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being the particle principal axes, 𝐱 = [x,y,z]; (3) a co-moving frame, 𝐱 = [x,𝑦,z], with its origin at the

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center-of-mass of the particle and its axes parallel to the corresponding axes of the inertial frame.

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Transformation of the coordinates between the particle and co-moving frames can be achieved by the

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orthonormal transformation matrix 𝐀 given by Goldstein:77

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1 ― 2(𝑞22 + 𝑞23) 2(𝑞1𝑞2 + 𝑞3𝑞0) 2(𝑞1𝑞3 ― 𝑞2𝑞0) 𝐀 = 2(𝑞2𝑞1 ― 𝑞3𝑞0) 1 ― 2(𝑞23 + 𝑞21) 2(𝑞2𝑞3 + 𝑞1𝑞0) , 2(𝑞3𝑞1 + 𝑞2𝑞0) 2(𝑞3𝑞2 ― 𝑞1𝑞0) 1 ― 2(𝑞21 + 𝑞22)

[

]


(5)

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where q0, q1, q2 and q3 are the four Euler parameters. The particle position can be described by tracking its

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center-of-mass in the inertial frame. The particle frame is mainly used to describe the rotational motions

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of the particles by following the three Euler angles (i.e., ,  and ) (Figure 2) or four Euler parameters

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(i.e., quaternions q0, q1, q2, q3). Euler angles and Euler parameters are also interconvertible.1

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Rotation Dynamics of an Ellipsoid in Shear Flow. Euler angles are convenient for visualization. Thus,

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they were used to assign particle initial orientation and to record the final orientation of a particle. Euler

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parameters are frequently employed in the simulations, since the use of Euler angles would involve a

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large number of trigonometric calculations, which are computationally expensive. As a consequence, the

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quaternions would evolve with time according to the angular velocity and describe the time evolution of

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particle orientation:78

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[ ][ 𝑑𝑞0/𝑑𝑡 𝑑𝑞1/𝑑𝑡 1 𝑑𝑞2/𝑑𝑡 = 2 𝑑𝑞3/𝑑𝑡

]

― 𝑞1𝜔𝑥 ― 𝑞2𝜔𝑦 ― 𝑞3𝜔𝑧 𝑞0𝜔𝑥 ― 𝑞3𝜔𝑦 + 𝑞2𝜔𝑧 𝑞3𝜔𝑥 + 𝑞0𝜔𝑦 ― 𝑞1𝜔𝑧 , ― 𝑞2𝜔𝑥 + 𝑞1𝜔𝑦 + 𝑞0𝜔𝑧

(6)

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where 𝜔𝑥, 𝜔𝑦, and 𝜔𝑧 were the components of particle angular velocity vector in the particle frame. This

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equation illustrates how eq 4 was implemented in our simulations.

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The angular velocity of a moving ellipsoidal particle is governed by the torques acting on the particle. For

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a homogeneously dense particle, only forces that do not pass through the center-of mass of the particle

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would generate torques, for example, hydrodynamic fluid drags and colloidal forces. As a result, for an

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ellipsoid moving in a shear flow, eq 3 can be decomposed as1, 35, 79 𝑑𝜔𝑥

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𝐼𝑥 𝑑𝑡 ― 𝜔𝑦𝜔𝑧(𝐼𝑦 ― 𝐼𝑧) = 𝑇ℎ𝑥 + 𝑇𝑐𝑜𝑙𝑙 𝑥 ,

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𝐼𝑦 𝑑𝑡 ― 𝜔𝑧𝜔𝑥(𝐼𝑧 ― 𝐼𝑥) = 𝑇ℎ𝑦 + 𝑇𝑐𝑜𝑙𝑙 𝑦 ,

𝑑𝜔𝑦


(7)

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𝑑𝜔𝑧

𝐼𝑧 𝑑𝑡 ― 𝜔𝑥𝜔𝑦(𝐼𝑥 ― 𝐼𝑦) = 𝑇ℎ𝑧 + 𝑇𝑐𝑜𝑙𝑙 𝑧 ,

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where the components of the moment of inertia are (using z-axis as the major axis here): 𝐼𝑥 = 𝐼𝑦 = (𝛽2 + 1)𝑎2 5

𝑚, 𝐼𝑧 =

2𝑎2 5 𝑚,

a is the semi-minor axis of the ellipsoid and  is the aspect ratio (equal to major

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𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 axis divided by minor axis); 𝑇ℎ𝑥,𝑇ℎ𝑦, 𝑇ℎ𝑧 and 𝑇𝑐𝑜𝑙𝑙 𝑥 ,𝑇𝑦 , 𝑇𝑧 are the components of torques due to

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hydrodynamic drag and colloidal forces in the particle frame, respectively.

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Heterogeneity Pattern Design. To determine how the location of charge heterogeneity over particle

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surface affects retention, the following three pattern designs of heterogeneity patches were employed in

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the simulations: heterogeneity on one-end, heterogeneity on two-ends, and heterogeneity on the middle

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band (Figure 3). The heterogeneous patches were assigned with positive charge, whereas the rest of

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particle surface was negatively charged (see Table 1 for zeta-potential values).

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Colloidal Forces and Torques. The heterogeneity patches on the ellipsoidal surface call for a discretized

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method to accurately account for the colloidal forces between an ellipsoid and the collector surface. The

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surface element integration (SEI) method80-81 was employed here to calculate colloidal interactions, e.g.,

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the van der Waals interactions and electrical double layers forces (Figure 4). At each separation distance,

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the particle surface was discretized into small elements (e.g., size of 0.2 nm to 60 nm). Each particle

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surface element was assigned a zeta potential value according to specified heterogeneity design pattern.

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The interactions between each discretized element from the particle and the entire collector surface were

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computed and then integrated over entire particle surface to obtain the overall interaction between the

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particle and the collector.1 The SEI method is capable of studying colloidal interactions for particles of

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arbitrary shape, arbitrary orientation and arbitrary heterogeneity pattern.

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Similarly, to calculate torques, the interaction forces for each discretized particle surface element were

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multiplied with the lever distance from the center-of-mass of the particle to the line connecting the

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particle surface element to its projected element over the collector surface. This operation was then

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repeated over entire particle surface to obtain the total torques produced from colloidal forces.1 In this

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particular case, to use SEI method, one needs to find the closest approach distance between the ellipsoid

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of an arbitrary orientation and the spherical collector surface, and the method to find such a distance can

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be found in our previous study.1

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Computational Algorithm and Simulation Details. A computer program for solving the time evolution

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of the three-dimensional translation and rotation of an ellipsoidal particle with particle surface charge

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heterogeneity in a shear field was developed. Briefly, at each time step, according to the total forces and

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torques acting on each particle, particle linear velocity and angular velocity were calculated based on eq 1

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and eq 3 (or eq 7 more specifically), respectively. Based on the obtained linear velocity (eq 2) and angular

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velocity (eq 4, or more specifically eq 6), particle position vector (x,y,z) and orientation matrix 𝐀 (given

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by q0, q1, q2, q3) were correspondingly determined. Detailed computational steps can be found in our

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previous work.1

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The two-step Adams scheme was used to advance the particle position based on eq 2. Gauss elimination

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method was used to solve simultaneously the coupled equations of quaternions (eq 6) to obtain the new

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orientation quaternions and also to solve the three components of eq 1 to obtain the new translational

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velocity on each direction. For the coupled non-linear equations in eq 7, a mixed approach was used,

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where the time derivatives on the left-hand sides of the equations were treated by forward differencing. In

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each equation, the particle angular velocities that corresponded to the dependent variable in the time


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derivative term was evaluated at the next time step, and the other angular velocity and velocity gradient

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terms were evaluated at the current time step.1

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We performed a series of simulations under unfavorable conditions (i.e., several different heterogeneity

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patterns as shown in Figure 3) with ellipsoidal particles of different sizes (diameters between 200 nm –

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6.6 µm) and shapes (aspect ratios from 1- 6) flowing within the Happel model under pore water velocity

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of 5 m/day. For each simulation, approximately 2000 – 9000 particles were injected randomly (both in

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position and orientation) into the model system as shown in Figure 1. Simulation parameter values were

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provided in Table 1. The computation algorithms described above were carried out to obtain the time

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evolution of particle translation and rotation. We considered particles were in “physical contact” with the

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collector surface once the separation distances were within a roughness layer 𝛿,1, 80 which, depending

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upon particle sizes, were typically between 2-10 nm for our simulations. Within this distance, particle-

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collector contact forces and torques were evaluated to investigate the dynamic adhesion of rolling or

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rotating particle.37 JFK theory82 was employed to obtain particle deformation area resulting from particle-

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surface contact. Based on this deformation, a non-hydrodynamic adhesive torque (e.g., resulted from

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gravity, colloidal forces) was obtained. A particle was considered to be attached when the adhesive torque

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was greater than the driving hydrodynamic torque.1 Based on simulated trajectory outcome, e.g., attached,

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exited, or remained in the system without direct attachment, collector efficiency (the extent of retention),

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attachment rate and attachment orientation were obtained.

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Validation of Simulation Model. We have validated our trajectory model in two steps: i) simulated

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retention trends for spherical particles agreed with existing filtration theory under favorable conditions;1 ii)

258

simulated rotation patterns of rod-shaped particles in a linear shear flow ignoring any external forces and

259

diffusion matched well with Jeffery’s prediction.41 Note that, in the trajectory simulations, we can easily


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260

turn off each mechanism including Brownian motion, and external forces and diffusion were turned off

261

for this particular simulation to match Jeffery’s theoretical conditions.

262 263

Results with Discussion

264

Colloidal retention onto collector surfaces is controlled by its ability to transport to the vicinity of

265

collector surfaces and the subsequent interactions between the colloid and the collector. The shape of a

266

colloid (e.g., rod-shaped) can greatly affect both the transport process and the colloid-surface interactions.

267

Under favorable conditions, since every collision of a colloid with the collector leads to attachment, the

268

collector efficiency is essentially the probability of particles colliding with the collector. Under

269

unfavorable conditions, not all collisions result in attachment; thus, the attachment efficiency (α) is

270

conventionally used to describe the probability of attachment upon collision. The attachment efficiency is

271

calculated by the ratio of simulated single collector efficiency under unfavorable condition to that under

272

favorable condition. Below, we will present and discuss how the location of heterogeneous patches,

273

particle aspect ratio, colloid size affect retention, distribution and orientation of attached colloids over the

274

collector surface under unfavorable conditions.

275 276

Effect of Heterogeneity Location over Colloid Surface on Retention

277

Heterogeneous Patches Located on One versus Two Ends. The attachment efficiencies () for the

278

heterogeneous patches located on one end versus two ends of colloids of 1 m size as a function of

279

colloid charge heterogeneity coverage (0 - 100) from the Happel model were provided in Figure 5

280

(porosity 0.36, grain diameter 390 µm, pore water velocity of 5 m/day). For all the rods (AR = 2 – 6) and

281

spheres (AR = 1) examined here, the minimum surface coverage required to capture a colloid for

282

attachment was the same for heterogeneity on one end versus two ends. Also, the retention trends for

283

heterogeneity on one end versus two ends as a function of colloid surface heterogeneity coverage were 12 ACS Paragon Plus Environment

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284

very similar (Figure 5 a versus b). The similar retention behaviors for heterogeneity located on one end

285

versus two ends were also observed from other particle sizes (e.g., 6.6 m in diameter as in Figure S2 in

286

the Supporting Information). Nonetheless, since heterogeneity located at the one versus two ends resulted

287

in similar retention behaviors, in what follows we will only examine the cases where heterogeneous

288

patched were located at the two ends of a colloid.

289 290

Also, from Figure 5, rods needed smaller patch coverage to initiate retention compared to spheres. This

291

reflected that the tumbling motions of rods in shear flow, as illustrated in our previous work,1 increased

292

the likelihood of colloid end(s) to intercept the collector surface. In addition, due to the relatively smaller

293

curvature at the end, a rod would experience less repulsion compared to a sphere,6,

294

discuss more shortly.

65

which we will

295 296

Heterogeneous Patches Located on the Ends versus the Middle Band. The attachment efficiencies for

297

the heterogeneous patches located on the ends versus the middle band of colloids for three representative

298

sizes (200 nm, 1 m and 6 m in diameter) as a function of colloid charge heterogeneity coverage (0 -

299

100) from the Happel model were presented in Figure 6. For spheres (aspect ratio = 1) undergoing

300

three-dimensional rotation, there was no distinction between the ends and the middle. Consequently, the

301

location of heterogeneity patches on a sphere surface (ends versus middle) did not affect retention. But for

302

rods, the location of heterogeneous patches over colloid surface affected retention dramatically. For

303

example, for 200 nm colloids, when charge heterogeneity was located on the ends, as shown in Figure 6a,

304

rods (with aspect ratio of 2 – 6) required considerably smaller patches compared to spheres to initiate

305

attachment. However, when charge heterogeneity was located on the middle band of the colloids (Figure

306

6b), rods required significantly larger patches compared to spheres to start attachment. The opposite

307

effects of heterogeneity location (ends versus middle) on retention of rods relative to that of spheres were

308

observed for other colloid sizes as well (see Figure 6c - 6d for 1 m colloids, and 5e - 5f for 6.6 m


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309

colloids).

310 311

In general, once colloid surface charge heterogeneity was greater than the minimum patch size (which

312

was calculated based on the minimum surface coverage times the total surface area of the particle)

313

required for retention, attachment efficiencies increased with increasing heterogeneity coverage, till

314

reaching retention for favorable conditions (where   1). This was true for both spheres and rods of all

315

the sizes examined. The smaller the colloid size, the faster the attachment efficiency transitioned from

316

completely unfavorable to favorable conditions (see Figure 6).

317 318

We have rendered the simulated 3D translational and rotational motions of representative rods with aspect

319

ratio 6 for the heterogeneous patches located on the ends (Video S2) versus the middle band (Video S3)

320

within the Happel model into videos and provided in the Supporting Information. One can observe from

321

these two videos that the near-surface rotation patterns for rod particles to become attached were different

322

for the heterogeneous patches located on the ends versus the middle band. For the attached rods with

323

heterogeneous patches on the middle band, one can see that they approached to the collector surface with

324

a side-on orientation and the plane of rotation was approximately parallel to the collector surface.

325

Whereas the attached rods with heterogeneous patches on the ends generally travelled near the surface

326

region with an end-on orientation and the plane of rotation was about normal to the collector surface.

327 328

Effect of Colloid Aspect Ratio on Retention under Unfavorable Conditions

329

As shown in Figure 6, the influences of aspect ratio on retention under unfavorable conditions strongly

330

depended upon the location of colloid surface charge heterogeneity. When charge heterogeneity was on

331

the ends of colloids, the minimum surface coverage required for attachment, as well as heterogeneous

332

coverage needed to transition from unfavorable to favorable retention, decreased with increasing aspect

333

ratio (Figures 6a, 6c, 6e). For instance, 200 nm rods with aspect ratio 6 started to attach at very small


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heterogeneity coverage (< 0.01%); at around 0.1% heterogeneity coverage, these rods could reach

335

retention for favorable conditions (α ≈ 1). As the aspect ratio decreased, rods required more heterogeneity

336

coverage to initiate attachment (e.g., ~ 0.1% and 0.5% for AR = 3 and 2, respectively) and to reach

337

attachment for favorable conditions (e.g., ~ 0.2% and 1% for AR = 3 and 2, respectively). Spheres needed

338

the most heterogeneity coverage – started to attach at 4-6% heterogeneity coverage and behaved like

339

favorable condition at 6-11% heterogeneity coverage.

340 341

However, when charge heterogeneity was on the middle section of colloids, the minimum surface

342

coverage required for attachment, along with the heterogeneous coverage needed to transition from

343

unfavorable to favorable retention, increased with increasing aspect ratio (Figures 6b, 6d, 6f). For instance,

344

for 200 nm particles, spheres started to attach and reached favorable retention with the smallest

345

heterogeneity coverage (14% and 17% respectively); and then followed by rods with aspect ratio of 2 and

346

3. Rods with aspect ratio 6 started to attach and reached favorable retention with the biggest heterogeneity

347

coverage (73% and 86% respectively). Besides plotting the attachment efficiency as a function of surface

348

heterogeneity coverage as shown in Figure 6, a similar figure showing the collector efficiency as a

349

function of surface heterogeneity coverage was also provided in Figure S3 in the Supporting Information.

350

Additionally, the minimum raw patch sizes required for particle capture changed with particle size, aspect

351

ratio and heterogeneity patterns (see Table S1 in the Supporting Information), but the retention trends

352

versus minimum surface coverage discussed above also held when raw patch sizes were used (see Table

353

S1 and discussion thereafter in the Supporting Information).

354 355

When colloids were relatively far away from the collector surfaces, the location of charge heterogeneity

356

over colloid surface would not affect the transport behavior of rods or spheres. In other words, regardless

357

of the patches on the ends or middle, a colloid would have similar propensity to transport to the vicinity


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358

of the collector surface. However, once a colloid was very close to the collector, the shape of colloid (i.e.,

359

aspect ratio here) and the location of colloid charge heterogeneity could greatly influence colloid-surface

360

interactions. Figure 7 illustrated the integrated DLVO (abbreviated after Derjaguin-Landau-Verwey-

361

Overbeek) interaction profiles computed from the SEI methods for 6.6 m colloids with the

362

heterogeneous patches on the ends versus the middle for various aspect ratios. When charge heterogeneity

363

was located on the ends, repulsive energy barriers decreased with increasing aspect ratio for a given

364

colloid charge heterogeneity coverage (Figure 7a). This was due to the decreasing radius of curvature

365

with increasing aspect ratio when a colloid approached the collector surface with an end-on orientation. In

366

addition, as the heterogeneity coverage increased, the energy barriers dropped faster for rods with larger

367

aspect ratio to reach no-barrier attachment conditions relative to spheres (Figure S4 (a) in the Supporting

368

Information). For this reason, rod particles with larger aspect ratio required less heterogeneity coverage to

369

initiate attachment and to reach favorable attachment. Likewise, when charge heterogeneity was located

370

on the middle band, repulsive energy barriers increased with increasing aspect ratio (Figure 7b), because

371

the radius of curvature increased with increasing aspect ratio when a colloid approached the collector

372

surface with a side-on orientation. Moreover, as the heterogeneity coverage increased, the energy barriers

373

decreased slower with increasing aspect ratio (Figure S4 (b) in the Supporting Information). Consequently,

374

relative to spheres, rods with higher aspect ratio required larger heterogeneous patches to start attachment

375

and reach favorable retention.

376 377

In Figure 6f, for large size rods with heterogeneous patches located on the middle band, we observed that

378

the attachment efficiencies reached a plateau, but well below favorable retention, over a wide percentage

379

of colloid surface coverage (SCOV). For example, for 6.6 m rods with aspect ratio 6,  was

380

approximately 0.25 when SCOV varied within 32 – 99%. Similar plateau in  was observed for 6.6 m

381

rods with aspect ratio of 3 and 2, but not for small rod sizes of any aspect ratio examined (Figure 6b and

382

6d). To be sure of this behavior, we also ran simulations for 6.6 m with aspect ratio 6 under the same 16 ACS Paragon Plus Environment

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383

heterogeneous pattern and coverage as in Figure 6f within the parallel plate geometry and observed

384

similar plateau behavior in  (data not shown). This can be partly explained by the concept of zone of

385

influence from the electrostatic double layer interaction for particles of differing size and curvature. Since

386

electrostatic interaction decays exponentially with distance away from the collector, for a given colloid-

387

collector separation distance, the relative portion of heterogeneous patches in the middle band out of the

388

entire colloid surface area contributing to the electrostatic interaction was smaller for large size rods (e.g.,

389

6.6 m) compared to small size rods (e.g., 1 m) (Figure S5 in the Supporting Information). Another

390

reason for the observed plateau in  arose from particle rotation, which controlled the likelihood of the

391

rod middle section facing the collector surface. As demonstrated above (see Video S3 in the Supporting

392

Information), for heterogeneity located on the middle band, only those trajectories that happened to be a

393

side-on orientation near the collector surface were able to attach. This plateau suggested that the rod

394

retention within this SCOV range was rate-limited by particle transport and rotation, not by colloid-

395

collector interaction.

396 397

These behaviors indicated that it was easier for rods to become attached to the collector when the

398

attractive heterogeneity was on the end(s) than on the side. This could explain why many rod-shaped

399

bacteria have attachment ligands particularly on the end(s).83-87 A traveling and rotating rod-shaped

400

bacterium would intercept with surrounding surfaces more likely with its end first and subsequently

401

become attached with an end-on orientation.66 On the other hand, a study from Afrooz et al.88 on

402

aggregation behaviors of poly-acrylic acid-coated gold nano-spheres versus nano-rods (12 nm, aspect

403

ratio = 5) showed that nano-rods were less likely to form aggregates in low NaCl concentrations

404

compared to nano-spheres. The relative stability of these nano-rods in suspension primarily resulted from

405

the stronger electro-steric repulsion when rods approached each other with side-on orientation to

406

aggregate compared to spheres due to curvature effect.88 These observations qualitatively agreed with our

407

simulation results when the heterogeneous patches were on the middle section of rods. Moreover, as 17 ACS Paragon Plus Environment

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408

mentioned earlier, contradictory results regarding the role of rod aspect ratio on retention had been

409

reported from experiments.15, 39-40 Our findings here elucidated that the influences of rod aspect ratio on

410

retention depended strongly on the location of colloid surface heterogeneity, which may provide a

411

possible explanation for those observed contradictions.

412 413

Effect of Colloid Size on Retention under Unfavorable Conditions

414

The size of a colloid (e.g., rod or sphere) greatly affects particle transport process as well as colloid-

415

surface interactions. The simulated single collector efficiencies () for rods (aspect ratio = 6) and their

416

counterpart spheres of equivalent volume were plotted for several representative sizes (0.1, 0.2, 1, 3, 6.6

417

µm in diameter) under different charge heterogeneity coverages at pore water velocity of 5 m/day with

418

the Happel model in Figure 8. For spheres, when surface charge heterogeneity coverage reached 30% or

419

above, the retention trends were similar to those under favorable conditions (Figure 8a), where retention

420

was at minimum for size ~1 – 3 µm, as expected from colloidal filtration theories.10, 71, 76, 89-90 As colloid

421

heterogeneity coverage decreased (e.g., 3.5%, 1.5%), the simulated collector efficiencies decreased and

422

the decreased amount in  strongly depended upon sphere size. For small size spheres (e.g., 0.1 and 0.2

423

m), our model predicted no retention because 3.5% coverage or lower was below the minimal patch

424

coverage needed to initialize attachment (see Figure 6a). For spheres from 1 to 6.6 µm, a local minimum

425

in retention was still observed around 1 – 3 µm, but the drop in  compared to favorable conditions was

426

more pronounced around this minimum (e.g., at 1.5%). It is worth to note that the gaps in retention,

427

especially around sphere sizes corresponding to local minimum attachment, between favorable and

428

unfavorable conditions have been observed by experiments.21, 52 For example, in an impinging jet cell to

429

study colloid retention with carboxylate-modified polystyrene latex fluorescent microspheres, Rasmuson

430

et al.52 reported that the minimal retention occurring at 1.1 µm microspheres dropped by two orders of

431

magnitude from favorable to unfavorable conditions, exhibiting the biggest gap compared with other


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432

sphere sizes. Our simulation results in Figure 8a can be explained by the colloidal forces between patched

433

colloids with the collector surface. To illustrate this, we compared the DLVO energy profiles for the

434

sphere sizes examined above (see Figure S6 in the Supporting Information) at a heterogeneity coverage

435

0.25%, and found that spheres around 3 µm in diameter, which corresponded to the size with the largest

436

gap in retention in Figure 8a, were subjected to the biggest energy barrier compared to other sizes. The

437

0.25% coverage was chosen because at this coverage, an energy barrier existed for all the sphere sizes

438

examined, even when the heterogeneous patch was facing the collector surface. In other words, at 1.5% or

439

3.5% colloid surface coverages, retention may occur when the heterogeneous patch faced toward the

440

collector at close contact due to lack of repulsion; but retention definitely would not occur when the patch

441

faced away from the collector surface. Essentially, retention for heterogeneously patched spheres

442

reflected the probability of the patch(s) facing the collector surface when spheres were transported near

443

collector surfaces, indicating the importance of particle rotation on retention.

444 445

For rods, when colloid surface charge heterogeneity coverage (located at the ends of rods) reached 6% or

446

above, the retention trends were similar to those under favorable conditions, where the dependency of 

447

showed a “W-shape” dependency on rod size (Figure 8b) – for particle size greater than 2 µm, the

448

attachments of rods increased with increasing size; for particle size smaller than 200 nm, the attachment

449

of rods decreased with increasing size; whereas for rod particle size from 200 nm to 2 µm, the attachment

450

first increased then decreased with increasing size (“hump” region).1 As colloid heterogeneity coverage

451

decreased, e.g., < 0.01% coverage with patchy ends, retention of rod-shaped particles still followed the

452

trend as observed under favorable condition, but shifted to smaller  values. As described above, when

453

charge heterogeneity was located at the ends of rods, only a very small coverage was required to initiate

454

retention due to the small radius of curvature for an end-on orientation. Thus, for rod-shaped particles

455

with heterogeneous patches on the end(s) under unfavorable conditions, retention was largely limited by


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456

the rotation dynamics as observed under favorable conditions,1 which controlled the rate of particles

457

being transported to the vicinity of collector surfaces.

458 459

Attachment Location and Orientation

460

Rod-shaped particles were observed to attach all over the Happel model surface (Figure 9a). In general,

461

more colloids were retained at the upstream collector surface relative to downstream. For heterogeneity

462

located on the ends versus on the middle band, the basic patterns of attachment locations were similar.

463

However, rod-shaped particles had higher tendency to attach at the downstream relative to spheres,

464

similar to what we have observed under favorable conditions.1 This was due to the drifting effect of rod

465

particles, which enabled rod particles to travel further downstream to attach. Also, small size particles

466

(e.g., 0.2 µm rods or spheres) were observed to more likely travel downstream for attachment compared

467

to large size particles (e.g., 6.6 µm rods or spheres), which was due to diffusion and was well discussed

468

by the general filtration theory.10-13

469 470

With regard to attachment orientation, for all the unfavorable conditions investigated (e.g., different

471

heterogeneous patterns over colloidal surfaces), both spheres and rods attached via the attractive

472

heterogeneity patches (the orientation distribution were shown for rods in Figure 9). Since it is easier to

473

visualize in the parallel plate geometry, we also performed simulations in that geometry under the same

474

conditions as those in the Happel model and presented the simulated orientation distribution patterns for

475

attached rods (aspect ratio 6) in Figure 9(b). The parameters for the parallel plate geometry were set with

476

same channel width (31.3 µm to fluid envelope, very narrow) and average pore water velocity as those in

477

the Happel model. Hence, aside from the curved collector surface from the Happel model, simulation

478

results obtained from the parallel plate geometry under similar conditions should allow us to clearly

479

visualize the orientations of attached rod colloids, as shown in Figure 9(b). One can see clearly that, in the 20 ACS Paragon Plus Environment

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480

parallel plate model, rod-shaped particles with heterogeneity on the ends adhered to the collector surface

481

with one end, whereas rods with heterogeneity on the middle band attached via a side-on orientation. Note

482

that, this observation did not contradict with previous statements wherein rod-shaped particles preferred

483

an end-on orientation for initial attachment, because for the same heterogeneity coverage (e.g. before

484

reaching favorable conditions), the attachment efficiencies for rods with heterogeneity on the ends were

485

often several orders of magnitude greater than those for rods with heterogeneity on the middle band. We

486

stress here that in this work, we do not intend to quantitatively compare the attachment efficiency or

487

collector efficiency obtained from the Happel model with those from the parallel plate flow chamber. The

488

main reason we included the parallel plate geometry was to validate our trajectory model development for

489

rods against Jeffery’s theory which was done in simple shear flow,41and to clearly illustrate rod

490

attachment orientation, due to its simple flow field and flat collector surfaces. That being said, since this

491

work focused on the near-surface laminar flow regions (less than 50 m away from the walls) from both

492

Happel and parallel plate models, the trends on attachment efficiency obtained from the Happel model

493

should qualitatively apply to the parallel plate geometries, based on our preliminary simulation data from

494

the latter (data not shown).

495 496

Effects of Initial Position and Orientation on Rod Retention

497

Particle initial position and orientation greatly affected the retention and orientation of rods, and this

498

effect depended upon model geometry dimensions relative to particle size. As described above, it is easier

499

to visualize rod orientation in the parallel plate geometry, so the effects of particle initial conditions on

500

retention will be discussed primarily in that geometry (unless stated otherwise). For the narrow and short

501

channel explored here (e.g., half channel height of 31.3 m, channel length of ~ 1 mm, similar to the

502

Happel model dimension), large size retained rods (e.g., 6.6 m) with heterogeneity located on the ends

503

had a wide range of initial orientations (e.g., Euler angle , which characterizes the angle between the rod


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504

major axis and the normal to the flat collector surface in the parallel plate geometry, varied from 0 to ),

505

whereas large size retained rods with heterogeneity located on the middle band tended to have an initial

506

orientation close to side-on relative to the channel walls (e.g., Euler angle  was close to /2). However,

507

small size retained rods (e.g., 1.0 m) with either side-on or end-on attachment orientation had a wide

508

range of initial orientations, regardless of the locations of heterogeneity (data not shown). Based on this

509

observation, the channel height and/or channel length were increased by 10 times, we then observed that

510

large size retained rods (e.g., 6.6 m) with heterogeneity located on the middle band no longer showed

511

any preferable initial orientations. Moreover, the effect of initial position on retention depended upon the

512

channel length. For instance, when the channel length was set to 1 mm (similar to the travel length in the

513

Happel model), rods with initial positions within ~10 um distance from the wall (limiting trajectory

514

distance) could become attached. This limiting trajectory distance increased to ~20 um when channel

515

length was increased to 3 mm. However, further increase in channel length would not increase the

516

limiting trajectory distance any more for the conditions examined here. As described earlier and in our

517

previous paper,1 rods tended to rotate in shear flow, and their positions would oscillate during rotation,

518

which led to the drift of rods across flow streamlines toward the channel walls. If confined within a

519

narrow and/or short channel, the oscillation and drifting effects of rods may not be fully observable.

520 521

Likewise, in the Happel model, rods injected close to the center of the injection plane had a higher

522

possibility to become attached, whereas particles injected outside of the limiting injection area were not

523

going to attach (Figure 1). In the Happel model, the limiting trajectories beyond which particles wouldn’t

524

attach were similar for rods for the two heterogeneity patterns (ends versus middle band). This reflected

525

the fact that the location of heterogeneity over the colloidal surface did not affect the rotational and

526

oscillational motions (shown in Video S1 in the Supporting Information) of the rods when far away from

527

the collector surface. But we did not observe any preferential initial orientation for the attached rods with

528

middle-band heterogeneity from the Happel model, which was probably due to the coupled effects from 22 ACS Paragon Plus Environment

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Langmuir

529

the complicated flow patterns as well as collector curvature in the Happel model relative to those in the

530

parallel plate geometry. For the retained rods with heterogeneity located on the ends, no preferential

531

initial orientation was observed from both geometric models.

532 533

Summary and Conclusions

534

In this work, we extended our three-dimensional particle trajectory model to investigate the influences of

535

particle shape and location of colloid surface charge heterogeneity on retention in porous media under

536

unfavorable conditions for a wide range of particle sizes. Our model accounted for the combined

537

influences of flow hydrodynamics, Brownian rotation and translation, colloid-surface interaction on the

538

rotation and retention dynamics of prolate spheroid particles during transport process. Retention of a

539

colloid onto collector surfaces was governed by its ability to transport to the vicinity of collector surfaces

540

and the subsequent interactions between the colloid and the collector. The shape of a colloid greatly

541

affected both transport process and colloid-surface interactions. The location of colloid heterogeneous

542

patches did not affect transport, but predominantly affect colloid-surface interactions by influencing the

543

likelihood of the heterogeneity patches facing the collector due to particle rotation.

544 545

Under unfavorable conditions, particles (rods or spheres) attached via attractive heterogeneity. For rod-

546

shaped colloids, it was more effective for retention to occur when heterogeneities were located on the

547

end(s) than on the middle band. This indicated that the initial retention of rod particles was controlled by

548

particle rotation, since the end of a rod had higher probability to intercept the collector surface during

549

rotation for attachment than its sides. In addition, a rod would experience much less repulsion with an

550

end-on orientation toward the collector surface than a side-on orientation due to curvature effect. This

551

observation could explain why rod-shaped bacteria have tethered ligands located on the end(s),83-87 and

552

why rod-shaped bacteria would intercept with surrounding surfaces more likely with its end first to 23 ACS Paragon Plus Environment

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553

establish initial contact.66 The preferential attachment by the end(s) of a rod-shaped particle could have

554

important implications for rod-shaped bacteria (or engineered rods) transport and application in biological

555

or colloidal science fields.

556 557

The influences of particle aspect ratio on retention under unfavorable conditions strongly depended upon

558

the location of colloid surface charge heterogeneity. For a given colloid surface coverage, retention

559

increased with increasing aspect ratio when charge heterogeneity was on the end(s) of colloids; whereas

560

retention decreased with increasing aspect ratio when charge heterogeneity was located on the middle

561

band of colloids. These interesting transport and retention behaviors for rod-shaped colloids could not be

562

captured if rods were simplified as “effective spheres” or assumed with “homogeneous surface”. Our

563

findings could provide useful guidance on engineering rod-shaped spheroids of differing aspect ratios

564

with appropriate surface properties to achieve desired transport and retention results. More importantly,

565

our study demonstrated that when studying the transport process of non-spherical colloids or spherical

566

particles with inhomogeneous surface properties – the typical scenarios encountered in the environment,

567

particle rotation becomes very important and needs to be considered.

568 569

Supporting Information. Additional information includes laminar flow field in the Happel model,

570

energy profiles for spheres and rods with patched surfaces, and short videos for the tumbling and

571

oscillational motion of rods in the parallel plate geometry, and transport of rods with different

572

heterogeneity patterns in the Happel geometry. This material is available free of charge via the Internet at

573

http://pubs.acs.org.

574


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Langmuir

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Acknowledgements. This article was based upon work supported by the National Science

576

Foundation Hydrologic Science Programs (EAR 1521421). Any opinions, findings, and conclusions or

577

recommendations expressed in this material are those of the authors and do not necessarily reflect the

578

views of the National Science Foundation. We are grateful for the technical and facility support provided

579

at the Center for High Performance Computing at the University of Utah. We thank Professor William P.

580

Johnson and his research group for kindly sharing their computational resources and for their technical

581

help in many subjects. We also thank Professors Yusong Li, Mathias Schubert and their research groups

582

for their constructive inputs throughout our meetings together.


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771

Langmuir

Table 1. Parameters Used in Lagrangian Trajectory Simulation parameter

value

collector diameter, 𝑎𝑔

390 µm

porosity, 𝜀

0.36

pore water velocity, v

5 m/day

particle density, 𝜌𝑝

1055 kg/m3

fluid density, 𝜌𝑓

998 kg/m3

dynamic fluid viscosity, 𝜇

9.98 × 10 ―4 kg ∙ m/s

Hamaker constant, 𝐴𝐻

3.84 × 10 ―21 𝐉

Ionic strength, IS

1 mM

colloid zeta potential, 𝜉𝑝𝑜

-20 mV

collector zeta potential, 𝜉𝑐

-53.5 mV

colloidal heterogeneity potential, 𝜉𝑝

20 mV

absolute temperature, T

298.2 K

time step, ∆𝑡

1-100 MRT

aspect ratio, 𝛽

1-6

shear rate at the wall

1.96 s ―1

Peclet number, Pe

0.1~100

Reynolds number, Re

< 0.001

772


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

773

Figure Captions.

774

Figure 1. Schematic of the Happel model. Particles were injected through the particle injection plane, and

775

only particles injected through the limiting injection area that were able to attach.

776

Figure 2. Coordinate systems and Euler angles used to describe particle translation and rotation.

777

Figure 3. Heterogeneity pattern designs over particle surfaces: (a) on one end, (b) on two ends, and (c) on

778

the middle. Particle surface heterogeneous coverage was varied from 0-100% of the total colloid surface

779

area.

780

Figure 4. Schematic of the SEI method. The local unit normal (n) of an ellipsoid surface element and its

781

local distance (h) to the surface were required to compute the interactions between this element and the

782

collector. The closest separation distance between the ellipsoid and the collector was represented by H.

783

Blue patches represented the heterogeneity location over particle surface.

784

Figure 5. Simulated attachment efficiencies for heterogeneity located on (a) one end versus (b) two ends

785

as a function of surface heterogeneity coverage (SCOV) and aspect ratio for 1 µm in diameter particles.

786

AR1, AR2, AR3 and AR6 represented ellipsoids with an aspect ratio of 1 (red circle), 2 (blue triangle), 3

787

(orange cross), and 6 (purple square), respectively. Simulation conditions were provided in Table 1. The

788

statistical errors for our simulation data depended on the total numbers of simulated particle trajectories

789

and the numbers of particles attached and typically were less than ±0.05 in α.

790

Figure 6. Simulated attachment efficiencies for heterogeneity located on the ends (a, c, e) versus on the

791

middle band (b, d, f) as a function of colloid surface heterogeneity coverage (SCOV) and aspect ratio for

792

representative particle sizes: (a-b) 200 nm; (c-d) 1 µm; (e-f) 6.6 µm. AR1, AR2, AR3 and AR6

793

represented ellipsoids with an aspect ratio of 1 (red circle), 2 (blue triangle), 3 (orange cross), and 6

794

(purple square), respectively. Simulation conditions were provided in Table 1. Note that the x-axis scales

795

for (a, c, e) and (b, d, f) were different. The statistical errors were the same as described in Figure 5.


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Langmuir

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Figure 7. Integrated DLVO interaction energy profiles from the SEI method as a function of colloid-

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collector separation distance when colloid surface heterogeneity was located (a) on the ends versus (b) on

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the middle band for 6.6 µm (in diameter) ellipsoids of different aspect ratios: AR1 - sphere (purple), AR2

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- aspect ratio 2 rod (red), AR3 - aspect ratio 3 rod (blue) and AR6 - aspect ratio 6 rod (green). Blue

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patches represented colloid surface charge heterogeneity. These energy profiles were generated when the

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heterogeneous patches were facing the collector with an end-on orientation (a, 0.004% coverage) or side-

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on orientation (b, 2.6% coverage). Other parameter values were identical as Table 1 unless specified

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otherwise.

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Figure 8. Simulated single collector efficiencies () as a function of particle size under unfavorable

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conditions with representative heterogeneity surface coverage (SCOV): (a) spheres colloids - 1.5% (red

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circle) , 3.5% ( green square), 30% (orange triangle), 100% (blue diamond), and representative theory

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predictions from MHJ prediction71 (blue line); (b) aspect ratio 6 rods -

Rotation and Retention Dynamics of Rod-Shaped ... - ACS Publications

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