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Deposition Dynamics of Rod-shaped Colloids during Transport in Porous Media under Favorable Conditions ke li, and Huilian Ma Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b03983 • Publication Date (Web): 05 Feb 2018 Downloaded from http://pubs.acs.org on February 6, 2018
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Deposition Dynamics of Rod-shaped Colloids during Transport in
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Porous Media under Favorable Conditions
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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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A 3D computational modeling study of the deposition dynamics of rod-shaped colloids during
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transport in porous media under favorable conditions (no energy barrier to deposition) was
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presented. The objective was to explore the influences of particle shape on colloid transport
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and retention. During the simulation, both translation and rotation of ellipsoidal particles were
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tracked and evaluated based on an analysis of all the forces and torques acting on the particle.
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We observed that shape was a key factor affecting colloid transport and attachment. Rod
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particles exhibited enhanced retention compared with spheres of equivalent volume at size
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range greater than ~ 200 nm. The shape effect was most pronounced for particles around 200
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nm to 1 µm under simulated conditions. Shape effect was also strongly dependent upon fluid
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velocity; it was most significant at high velocity, but not so at very low velocity. The above-
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described shape effect on retention was directly related to particle rotation dynamics, due to
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the coupled effects from rotational diffusion and flow hydrodynamics. Rotational diffusion
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changed particle orientation randomly, which caused rod particles to drift considerably across
∗
Corresponding author. E-mail: huilian.
[email protected]; Tel: (801)585-5976; Fax: (801)581-7065.
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flow streamlines for attachment at size range 200 nm ~ 1µm. The hydrodynamic effect
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induced periodic particle rotation and oscillation, which rendered large size rod particles to
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behave like “spinning bodies” prescribed by their long axes so to easily intercept with
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collector surface for retention. Our findings demonstrated that the practice of using equivalent
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spheres to approximate rods is inadequate in predicting the transport fate and adhesion
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dynamics of rod-shaped colloids in porous media.
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1. Introduction
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The transport and deposition of colloids in saturated porous media is essential to manage and
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remediate various environmental processes. For example, an accurate understanding of
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colloid transport is necessary to assess surfactant loss in enhanced oil recovery,1 to track
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contaminant fate,2 and especially to produce safe drinking water.3 Almost all colloids in the
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environment are non-spherical in shape, such as clay particles, different types of bacteria (e.g.
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Bacilli), human blood cells, polymer molecules, agglomerates and asbestos particles.
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However, existing filtration theories often simplify natural colloids as spheres of certain
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effective diameters, which have been demonstrated by increasing evidence4-9 to be insufficient
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in describing their transport and retention behaviors.
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Unlike spheres in a force field, for each different orientation, the forces acting on non-
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spherical particles are different. Various researchers have proved that ellipsoidal particles
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have different transport and retention behaviors compared with spheres.10-11 For example, rod-
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shaped colloids exhibited elevated retention in column experiments compared to their 2 ACS Paragon Plus Environment
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equivalent-volume spheres under otherwise identical conditions.12-13 A few studies also
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showed that the non-spherical colloid shape affected not only the extent, but the pattern of
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colloidal retention.10,
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straining kinetics and found that as those particles approached pore-space constrictions, a
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preferred orientation was adopted and theirs major axes tended to align with the local flow
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direction. Yoo et al.17 used synthesized PLGA particles with spherical and elliptical disk shape
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to investigate the rate of particle endocytosis and their distribution in endothelial cells. They
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found that elliptical disks were endocytosed slower compared to spheres, and that disks
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oriented tangentially along the nuclear membrane. Zhao et al.10,
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simulation work of ellipsoids that those particles aligned in the stream-wise direction in the
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near-wall region during channel flow, and that the particles affected the flow by altering the
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turbulence structure near and around the kinetic energy peak. In the process of particles
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depositing at the edge of evaporating suspension, Yunker et al.15 found that spheres deposited
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randomly with surface diffusion, while ellipsoids induced strong capillary attraction on the
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air-water interface and the deposition appeared to be ‘sticky’, i.e., clustering and creating void
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regions and particle-rich regions. The contrasting behaviors between non-spherical and
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spherical particles reflect different mechanisms by which non-spherical and spherical particles
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transport and attach.
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Xu et al.16 explored the effects of peanut-shaped particles on
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concluded in their
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Jeffrey (1922) predicted that rigid ellipsoidal particles tended to rotate with a certain period in
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a linear shear field.18 This prediction was validated by Zia via his experiments,19 and has been
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generally accepted. Although spheres also rotated in Jeffrey’s prediction, homogeneous 3 ACS Paragon Plus Environment
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spheres tended to follow streamlines under conditions when external forces or Brownian
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diffusion were absent.18,
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rotation of non-spherical particles, especially inertial particles, may also affect significantly
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its translational motion.20 Broday21 calculated that inertial particles in vertical shear flow
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migrated across the streamlines. Huang and Joseph22 also showed that elliptical cylinders
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settling vertically in a channel drifted laterally due to the orientation-dependent hydrodynamic
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interactions. These pioneer studies into shape effect introduced a valuable perspective on non-
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spherical particle transport and retention. However, the above-mentioned studies did not
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consider diffusion process, which is especially important for micro- and nano-meter size
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ranged particles.
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However, this is not the case for non-spherical particles. The
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With respect to homogeneous sphere transport, Brownian rotation may be irrelevant; hence, it
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is generally overlooked in general filtration theory. In contrast, rotational diffusion is a
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fundamental characteristic describing the tumbling of natural anisotropic (non-spherical)
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particles in solution. Rotational diffusion (or angular diffusion) influences non-spherical
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particle transport through the orientation dependency of drag forces.23-24 Dresher et al.25
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observed in their study that rotational diffusion diminished the hydrodynamic interactions
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between two E. coli bacteria and altered their travelling paths. Kuzhir et al.11 found that the
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magnetically induced rotational diffusion process caused the misalignment of particle
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aggregates from the streamlines, which further illustrated that the long-range non-
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hydrodynamic interactions of rotary diffusion process was also indispensable in studying
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spheroid transport phenomenon. However, most of these studies were limited to particle sizes 4 ACS Paragon Plus Environment
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in the Brownian regime, and they did not consider the influences of hydrodynamic
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interactions induced by non-spherical shape.
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Realizing the unique physical properties of non-spherical particles that can regulate transport
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process, increasing attention has been attracted to the utilization of such particles in
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biomedical field.26-28 For instances, many studies in drug-delivery systems showed that non-
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spherical particles (e.g., rod-shaped or disc-shaped) exhibited better migration propensity or
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adhesion onto blood vessel walls than spheres (including nano-spheres).29-35 But most of these
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investigations, experimental or simulation, were carried out in parallel plate flow chambers
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(PPFC).32-35 In contrast, studies on the influences of particle shape on transport and retention
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in colloidal filtration field have been very limited.5, 36-37 Those few existing studies were all
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from experimental investigations; and to the best knowledge of the authors, there is no
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simulation work available yet from literature on non-spherical particle transport in porous
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media.
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In this work, we develop a three-dimensional particle transport model that incorporates the
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non-spherical shape (e.g., rod-shaped particles as our model colloids here). This model will
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simulate both the translation and orientation of rod-shaped particles in representative flow
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fields based on an analysis of all the forces and torques acting the particle. From this
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trajectory model analysis, we will obtain: i) 3D translation and rotation trajectory of
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individual particles; ii) retention probability and distribution; iii) attachment orientation; iv)
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transient particle position distribution; v) transient linear and angular velocities. Simulating 5 ACS Paragon Plus Environment
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both translation and rotation is very challenging, because non-spherical colloids may have
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several different axes of rotation and also because its translation may depend upon its real-
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time orientation. That is probably why quite a number of existing numerical studies on non-
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spherical particle transport from biomedical field chose to simplify the problem with a 2D
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model.20, 38-40 However, due to the complexities involved, we have two main assumptions in
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our model that we believe are reasonable: i) since what we obtain here was clean single
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collector efficiency, particles injected into our model systems were of dilute concentration
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such that particle-particle collisions and aggregate formations could be neglected; ii) fluid
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field disturbance due to the presence of particles was ignored. The model thus developed for
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non-spherical particle transport can be modified and applied to various different geometric
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flow models, because the essential simulation methodology stays the same. In this work, we
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will present results obtained during our systematic investigations of rod-shaped colloids in
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Happel sphere-in-cell model (i.e., a well-known model to represent porous media in classic
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colloid filtration theory) under favorable attachment conditions (i.e., particle-collector
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repulsion is absent). The simulation parameters spanned a wide range of ellipsoidal particle
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sizes and shapes flowing under different pore water velocity. The main goals from these
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simulations are to answer the following questions: i) under what conditions, non-spherical
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particles can be treated as “effective spheres” and under what conditions, they can’t; ii) how
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the shape of a particle affects its transport and retention? By what mechanisms?
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2. Simulating Transport of Ellipsoidal Particles
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Particle Transport Model. A Lagrangian trajectory approach was developed in this work to 6 ACS Paragon Plus Environment
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simulate the time evolution of particle translation and rotation in representative porous media,
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e.g., the well-known Happel sphere-in-cell model. The Happel model represents porous media
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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.41-42 The general equations
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of motion to account for the translation and rotation of a rigid particle are given by
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= ∑
=
∙ = ∑ =
where t is time, m is the mass of the particle, is the moment of inertia of the particle; x and u
are the linear position and velocity vectors of the particle, respectively; Ω and are the
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respective angular position and velocity vectors of the particle; and ∑ and ∑ represent the
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total forces and torques acting on the particle, respectively. Our previous work simulated the
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linear translations (i.e., eqs 1-2) of spherical particles in representative porous media models
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extensively,4, 43-47 but did not explicitly account for particle orientation excepting converting
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rotational motion of a sphere to linear translation.48 Here, we will focus on how to incorporate
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both the rotational and linear motions of particles into simulation, using prolate ellipsoid
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particles as model colloids. Due to symmetry for ellipsoids, the coupling tensor between
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translation and rotation in eqs 1 and 3 vanishes based on Brenner’s research.49 Also in this work,
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we treat Brownian motions differently from our previous work, where Brownian forces were
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included into eq 1 to account for Brownian translation but Brownian rotation was neglected.
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Here, we first determined the linear and rotational displacements from the total forces and
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torques in eqs 1 and 3 that included only the deterministic forces and torques (i.e., excluding 7 ACS Paragon Plus Environment
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those from Brownian motions). We then added Brownian translation and rotation, which were
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modeled in each degree of freedom as a stochastic process using Einstein’s equations, to those
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resulted from the deterministic motions, respectively. Below, we shall describe in details how
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to numerically solve the above equations of motion for an ellipsoid moving in shear flow.
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Coordinate Systems. In order to describe the three-dimensional rotation and translation of
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ellipsoidal particles, the following coordinate systems were employed (Figure 1): i) an inertial
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frame of reference, = x, y, z, with its origin located at the collector (e.g., the origin of the
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Happel sphere-in-cell model); ii) a particle frame, = x , y , z , with its origin at the center-of-
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mass of the particle and its axes being the particle principal axes; and iii) a co-moving frame,
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= x , , z , with its origin at the center-of-mass of the particle and its axes parallel to the
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corresponding axes of the inertial frame. The translational motions of the ellipsoid were
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described by tracking the center-of-mass of the particle in the inertial frame. The rotational
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motions were described by following the three Euler angles (Figure 1) or the four Euler
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parameters (i.e., quaternions) in the particle frame.
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Transformation of the coordinates between the particle and co-moving frames is given by 50-51
= ,
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(5)
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where the orthonormal transformation matrix A can be expressed in terms of either Euler
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angles or the four Euler parameters (i.e., q0, q1, q2, q3). Below, matrix A is given using Euler
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parameters as
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Langmuir
1 − 2 $%% + $'% 2 $( $% + $' $) 2 $( $' − $% $) = 2 $% $( − $' $) 1 − 2 $'% + $(% 2 $% $' + $( $) *. 2 $' $( + $% $) 2 $' $% − $( $) 1 − 2 $(% + $%%
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(6)
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The four Euler parameters are subject to the following constraint to preserve the three degrees
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of freedom for a rotating rigid body: $)% + $(% + $%% + $'% = 1.
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(7)
Euler angles are related to the Euler parameters as $) = cos .
/01 3 2 cos . % 2 , $( %
/01 3 2 cos . % 2. %
sin .
= cos .
/41 3 2 sin .% 2, $% %
= sin .
/41 3 2 sin .% 2, $' %
=
(8)
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The ranges for Euler angles used in this study are ϕ ∈[-π,π], θ ∈[0,π], and ψ ∈[-π,π].
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Conversion from Euler parameters to Euler angles is provided in the Supporting Information.
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Rotation Dynamics of an Ellipsoid in Shear Flow. In this work, we chose the Euler four
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parameters to describe the time evolution of particle angular displacements, because the use
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of Euler angles would involve a large number of trigonometric calculations, which are
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computationally expensive, and also because there is a singularity in Euler angles (see
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Supporting Information). However, Euler angles were used to assign the initial orientation of
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a particle (most of simulation data here with random initial orientations). The initial Euler
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parameters were then determined from eq 8. At the subsequent time steps, the Euler
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parameters evolved with time as51
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−$(
