Concurrent Modeling of Hydrodynamics and Interaction Forces

Article pubs.acs.org/est

Concurrent Modeling of Hydrodynamics and Interaction Forces Improves Particle Deposition Predictions Chao Jin,† Carolyn L. Ren,‡ and Monica B. Emelko*,† †

Department of Civil and Environmental Engineering, University of Waterloo, 200 University Ave W., Waterloo, Ontario N2L 3G1, Canada ‡ Department of Mechanical and Mechatronics Engineering, University of Waterloo, 200 University Ave W., Waterloo, Ontario N2L 3G1, Canada S Supporting Information *

ABSTRACT: It is widely believed that media surface roughness enhances particle depositionnumerous, but inconsistent, examples of this effect have been reported. Here, a new mathematical framework describing the effects of hydrodynamics and interaction forces on particle deposition on rough spherical collectors in absence of an energy barrier was developed and validated. In addition to quantifying DLVO force, the model includes improved descriptions of flow field profiles and hydrodynamic retardation functions. This work demonstrates that hydrodynamic effects can significantly alter particle deposition relative to expectations when only the DLVO force is considered. Moreover, the combined effects of hydrodynamics and interaction forces on particle deposition on rough, spherical media are not additive, but synergistic. Notably, the developed model’s particle deposition predictions are in closer agreement with experimental observations than those from current models, demonstrating the importance of inclusion of roughness impacts in particle deposition description/simulation. Consideration of hydrodynamic contributions to particle deposition may help to explain discrepancies between model-based expectations and experimental outcomes and improve descriptions of particle deposition during physicochemical filtration in systems with nonsmooth collector surfaces.

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INTRODUCTION Particle deposition on surfaces is critical to numerous applications ranging from drinking water treatment1 to semiconductor manufacturing,2 accordingly, significant research effort has focused on improved understanding of colloidal and nanoparticle transport and deposition phenomena in a variety of natural and industrial environments.1,3−6 Particle deposition on target (or collector) surfaces can be numerically described by Eulerian or Lagrangian methods, assuming the collectors have ideal geometries such as spheres,7 cylinders,8 rotation disks,9 or parallel plates.8 Typical approaches to modeling particle deposition on collector surfaces have involved quantification and subsequent summation of van der Waals forces (VDW) and electrostatic double layer interaction forces (EDL) (i.e., calculation of DLVO force) to describe physicochemical interactions between particles and collector surfaces.10−13 Classic filtration theory can reasonably predict particle deposition on smooth surfaces when particle and collector surfaces are oppositely charged or when there is no substantial energy barrier between the approaching particles and target surfaces.10,11,14 In contrast, vast discrepancies between theoretical predictions and experimental outcomes have been reported when particles and collectors are similarly charged and a large interaction energy barrier is present.11,15−17 For © XXXX American Chemical Society

example, although several theoretical models have predicted “sharp” decreases in particle deposition flux when background ionic strength is below a critical threshold concentration,7,11,18 experimental investigations have frequently demonstrated gradually decreasing particle deposition flux with decreasing background ionic strength.11,19,20 At low ionic strength, experimentally observed particle deposition fluxes several orders of magnitude higher than model predictions have been commonly reported.21−24 Although parameters such as attachment efficiency (α) have been introduced to enable improved description of experimental observations of particle deposition on collector surfaces,1,14,25 existing filtration theory remains too sensitive to changes in ionic strength and ζ-potential, in some cases; thereby rendering many model predictions of particle deposition flux inadequate. Moreover, α cannot be mechanistically predicted. Discrepancies between theoretical predictions of particle deposition on collector surfaces and experimental observations have been attributed to (1) roughness on particle and collector surfaces,11 (2) chemical charge heterogeneity26 Received: January 20, 2016 Revised: March 22, 2016 Accepted: March 23, 2016

A

DOI: 10.1021/acs.est.6b00218 Environ. Sci. Technol. XXXX, XXX, XXX−XXX

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Environmental Science & Technology and deposition due to the secondary energy minima,27 (3) hydrophobicity,25 and short-range forces,28 and (4) physical mechanisms, such as straining,29 wedging,17 and blocking.24 Regardless of the exact factors that contribute to these discrepancies, it is critical to recognize that some of the original assumptions on which many traditional and current models of particle deposition are based (e.g., smooth collector surfaces) may not be reasonable for many natural and engineered particle deposition scenarios. Although enhanced particle deposition due to collector surface roughness has been frequently reported,30−33 decreased particle deposition also has been reported.34,35 Recent investigations have suggested that collector surface roughness affects particle deposition in a nonlinear, nonmonotonic manner that depends on the relative relationship between surface roughness, and particle and collector sizes.36 Although several studies have recently reported that collector surface roughness can significantly alter the shape and magnitude of particle-surface interaction energy profiles,28,37,38 it also affects hydrodynamic factors such as flow field and retardation, which also contribute to particle deposition. Despite extensive empirical evidence of collector surface roughness impacts on particle deposition, a mathematical framework for describing particle deposition on rough collector surfaces in packed beds is currently lacking, thereby precluding reasonable application of particle deposition models to many natural and engineered systems. Here, an Eulerian approach was used to develop a new mathematical model for describing particle deposition on spherical collectors with nanoscale surface roughness in absence of an energy barrier. Clean bed filtration conditions were assumed and the classic Happel sphere-in-cell model was used to describe the flow field profile around the collector.7 In addition to quantification of DLVO force in the presence of nanoscale surface roughness (i.e., without assumption of smooth surfaces), the model incorporates recently developed numerical approaches that provide improved description of the flow field profiles and hydrodynamic retardation.36,39−41 This model was used to quantitatively evaluate each of the individual impacts of these factors on particle deposition on spherical collectors with several sizes (i.e., heights) of surface roughness. To validate the developed model and underscore the importance of inclusion of roughness impacts on particle deposition in describing and simulating these phenomena, the theoretical solution of single collector efficiency (η) using the developed model was compared to experimental and theoretical outcomes published by our team36,42 and others.14,43

surface-to-surface distance of Si (Figure 1).32,42 Following the approach of Kemps and Bhattacharjee(2005),46 the roughness asperities were described by stacked spheres with radius of ar; thus, the height of the roughness asperities was equal to 2nsar where ns represents the number of stacked spheres. The

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MODEL DEVELOPMENT Representation of a Particle Approaching a Rough Surface. Modeling particle deposition flux on spherical collectors with surface roughness requires description of not only roughness size, but also its coverage (or distribution) on collector surfaces. Detailed representation of surface roughness topography can be achieved with Fourier transform44 or fractal analysis approaches;45 however, integration of these approaches with Convective-Diffusion models becomes exceedingly computationally intensive. Here, a previously described,32,42 simplified geometric representation of rough surfaces that was developed and validated within a parallel plate system was used. The rough collector surface was represented as a smooth bottom surface with protruding planar roughness asperities that are vertically distributed as elongated filaments separated by a

Figure 1. Schematic of (a) flow around a spherical collector with rough (left hemisphere) and smooth (right hemisphere) surfaces; the flow fields and diffusion coefficients for (b) smooth and (c) rough collector surfaces. B


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roughness asperities are relatively small as compared to the bottom surface, which can be assumed as an infinite plate for Derjaguin approximation. Mass Transfer in Spherical Geometry. At steady state and in absence of chemical reactions, the general ConvectionDiffusion equation for particle mass transfer due to diffusion, convection, and external forces can be expressed as

composition of the roughness asperities was essentially identical to that of the smooth bottom surface; therefore, the collector surface and roughness asperities had the same ζ-potential, Hamaker constant, and hydrophilic/hydrophobic characteristics. Given the aquatic conditions implemented herein (favorable conditions for particle deposition), the pairwise summation method was used to evaluated the degree of DLVO interaction energy.46,47 Other more accurate approaches such as grid-surface integration48 and surface element integration49 for describing DLVO interactions are available; however, the use of these approaches was too computationally intensive when incorporated into the present modeling framework for particle deposition using the finite element method. It should be noted that this study is limited to the case that (a) the approaching particle can only contact and attach on the tops of roughness asperities, (b) the roughness asperities are adequately close to one another such that particle deposition on the bottom smooth surface between them is precluded, and (c) the

⎛ D·F ⎞ ∇·(uC) = ∇·(D·∇C) − ∇·⎜ C⎟ ⎝ kbT ⎠

where u is the fluid velocity, T is the absolute temperature, kb is Boltzmann’s constant, C is the particle concentration, D is the diffusion tensor, and F represents external forces that can act as sources or sinks of momentum. eq 1 can be solved numerically when the flow velocity field, D, and F are known. In spherical geometry, the full, dimensionless expression of the ConvectiveDiffusion equation is described as

N ∂c ∂c + NR 2f4 (H )cot θ − f4 (H )FG*sin θNR − Pe f3 (H )VθNR NR 2f4 (H ) ∂θ ∂θ 2

{

+

}

⎡ ∂f1 (H ) ⎞ ⎛ ∂φ N ⎤⎫ ∂c ⎧ ∂c *cos θ ⎟ − f (H )f (H )Vr Pe ⎥⎬ ⎜ ⎨ ⎢ + + + + f ( H ) 2 N f ( H ) f ( H ) F R G 1 1 1 1 2 ⎝ ∂H ⎠ ∂H ⎪ ∂H ∂H 2 ⎦⎪ ⎣ ⎩ ⎭ ⎪

⎪

⎧ ⎡⎛ ⎤⎫ f1 (H )∂FG*cos θ ∂f (H ) ⎞⎛ ∂φ ⎞ ∂ 2φ ⎪ ⎢⎜2N f (H ) + 1 * * ⎥⎪ ⎜ ⎟ + f (H ) ⎟ f H N F θ θ + F cos 2 ( ) cos + − G R G 1 4 ⎠ ⎪ ⎢⎣⎝ R 1 ∂H ⎠⎝ ∂H ∂H ⎥⎦ ⎪ ∂H2 ⎪ ⎪ ⎪ ⎪ ⎡ f H f H ( ) ( ) ∂ ∂ ∂Vr ⎬ = c⎨− NPe ⎢2N f (H )f (H )V + 1 2 f (H )Vr + f (H )Vr + f1 (H )f2 (H ) r 2 ⎪ 2 ⎣ R1 ⎪ ∂H 2 ∂H 1 ∂H ⎪ ⎪ ⎤ ⎪ ⎪ ⎪ + 2NRf3 (H )cot θVθ ⎥ ⎪ ⎦ ⎩ ⎭

C(H = 0, θ ) = 0

(3)

∂C = −UC0cos θ at r = b , 0 ≤ θ ≤ 0.5π ∂r (4a)

∂C = 0 at r = b , 0.5π ≤ θ ≤ π ∂r ⎛ ∂c ⎞ ⎜ ⎟ =0 ⎝ ∂θ ⎠θ= 0

(2)

The term − UC0cos θ refers to the flux into Happel’s sphere ∂C where the vrC − Dr ∂r refers to the particle flux within the cell due to convection and diffusion. eq 5 arises from the symmetry around the forward stagnation path within the Happel sphere.50 The complete form of the Convective-Diffusion equation derived from first-principles was utilized to describe particle deposition. In contrast to other versions of the equation7 that have been commonly utilized as a foundation for modeling particle deposition on porous media,14,51,52 eq 2 includes three additional terms that were previously assumed negligible:7

The list of dimensionless parameters used in eq 2 is provided in Table 1. To solve eq 2 numerically, the implemented boundary conditions were

vrC − Dr

(1)

(4b)

∂ 2c

∂c

f (H )∂F *cos θ

G NR 2f4 (H ) 2 , NR 2f4 (H )cot θ ∂θ , and 1 . In the present ∂θ ∂H work, these three terms were included in the simulation for completeness. When dimensionless particle concentration is determined using eqs 2−5, particle flux perpendicular to collector surfaces J(Hδ,θ), cumulative colloid deposition flux I, local dimensionless deposition rate Sh, and single collector efficiency η can be calculated at a cut off distance δ by implementing the perfect sink boundary condition that has been applied in classic, previously reported works focused on describing particle deposition in porous media at favorable conditions7(Supporting Information). DLVO Force Modifications for Approaching Particles. In contrast to typical approaches that assume smooth collector surfaces when quantifying DLVO force between approaching particles and spherical collectors,10,14,53 in the present study,

(5)

Equation 3 describes the perfect sink boundary condition at the collector surface. This applies to the clean bed period of filtration and does not account for particle to particle interactions, rolling, shear lift, drag, and other factors that may be significant to particle detachment.1,7,14 To avoid overestimated deposition at low Peclet numbers, modified boundary conditions at the outer shell of Happel’s sphere at r = b (eqs 4a and 4b) were implemented to describe the continuous net radial flux (without singularity for simulation) toward the collector surface at the outer boundary of the upper and lower half shells of the unit Happel’s sphere, respectively. C


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(φtotal DLVO|rough ) and particle-smooth collector interactions

Table 1. Dimensionless Groups and Parameters Used in the Convection-Diffusion Equation dimensionless group

(φtotal DLVO|smooth ) as

expression

center-to-center distance between particle and collector

r = r = ac + ap + Hap

radial velocity component of colloidal particle

− 1 ∂ψ vr = 2 r sin θ ∂θ

tangential velocity component of colloidal particle

vθ =

1 ∂ψ r sin θ ∂r

scaled surface-to-surface distance for h

H=

h ap

scaled particle ratio

NR =

scaled radial velocity component of colloidal particle scaled tangential velocity component of colloidal particle

v Vr = r U

Peclet number

NPe =

Sherwood number

Sh =

scaled interaction energy

ap r

φPS VDW = −

D∞ J ·Udc D∞

φij VDW = −

FGap

kT φ Φ= kBT

+

rroughness

rslip =

scaled hydrodynamic retardation functions scaled velocity vector of colloidal particle scaled average interaction energy

fi (h) = Fi(h + rslip)

φtotal DLVO = φtotal VDW + φtotal EDL

∑ φPR VDW + φPS EDL

⎤ + (ψP2 + ψS2)ln(1 − exp( −2κH ))⎥ ⎥⎦

i=1 n i=1

(9)

⎡ ⎛ 1 + exp( −κH ) ⎞ φPS EDL = πεε0ap⎢2ψPψS ln⎜ ⎟ ⎢⎣ ⎝ 1 − exp( −κH ) ⎠

n

∑ φPR EDL

R ij2 − (ai + aj)2 + 4aiaj

where i and j represent the rough sphere and approaching particle, Rij is the center-to-center distance between two spheres, Aij is the Hamaker constant between i and j in the background solution, λ is the characteristic wavelength, h and H are the true and dimensionless surface-to-surface distances, respectively. The EDL interaction energy between the approaching particle and the rough collector surface was assessed using the linear approximation of the Poisson−Boltzmann equation.37,38 Accordingly, the individual EDL interaction energies between the particle and bottom surface or spheres comprising the roughness asperities were described by

φ = χUtotal DLVO|rough + (1 − χ )Utotal DLVO|smooth

particle-collector DLVO interaction energies in the presence of surface roughness were approximated by the pairwise summation method37,54 under the assumption of constant ζpotential. Here, the total interaction energy arising for rough surfaces from VDW and EDL forces (φtotal DLVO) was described as

+

(8)

Aij ⎡ 2aiaj ⎢ 6 ⎢⎣ R ij2 − (ai + aj)2 2aiaj

⎛ ⎞⎤ R ij2 − (ai + aj)2 ⎟⎥ + ln⎜⎜ 2 ⎟ 2 ⎝ R ij − (ai + aj) + 4aiaj ⎠⎥⎦

ap

V (r ) = V (r + rslip)

= φPS VDW +

⎞ AaP ⎛ 1 ⎜ ⎟ 6h ⎝ 1 + 14h/λ ⎠

Equation 9 accounts for the unretarded effect of VDW interaction between the approaching particle and roughness asperities, because the approaching particle can attach on the top of roughness asperities and then be removed from the bulk suspension.

2Uap

scaled roughness height

(7)

Here, χ is the fraction of rough surface and (1 − χ) is the fraction of smooth surface over the entire collector surface. Hamaker’s approach was used to evaluate the net VDW interaction energy (φtotal VDW) between the approaching particle and the rough collector surface.7 eq 8 accounts for the retardation effect of VDW interaction between the approaching particle and bottom surface. That is because the approaching particle cannot contact/attach on the bottom surface; thus, the closest distance between the approaching particle and bottom surface is the height of the roughness asperities (2nsar).

v Vθ = θ U

FG* =

scaled gravity force

φ = χφtotal DLVO|rough + (1 − χ )φtotal DLVO|smooth

(6)

where φVDW and φEDL PS PS represent the VDW and EDL interaction energies between the approaching particle and smooth surface of the collector. The VDW and EDL interaction energies between the particle and the roughness elements on the VDW VDW collector surface were described by φPR and φPR , 37,38,46,55 respectively. Assuming that roughness covers a portion of the entire collector surface, the average magnitude of particle-collector interaction energy (φ) can be calculated from the weighted sum of particle-rough collector interactions

φij EDL =

(10)

⎛ 1 + exp( −κH ) ⎞ πεε0aiaj ⎡ ⎢2ψψ ln⎜ ⎟ i j (ai + aj) ⎢⎣ ⎝ 1 − exp( −κH ) ⎠ ⎤ + (ψi2 + ψ j2)ln(1 − exp( −2κH ))⎥ ⎥⎦

(11)

where ψ1 and ψi are the respective ζ-potentials of the particle and roughness asperity spheres. Consistent with previously reported approaches for simulating particle deposition on D


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Figure 2. (a) DLVO interaction energy curves and the absolute values of interaction energy at a cutoff distance of 1 nm for different roughness element sizes; (b) Particle deposition flux (Sh) for different roughness element sizes solely impacted by DLVO force, and (c) the concurrent impacts of modified flow field, hydrodynamic retardation functions and corresponding DLVO interactions. The inset figures are the corresponding single collector efficiency (η) over the entire collector surface (π). The curves correspond to rroughness: 0, 6, 30, 60, 120, 240, and 360 nm; Particle and collector surface potentials = −20 mV and 20 mV respectively; ionic strength = 100 mM KCl; particle radius =200 nm; collector diameter = 200 μm; temperature = 25 °C; Gravity number = 0 and Hamaker constant = 1 × 10−20 J.

collector, b was scaled by the size of the roughness elements to enable the use of dimensionless slip length (rslip) to properly describe the modified flow field velocity; thus, the original assumptions of the Happel’s sphere-in-cell model were not changed (Figure 1a).42 The flow field velocity component was described as

spherical collectors, the approaching particle and collector surface ζ-potentials were assumed to be 20 mV and −20 mV respectively.7,37The net force between the particles and collector surfaces was attractive and without a large interaction energy barrier in all cases. Nanoscale surface roughness sizes from 0 to 600 nm were evaluated. Flow Field Description. When collector surfaces are rough, the traditional implementation of the no-slip boundary condition at the top of the contact surfaces to describe flow field profiles is inappropriate (Figure 1a).56,57 Slip or partial-slip boundary conditions are more appropriate in these situations, depending on the absolute height and extent of coverage of the roughness elements.39,40 Alternatively, when surface roughness size is less than the approaching particle radius and the extent of surface coverage with roughness features exceeds 50%, a hydrodynamically equivalent smooth plane (i.e., the “effective target surface”) located between the top and bottom (i.e., collector) surfaces of the roughness elements can be utilized and the no-slip boundary condition can be shifted from the top of the roughness elements to the bottom (collector) surface (Figure 1b and c).41,58 Thus, the appropriately modified flow field can be approximated using slip length (i.e., dimensionless roughness size) (Figure 1c).41,58 This was done in the present investigationthe thickness of the fluid shell around the

V′(r ) = V(r + rslip)

(12)

rslip = rroughness/a p

(13)

where the V(r) and V′(r) was the smooth and modified velocity vector which can be described as V(r) = iVθ + jVr. As a result of this adjustment, the radial velocity was higher for rough surfaces than for smooth ones. Notably, when rroughness is equal to zero (indicating no roughness on the collector surface), the velocity component should be equal to that of a smooth surface (Figure 1b). Hydrodynamic Retardation. Surface roughness reduces hydrodynamic retardation; therefore, the associated descriptive functions must be appropriately modified.58,59 Analogous to the modification for the flow field, slip length can also be used in describing the influence of surface roughness on hydrodynamic retardationthis approach was utilized herein (Figure 1c). Proposed and validated by Vinogradova and Belyaev (2011),59 E


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Elimelech and Song (1998)7 and demonstrated excellent agreement (Figure S-4).

a hydrodynamically equivalent plane between the top and bottom(collector) of the roughness elements can be assumed as the effective target surface for calculation of the hydrodynamic retardation functions (Figure 1c and Figure S-1). This approach has been utilized for describing surface roughness impacts on particle deposition in parallel plate systems, in which good agreement between numerical solutions and experimental results was rigorously demonstrated.42 The modified hydrodynamic retardation functions incorporating roughness impacts on particle deposition can be expressed as fi (h) = Fi(h + rslip)

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RESULTS AND DISCUSSION DLVO Interaction Energy. The DLVO interaction energy curves between an approaching particle and collector surface with different roughness sizes are presented as a function of the dimensionless separation distance (H) in Figure 2a. Notably, all of the calculated interaction energies were negative due to the applied physicochemical conditions (indicating a net attractive force between the particle and collector surface), as would be expected in absence of an energy barrier to particle deposition. To quantitatively compare the DLVO interaction energies for different roughness sizes, the value of interaction energy at a cut off distance of 1 nm is presented in the Figure 2a inset. When collector surface roughness size increased from 0 to 360 nm, the absolute value of interaction energy at the cutoff distance first decreased to a minimum (at a roughness size of ∼60 nm) and then increased. This nonmonotonic effect of collector surface roughness on interaction energy has been described elsewhere.33,37,42 Dimensionless particle deposition flux (Sh) on the collector surface as a function of the radial location on spherical collectors (θ) with different roughness sizes, and impacted only by the DLVO force, is presented in Figure 2b; the cumulative dimensionless flux (i.e., single collector efficiency (η) over the simulation domain [0, 2π]), is presented in the inset. When DLVO interaction was the sole mechanism driving deposition, the maximum amount of particle deposition that was observed occurred on smooth surfaces (i.e., rroughness = 0); accordingly, over the range of conditions investigated, the DLVO force alone consistently resulted in decreased total particle deposition flux to rough surfaces relative to that observed on smooth surfaces (Figure 2b). When surface roughness size increased from 0 to 360 nm, deposition flux decreased to a minimum value (at a corresponding roughness size of ∼60 nm), then increased again. It should be noted that although the deposition curves in Figure 2b also followed a nonlinear, nonmonotonic pattern similar to the DLVO interaction energy profiles in Figure 2a, the determined η (the cumulative deposition flux on a single collector; Figure 2b inset) is not directly proportional to the DLVO profiles. In this case, the maximum difference in single collector efficiency (η) among the various roughness scenarios was only ∼19%. As demonstrated by Jin et al. (2015),36 such small variations in η are too small to solely explain reported differences in observed particle deposition in the presence of surface roughness in packed beds and parallel plate systems.30,32,36,42 This analysis and conclusion are consistent with the extensive investigation of Elimelech (1990),63 who concluded that “...equilibrium (static) aspects of surface roughness [described by DLVO interactions only] cannot explain the observed discrepancies.” The inability to accurately describe particle deposition in the presence of collector surface roughness by using DLVO force alone is not surprising because interaction energy is not a direct measure of interaction force(s) acting on depositing particles, but only a surrogate measure of long-range forces that is often used to draw inferences about particle deposition flux. The DLVO force acting on depositing particles is the derivative of the interaction energy between the particles and the surface over distance.64 Although the interaction energy profiles in Figure 2a are considered significantly different from one another, they are generally similar in shape and therefore yield

(14)

where f i(h) represents the original individual hydrodynamic retardation functions f1−4(h) and Fi(h) represents the modified hydrodynamic retardation functions. Implementation of the analytical solution for the f1−4 functions over the entire simulation domain is computationally intensive.53,60 To overcome this barrier, f1−4 were calculated as a blend of ⎡ (f f )n ⎤1/ n asymptotic solutions described by fi = ⎢ f ni0 +i∞f n ⎥ where n is a ⎣ i0 i ∞ ⎦ fitting parameter, f i0 and f i∞are the analytical solutions for f1−4 when the dimensionless distance is approaching 0 and infinity (Table S-1).42 Compared to other previously reported numerical solutions,7,12,61 the new implemented functions of f1−4 are simple and computationally efficient, providing best fit to the analytical solutions with no more than 2% relative difference over the entire simulation domain (Figure S-2). Numerical Methods. The numerical solution of eq 2−5 was solved using the finite element method (COMOSL 3.5a, Inc., Canada). The simulation domain was discretized using quadrilateral meshing. Highly refined meshes were required at the regions with high concentration gradients or large tensors of applied forces (Figure S-3). Because a steep particle concentration gradient existed in the vicinity of the collector surface, extremely fine mesh elements smaller than the Debye− Hückel length (e.g., 10−4 in dimensionless size) were utilized at the bottom of the collector surface. The implemented numerical methods were validated using three approaches. First, the numerical solution to the Convective-Diffusion equation obtained using eq 2 was compared to the analytical solution (i.e., the SmoluchowskiLevich approximation) without hydrodynamic retardation effects (i.e., f1−4 equal to 1.0) and without interactions between the approaching particle and collector surface (i.e., φtotal DLVO equal to 0). The relative difference between the determined numerical solution and the analytical solution was 1.02% (at NPe = 0.63 when particle transport is controlled by pure convective diffusion). The numerical model was further validated following the approaches of Prieve and Ruckenstein (1974)62 and Elimelech (1991),43 in which the overall rate of particle deposition is obtained from integration of local particle flux (H = δ) over the entire surface of a spherical collector (0 to 2π) and η within the range of attractive interaction forces should be the same at steady state. Accordingly, particle flux was integrated over the entire collector surface for different values of H (δ to 100δ). The calculated values of Sh and η for various H were very similar (i.e., within less than 1%) to the values determined at H = δ, thus confirming the accuracy of the numerical solution of the transport equation. Finally, profiles of particle deposition on smooth collectors (i.e., in absence of collector surface roughness) obtained using the numerical solution were compared to the previously reported results of F


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Environmental Science & Technology similar DLVO force. Accordingly, the particle deposition flux profiles obtained when only the DLVO interactions were considered were also similar to one another, despite changes in nanoscale surface roughness size (Figure 2b). Thus, the relatively small differential impact of changes on particle deposition due to DLVO interaction energy (i.e., VDW and EDL forces) cannot be used solely to quantitatively interpret, describe, or predict particle deposition in the presence of surface roughness, even at nanoscalethis has been previously shown both experimentally and numerically in parallel plate systems.42 Concurrent Impacts of Hydrodynamics and Interaction Forces on Particle Deposition. Figure 2c describes the concurrent impacts of flow field profile, hydrodynamic retardation, and DLVO interactions on particle deposition on spherical collector surfaces due to the presence of nanoscale roughness. The dimensionless deposition rate (Sh) and single collector efficiency (η) changed significantly with increasing nanoscale roughness (Figure 2c); consistent with previous reports,36,42 a nonlinear impact of surface roughness on particle deposition was suggested (Figure 2c inset). When collector surface roughness size was smaller than ∼30 nm, η was not substantially enhanced by surface roughness. More substantial increases in total particle deposition flux were observed at roughness sizes greater than 60 nm, at which changes in hydrodynamic effects on the flow field and retardation that were attributable to surface roughness enhanced particle deposition and overshadowed DLVO interaction impacts in magnitude and in a manner consistent with previous work in a parallel plate system.42 Here, significantly higher (up to ∼330% more) particle deposition was observed when both factors were represented (Figure 2c), as compared to the scenario in which only the impacts of the DLVO force were considered (Figure 2b). It should be noted that this simulation was conducted assuming that the slip-lengths for flow field and hydrodynamic retardation are both linearly proportional to roughness height. In other cases, depending on surface hydrophobicity, shortrange forces, chemical heterogeneity, and the extent of surface coverage by roughness, the most appropriate slip-length(s) for describing the particle flow field and hydrodynamic retardation functions might not have the same values.40,41,56,65 Regardless, this work may shed light on the classic findings of Yao and O’Melia (1971),1,63 who concluded that the single collector efficiency developed for smooth surfaces underestimates particle deposition. That and other reported work has suggested that nanoscale roughness may explain the associated lack of agreement between reported experimental observations and simulation outcomes.1,18,63 Sensitivity Analysis. The relative sensitivity of particle deposition flux to DLVO energy was discussed above. The relative contributions of the introduced non-DLVO factors (i.e., hydrodynamics, as described by flow field and hydrodynamic retardation) to particle deposition on spherical collectors and the sensitivity of deposition flux in response to modified flow field and hydrodynamic retardation due to surface roughness also were evaluated. To illustrate the impact of changes in flow field profile on particle deposition, slip length (used to modify flow profile) as a function of radial location on spherical collectors (θ) was varied and the resulting tangential (vθ) and radial (vr) flow velocities and dimensionless particle deposition flux (Sh) were evaluated (Figure 3a−c respectively). The tangential velocity approached zero when θ = 0 or π and achieved maximum values when θ = 0.5π in all cases.

Figure 3. (a) Tangential (vθ) and (b) radial (vr) flow velocity components and particle deposition flux (Sh) sensitivity to slip-lengths used to modify (c) only the flow field around a spherical collector, (d) only the hydrodynamic retardation functions, and (e) both the flow field and hydrodynamic retardation functions as a function of the radial location on a spherical collector (θ) at different slip-lengths. The particle and collector surface potentials equal −0 mV and 0 mV, respectively. The curves correspond to the following dimensionless slip lengths: 0, 0.0001, 0.001, 0.01, 0.1, and 1. The other parameters are same as those used in Figure 2.

Dimensionless slip lengths from 0 to 1, corresponding to roughness sizes of 0−200 nm were evaluated. Bigger dimensionless slip-lengths resulted in higher tangential (Figure 3a) and radial (Figure 3b) flow velocities; particularly for slip lengths greater than 0.1, corresponding to roughness sizes greater than 20 nm. Surface roughness-associated changes in the flow fields resulted in changes in particle deposition flux, which followed patterns similar to those of the flow velocity components (i.e., vθ and vr). When surface roughness size (and associated dimensionless slip length) increased, total particle deposition flux on the spherical collector surfaces (i.e., area under the curve) increased (Figure 3c) due to changes in the flow velocity profile. The incremental changes in deposition flux due to flow field changes were relatively small when slip length was less than 0.1, but became more substantial at larger slip lengths. This is because higher flow velocities (vθ and vr) resulting from collector surface roughness enable suspension fluids to carry more particles to the vicinity of collector surfaces per unit time; thus, for a fixed attachment efficiency (α), more G


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Environmental Science & Technology available particles equate to higher deposition flux.11,43,66 Notably, it is the radial component of the velocity that results in a higher flux toward the surface.11 By definition, in the Happel sphere-in-cell model, a higher angular velocity might result in a lower flux toward the surface. At nonattractive DLVO conditions, higher angular velocity may reduce particle deposition to a greater extent than it increases by higher radial velocity. This relationship may reverse at attractive DLVO conditions because the separation distance under the influence of the attractive DLVO force is increased; thus, resulting in greater particle deposition.11,67 Hydrodynamic retardation and the associated functions that describe it ( f1−4(h)) as a function of radial location on spherical collectors (θ) were also modified using dimensionless slip length to describe surface roughness effects. Dimensionless slip lengths from 0 to 1, corresponding to roughness sizes of 0 to 200 nm were evaluated. The sensitivity of dimensionless particle deposition flux (Sh) to hydrodynamic retardation modified by slip length is presented in Figure 3d. Total dimensionless particle deposition flux (i.e., area under the curve) increased with increasing slip length because there is less hydrodynamic retardation of particles when they approach rough surfaces as compared to smooth ones (Figure S-2), making it easier for particles to deposit on rough collector surfaces.37,68 The majority of the increase in particle flux occurs on the first half of the collector surface (i.e., the surface facing flow) where θ is between 0 and 0.5π; this observation is consistent with other reported studies.7,11,43 In contrast to the relative insensitivity to changes in flow field (Figure 3c), particle deposition was quite sensitive to changes in hydrodynamic retardation associated with surface roughness size (Figure 3d; as described by dimensionless slip length). These contributions to particle deposition were equal to or greater than DLVO contributions in some cases, such as the larger roughness sizes investigated (Figure 2d compared to Figure 2b). The overall effect of hydrodynamics (i.e., increased flow velocity and reduced hydrodynamic retardation) on particle deposition is presented in Figure 3e. Here, values of the slip length used to modify the flow field profile and hydrodynamic retardation functions for a specific surface roughness element size are the same. Notably, this figure demonstrates that the total increase in particle deposition flux (ΔSh = Shslip − Shslip=0) was not additive; specifically, it was synergistic and greater than the value obtained by adding the ΔSh obtained from the data presented in Figure 3c and d for the corresponding slip length. For example, when the slip length increased from 0 to 1, the total particle deposition flux over the collector surface increased by 3.9% and 80.3% due to the respective individual changes in the flow field profile and hydrodynamic retardation functions (summing to an 84.2% increase); however, the concurrent, synergistic impact of these factors increased total particle deposition flux by 99.9%, as presented in Figure 3e. In other words, in these cases, nanoscale collector surface roughness at the sizes presented in Figure 3e resulted in higher flow velocity and lower hydrodynamic retardation effects that synergistically increased particle deposition flux (Sh) to collector surfaces. Flow field profiles and hydrodynamic retardation are not independent; thus, an additive relationship between their effects on particle deposition would not necessarily be expected. Here, we observed a synergistic effect of flow field profiles and hydrodynamic retardation on particle deposition flux.

Figures 2 and 3 collectively demonstrate several important outcomes regarding particle deposition on spherical collectors with nanoscale surface roughness. These are (1) flow field and hydrodynamic retardation synergistically, but not independently drove trends in particle deposition flux due to changes of collector surface roughness; (2) flow field and hydrodynamic retardation (especially the latter) were more sensitive to changes in collector surface roughness size described by modification of slip length, as compared to DLVO force; and (3) hydrodynamic (i.e., flow field and hydrodynamic retardation) contributions to particle deposition were equal to or greater than DLVO contributions in some cases; specifically, they resulted in insignificant changes to particle deposition when rroughness < 30 nm, but substantially increased particle deposition flux when rroughness > 30 nm as compared to the scenario when DLVO forces were the sole driver of particle deposition. The individual impacts of particle size and loading rate on particle deposition on spherical collectors with various roughness sizes were also quantitatively evaluated (Supporting Information; Figure S-6). In the presence of surface roughness, loading rates and particle size consistently had a nonmonotonic impact on the cumulative particle deposition flux ratio η/η0 (i.e., the ratio of dimensionless flux under the influence of all physicochemical interactions to the flux impacted only by Brownian motionthe Smoluchowski-Levich approximation). Moreover, the impacts of surface roughness on particle deposition were quite pronounced at some operational conditions (e.g., larger particles sizes, higher loading rates). These results underscore the importance of not only particle size (as it is in most contemporary modeling approaches), but also collector surface roughness attributes when describing particle deposition. Model Validation. The theoretical solution of the Convective-Diffusion Equation using the proposed method was compared to reported data from well-controlled laboratoryscale column experiments reported by Elimelech (1991)43 and Jin et al. (2015).36 Those data were appropriate for validating the model developed herein because the studies were conducted at conditions in which EDL interaction energies between the particles and collector surface were attractive43 or negligible (i.e., the attachment efficiency (α) approached unity);36 moreover, one of the studies included investigation of particle deposition on collectors with variable surface roughness.36 To compare the experimental and simulation outcomes, the experimentally derived single collector efficiency (ηexp) was described by ηexp = −

dc 2 ln(C /C0) 3 α(1 − f )L

(15)

The theoretical and experimental single collector efficiencies of Elimelech (1991)43 are shown as a function of ionic strength in Figure 4a. That approach assumed no slip conditions and smooth collector surfaces,43 which are respectively equivalent to rslip = 0 and rroughness = 0 in the model developed herein. Although a relatively smooth medium was used during those experiments, it is reasonable to assume that some degree of nanoscale roughness ( 10−5 (by approximately 20%) and overestimated it at IS < 10−5 (by approximately 15%) (Figure 4a). Notably, those results are insensitive to variable DLVO interaction energies that would result from differing degrees of surface coverage by nanoscale roughness (Figure S-5). Thus, that work was consistent with the earlier conclusions of Elimelech and O’Melia (1990),59 who also showed that accounting for DLVO force alone improved estimation of single collector efficiency, though it could not I


Article

Environmental Science & Technology Ψ EDL VDW DLVO

the “effective contact surface” (rslip) because factors such as hydrophobicity,25 heterogeneity,26 short-range forces and adhesion/detachment40,69 that contribute to roughness complexity in real systems complicate that analysis. We stress that our goal here was not to exactly, quantitatively describe effective contact surfaces; rather, it was to demonstrate that inclusion/description of hydrodynamic contributions to particle deposition can significantly improve single collector efficiency predictions and may thus contribute to (1) helping to explain discrepancies between model-based expectations and experimental outcomes and (2) ultimately developing a filtration model that is more universally relevant and predictive. Future research focused on improved quantitative description of the hydrodynamic effects on particle transport in systems with more complex geometry will contribute to a more thorough understanding of particle deposition in porous media.

■

χ φ

Symbols and Notations Used in This Work

A ac ap b C C0 D∞ D e f F f i(h) Fi(h) f i∞(h) f i0(h) h H I J kb L ns Pe Q Rij rslip rroughness Si,j

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.est.6b00218. Schematic of roughness surface, developed f1−4(H) functions, meshed domain for numerical simulation in COMSOL, and model validation and dimensionless groups used in the Convection-Diffusion Equation. (PDF)

■

stream function electrostatic double layer force Van deer Waal force Derjaguin−Landau−Verwey−Overbeek interfacial force fraction of roughness surface over the collector weighted sum of DLVO interaction energy for rough and smooth surfaces

AUTHOR INFORMATION

Corresponding Author

*Phone: +1-519-888-4567, ×32208; e-mail: mbemelko@ uwaterloo.ca. Author Contributions

C.J. and M.E. envisioned the work. C.J. conducted the experiments and modeling. C.J. and M.E. wrote the paper. All of the authors discussed the experimental plan and numerical simulations, contributed to reviewing the manuscript, and gave approval for the final version of the paper.

Sh T t U u uθ ur V vθ vr Vr Vθ Vr′ Vθ′

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canadian Water Network for financial support. We are thankful to Dr. T. Glawdel for useful discussions on numerical simulation.

■

GREEK LETTERS ψi ζ-potential of surface, i κ−1 Inverse Debye-Hϋckel length α colloid attachment efficiency ε0 dialectic permittivity in vacuum εr relative dialectic permittivity of water λ characteristic wavelength of electron oscillation μ absolute viscosity of bulk fluid δ dimensionless closest distance η single collector efficiency ηexp experimental determined Single collector efficiency ηTE values of η from Tufenkji and Elimelech, 2004 ηsim−rough theoretical solution of η for rough surface ηsim−smooth theoretical solution of η for smooth surface ηno−EDL theoretical solution of η without EDL force

■

Hamaker constant collector radius colloid radius radius of the fluid thickness colloid concentration bulk colloid concentration diffusion coefficient in an infinite medium diffusion coefficient tensor charge of an electron porosity of packed media external forces act as source hydrodynamic retardation functions, i= 1 ∼ 4 modified hydrodynamic retardation function, i= 1−4 analytical solution of f i(h)when h= ∞ analytical solution of f i(h) when h = 0 true distance between surface to surface dimensionless distance cumulative colloid deposition flux local colloid deposition flux Boltzmann constant packed column length number of stacked rough sphere for roughness Peclet number source term Center to center distance between sphere i and j dimensionless slip length dimensionless roughness height surface to surface distance between two roughness element Sherwood number absolute temperature time superficial velocity particle velocity vector colloid tangential velocity component colloid radial velocity component the radial velocity component vector, V(r) = iVθ + jVr fluid tangential velocity component fluid radial velocity component dimensionless fluid radial velocity dimensionless fluid tangential velocity modified dimensionless radial velocity modified dimensionless tangential velocity

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