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Contributions of Nanoscale Roughness to Anomalous Colloid Retention and Stability Behavior Scott A. Bradford, Hyunjung Kim, Chongyang Shen, Salini Sasidharan, and Jianying Shang Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02445 • Publication Date (Web): 28 Aug 2017 Downloaded from http://pubs.acs.org on September 3, 2017
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Contributions of Nanoscale Roughness to
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Anomalous Colloid Retention and Stability Behavior
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Scott A. Bradford1, Hyunjung Kim2, Chongyang Shen3, Salini Sasidharan4, and Jianying Shang3
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US Salinity Laboratory, USDA, ARS, Riverside, CA 92507
Department of Mineral Resources and Energy Engineering, Chonbuk National University, 66414 Duckjin, Jeonju, Jeonbuk 561-756, Republic of Korea
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Department of Soil and Water Sciences, China Agricultural University, Beijing, China 100193 4
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Department of Environmental Sciences, University of California, Riverside, CA 92521
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Revised
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August 23, 2017
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Corresponding Author. Phone:951-369-4857; Email:
[email protected] 1 ACS Paragon Plus Environment
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Abstract - All natural surfaces exhibit nanoscale roughness (NR) and chemical heterogeneity
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(CH) to some extent. Expressions were developed to determine the mean interaction energy
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between a colloid and a solid-water interface, as well as for colloid-colloid interactions, when
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both surfaces contain binary NR and CH. The influence of heterogeneity type, roughness
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parameters, solution ionic strength (IS), mean zeta potential, and colloid size on predicted
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interaction energy profiles was then investigated. The role of CH was enhanced on smooth
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surfaces with larger amounts of CH, especially for smaller colloids and higher IS. However,
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predicted interaction energy profiles were mainly dominated by NR, which tended to lower the
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energy barrier height and the magnitudes of both the secondary and primary minima, especially
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when the roughness fraction was small. This dramatically increased the relative importance of
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primary to secondary minima interactions on net electrostatically unfavorable surfaces,
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especially when roughness occurred on both surfaces and for conditions that produced small
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energy barriers (e.g., higher IS, lower pH, lower magnitudes in the zeta potential, and for smaller
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colloid sizes) on smooth surfaces. The combined influence of roughness and Born repulsion
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frequently produced a shallow primary minimum that was susceptible to diffusive removal by
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random variations in kinetic energy, even under electrostatically favorable conditions.
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Calculations using measured zeta potentials and hypothetical roughness properties demonstrated
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that roughness provided a viable alternative explanation for many experimental deviations that
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have previously been attributed to electrosteric repulsion (e.g., a decrease in colloid retention
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with an increase in solution IS; reversible colloid retention under favorable conditions; and
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diminished colloid retention and enhanced colloid stability due to adsorbed surfactants,
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polymers, and/or humic materials).
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Introduction An understanding of factors that influence interactions of colloids with solid surfaces in
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the presence of water is needed for many industrial and environmental applications, including:
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the design of surfaces to enhance or diminish colloid retention,1 wastewater treatment,2
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remediation of hazardous waste sites,3 risk assessment of pathogenic microorganisms and
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engineered nanoparticles,4 colloid-facilitated contaminant transport,5 petroleum production,6 and
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managed aquifer storage and recovery.7 Conventional filtration theory considers that the
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retention rate coefficient depends on the mass transfer and immobilization of colloids on the
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solid surface.8 Colloid immobilization has commonly been assumed to be controlled by the
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adhesive interaction between a colloid and a solid-water interface (SWI).8 Similarly, the
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adhesive interaction between colloids determines the stability of a suspension.9
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The adhesive interaction is commonly derived from interaction energy calculations that
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consider electrostatic double layer and van der Waals interactions,10, 11 but may also be extended
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to include short range interactions such as Born repulsion, steric, and Lewis acid-base
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interactions that are still incompletely understood.9, 12 Colloid and natural surfaces always
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exhibit some degree of nanoscale roughness (NR) and chemical heterogeneity (CH),13, 14 and
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interaction energy calculations based on homogeneous surfaces have frequently been shown to
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provide an inadequate description of colloid retention on the SWI or colloid stability.14-16 A
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number of researchers have modified interaction energy calculations to account for NR and/or
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CH on surfaces in order to overcome these limitations.14-22 For example, the surface element
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integration approach allows for the consideration of NR and/or CH on the SWI or the colloid.16
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Results have demonstrated that the mean interaction energy on a heterogeneous surface can be
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represented as a linear expression of interactions energies associated with the various 3 ACS Paragon Plus Environment
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heterogeneity combinations within the electrostatic zone of influence.15, 16, 22 For example,
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Bradford and Torkzaban22 demonstrated that the influence of binary NR and CH on the SWI
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could be accurately accounted for using a linear expression of the various interaction energies. Nanoscale heterogeneities have been shown to have a large influence on predicted
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interaction energy profiles.14-23 In brief, electrostatically unfavorable surfaces can exhibit
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localized reductions and/or elimination of the energy barrier due to the presence of nanoscale
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heterogeneity in zeta potential, depending on the amount, size, and charge of the CH within the
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electrostatic zone of influence, the solution ionic strength (IS), and the colloid size.16, 23 In
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general, the influence of a given CH increases at higher IS and for smaller colloid size.23 The
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energy barrier on electrostatically unfavorable surfaces can also be locally reduced or eliminated
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by NR.14-22 In addition, the depths of the secondary and primary minima can also be reduced by
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NR depending on the amount, size, and height of the NR within the electrostatic zone of
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influence.22 Some interacting colloids are therefore susceptible to diffusive and/or
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hydrodynamic removal on rough surfaces,24 even under electrostatically favorable conditions.25
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When both NR and CH are considered on the SWI, results indicate that NR tends to control the
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interaction energy profile shape, as well as colloid retention and release.16, 22, 24 To date, no published studies have considered the influence of NR and CH on both the
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SWI and the colloid. Consequently, it is presently unclear whether heterogeneity needs to be
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considered only on the SWI, only on the colloid, or on both the SWI and colloid. Furthermore,
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the predicted influence of NR on interaction energy profiles depends not only on roughness
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parameters, but also on parameters that influence electrostatic and van der Waals interactions.16,
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22
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chemistry, and colloid size) parameters has not yet been systematically studied when
However, the coupled influence of NR and interaction energy (e.g., zeta potentials, solution
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heterogeneity occurs on both the colloid and the SWI. Consideration of these effects may
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provide a viable alternative explanation for experimental observations that have typically been
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attributed to electrosteric repulsion.26-32 For example, electrosteric repulsion has been invoked to
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explain: (i) a decrease in colloid retention with an increase in solution IS;26-27 (ii) reversible
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colloid retention under favorable conditions;28-29 (iii) diminished colloid retention due to
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adsorbed humic materials;30 and (iv) enhance stability of colloid suspensions by adsorbed
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polymers and surfactants.31-32 However, this literature has not explicitly considered the role of
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NR on the SWI or the colloid in their interaction energy calculations.
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In this work we present simple linear expressions to determine the mean interaction
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energy between a colloid and a SWI, as well as for colloid-colloid interactions, when both
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surfaces contain binary NR and CH. Next, we investigate the relative importance of NR and CH
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on predicted interaction energy profiles. We then systematically study the coupled role of NR on
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the colloid and SWI with various interaction energy parameters in order to test the hypothesis
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that it can explain many experimental deviations that have previously been attributed to
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electrosteric repulsion. Results from this research provide valuable insight on: (i) the relative
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importance of various nanoscale heterogeneity types on interaction energies; (ii) the influence of
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NR with various interaction energy parameters; and (iii) a viable alternative explanation for
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experimental deviations that have previously been attributed to electrosteric repulsion.
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Furthermore, this approach can be used to design surfaces with enhanced or diminished colloid
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interactions by selecting the optimum NR and CH parameters.
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Materials and Methods
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Interaction Energies 5 ACS Paragon Plus Environment
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In this research we assume that a spherical colloid is suspended in a monovalent
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electrolyte solution with a given IS and interacts with the surface of a porous medium. Both
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interacting surfaces may exhibit binary NR and CH within the area of the electrostatic zone of
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influence (Az, L2 where L denotes units of length). The value of Az refers to the projected area of
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the colloid on the SWI that effectively contributes to the colloid-SWI interaction energy and is
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proportional to the colloid radius and the Debye length.33 Symmetry implies that the same value
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of Az also acts on the colloid. This assumption (i.e., pairwise interaction) is also invoked in
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commonly employed Derjaguin and surface element integration methods.34-35 Figure 1 provides
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a schematic illustrating NR and CH on the colloid and SWI within Az. Note that Az on the SWI
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contains a NR fraction (fsr) with a height equal to hsr and a positive zeta potential fraction (fs+)
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that is equal to ζs+ . The complementary fractions (1-fsr) and (1-fs+) correspond to a smooth
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surface and a negative zeta potential ζs− , respectively. Similar NR and CH parameters were
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defined within Az for the colloid as for the SWI. In this case, parameters fsr, fs+, hsr,
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for the SWI correspond to fcr, fc+, hcr,
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ζs+ , and ζs−
ζc+, and ζc− for the colloid, respectively.
It has previously been demonstrated that the mean value of the dimensionless (divided by
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the product of the Boltzmann constant and absolute temperature) interaction energy (Φ), from
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many realizations of the same heterogeneity parameters, can be determined as a linear
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combination of interaction energies associated with the NR and CH fractions within Az.15, 16, 22
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This approach is extended in this research to account for binary NR and CH on both the SWI and
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colloid as:
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Φ ( h ) = ar1Φ s ( h + hsr + hcr ) + ar 2Φ s ( h + hsr ) + ar 3Φ s ( h + hcr ) + ar 4 Φ s ( h )
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[1]
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where h [L] is the separation distance from the center of Az at a height hsr from the SWI to the
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leading face of the colloid center at a height hcr (see Figure 1), Φs is the dimensionless interaction
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energy associated with a smooth, nanoscale chemically heterogeneous surface, and ar1 [-], ar2 [-],
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ar3 [-], and ar4 [-] are constants that determine the contributions of the various possible roughness
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combinations (e.g., smooth SWI and smooth colloid, smooth SWI and rough colloid, rough SWI
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and smooth colloid, and rough SWI and rough colloid, respectively) that are equal to:
( )( ) = (1 − f ) f = f (1 − f )
ar1 = 1 − f sr 1 − f cr 142
ar 2 ar 3
sr
sr
cr
[2]
cr
ar 4 = f sr f cr 143
The value of Φs is given as:23
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Φs ( h) = ac1Φs−c− ( h) + ac2Φs−c+ ( h) + ac3Φs+c− ( h) + ac4Φs+c+ ( h)
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where Φs-c-, Φs-c+, Φs+c-, and Φs+c+ are dimensionless interaction energies that determine the
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contributions of the possible CH combinations (e.g., solid and colloid zeta potentials of: ζs− and
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ζc−; ζs− and ζc+; ζs+ and ζc−; and ζs+ and ζc+, respectively), and ac1 [-], ac2 [-], ac3 [-], and ac4 [-]
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are constants that are equal to:
( )( = (1 − f ) f = f (1 − f )
a c1 = 1 − f s + 1 − f c + 149
ac 2 ac 3
s+
s+
[3]
)
c+
[4]
c+
ac 4 = f s + f c + 150
Equations [1]-[4] were also applied for colloid-colloid interactions. All of the above interaction
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energies considered electrostatics,36 retarded London-van der Waals attraction,37 and Born
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repulsion38-39 that are summarized in section S1 of the Supporting Information for sphere-plate
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and sphere-sphere geometries. It should be mentioned that Eqs. [1]-[4] assumed that h>0 such
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that roughness on the colloid and SWI (or colloid) do not overlap (Figure 1). Furthermore,
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lateral interactions between NR on the colloid and the SWI (or colloid) were assumed to be
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distributed (e.g., randomly or exhibited symmetry) such that their average contribution to the
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interaction energy cancels out.
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Equations [1]-[4] therefore allow the influence of NR and CH to be systematically
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investigated for various heterogeneity combinations. In this work, the following heterogeneity
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combinations were considered: physically and chemically homogeneous characteristics on both
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the SWI and colloid (denoted as H-Both); NR on the SWI (denoted as NR-SWI); NR on the
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colloid (denoted as NR-Colloid); NR on both the SWI and colloid (NR-Both); CH on both the
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SWI and colloid (CH-Both); and CH and NR on both the colloid and SWI (CH+NR-Both). All
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interaction energy profiles were analyzed to determine the energy barrier height (Φmax), and the
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depths of the secondary (Φ2min) and primary (Φ1min) minima.
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Energy Balance The kinetic energy model was used to determine the probability that a colloid interacting
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in a primary or secondary minimum would be immobilized in the presence of random kinetic
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energy fluctuations of a diffusing colloid. In this case, the immobilization probability (εj) was
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determined by using the Maxwellian cumulative density function to account for the kinetic
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energy distribution of a diffusing colloid and evaluating it over specific ranges in the interaction
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energy profile:24, 40-41
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εj =∫
B
A
exp(−Φ)dΦ = erf π
2 Φ
( B )−
exp(− B) − erf π
4B
( A)−
exp(− A) π
4A
[5]
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where A is the lower integration limit, B is the upper integration limit, and the subscript j on ε
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indicates whether the immobilization was associated with a primary or secondary minimum (j=1
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or 2). Values of A and B are equal to the interaction energies to enter and escape from a
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minimum, respectively.24 Equation [5] was also used to determine the probability that
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interacting colloids would be released from secondary and primary minima (εr1 and εr2) by
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diffusion. In this case, A is equal to the minimum kinetic energy to escape from the minimum
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and B=∞ (e.g., the maximum kinetic energy).24
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Figure S1 presents a plot of ε1 as a function of Φmax when Φ2min=0 and Φ1min=-∞ (unfavorable
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electrostatic conditions), and εr1 as a function of Φ1min when Φ2min=0 and Φmax=0 (favorable
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electrostatic conditions). Note that ε1 rapidly increases with decreasing Φmax when Φmax -5.6. Consequently, long-term
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colloid immobilization in a primary minimum is only possible when Φmax 0.25, such that the oocyst interaction was
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mainly limited to the secondary minimum, which became shallower for smaller fsr=fcr. The
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insensitivity of interaction energy parameters on hsr=hcr at ≥ 10 nm is because the bulk colloid
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and collector surfaces receded very far, which had little influence on the total interaction energy.
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Similar to Fig. 3B, variations in CH parameters were found to have a greater influence on
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interaction energy parameters when fsr=fcr >0.25 (data not shown).
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An alternative way of examining the roles of NR and CH on colloid immobilization is to
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present the energy balance results in terms of ε1, ε2, β1, and β2. For example, interaction energy
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parameters shown in Fig. 4 were utilized in Eqs. [5] and [6] to determine ε1, ε2, and β1 shown in
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Fig. 5 as a function fsr=fcr and of hsr=hcr. Consistent with Fig. 4, ε1>0 occurred when fsr=fcr was
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≤0.25 and hsr=hcr was ≥ 10 nm, whereas ε2>0 dominated oocyst immobilization for the opposite
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roughness conditions. Interestingly, values of β2 were always zero and this indicates that oocyst
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interaction in the secondary minimum was always reversible. In contrast, some values of β1
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were irreversible for the selected solution chemistry under certain roughness conditions. In
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particular, values of β1=1 when 0.025≤fsr=fcr≤0.25 and hsr=hcr≥ 10 nm. It should be mentioned
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that Eq. [6] can be modified to incorporate the influence of flowing water on the torque balance
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and spatial variations in NR and CH heterogeneity parameters on the SWI.24 Consequently, this
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information can be used to design surfaces with specific roughness characteristics for reversible
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or irreversible colloids retention under selected physicochemical conditions, or to study colloid
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retention and release on natural heterogeneous surfaces.
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The electrostatic double layer and van der Waals interactions are proportional to the
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colloid radius (Eqs. [S2]-[S3]). Consequently, if other parameter values remain unchanged, then
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the magnitudes of Φ2min, Φmax, and Φ1min will decrease with decreasing colloid size. This
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information implies that the influence of NR and CH on interaction energy parameters is
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enhanced for small colloid sizes. To illustrate this point Fig. 6 presents contour plots of ε1 and β1
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for a 100 nm colloid as a function of fsr=fcr and hsr=hcr for the same conditions shown in Fig. 5.
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In comparison to Fig. 5, values of ε1>0 and β1=1 now occur over a wider range of roughness
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conditions. In particular, values of β1=1 occurred when fsr=fcr≥0.25 and hsr=hcr≥2.5 nm.
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Reversible primary and secondary minimum interactions were predicted when the IS=10 mM
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when fsr=fcr
