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Interactions between nanoparticles and fractal surfaces
Water Research 151 (2019) 296e309
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Interactions between nanoparticles and fractal surfaces Hong Wang a, Wei Zhang b, Saiqi Zeng c, Chongyang Shen a, *, Chao Jin d, Yuanfang Huang a, ** a
Department of Soil and Water Sciences, China Agricultural University, Beijing, 100193, China Department of Plant, Soil and Microbial Sciences, and Environmental Science and Policy Program, Michigan State University, East Lansing, MI, 48824, United States c Department of Plant and Soil Sciences, University of Delaware, Newark, DE, 19716, United States d School of Environmental Science and Engineering, Sun Yat-sen University, Guangzhou, Guangdong, 510006, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 September 2018 Received in revised form 13 December 2018 Accepted 15 December 2018 Available online 27 December 2018
This study evaluated attachment of a 30-nm nanoparticle to and detachment from fractal surfaces by calculating Derjaguin-Landau-Verwey-Overbeek (DLVO) interaction energies in three-dimensional space using the surface element integration technique. The fractal surfaces were generated using the Weierstass-Mandelbrot function with varying values of fractal dimension D (2.3 D 2.7) and fractal roughness G (0.000136 G 0.136). Results show that maximum energy barrier is reduced at peak areas of a fractal surface, and hence attachment in primary minima is favored. Some nanoparticles attached in primary minima at the peak areas can be detached by decreasing ionic strength (IS) due to monotonic decrease of interaction energy with increasing separation distance at low ISs. While the attachment in primary minima at valley areas is irreversible to IS reduction, the attachment is inhibited due to enhanced maximum energy barrier at these areas. A nonmonotonic variation of attachment efficiency in primary minimum (AEPM) with IS is present at high fractal dimension (D 2.4) or low fractal roughness (G < 0.00136), whereas the AEPM decreases monotonically with decreasing IS at low fractal dimension (D < 2.4) or high fractal roughness (G 0.00136). The AEPM decreases monotonically with increasing D or decreasing G at ISs from 1 mM to 200 mM. The decrease of AEPM with D or G is much slower at 10 mM compared to other ISs. These theoretical findings can explain various experimental observations in the literature, and can have important utility to development of water filtration techniques in engineered systems and to assessment of environmental risks of nanoparticles. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Nanoparticle Fractal surface Interaction energy Attachment Detachment
1. Introduction Transport of hazardous particles (e.g., pathogenic viruses, bacteria, and protozoa, and toxic nanomaterials) in porous media such as soil has been extensively investigated during the past two decades due to intense interest in preventing groundwater pollution, and developing water and wastewater filtration technologies (Ryan and Elimelech, 1996; de Jonge et al., 2004; Batley et al., 2013). Attachment on and detachment from collector surfaces are two primary processes that inhibit and facilitate particle transport in porous media, respectively (Bradford and Torkzaban, 2015; Molnar
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (C. Shen), [email protected] (Y. Huang). https://doi.org/10.1016/j.watres.2018.12.029 0043-1354/© 2018 Elsevier Ltd. All rights reserved.
et al., 2015). Whether particles attach to or detach from collector surfaces is dependent on the energy balance between adhesive energy and Brownian diffusion and/or the torque balance between adhesive torque and hydrodynamic torque (Bergendahl and Grasso, 2003; Pazmino et al., 2014a, 2014b; Bradford et al., 2017). Both energy and torque balance analysis require accurate determination of the adhesive force/energy that arises due to colloidal interactions between particles and surfaces. The colloidal interactions include van der Waals (VDW) attraction, electrical double layer (DL) interaction, and short-range repulsion (e.g., hydration and steric repulsion) (Hoek and Agarwal, 2006; Israelachvili, 2011). The Derjaguin-Landau-Verwey-Overbeek (DLVO) theory quantitatively describes the interaction energy profile as a function of separation distance between a particle and a surface (Ryan and Elimelech, 1996). When particles and collector surfaces are like-charged, both primary and secondary minima are present in the energy profile, between which an energy barrier
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exists. If the two surfaces are oppositely charged, both secondary minimum and energy barrier disappear from the profile and only primary minimum exists. The primary and secondary energy minima are considered as the locations for particle attachment (Grasso et al., 2002; Hahn and O’melia, 2004; Hahn et al., 2004). In the classic DLVO theory, calculation of interaction energies is based on the assumption that particle and collector surfaces are perfectly smooth. In real systems, surfaces of natural particles and collectors, however, all contain some degree of physical nonuniformity at small scales (Shellenberger and Logan, 2002). The influence of surface roughness has been frequently taken as the cause of discrepancies between the theoretical prediction and experimental observations of particle deposition (Fischer et al., 2014; Jin et al., 2015, 2017; Rasmuson et al., 2017). Theoretical studies (Huang et al., 2010; Bradford and Torkzaban, 2015; Bendersky et al., 2015) have shown that the nanoscale protruding asperities can increase particle attachment at primary minima by reducing energy barrier under the unfavorable conditions. However, the favorably attached particles are readily to be detached by perturbation of solution chemistry and/or hydrodynamics because the depths of the primary minima are also decreased by nanoscale protruding asperities (Shen et al., 2012, 2014). Such reduction of primary minimum depth by nanoscale protruding asperities can also decrease particle attachment at primary minima under favorable conditions (Rasmuson et al., 2017). While various mechanisms controlling interactions of particles with rough surfaces have been elucidated as shown above, the aforementioned theoretical results were obtained by modeling the rough surfaces as smooth surfaces carrying geometrically regular asperities (e.g., pillars, hemispheres, and cones). This technique is capable of characterizing the actual morphologies of rough surfaces to some extent by using the measured values of rough parameters (e.g., root-means-square roughness and surface area difference) from atomic force microscopy (AFM) scans to constrain the distribution of asperities on the smooth surfaces (Hoek et al., 2003). The modeling of rough surface was further improved in recent studies (Jaiswal et al., 2009; Siegismund et al., 2014) by using fast Fourier transform algorithm to model the roughness based on the AFM measurement. This technique can generate more realistic roughness profiles than using the regular geometries. However, all of these modeled surfaces are stationary because the rough parameter values of the surfaces (i.e., those from AFM scans) are fixed, which do not change with the length scale used to measure the modeled surfaces (Eichenlaub et al., 2004). Many natural and engineered surfaces such as sand and membrane surfaces are fractal and their rough parameter values change with the resolution and AFM scan size (Yang et al., 2016; Cai et al., 2017). A fractal surface bears the property of self-affinity. Specifically, when a part of the fractal surface is magnified, the magnified image
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function which is a superposition of sinusoids with geometrically spaced frequencies and amplitudes that follow a power law (Yan and Komvopoulos, 1998; Eichenlaub et al., 2004). The W-M function has been used to represent fractal surfaces for investigating influence of surface topography on engineered processes such as gaseous and fluid flow and thermal contact resistance (Majumdar and Tien, 1990; Yan et al., 2015). The W-M function has also been employed recently to model membrane surfaces for examining the interactions between a particle and the surfaces (Cai et al., 2017; Chen et al., 2017; Feng et al., 2017; Zhang et al., 2017). It was found that the adhesions between particles and fractal surfaces are significantly lowered compared to those of particle-flat surface interactions. While these studies advanced understanding about influence of fractal on colloidal interaction, they only investigated the variation of interaction energy with separation distance for a particle at a single horizontal location above a fractal surface. The interaction energy that acts on the particle should change with its horizontal location due to the randomness of fractal surface. Moreover, the influence of environmental factors such as ionic strength (IS) on interaction energies between particles and fractal surfaces has not been investigated to date. The detachment of particles from fractal surfaces is also unclear. This study systematically evaluated the attachment and detachment of nanoparticles (NPs) on/from fractal surfaces represented using the W-M function by calculating interaction energies at various horizontal and vertical locations. The surface element integration (SEI) technique (Bhattacharjee and Elimelech, 1997; Siegismund et al., 2014) was used for the energy calculations and the interaction energy maps were obtained to describe the variation of interaction energy in three dimensional space for various ISs and fractal roughness parameter values. We showed that the fractal dimension and fractal roughness are more effective for predicting attachment and detachment on/from primary and secondary minima than traditional roughness parameters such as average roughness height. The findings in this study can help explain various observations in the literature. For example, our results could explain why a nonmonotonic variation of attachment with IS for NPs was predicted in a theoretical study (Lin and Wiesner, 2012) through exact interaction energy calculations but not observed experimentally. 2. Theory 2.1. The W-M function The modified two variable W-M function by Yan and Komvopoulos (1998) was employed to generate fractal surfaces in this study:
ðD2Þ G ln g 1=2 zðx; yÞ ¼ L L M M n¼∞ X X m¼1 n¼0
( ðD3Þn
g
"
cosfm;n cos
1=2 2pgn x2 þ y2 L
y pm io þ fm;n cos tan1 x M
looks very similar to the original surface (Chen et al., 2009). In addition, the fractal surface possesses the properties of continuity and non-differentiability (Mandelbrot, 1983). These mathematical properties are satisfied by the Weierstrass-Mandelbrot (W-M)
(1)
where z(x, y) is the three-dimensional surface profile, D (2 < D < 3) is the fractal dimension of surface profile, G is a height scaling parameter (commonly termed as the fractal roughness) that is independent of frequency, L is the sample length, g (g > 1) is a scaling
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variable that determines the density of the frequency spectrum, M (M > 3) is the number of superposed ridges (commonly taken as 10), n is the spatial frequency index, and 4m,n is the random phase. The fractal roughness (G) and the fractal dimension (D) are two critical parameters that determine the fractal topography. The fractal dimension determines the relative contributions of highand low-frequency components in the fractal surface profile. Higher value of D means that high-frequency components in the surface profile are more dominant than low-frequency components (Yin and Komvopoulos, 2010). The fractal roughness is a height scaling parameter (i.e., amplitude coefficient) and the amplitudes of wavelengths comprising the surface profile increase with increasing the value of G. Both values of D and G are independent of the measurement scale for a given fractal surface. The spatial index n assumes finite values in practice (Yan and Komvopoulos, 1998). The lower limit of n is commonly taken as zero, and the higher limit of n is given by n ¼ int[log(L/Ls)/logg], where Ls is the cut-off wavelength (i.e., the smallest allowed length scale) determined by the instrument resolution limit, which was taken as 1 nm (Yin and Komvopoulos, 2010). g is commonly taken as 1.5 for most surfaces based on surface flatness and frequency distribution density considerations (Majumdar and Tien, 1990). A random number generator in Matlab was used to uniformly distribute 4m,n in the interval [0, 2p] to prevent the coincidence of frequencies at any point of surface profile.
corresponding area element on fractal surface (dSc) is calculated as (Bhattacharjee et al., 1998; Hoss et al., 2017)
dU ¼ ðn1 ,k1 Þðn2 ,k2 ÞEðhÞdSc
(2)
where k1 and k2 are unit vectors directed towards the positive and negative z axis, respectively, n1 and n2 are outward units normal to the NP and collector surfaces, respectively, E(h) is the differential interaction energy between element dSp and dSc, and h is the local separation distance between the two elements. The total interaction energy (U) that the NP experiences is then obtained by summation of the differential interaction energy (dU) for each dSc on the projected area of the NP on the fractal surface. The equation used to calculate the total interaction energy can be expressed as
U¼
X ðn1 ,k1 Þðn2 ,k2 ÞEðhÞdSc
(3)
Sc
where Sc is the project area of the NP on the fractal surface. It is very difficult to directly determine the value of n2 ,k2 due to the randomness of the fractal surface. However, the term of n2 ,k2 can be eliminated from Equation (3) by expressing the element dSc in terms of its projection dA on the xy plane as
dSc ¼
1 dA jn2 ,k2 j
(4)
Substituting Equation (4) into Equation (3), we obtain 2.2. Calculation of DLVO interaction energies Fig. 1 schematically illustrates the interaction between a spherical NP and a fractal surface generated by the W-M function. Detailed procedure for calculating interaction energy between a spherical particle and a rough surface using the SEI technique can be referred to previous studies (Bhattacharjee et al., 1998; Duffadar and Davis, 2007; Hoss et al., 2017). Briefly, the Cartesian coordinate system was employed for the model system in Fig. 1. The xy plane of the coordinate system is oriented parallel to the zero plane of the fractal surface, which divides the NP into two identical hemispheres. The z axis passes through the NP center and faces toward the fractal surface. Similar to the grid-surface integration technique (Duffadar and Davis, 2007), both NP and fractal surfaces were discretized into small area elements. The differential interaction energy dU between an area element on NP surface (dSp) and the
U¼
X n ,k ðn1 ,k1 Þ 2 2 EðhÞdA jn 2 ,k2 j A
When the unit normal to the fractal surface at a given point n2 2 makes an obtuse or acute angle with k2, the value of jnn2 ,k is equal 2 ,k2 j to 1 or 1. Hence, the total interaction energy can be obtained by integrating the differential interaction energy over the projected area of the NP on the xy plane without determining the value of z n2 ,k2 . By using n1 ,k1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Equation (5) is changed to 2 2 2 x þy þz
X z U ¼ ± pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EðhÞdA 2 þ y2 þ z 2 x A
(6)
The differential interaction energy between element dS and dA [i.e., E(h)] was calculated by summing van der Waals (VDW), constant potential double layer (DL), and Born (BR) energies per unit area [i.e., EðhÞ ¼ EVDW ðhÞ þ EDL ðhÞ þ EBR ðhÞ]. The expressions used to calculate EVDW(h), EDL(h), and EBR(h) are following
EVDW ðhÞ ¼
Fig. 1. Schematic illustration of a spherical NP interacting with a fractal collector surface. dSp and dSc are differential area element on the NP and collector surfaces, respectively, n1 and n2 are unit outwards normal to the NP and collector surfaces, respectively, k1 and k2 are unit vectors directed towards the positive and negative zaxis, respectively.
(5)
AH 12ph2
(7)
EDL ðhÞ ¼
2j j εε0 k 2 jp þ j2c ð1 cothkhÞ þ p c 2 sinhkh
(8)
EBR ðhÞ ¼
AH H60 48ph8
(9)
where AH is the Hamaker constant, 3 0 is the dielectric permittivity of vacuum, 3 is dielectric permittivity of vacuum, k is the inverse Debye screening length, jp and jc are the surface potentials of NP and collector, respectively, H0 is the minimum separation distance between the NP and the fractal surface (0.158 nm) (Hoek and Agarwal, 2006). Both NP and fractal surfaces were discretized into grid elements with projection areas on the xy plane of 0.1 nm 0.1 nm. A Matlab program was developed to calculate DLVO interaction energies for
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the aforementioned interaction configuration, and the calculated energies were expressed in terms of kT (k is Boltzmann constant and T is absolute temperature). A horizontal interaction energy map (denoted as HEP) was obtained by varying the horizontal positions (x, y) of the NP center in a rasterized manner and keeping the vertical distance between the NP center and the rough surface fixed. Various HEPs were produced at different vertical distances between the NP center and the rough surface. Vertical interaction energy maps (VEPs) were also calculated by varying the vertical positions (x, z) or (y, z) of the NP center for a given value of y or x. Note that the interaction energy maps have also been calculated by varying the horizontal positions (x, y) of a particle center and keeping the vertical distance between the NP center and the zero plane of a rough surface fixed (Hoek et al., 2003). We did not adopt this approach because physical overlaps can occur between the particles and protruding asperities on the rough surfaces. A number of studies (Siegismund et al., 2014; Feng et al., 2017; Zhang et al., 2017) have examined the interaction of a particle and a surface with random roughness. However, these studies only examined adhesion or primary minimum depth for the particle at a single location of the surface. In contrast, by using the HEPs and VEPs, the maps of primary minimum, maximum energy barrier, and secondary minimum were obtained in this study. As such, the distribution of attachment in primary and secondary minima and subsequent detachment as well as the variation of the attachment and detachment with surface morphology (e.g., height and curvature) can be obtained. For theoretical calculation of interaction energies, the NPs and fractal collector surfaces were assumed to be 30-nm polystyrene latex particles and quartz sand, respectively, which have the same properties as those used in Shen et al. (2007). Table 1 presents the zeta potentials of the latex NPs and quartz sand used to calculate the interaction energies at different solution ISs. A value of 1 1020 J was taken as the Hamaker constant for latex-water-sand system (Elimelech and O'Melia, 1990a, 1990b; Shen et al., 2007).
3. Results and discussion 3.1. Generated fractal surfaces Fig. 2 presents simulated fractal surfaces using the W-M function with projected surface areas on xy plane of 150 nm 150 nm for varying D values (2.3, 2.4, 2.5, and 2.6) and a G value of 0.0136 nm. The scale bar to the right of each plot represents height in units of nm. We adopted these values of D and G because for most fractal surfaces, the values of D range from 2.3 to 2.7 and the values of G range from 0.000136 nm to 0.136 nm (Sahoo and Ghosh, 2007). Fig. 2 shows that for a given value of G, larger value of D yields smoother surface. This is because high frequency components are more dominant than low frequency components in the surface topography profile at larger D values (Chatterjee and Sahoo, 2014). For a given value of D, larger G value results in rougher surfaces due to an increase in the amplitude of surface wavelength with G, as shown in Supplementary Information (SI) Fig. S1. The variation of
Table 1 Zeta potentials of sand and NPs at different solution ISs. IS (M)
0.001 0.01 0.1 0.2
Zeta Potential (mV) Sand
NPs
39.27 39.13 32.08 27.49
44.3 43.25 38.69 36.86
299
surface roughness with D and G are in agreement with those in previous studies (Yan and Komvopoulos, 1998; Chatterjee and Sahoo, 2014; Yan et al., 2015). 3.2. Attachment in primary minima Fig. 3 presents HEPs for interactions of a 30-nm NP with a fractal surface with G ¼ 0.0136 nm and D ¼ 2.3 or D ¼ 2.6 (i.e., the surface in Fig. 2a or 2d) at 0.001 M at varying separation distances. The scale bar to the right of each plot represents interaction energy in units of kT (k is the Boltmann constant, T ¼ 297.13 K is absolute temperature). The interaction energy is non-uniformly distributed (plaque-like) in the horizontal direction due to the random variation of surface curvature (Shen et al., 2015). The magnitude of interaction energy is reduced and enhanced at peak and valley regions, respectively, as shown by the VEPs in SI Fig. S2. The variation of interaction energy with surface curvature can be explained by the concept of “interaction volume” developed by Huang et al. (2010). The interaction volume is defined as the volume between the leading half of a spherical particle and a substrate surface. The interaction volume is smaller for a NP interacting with a peak surface than with a valley surface. Huang et al. (2010) illustrated that smaller interaction volume between a particle and a substrate surface causes larger interaction energy because the mean plane of the surface is closer to the apex of the particle. Therefore, the magnitude of interaction energy is reduced at peak areas whereas enhanced at valley areas. The interaction energy is more uniformly distributed at larger separation distance because the influence of local variation of surface curvature on interaction energy decreases with increasing separation distance (cf., Fig. 3). The energy plaques on the maps are larger for smaller D value because of more dominance of high frequency components than low frequency components. Due to a similar reason, increasing the value of G increases the sizes of the plaques (cf., Fig. 4). For a given fractal surface, the interaction energy decreases with increasing IS (SI Fig. S3). Fig. 5 shows maximum energy barrier maps for interactions of the 30-nm NP with a fractal surface with G ¼ 0.0136 nm and D ¼ 2.3 or D ¼ 2.6 at different solution ISs. The plaques on the energy barrier maps are larger for smaller value of D (i.e., smoother surface). The value of maximum energy barrier is smaller for smaller value of D. Interestingly, white spots are present on the HEPs for D ¼ 2.3 at IS 10 mM. These spots are located at peak areas where the maximum energy barrier is absent due to a monotonic decrease of interaction energy with increasing separation distance. In this case, the primary energy minimum between the NP and a peak is eliminated by the repulsion from the rest of the surface, causing the monotonic decrease of interaction energy with increasing separation distance (Shen et al., 2012). Inspection of Fig. 5 also shows that while the maximum energy barrier increases monotonically with decreasing solution IS for D ¼ 2.3, a nonmonotonic variation of energy barrier with IS exists for D ¼ 2.6. The energy barrier increases with decreasing IS from 0.2 M and reaches maximum at 0.01 M for D ¼ 2.6, and increases with increasing IS at IS 0.01 M. The increase of energy barrier with increasing solution IS at the low IS range has also been observed in Lin and Wiesner (2012) by exactly examining the interaction energy of a NP with a planar surface. Similar to the SEI technique, Lin and Wiesner (2012) considered both screening effect and the osmotic pressure between the NP and the planar surface in electrolyte solutions. Increasing IS increases both screening effect and osmotic pressure, which decreases and increases energy barrier, respectively. A net increase of the energy barrier with increasing IS will occur if the enlargement of the energy barrier by the osmotic pressure exceeds the energy barrier reduction by the screening effect. The maximum energy barrier increases monotonically with decreasing solution IS
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Fig. 2. Generated fractal surfaces using the W-M function with G ¼ 0.0136 nm and different values of D (a, D ¼ 2.3; b, D ¼ 2.4; c, D ¼ 2.5; d, D ¼ 2.6).
for D ¼ 2.3 because the wide distribution of peaks on the surface significantly reduces the osmotic pressure effect between the NP and the surface. This could explain why monotonic increase of attachment in primary minima with IS for NPs was frequently observed in experimental studies (Shen et al., 2015; Jin et al., 2017; Rasmuson et al., 2017) since roughness always exists on natural collector surfaces. Due to a similar reason, the maximum energy barrier increases monotonically with decreasing IS at large values of G, whereas a nonmonotonic variation of energy barrier with IS exists at small values of G (cf., Fig. 6). Fig. 7 presents maximum energy barrier maps for the 30-nm NP interacting with a fractal surface with G ¼ 0.0136 nm and different values of D at 0.2 M. The blue plaques mean that the values of energy barrier are negative, indicating favorable attachment at primary minima. The blue plaques cover most fraction of the map at D ¼ 2.3. Consequently, favorable attachment at primary minima occurs at most locations of the fractal surface. The fraction of blue plaques on a map is less for larger D values. The blue plaques completely disappear at the largest D value considered (i.e., D ¼ 2.6). Hence, the attachment is unfavorable on the entire fractal surface. Fig. 8 shows that for a given value of D, the value of maximum energy barrier increases with decreasing the value of G, demonstrating that attachment at primary minima is inhibited at lower value of G (i.e., smoother surface). Note that smaller D value or larger G value means larger average height. If the surface is modeled as a planar surface covered with regular asperities such as hemispheres, the maximum energy barrier decreases with increasing the average roughness height or hemisphere radius at large height range (Shen et al., 2011). Our results, however, show
that the maximum energy barrier is smallest for the largest average height (i.e., the smallest D or largest G). This is because asperities on a fractal surface are covered by smaller asperities due to selfaffinity. The fractal surface with smaller D or larger G has more nanoscale protruding asperities, which can significantly decrease maximum energy barrier. The use of simple rough models composed of regular geometries cannot capture this structure and accordingly can cause incorrect prediction of colloid attachment on natural collector surfaces where fractal roughness is frequently present. The obtained maximum energy barrier maps were used to determine the average attachment efficiency at primary minima (a) for interaction of the NP with a fractal surface via Boltzmann factor equation (Elimelech and O'Melia, 1990a, 1990b; Shen et al., 2007). The expression used the determine the value of a is given by a ¼ P exp½Umax ðx; yÞ dA A , where Umax is maximum energy barrier. A Fig. 9 presents calculated value of a for interaction between the 30 nm NP and a fractal surface as a function of D or G at different solution ISs. The value of a decreases monotonically with increasing D or decreasing G at all the ISs considered. The value of a decreases more slowly with D or G at 0.01 M compared to other ISs. It is noted that previous studies (Shen et al., 2012, 2014; Tian et al., 2012; Jin et al., 2015, 2017; Torkzaban and Bradford, 2016; Bradford et al., 2017; Li et al., 2017) frequently used roughness height to describe the degree of roughness. However, both monotonic and nonmonotonic variations of attachment in primary minima with roughness height existed. For example, Torkzaban and Bradford (2016) modeled rough surfaces as planar surfaces covered with pillars and showed that the primary minimum attachment
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Fig. 3. Horizontal energy profiles between a 30 nm NP and a fractal surface with G ¼ 0.0136 nm and (a) D ¼ 2.3 or (b) D ¼ 2.6 at 0.001 M at different separation distances (a1 and b1, 5 nm; a2 and b2, 1 nm; a3 and b3, 0.5 nm, a4 and b4, 0.1 nm).
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Fig. 4. Horizontal interaction energy maps between a 30 nm NP and a fractal surface with D ¼ 2.4 and (a) G ¼ 0.136 nm or (b) G ¼ 0.000136 nm at 0.001 M at different separation distances (a1 and b1, 5 nm; a2 and b2, 1 nm; a3 and b3, 0.5 nm; a4 and b4, 0.1 nm).
increased monotonically with increasing pillar height and decreasing pillar density. Jin et al. (2017) conducted column experiments to investigate influence of collector surface roughness on particle attachment under unfavorable conditions. They showed
that a critical roughness height associated with minimum particle attachment at primary minima was consistently observed. Particle attachment decreased with increasing roughness height when the roughness is smaller than the critical height and it increased with
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303
Fig. 5. Maximum energy barrier maps for interaction of the 30 nm NP with a fractal surface with G ¼ 0.0136 nm and (a) D ¼ 2.3 or (b) D ¼ 2.6 at different solution ISs (1, 0.001 M; 2, 0.01 M; 3, 0.1 M; 4, 0.2 M).
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Fig. 6. Maximum energy barrier maps for interactions between a 30 nm NP and a fractal surface with D ¼ 2.4 and (a) G ¼ 0.136 nm or (b) G ¼ 0.000136 nm at different solution ISs (1, 0.001 M; 2, 0.01 M; 3, 0.1 M; 4, 0.2 M).
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Fig. 7. Maximum energy barrier maps for the 30 nm NP with a fractal surface with G ¼ 0.0136 nm and different values of D (a, 2.3; b, 2.4; c, 2.5; d, 2.6) at 0.2 M.
Fig. 8. Maximum energy barrier maps between a 30 nm NP and a fractal surface with D ¼ 2.4 and different values of G (a, 0.136 nm; b, 0.0136 nm; c, 0.00136 nm; d, 0.000136 nm) at 0.2 M.
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Fig. 9. Calculated attachment efficiencies for interaction between the 30 nm NP and a fractal surface as a function of (a) D and (b) G at different solution ISs. The value of G is 0.0136 nm in (a) and the value of D is 2.4 in (b).
increasing roughness when its size was greater the critical value. The inconsistent variation of a with roughness height is likely because the variation of roughness height can also cause change of other roughness parameters such as asperity density and surface curvature. Therefore, the fractal dimension and fractal roughness are more robust for predicting attachment of colloids on natural collector surfaces with fractal roughness compared to traditional roughness parameters such as roughness height. 3.3. Detachment from primary minima Fig. 10 presents maps of detachment energy barrier from primary minima for interaction of the 30-nm NP with a fractal surface for G ¼ 0.0136 nm and D ¼ 2.3 or D ¼ 2.6 at different ISs. The detachment energy barrier was calculated by subtracting primary minimum from maximum energy barrier. Similar to the maximum energy barrier maps, white spots are also present at 0.01 and 0.001 M for D ¼ 2.3, where the detachment energy barrier (or primary minimum well) is absent due to monotonic decrease of interaction energy with increasing separation distance. The white spots are located at peak areas and the NPs attached at these locations via primary minimum association at a higher IS will be detached upon IS reduction. As mentioned previously, the maximum energy barrier is also reduced at the peak areas. Therefore, the peaks on rough surfaces facilitates both particle attachment at primary minima and subsequent detachment by IS reduction (Fang et al., 2014). At other locations, the detachment energy barrier increases with decreasing solution IS, illustrating
that the NP attachment is irreversible to reduction of IS. However, the detachment energy barriers are low at all locations of a map at D ¼ 2.3 for all solution ISs, illustrating that the NPs attached at primary minima experience weak attractions. These weakly attached NPs could be detached by increasing flow velocity if the hydrodynamic torques that act on the NPs exceed the adhesive torques (Bergendahl and Grasso, 2003; Bradford and Torkzaban, 2015). The detachment from primary minima by increasing flow velocity is more significant for large particles (e.g., microparticles) than NPs because they experience much greater hydrodynamic torques at a given flow velocity (Bergendahl and Grasso, 2003). Fig. 10 shows that the detachment energy barrier increases with increasing the value of D from 2.3 to 2.6. The detachment energy barrier is distributed more nonuniformly at D ¼ 2.6 compared to the case of D ¼ 2.3. The blue subdomains (low detachment energy barriers) are circumscribed by yellow or red nets (high detachment energy barriers) at D ¼ 2.6. The yellow or red nets are located at the valleys of the fractal surface. The detachment energy barriers at the valleys increases with decreasing solution IS, illustrating that attachment in primary minima at the valleys is irreversible to IS reduction. Li et al. (2017) has experimentally observed irreversible attachment in primary minima at valleys of rough surfaces via scanning electron microscope examinations. SI Fig. S4 shows that the variation of detachment energy barrier with decreasing G is similar to that with increasing D. Particularly, the detachment from primary minima by IS reduction occurs more at a high value of G (i.e., with peaks widely existing on the fractal surface). Decreasing the value of G results in more irreversible attachments at the primary minima. Note that the maximum energy barrier is also enhanced at the valleys, as have been shown previously. Therefore, the particles are less readily to access the primary minima at the valleys than at the peaks although the primary minimum attachment at the valleys is very stable.
3.4. Role of secondary minimum Fig. 11 presents secondary energy minimum maps for interactions of the 30-nm NP and a fractal surface with G ¼ 0.0136 nm and different values of D at 0.2 M. The secondary minimum is enhanced at the valley areas whereas reduced at the peak areas. The secondary minimum increases with increasing the value of D. For a given value of D, the secondary minimum increases with decreasing the value of G (cf., SI Fig. S5). The values of secondary energy minimum are small, which are comparable to the average kinetic energy of a colloid (i.e., 1.5 kT) at both valleys and peaks at 0.2 M. Note that the value of secondary energy minimum will further decrease with decreasing solution IS. Therefore, the 30-nm NPs cannot be attached on the fractal surfaces via secondary minimum association due to Brownian diffusion. However, as the value of secondary minimum is proportional to the size of a particle (Hahn et al., 2004), the enhancement of secondary minimum may play a significant role in immobilization of large colloids such as microparticles. The attachment of microparticles at valleys via secondary minimum associated could be further facilitated due to reduced hydrodynamic shear at these areas. It has been traditionally believed that particle attachment at secondary minimum can occur only at the stagnation point regions of a porous medium and particles at other locations of a collector surface will translate and rotate along the surface until they reach those areas (Elimelech and O'Melia, 1990a, 1990b; Hahn and O’melia, 2004; Hahn et al., 2004). The presence of roughness on surface, however, can cause attachment of particles at secondary minima distributed on an open collector surface.
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Fig. 10. Detachment energy barrier maps for interaction of the 30 nm NP with a fractal surface with G ¼ 0.0136 nm and (a) D ¼ 2.3 or (b) D ¼ 2.6 at different solution ISs (1, 0.001 M; 2, 0.01 M; 3, 0.1 M; 4, 0.2 M).
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Fig. 11. Secondary minimum energy maps for interaction between the 30 nm NP and a fractal surface with G ¼ 0.0136 nm and different values of D (a, 2.3; b, 2.4; c, 2.5; d, 2.6) at 0.2 M.
4. Conclusions
Declaration of interests
The interaction energy between a NP and fractal surfaces modeled using the W-M function was examined in threedimensional space. The roughness of a fractal surface increases with increasing the fractal roughness G and decreasing the fractal dimension D. A nonmonotonic variation of primary minimum attachment with solution IS exists for the NP on fractal surfaces with high values of D or low values of G, whereas the attachment decreases monotonically with decreasing IS at low values of D or high values of G. A consistent decrease of primary-minimum attachment with D or G exists at all ISs. The primary minimum attachment occurs at peak areas where the repulsive energy barriers are reduced or eliminated. A fraction of NPs attached at the peak areas via primary minimum association can detached by IS reduction due to monotonic decrease of interaction energy with increasing separation distance at low ISs. Although the primary minimum attachment at valley regions is irreversible to IS reduction, the NPs are difficult to be attached in these areas due to enhanced repulsive energy barriers. In contrast, the secondary minimum depths are enhanced at the valley regions, which may play an important role in immobilization of particles (particularly microparticles). Our results indicate that the fractal dimension and fractal roughness are more robust for predicting attachment and detachment on/from natural collector surface surfaces than traditional roughness parameters (e.g., average roughness height). These findings have important implication to accurate prediction of NP transport in subsurface environments and design of surfaces for particle removal in engineered applications such as water and wastewater treatment.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was supported by the National Natural Science Foundation of China (41671222, 41271009), Beijing Nova Program (Z161100004916116), Sichuan Agricultural Academy Youth Found (2018QNJJ-018), and the National Key Research and Development Program of China (2017YFD0800300, 2018YFD0800605). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.watres.2018.12.029. References Batley, G.E., Kirby, J.K., McLaughlin, M.J., 2013. Fate and risks of nanomaterials in aquatic and terrestrial environments. Acc. Chem. Res. 46, 854e862. Bendersky, M., Santore, M.M., Davis, J.M., 2015. Statistically-based DLVO approach to the dynamic interaction of colloidal microparticles with topographically and chemically heterogeneous collectors. J. Colloid Interface Sci. 449, 443e451. Bergendahl, J.A., Grasso, D., 2003. Mechanistic basis for particle detachment from granular media. Environ. Sci. Technol. 37, 2317e2322. Bhattacharjee, S., Elimelech, M., 1997. Surface element integration: a novel technique for evaluation of DLVO interaction between a particle and a plat plate. J. Colloid Interface Sci. 193, 273e285. Bhattacharjee, S., Ko, C.H., Elimelech, M., 1998. DLVO interaction between rough surfaces. Langmuir 14, 3365e3375.
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Water Res. 114, 286e295. Li, T., Jin, Y., Huang, Y., Li, B., Shen, C., 2017. Observed dependence of colloid detachment on the concentration of initially attached colloids and collector surface heterogeneity in porous media. Environ. Sci. Technol. 51, 2811e2822. Lin, S., Wiesner, M.R., 2012. Paradox of stability of nanoparticles at very low ionic strength. Langmuir 28, 11032e11041. Majumdar, A., Tien, C.L., 1990. Fractal characterization and simulation of rough surfaces. Wear 136 (2), 313e327. Mandelbrot, B.B., 1983. The Fractal Geometry of Nature. Freeman, New York. Molnar, I.R., Johnson, W.P., Gerhard, J.I., Willson, C.S., O'Carroll, D.M., 2015. Predicting colloid transport through saturated porous media: a critical review. Water Resour. Res. 51, 6804e6845. Pazmino, E., Trauscht, J., Dame, B., Johnson, W.P., 2014a. Power law size-distributed heterogeneity explains colloid retention on soda lime glass in the presence of energy barriers. Langmuir 30, 5412e5421. Pazmino, E., Trauscht, J., Johnson, W.P., 2014b. Release of colloids from primary minimum contact under unfavorable conditions by perturbations in ionic strength and flow rate. Environ. Sci. Technol. 48, 9227e9235. Rasmuson, A., Pazmino, E., Assemi, S., Johnson, W.P., 2017. Contribution of nano- to microscale roughness to heterogeneity: closing the gap between unfavorable and favorable colloid attachment conditions. Environ. Sci. Technol. 51, 2151e2160. Ryan, J.N., Elimelech, M., 1996. Colloid mobilization and transport in groundwater. Colloids Surf., A 107, 1e56. Sahoo, P., Ghosh, N., 2007. Finite element contact analysis of fractal surfaces. J. Phys. D Appl. Phys. 40, 4245e4252. Shellenberger, K., Logan, B.E., 2002. Effect of molecular scale roughness of glass beads on colloidal and bacterial deposition. Environ. Sci. Technol. 36, 184e189. Shen, C., Li, B., Huang, Y., Jin, Y., 2007. Kinetics of coupled primary- and secondaryminimum deposition of colloids under unfavorable chemical conditions. Environ. Sci. Technol. 41, 6976e6986. Shen, C., Li, B., Wang, C., Huang, Y., Jin, Y., 2011. Surface roughness effect on deposition of nano- and micro-sized colloids in saturated columns at different solution ionic strengths. Vadose Zone J. 10, 1071e1081. Shen, C., Wang, L.P., Li, B., Huang, Y., Jin, Y., 2012. Role of surface roughness in chemical detachment of colloids deposited at primary energy minima. Vadose Zone J. 11 https://doi.org/10.2136/vzj2011.0057. Shen, C., Jin, Y., Li, B., Zheng, W., Huang, Y., 2014. Facilitated attachment of nanoparticles at primary minima by nanoscale roughness is susceptible to hydrodynamic drag under unfavorable chemical conditions. Sci. Total Environ. 466e467, 1094e1102. Shen, C., Zhang, M., Zhang, S., Wang, Z., Zhang, H., Li, B., Huang, Y., 2015. Influence of surface heterogeneities on reversibility of fullerene (nC60) nanoparticle attachment in saturated porous media. J. Hazard Mater. 290, 60e68. Siegismund, D., Undisz, A., Germerodt, S., Schuster, S., Rettenmayr, M., 2014. Quantification of the interaction between biomaterial surfaces and bacteria by 3-D modeling. Acta Biomater. 10, 267e275. Tian, Y., Gao, B., Morales, V.L., Wang, Y., Wu, L., 2012. Effect of surface modification on single-walled carbon nanotube retention and transport in saturated and unsaturated porous media. J. Hazard Mater. 239e240, 333e339. Torkzaban, S., Bradford, S.A., 2016. Critical role of surface roughness on colloid retention and release in porous media. Water Res. 88, 274e284. Yan, H., Zhang, W., Peng, Z., Meng, G., 2015. Effect of random surface topography on the gaseous flow in microtubes with an extended slip model. Microfluid. Nanofluidics 18, 897e910. Yan, W., Komvopoulos, K., 1998. Contact analysis of elastic-plastic fractal surfaces. J. Appl. Phys. 84 (7), 3617e3624. Yang, H.W., Baudet, B.A., Yao, T., 2016. Characterization of the surface roughness of sand particles using an advanced fractal approach. Proc. R. Soc. A 472, 20160524. https://doi.org/10.1098/rspa.2016.0524. Yin, X., Komvopoulos, K., 2010. An adhesive wear model of fractal surfaces in normal contact. Int. J. Solid Struct. 47, 912e921. Zhang, M., Zhou, X., Shen, L., Cai, X., Wang, F., Chen, J., Lin, H., Li, R., Wu, X., Liao, B.eL., 2017. Quantitative evaluation of the interfacial interactions between a randomly rough sludge floc and membrane surface in a membrane bioreactor based on fractal geometry. Bioresour. Technol. 234, 198e207.
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Water Research journal homepage: www.elsevier.com/locate/watres
Interactions between nanoparticles and fractal surfaces Hong Wang a, Wei Zhang b, Saiqi Zeng c, Chongyang Shen a, *, Chao Jin d, Yuanfang Huang a, ** a
Department of Soil and Water Sciences, China Agricultural University, Beijing, 100193, China Department of Plant, Soil and Microbial Sciences, and Environmental Science and Policy Program, Michigan State University, East Lansing, MI, 48824, United States c Department of Plant and Soil Sciences, University of Delaware, Newark, DE, 19716, United States d School of Environmental Science and Engineering, Sun Yat-sen University, Guangzhou, Guangdong, 510006, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 September 2018 Received in revised form 13 December 2018 Accepted 15 December 2018 Available online 27 December 2018
This study evaluated attachment of a 30-nm nanoparticle to and detachment from fractal surfaces by calculating Derjaguin-Landau-Verwey-Overbeek (DLVO) interaction energies in three-dimensional space using the surface element integration technique. The fractal surfaces were generated using the Weierstass-Mandelbrot function with varying values of fractal dimension D (2.3 D 2.7) and fractal roughness G (0.000136 G 0.136). Results show that maximum energy barrier is reduced at peak areas of a fractal surface, and hence attachment in primary minima is favored. Some nanoparticles attached in primary minima at the peak areas can be detached by decreasing ionic strength (IS) due to monotonic decrease of interaction energy with increasing separation distance at low ISs. While the attachment in primary minima at valley areas is irreversible to IS reduction, the attachment is inhibited due to enhanced maximum energy barrier at these areas. A nonmonotonic variation of attachment efficiency in primary minimum (AEPM) with IS is present at high fractal dimension (D 2.4) or low fractal roughness (G < 0.00136), whereas the AEPM decreases monotonically with decreasing IS at low fractal dimension (D < 2.4) or high fractal roughness (G 0.00136). The AEPM decreases monotonically with increasing D or decreasing G at ISs from 1 mM to 200 mM. The decrease of AEPM with D or G is much slower at 10 mM compared to other ISs. These theoretical findings can explain various experimental observations in the literature, and can have important utility to development of water filtration techniques in engineered systems and to assessment of environmental risks of nanoparticles. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Nanoparticle Fractal surface Interaction energy Attachment Detachment
1. Introduction Transport of hazardous particles (e.g., pathogenic viruses, bacteria, and protozoa, and toxic nanomaterials) in porous media such as soil has been extensively investigated during the past two decades due to intense interest in preventing groundwater pollution, and developing water and wastewater filtration technologies (Ryan and Elimelech, 1996; de Jonge et al., 2004; Batley et al., 2013). Attachment on and detachment from collector surfaces are two primary processes that inhibit and facilitate particle transport in porous media, respectively (Bradford and Torkzaban, 2015; Molnar
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (C. Shen), [email protected] (Y. Huang). https://doi.org/10.1016/j.watres.2018.12.029 0043-1354/© 2018 Elsevier Ltd. All rights reserved.
et al., 2015). Whether particles attach to or detach from collector surfaces is dependent on the energy balance between adhesive energy and Brownian diffusion and/or the torque balance between adhesive torque and hydrodynamic torque (Bergendahl and Grasso, 2003; Pazmino et al., 2014a, 2014b; Bradford et al., 2017). Both energy and torque balance analysis require accurate determination of the adhesive force/energy that arises due to colloidal interactions between particles and surfaces. The colloidal interactions include van der Waals (VDW) attraction, electrical double layer (DL) interaction, and short-range repulsion (e.g., hydration and steric repulsion) (Hoek and Agarwal, 2006; Israelachvili, 2011). The Derjaguin-Landau-Verwey-Overbeek (DLVO) theory quantitatively describes the interaction energy profile as a function of separation distance between a particle and a surface (Ryan and Elimelech, 1996). When particles and collector surfaces are like-charged, both primary and secondary minima are present in the energy profile, between which an energy barrier
H. Wang et al. / Water Research 151 (2019) 296e309
exists. If the two surfaces are oppositely charged, both secondary minimum and energy barrier disappear from the profile and only primary minimum exists. The primary and secondary energy minima are considered as the locations for particle attachment (Grasso et al., 2002; Hahn and O’melia, 2004; Hahn et al., 2004). In the classic DLVO theory, calculation of interaction energies is based on the assumption that particle and collector surfaces are perfectly smooth. In real systems, surfaces of natural particles and collectors, however, all contain some degree of physical nonuniformity at small scales (Shellenberger and Logan, 2002). The influence of surface roughness has been frequently taken as the cause of discrepancies between the theoretical prediction and experimental observations of particle deposition (Fischer et al., 2014; Jin et al., 2015, 2017; Rasmuson et al., 2017). Theoretical studies (Huang et al., 2010; Bradford and Torkzaban, 2015; Bendersky et al., 2015) have shown that the nanoscale protruding asperities can increase particle attachment at primary minima by reducing energy barrier under the unfavorable conditions. However, the favorably attached particles are readily to be detached by perturbation of solution chemistry and/or hydrodynamics because the depths of the primary minima are also decreased by nanoscale protruding asperities (Shen et al., 2012, 2014). Such reduction of primary minimum depth by nanoscale protruding asperities can also decrease particle attachment at primary minima under favorable conditions (Rasmuson et al., 2017). While various mechanisms controlling interactions of particles with rough surfaces have been elucidated as shown above, the aforementioned theoretical results were obtained by modeling the rough surfaces as smooth surfaces carrying geometrically regular asperities (e.g., pillars, hemispheres, and cones). This technique is capable of characterizing the actual morphologies of rough surfaces to some extent by using the measured values of rough parameters (e.g., root-means-square roughness and surface area difference) from atomic force microscopy (AFM) scans to constrain the distribution of asperities on the smooth surfaces (Hoek et al., 2003). The modeling of rough surface was further improved in recent studies (Jaiswal et al., 2009; Siegismund et al., 2014) by using fast Fourier transform algorithm to model the roughness based on the AFM measurement. This technique can generate more realistic roughness profiles than using the regular geometries. However, all of these modeled surfaces are stationary because the rough parameter values of the surfaces (i.e., those from AFM scans) are fixed, which do not change with the length scale used to measure the modeled surfaces (Eichenlaub et al., 2004). Many natural and engineered surfaces such as sand and membrane surfaces are fractal and their rough parameter values change with the resolution and AFM scan size (Yang et al., 2016; Cai et al., 2017). A fractal surface bears the property of self-affinity. Specifically, when a part of the fractal surface is magnified, the magnified image
297
function which is a superposition of sinusoids with geometrically spaced frequencies and amplitudes that follow a power law (Yan and Komvopoulos, 1998; Eichenlaub et al., 2004). The W-M function has been used to represent fractal surfaces for investigating influence of surface topography on engineered processes such as gaseous and fluid flow and thermal contact resistance (Majumdar and Tien, 1990; Yan et al., 2015). The W-M function has also been employed recently to model membrane surfaces for examining the interactions between a particle and the surfaces (Cai et al., 2017; Chen et al., 2017; Feng et al., 2017; Zhang et al., 2017). It was found that the adhesions between particles and fractal surfaces are significantly lowered compared to those of particle-flat surface interactions. While these studies advanced understanding about influence of fractal on colloidal interaction, they only investigated the variation of interaction energy with separation distance for a particle at a single horizontal location above a fractal surface. The interaction energy that acts on the particle should change with its horizontal location due to the randomness of fractal surface. Moreover, the influence of environmental factors such as ionic strength (IS) on interaction energies between particles and fractal surfaces has not been investigated to date. The detachment of particles from fractal surfaces is also unclear. This study systematically evaluated the attachment and detachment of nanoparticles (NPs) on/from fractal surfaces represented using the W-M function by calculating interaction energies at various horizontal and vertical locations. The surface element integration (SEI) technique (Bhattacharjee and Elimelech, 1997; Siegismund et al., 2014) was used for the energy calculations and the interaction energy maps were obtained to describe the variation of interaction energy in three dimensional space for various ISs and fractal roughness parameter values. We showed that the fractal dimension and fractal roughness are more effective for predicting attachment and detachment on/from primary and secondary minima than traditional roughness parameters such as average roughness height. The findings in this study can help explain various observations in the literature. For example, our results could explain why a nonmonotonic variation of attachment with IS for NPs was predicted in a theoretical study (Lin and Wiesner, 2012) through exact interaction energy calculations but not observed experimentally. 2. Theory 2.1. The W-M function The modified two variable W-M function by Yan and Komvopoulos (1998) was employed to generate fractal surfaces in this study:
ðD2Þ G ln g 1=2 zðx; yÞ ¼ L L M M n¼∞ X X m¼1 n¼0
( ðD3Þn
g
"
cosfm;n cos
1=2 2pgn x2 þ y2 L
y pm io þ fm;n cos tan1 x M
looks very similar to the original surface (Chen et al., 2009). In addition, the fractal surface possesses the properties of continuity and non-differentiability (Mandelbrot, 1983). These mathematical properties are satisfied by the Weierstrass-Mandelbrot (W-M)
(1)
where z(x, y) is the three-dimensional surface profile, D (2 < D < 3) is the fractal dimension of surface profile, G is a height scaling parameter (commonly termed as the fractal roughness) that is independent of frequency, L is the sample length, g (g > 1) is a scaling
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variable that determines the density of the frequency spectrum, M (M > 3) is the number of superposed ridges (commonly taken as 10), n is the spatial frequency index, and 4m,n is the random phase. The fractal roughness (G) and the fractal dimension (D) are two critical parameters that determine the fractal topography. The fractal dimension determines the relative contributions of highand low-frequency components in the fractal surface profile. Higher value of D means that high-frequency components in the surface profile are more dominant than low-frequency components (Yin and Komvopoulos, 2010). The fractal roughness is a height scaling parameter (i.e., amplitude coefficient) and the amplitudes of wavelengths comprising the surface profile increase with increasing the value of G. Both values of D and G are independent of the measurement scale for a given fractal surface. The spatial index n assumes finite values in practice (Yan and Komvopoulos, 1998). The lower limit of n is commonly taken as zero, and the higher limit of n is given by n ¼ int[log(L/Ls)/logg], where Ls is the cut-off wavelength (i.e., the smallest allowed length scale) determined by the instrument resolution limit, which was taken as 1 nm (Yin and Komvopoulos, 2010). g is commonly taken as 1.5 for most surfaces based on surface flatness and frequency distribution density considerations (Majumdar and Tien, 1990). A random number generator in Matlab was used to uniformly distribute 4m,n in the interval [0, 2p] to prevent the coincidence of frequencies at any point of surface profile.
corresponding area element on fractal surface (dSc) is calculated as (Bhattacharjee et al., 1998; Hoss et al., 2017)
dU ¼ ðn1 ,k1 Þðn2 ,k2 ÞEðhÞdSc
(2)
where k1 and k2 are unit vectors directed towards the positive and negative z axis, respectively, n1 and n2 are outward units normal to the NP and collector surfaces, respectively, E(h) is the differential interaction energy between element dSp and dSc, and h is the local separation distance between the two elements. The total interaction energy (U) that the NP experiences is then obtained by summation of the differential interaction energy (dU) for each dSc on the projected area of the NP on the fractal surface. The equation used to calculate the total interaction energy can be expressed as
U¼
X ðn1 ,k1 Þðn2 ,k2 ÞEðhÞdSc
(3)
Sc
where Sc is the project area of the NP on the fractal surface. It is very difficult to directly determine the value of n2 ,k2 due to the randomness of the fractal surface. However, the term of n2 ,k2 can be eliminated from Equation (3) by expressing the element dSc in terms of its projection dA on the xy plane as
dSc ¼
1 dA jn2 ,k2 j
(4)
Substituting Equation (4) into Equation (3), we obtain 2.2. Calculation of DLVO interaction energies Fig. 1 schematically illustrates the interaction between a spherical NP and a fractal surface generated by the W-M function. Detailed procedure for calculating interaction energy between a spherical particle and a rough surface using the SEI technique can be referred to previous studies (Bhattacharjee et al., 1998; Duffadar and Davis, 2007; Hoss et al., 2017). Briefly, the Cartesian coordinate system was employed for the model system in Fig. 1. The xy plane of the coordinate system is oriented parallel to the zero plane of the fractal surface, which divides the NP into two identical hemispheres. The z axis passes through the NP center and faces toward the fractal surface. Similar to the grid-surface integration technique (Duffadar and Davis, 2007), both NP and fractal surfaces were discretized into small area elements. The differential interaction energy dU between an area element on NP surface (dSp) and the
U¼
X n ,k ðn1 ,k1 Þ 2 2 EðhÞdA jn 2 ,k2 j A
When the unit normal to the fractal surface at a given point n2 2 makes an obtuse or acute angle with k2, the value of jnn2 ,k is equal 2 ,k2 j to 1 or 1. Hence, the total interaction energy can be obtained by integrating the differential interaction energy over the projected area of the NP on the xy plane without determining the value of z n2 ,k2 . By using n1 ,k1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Equation (5) is changed to 2 2 2 x þy þz
X z U ¼ ± pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EðhÞdA 2 þ y2 þ z 2 x A
(6)
The differential interaction energy between element dS and dA [i.e., E(h)] was calculated by summing van der Waals (VDW), constant potential double layer (DL), and Born (BR) energies per unit area [i.e., EðhÞ ¼ EVDW ðhÞ þ EDL ðhÞ þ EBR ðhÞ]. The expressions used to calculate EVDW(h), EDL(h), and EBR(h) are following
EVDW ðhÞ ¼
Fig. 1. Schematic illustration of a spherical NP interacting with a fractal collector surface. dSp and dSc are differential area element on the NP and collector surfaces, respectively, n1 and n2 are unit outwards normal to the NP and collector surfaces, respectively, k1 and k2 are unit vectors directed towards the positive and negative zaxis, respectively.
(5)
AH 12ph2
(7)
EDL ðhÞ ¼
2j j εε0 k 2 jp þ j2c ð1 cothkhÞ þ p c 2 sinhkh
(8)
EBR ðhÞ ¼
AH H60 48ph8
(9)
where AH is the Hamaker constant, 3 0 is the dielectric permittivity of vacuum, 3 is dielectric permittivity of vacuum, k is the inverse Debye screening length, jp and jc are the surface potentials of NP and collector, respectively, H0 is the minimum separation distance between the NP and the fractal surface (0.158 nm) (Hoek and Agarwal, 2006). Both NP and fractal surfaces were discretized into grid elements with projection areas on the xy plane of 0.1 nm 0.1 nm. A Matlab program was developed to calculate DLVO interaction energies for
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the aforementioned interaction configuration, and the calculated energies were expressed in terms of kT (k is Boltzmann constant and T is absolute temperature). A horizontal interaction energy map (denoted as HEP) was obtained by varying the horizontal positions (x, y) of the NP center in a rasterized manner and keeping the vertical distance between the NP center and the rough surface fixed. Various HEPs were produced at different vertical distances between the NP center and the rough surface. Vertical interaction energy maps (VEPs) were also calculated by varying the vertical positions (x, z) or (y, z) of the NP center for a given value of y or x. Note that the interaction energy maps have also been calculated by varying the horizontal positions (x, y) of a particle center and keeping the vertical distance between the NP center and the zero plane of a rough surface fixed (Hoek et al., 2003). We did not adopt this approach because physical overlaps can occur between the particles and protruding asperities on the rough surfaces. A number of studies (Siegismund et al., 2014; Feng et al., 2017; Zhang et al., 2017) have examined the interaction of a particle and a surface with random roughness. However, these studies only examined adhesion or primary minimum depth for the particle at a single location of the surface. In contrast, by using the HEPs and VEPs, the maps of primary minimum, maximum energy barrier, and secondary minimum were obtained in this study. As such, the distribution of attachment in primary and secondary minima and subsequent detachment as well as the variation of the attachment and detachment with surface morphology (e.g., height and curvature) can be obtained. For theoretical calculation of interaction energies, the NPs and fractal collector surfaces were assumed to be 30-nm polystyrene latex particles and quartz sand, respectively, which have the same properties as those used in Shen et al. (2007). Table 1 presents the zeta potentials of the latex NPs and quartz sand used to calculate the interaction energies at different solution ISs. A value of 1 1020 J was taken as the Hamaker constant for latex-water-sand system (Elimelech and O'Melia, 1990a, 1990b; Shen et al., 2007).
3. Results and discussion 3.1. Generated fractal surfaces Fig. 2 presents simulated fractal surfaces using the W-M function with projected surface areas on xy plane of 150 nm 150 nm for varying D values (2.3, 2.4, 2.5, and 2.6) and a G value of 0.0136 nm. The scale bar to the right of each plot represents height in units of nm. We adopted these values of D and G because for most fractal surfaces, the values of D range from 2.3 to 2.7 and the values of G range from 0.000136 nm to 0.136 nm (Sahoo and Ghosh, 2007). Fig. 2 shows that for a given value of G, larger value of D yields smoother surface. This is because high frequency components are more dominant than low frequency components in the surface topography profile at larger D values (Chatterjee and Sahoo, 2014). For a given value of D, larger G value results in rougher surfaces due to an increase in the amplitude of surface wavelength with G, as shown in Supplementary Information (SI) Fig. S1. The variation of
Table 1 Zeta potentials of sand and NPs at different solution ISs. IS (M)
0.001 0.01 0.1 0.2
Zeta Potential (mV) Sand
NPs
39.27 39.13 32.08 27.49
44.3 43.25 38.69 36.86
299
surface roughness with D and G are in agreement with those in previous studies (Yan and Komvopoulos, 1998; Chatterjee and Sahoo, 2014; Yan et al., 2015). 3.2. Attachment in primary minima Fig. 3 presents HEPs for interactions of a 30-nm NP with a fractal surface with G ¼ 0.0136 nm and D ¼ 2.3 or D ¼ 2.6 (i.e., the surface in Fig. 2a or 2d) at 0.001 M at varying separation distances. The scale bar to the right of each plot represents interaction energy in units of kT (k is the Boltmann constant, T ¼ 297.13 K is absolute temperature). The interaction energy is non-uniformly distributed (plaque-like) in the horizontal direction due to the random variation of surface curvature (Shen et al., 2015). The magnitude of interaction energy is reduced and enhanced at peak and valley regions, respectively, as shown by the VEPs in SI Fig. S2. The variation of interaction energy with surface curvature can be explained by the concept of “interaction volume” developed by Huang et al. (2010). The interaction volume is defined as the volume between the leading half of a spherical particle and a substrate surface. The interaction volume is smaller for a NP interacting with a peak surface than with a valley surface. Huang et al. (2010) illustrated that smaller interaction volume between a particle and a substrate surface causes larger interaction energy because the mean plane of the surface is closer to the apex of the particle. Therefore, the magnitude of interaction energy is reduced at peak areas whereas enhanced at valley areas. The interaction energy is more uniformly distributed at larger separation distance because the influence of local variation of surface curvature on interaction energy decreases with increasing separation distance (cf., Fig. 3). The energy plaques on the maps are larger for smaller D value because of more dominance of high frequency components than low frequency components. Due to a similar reason, increasing the value of G increases the sizes of the plaques (cf., Fig. 4). For a given fractal surface, the interaction energy decreases with increasing IS (SI Fig. S3). Fig. 5 shows maximum energy barrier maps for interactions of the 30-nm NP with a fractal surface with G ¼ 0.0136 nm and D ¼ 2.3 or D ¼ 2.6 at different solution ISs. The plaques on the energy barrier maps are larger for smaller value of D (i.e., smoother surface). The value of maximum energy barrier is smaller for smaller value of D. Interestingly, white spots are present on the HEPs for D ¼ 2.3 at IS 10 mM. These spots are located at peak areas where the maximum energy barrier is absent due to a monotonic decrease of interaction energy with increasing separation distance. In this case, the primary energy minimum between the NP and a peak is eliminated by the repulsion from the rest of the surface, causing the monotonic decrease of interaction energy with increasing separation distance (Shen et al., 2012). Inspection of Fig. 5 also shows that while the maximum energy barrier increases monotonically with decreasing solution IS for D ¼ 2.3, a nonmonotonic variation of energy barrier with IS exists for D ¼ 2.6. The energy barrier increases with decreasing IS from 0.2 M and reaches maximum at 0.01 M for D ¼ 2.6, and increases with increasing IS at IS 0.01 M. The increase of energy barrier with increasing solution IS at the low IS range has also been observed in Lin and Wiesner (2012) by exactly examining the interaction energy of a NP with a planar surface. Similar to the SEI technique, Lin and Wiesner (2012) considered both screening effect and the osmotic pressure between the NP and the planar surface in electrolyte solutions. Increasing IS increases both screening effect and osmotic pressure, which decreases and increases energy barrier, respectively. A net increase of the energy barrier with increasing IS will occur if the enlargement of the energy barrier by the osmotic pressure exceeds the energy barrier reduction by the screening effect. The maximum energy barrier increases monotonically with decreasing solution IS
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Fig. 2. Generated fractal surfaces using the W-M function with G ¼ 0.0136 nm and different values of D (a, D ¼ 2.3; b, D ¼ 2.4; c, D ¼ 2.5; d, D ¼ 2.6).
for D ¼ 2.3 because the wide distribution of peaks on the surface significantly reduces the osmotic pressure effect between the NP and the surface. This could explain why monotonic increase of attachment in primary minima with IS for NPs was frequently observed in experimental studies (Shen et al., 2015; Jin et al., 2017; Rasmuson et al., 2017) since roughness always exists on natural collector surfaces. Due to a similar reason, the maximum energy barrier increases monotonically with decreasing IS at large values of G, whereas a nonmonotonic variation of energy barrier with IS exists at small values of G (cf., Fig. 6). Fig. 7 presents maximum energy barrier maps for the 30-nm NP interacting with a fractal surface with G ¼ 0.0136 nm and different values of D at 0.2 M. The blue plaques mean that the values of energy barrier are negative, indicating favorable attachment at primary minima. The blue plaques cover most fraction of the map at D ¼ 2.3. Consequently, favorable attachment at primary minima occurs at most locations of the fractal surface. The fraction of blue plaques on a map is less for larger D values. The blue plaques completely disappear at the largest D value considered (i.e., D ¼ 2.6). Hence, the attachment is unfavorable on the entire fractal surface. Fig. 8 shows that for a given value of D, the value of maximum energy barrier increases with decreasing the value of G, demonstrating that attachment at primary minima is inhibited at lower value of G (i.e., smoother surface). Note that smaller D value or larger G value means larger average height. If the surface is modeled as a planar surface covered with regular asperities such as hemispheres, the maximum energy barrier decreases with increasing the average roughness height or hemisphere radius at large height range (Shen et al., 2011). Our results, however, show
that the maximum energy barrier is smallest for the largest average height (i.e., the smallest D or largest G). This is because asperities on a fractal surface are covered by smaller asperities due to selfaffinity. The fractal surface with smaller D or larger G has more nanoscale protruding asperities, which can significantly decrease maximum energy barrier. The use of simple rough models composed of regular geometries cannot capture this structure and accordingly can cause incorrect prediction of colloid attachment on natural collector surfaces where fractal roughness is frequently present. The obtained maximum energy barrier maps were used to determine the average attachment efficiency at primary minima (a) for interaction of the NP with a fractal surface via Boltzmann factor equation (Elimelech and O'Melia, 1990a, 1990b; Shen et al., 2007). The expression used the determine the value of a is given by a ¼ P exp½Umax ðx; yÞ dA A , where Umax is maximum energy barrier. A Fig. 9 presents calculated value of a for interaction between the 30 nm NP and a fractal surface as a function of D or G at different solution ISs. The value of a decreases monotonically with increasing D or decreasing G at all the ISs considered. The value of a decreases more slowly with D or G at 0.01 M compared to other ISs. It is noted that previous studies (Shen et al., 2012, 2014; Tian et al., 2012; Jin et al., 2015, 2017; Torkzaban and Bradford, 2016; Bradford et al., 2017; Li et al., 2017) frequently used roughness height to describe the degree of roughness. However, both monotonic and nonmonotonic variations of attachment in primary minima with roughness height existed. For example, Torkzaban and Bradford (2016) modeled rough surfaces as planar surfaces covered with pillars and showed that the primary minimum attachment
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Fig. 3. Horizontal energy profiles between a 30 nm NP and a fractal surface with G ¼ 0.0136 nm and (a) D ¼ 2.3 or (b) D ¼ 2.6 at 0.001 M at different separation distances (a1 and b1, 5 nm; a2 and b2, 1 nm; a3 and b3, 0.5 nm, a4 and b4, 0.1 nm).
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Fig. 4. Horizontal interaction energy maps between a 30 nm NP and a fractal surface with D ¼ 2.4 and (a) G ¼ 0.136 nm or (b) G ¼ 0.000136 nm at 0.001 M at different separation distances (a1 and b1, 5 nm; a2 and b2, 1 nm; a3 and b3, 0.5 nm; a4 and b4, 0.1 nm).
increased monotonically with increasing pillar height and decreasing pillar density. Jin et al. (2017) conducted column experiments to investigate influence of collector surface roughness on particle attachment under unfavorable conditions. They showed
that a critical roughness height associated with minimum particle attachment at primary minima was consistently observed. Particle attachment decreased with increasing roughness height when the roughness is smaller than the critical height and it increased with
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303
Fig. 5. Maximum energy barrier maps for interaction of the 30 nm NP with a fractal surface with G ¼ 0.0136 nm and (a) D ¼ 2.3 or (b) D ¼ 2.6 at different solution ISs (1, 0.001 M; 2, 0.01 M; 3, 0.1 M; 4, 0.2 M).
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Fig. 6. Maximum energy barrier maps for interactions between a 30 nm NP and a fractal surface with D ¼ 2.4 and (a) G ¼ 0.136 nm or (b) G ¼ 0.000136 nm at different solution ISs (1, 0.001 M; 2, 0.01 M; 3, 0.1 M; 4, 0.2 M).
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Fig. 7. Maximum energy barrier maps for the 30 nm NP with a fractal surface with G ¼ 0.0136 nm and different values of D (a, 2.3; b, 2.4; c, 2.5; d, 2.6) at 0.2 M.
Fig. 8. Maximum energy barrier maps between a 30 nm NP and a fractal surface with D ¼ 2.4 and different values of G (a, 0.136 nm; b, 0.0136 nm; c, 0.00136 nm; d, 0.000136 nm) at 0.2 M.
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Fig. 9. Calculated attachment efficiencies for interaction between the 30 nm NP and a fractal surface as a function of (a) D and (b) G at different solution ISs. The value of G is 0.0136 nm in (a) and the value of D is 2.4 in (b).
increasing roughness when its size was greater the critical value. The inconsistent variation of a with roughness height is likely because the variation of roughness height can also cause change of other roughness parameters such as asperity density and surface curvature. Therefore, the fractal dimension and fractal roughness are more robust for predicting attachment of colloids on natural collector surfaces with fractal roughness compared to traditional roughness parameters such as roughness height. 3.3. Detachment from primary minima Fig. 10 presents maps of detachment energy barrier from primary minima for interaction of the 30-nm NP with a fractal surface for G ¼ 0.0136 nm and D ¼ 2.3 or D ¼ 2.6 at different ISs. The detachment energy barrier was calculated by subtracting primary minimum from maximum energy barrier. Similar to the maximum energy barrier maps, white spots are also present at 0.01 and 0.001 M for D ¼ 2.3, where the detachment energy barrier (or primary minimum well) is absent due to monotonic decrease of interaction energy with increasing separation distance. The white spots are located at peak areas and the NPs attached at these locations via primary minimum association at a higher IS will be detached upon IS reduction. As mentioned previously, the maximum energy barrier is also reduced at the peak areas. Therefore, the peaks on rough surfaces facilitates both particle attachment at primary minima and subsequent detachment by IS reduction (Fang et al., 2014). At other locations, the detachment energy barrier increases with decreasing solution IS, illustrating
that the NP attachment is irreversible to reduction of IS. However, the detachment energy barriers are low at all locations of a map at D ¼ 2.3 for all solution ISs, illustrating that the NPs attached at primary minima experience weak attractions. These weakly attached NPs could be detached by increasing flow velocity if the hydrodynamic torques that act on the NPs exceed the adhesive torques (Bergendahl and Grasso, 2003; Bradford and Torkzaban, 2015). The detachment from primary minima by increasing flow velocity is more significant for large particles (e.g., microparticles) than NPs because they experience much greater hydrodynamic torques at a given flow velocity (Bergendahl and Grasso, 2003). Fig. 10 shows that the detachment energy barrier increases with increasing the value of D from 2.3 to 2.6. The detachment energy barrier is distributed more nonuniformly at D ¼ 2.6 compared to the case of D ¼ 2.3. The blue subdomains (low detachment energy barriers) are circumscribed by yellow or red nets (high detachment energy barriers) at D ¼ 2.6. The yellow or red nets are located at the valleys of the fractal surface. The detachment energy barriers at the valleys increases with decreasing solution IS, illustrating that attachment in primary minima at the valleys is irreversible to IS reduction. Li et al. (2017) has experimentally observed irreversible attachment in primary minima at valleys of rough surfaces via scanning electron microscope examinations. SI Fig. S4 shows that the variation of detachment energy barrier with decreasing G is similar to that with increasing D. Particularly, the detachment from primary minima by IS reduction occurs more at a high value of G (i.e., with peaks widely existing on the fractal surface). Decreasing the value of G results in more irreversible attachments at the primary minima. Note that the maximum energy barrier is also enhanced at the valleys, as have been shown previously. Therefore, the particles are less readily to access the primary minima at the valleys than at the peaks although the primary minimum attachment at the valleys is very stable.
3.4. Role of secondary minimum Fig. 11 presents secondary energy minimum maps for interactions of the 30-nm NP and a fractal surface with G ¼ 0.0136 nm and different values of D at 0.2 M. The secondary minimum is enhanced at the valley areas whereas reduced at the peak areas. The secondary minimum increases with increasing the value of D. For a given value of D, the secondary minimum increases with decreasing the value of G (cf., SI Fig. S5). The values of secondary energy minimum are small, which are comparable to the average kinetic energy of a colloid (i.e., 1.5 kT) at both valleys and peaks at 0.2 M. Note that the value of secondary energy minimum will further decrease with decreasing solution IS. Therefore, the 30-nm NPs cannot be attached on the fractal surfaces via secondary minimum association due to Brownian diffusion. However, as the value of secondary minimum is proportional to the size of a particle (Hahn et al., 2004), the enhancement of secondary minimum may play a significant role in immobilization of large colloids such as microparticles. The attachment of microparticles at valleys via secondary minimum associated could be further facilitated due to reduced hydrodynamic shear at these areas. It has been traditionally believed that particle attachment at secondary minimum can occur only at the stagnation point regions of a porous medium and particles at other locations of a collector surface will translate and rotate along the surface until they reach those areas (Elimelech and O'Melia, 1990a, 1990b; Hahn and O’melia, 2004; Hahn et al., 2004). The presence of roughness on surface, however, can cause attachment of particles at secondary minima distributed on an open collector surface.
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Fig. 10. Detachment energy barrier maps for interaction of the 30 nm NP with a fractal surface with G ¼ 0.0136 nm and (a) D ¼ 2.3 or (b) D ¼ 2.6 at different solution ISs (1, 0.001 M; 2, 0.01 M; 3, 0.1 M; 4, 0.2 M).
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Fig. 11. Secondary minimum energy maps for interaction between the 30 nm NP and a fractal surface with G ¼ 0.0136 nm and different values of D (a, 2.3; b, 2.4; c, 2.5; d, 2.6) at 0.2 M.
4. Conclusions
Declaration of interests
The interaction energy between a NP and fractal surfaces modeled using the W-M function was examined in threedimensional space. The roughness of a fractal surface increases with increasing the fractal roughness G and decreasing the fractal dimension D. A nonmonotonic variation of primary minimum attachment with solution IS exists for the NP on fractal surfaces with high values of D or low values of G, whereas the attachment decreases monotonically with decreasing IS at low values of D or high values of G. A consistent decrease of primary-minimum attachment with D or G exists at all ISs. The primary minimum attachment occurs at peak areas where the repulsive energy barriers are reduced or eliminated. A fraction of NPs attached at the peak areas via primary minimum association can detached by IS reduction due to monotonic decrease of interaction energy with increasing separation distance at low ISs. Although the primary minimum attachment at valley regions is irreversible to IS reduction, the NPs are difficult to be attached in these areas due to enhanced repulsive energy barriers. In contrast, the secondary minimum depths are enhanced at the valley regions, which may play an important role in immobilization of particles (particularly microparticles). Our results indicate that the fractal dimension and fractal roughness are more robust for predicting attachment and detachment on/from natural collector surface surfaces than traditional roughness parameters (e.g., average roughness height). These findings have important implication to accurate prediction of NP transport in subsurface environments and design of surfaces for particle removal in engineered applications such as water and wastewater treatment.
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