Nadia Von Moos

Volodymyr B Koman

Christian Santschi
email: christian.santschi@epfl.ch

Olivier J F Martin

Lionel Maurizi

Amarnath Jayaprakash

Paul Bowen

Vera I Slaveykova

Pro-oxidant effects of nano-TiO 2 on Chlamydomonas reinhardtii during short-term exposure

Keywords: nanoecotoxicity, microalgae, hydrogen peroxide, oxidative stress, oxidative damage, biosensing, cytochrome c, flow cytometry

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INTRODUCTION

With the increasingly pervasive use of engineered nanomaterials (ENMs) in modern society, the aquatic system has been recognized as a primary environmental entry point and sink for ENMs inevitably discharged by anthropogenic activity [1][2][3] . Yet, the associated inadvertent implications for the overall ecotoxicological risk remain uncertain [3][4][5][6] . The established consensus maintains that the dominant toxicity factor of inorganic ENMs lies in their ability to generate reactive oxygen species (ROS) and thereby cause oxidative stress and damage 5,[7][8][9][10][11][12][13][14][15] . Thus, from a nanoecotoxicological perspective seeking the elucidation of the environmental hazards of ENMs, it follows that an in depth understanding of normal and ENM-stimulated ROS production as well as antioxidant levels in aquatic organisms is required to provide risk assessors the necessary knowledge to address the following key, yet unresolved challenges of aquatic toxicology formulated by Livingstone 16 more than a decade ago: (i) identification of pro-oxidant species, (ii) design of novel toxicity assays for the detection of pro-oxidant activity, (iii) quantitative assessment of contaminant-mediated pro-oxidant processes compromising biological fitness and lastly (iv) identification of environmental and biological factors that modulate ENM-stimulated ROS generation and oxidative damage 16 .

Against this background, the purpose of this research study is twofold: Firstly, it assesses the toxicity of nano-TiO2 to the model aquatic microorganism Chlamydomonas reinhardtii by investigating its pro-oxidant potential and possible modifying factors thereof by wellestablished fluorescent probes for oxidative stress and damage. Secondly, this is the first indepth, systematic nano-ecotoxicological study to use extracellular [H2O2] as a complementary endpoint in achieving the first aim, in an attempt to validate this new method based on a cytochrome c biosensor, which was recently developed by a subset of the present co-authors.

Unlike most other well-established approaches commonly employed in nanotoxicity testing, this novel biosensor is non-invasive and provides quantitative measurements of extracellular [H2O2] in real time. Given H2O2 is a relatively long-lived ROS, thanks to which it has a certain mobility within cells and across membranes, it is a highly suitable indicator of pro-oxidant responses in biological systems [START_REF]Free radicals in biology and medecine[END_REF] . Stress-induced H2O2 can rapidly diffuse across the plasma membrane passively or through aquaporin channels [START_REF] Dynowski | [END_REF]19 and can be detected from as early as a few seconds to as long as a few days after stress application 20 . Therefore, the cytochrome c biosensor has been put forth a promising new method to assess and compare the pro-oxidant potential of stressors, such as nanomaterials, based on the paradigm of oxidative stress. The hypothesis to be evaluated here is the assumption that extracellular [H2O2] can serve as a biomarker for oxidative stress and damage in the cell originating from exposure to nano-TiO2. In other words, we are proposing to exploit H2O2, a stable representative of ROS, as a possible biomarker of pro-oxidant stress. The rationale for our working hypothesis assumes that oxidative stress in cells occurs when a stressor either elevates ROS generation or weakens the antioxidant response, in which case homeostatic ROS levels are breached and H2O2, as the most stable ROS, is expected to be excreted by stressed cells. Thus, the present study attempts to link extracellular and cellular pro-oxidant processes in a well-described system consisting of a well-known model ENM and exposure organism.

For this purpose, the microalga C. reinhardtii was exposed to two series of nano-TiO2 suspensions at nominal concentrations of 10, 50, 100 and 200 mg L -1 , one of which previously received a 20 min UVA illumination, in two exposure media comprising a common laboratory testing buffer and lake Geneva water. In this way, we investigate the impact of the factors "exposure medium", "exposure concentration", "exposure time" and "UV treatment" on the pro-oxidant potential of nano-TiO2. The pro-oxidant potential of nano-TiO2 on the model aquatic microorganism Chlamydomonas reinhardtii was assessed with fluorescent probes for oxidative stress and oxidative membrane damage measured by flow cytometry (CellRox Green reagent and propidium iodide, respectively). The cellular endpoints were complemented with nano-TiO2-induced abiotic ROS measured by H2DCF-DA fluorescence as well as continuous measures of extracellular H2O2 generation measured in abiotic and biotic conditions using the cytochrome c biosensor.

The microalga C. reinhardtii employed in this study lends itself well as a model microorganism since it represents the most sensitive class of aquatic organisms to metal oxide ENMs 21 that can serve as early sentinel for potential environmental hazards in aquatic systems 22 .

Nano-TiO2 is the most abundantly produced, most widely applied and investigated ENM that assumes the role of a benchmark against which other particles can be compared 23 . Most common applications are in the fields of photovoltaics, photocatalysis and sensing but also include its use as a white pigment in paints, cosmetics, personal care products and as E-171 in food [24][25][26][27] . As a semiconductor, energies equal to or higher than its band gap around 3.2 eV (photons with wavelengths < 385 nm) generate electron-hole pairs (hVB + /eCB -) on its surface that, by reacting with surface H2O and O2 in aqueous media, drive the formation of various ROS, including superoxide anions (O2 -), hydrogen peroxide (H2O2), free hydroxyl radicals (OH  ) and singlet oxygen ( 1 O2) [28][29][30][31] . With reported EC50 values of nano-TiO2 for microalgae broadly varying from approximately 5 mg L -1 to 241 mg L -1 32 its relative potency lies at the lower end of the ENM toxicity scale. An augmented photocatalytic inhibition of algae is known 33 but it has been shown that ROS mediated nano-TiO2 toxicity on microalgae also occurs in normal light conditions and does not significantly differ from UV treatments [34][35][36][37] .

MATERIALS AND METHODS

Experimental design.

Exposure experiments with algae were performed in two different exposure media and in two series of four different nano-TiO2 concentrations (10, 50, 100 and 200 mg L -1 ), one of which was performed with untreated particles and the other of which received a 20 min pre-illumination with long wave UV before contact with cells. Extracellular ROS was then monitored during 1 h with a novel portable oxidative stress sensor (POSS). To assess intracellular ROS levels and membrane integrity, the same exposure conditions were repeated separately and the samples stained with fluorescent probes for measurements by flow cytometry at the beginning of exposure (t = 0 h) and after 1 h (t = 1 h). All exposure conditions were replicated at least three times. All reagents (analytical grade) were purchased from Sigma Aldrich (Buchs, Switzerland), unless stated otherwise.

Algal culture. Axenic cultures of Chlamydomonas reinhardtii (CPCC 11) from the Canadian Phycological Culture Center (CPCC, Department of Biology, University of Waterloo, Canada) were grown in four times diluted Tris-Acetate-Phosphate (TAP×4) liquid growth medium [START_REF] Harris | The Chlamydomonas Sourcebook -a Comprehensive Guide to Biology and Laboratory Use[END_REF] and maintained in an incubator (Infors, Bottmingen, Switzerland) at 20°C with a 24 h illumination regime (114.2 µmol phot m -2 s -1 ) and constant rotary shaking (100 rpm). The culture was regularly re-inoculated in fresh growth medium and cells were harvested in mid-exponential phase. For exposure experiments, cells were gently transferred to the respective exposure media by centrifugation (twice 805 g for 5 min, Sigma 3K10) and adjusted to a final concentration of approximately 10 6 cells mL -1 . All laboratory ware used for culturing was previously soaked in 5% v/v HNO3 for at least 24h, thoroughly rinsed with MilliQ water (MilliQ Direct system, Merck Millipore, Darmstadt, Germany) and sterilized in the autoclave (Steam Sterilizer,Nüve). All crucial manipulations were performed in a sterile, laminar flow hood.

Exposure media. The exposure media included a 10 -2 M solution of the Good's buffer 3-(Nmorpholino)propanesulfonic acid (MOPS, pH = 7 ± 0.2) and Lake Geneva water (pH = 8.1 ± 0.2, physicochemical parameters provided in Table S1). Surface lake water was sampled from Lake Geneva at 46.2824° N, 6.1661° E from ca. 1.5 m depth and filter sterilized (1.2 µm with a PolyPro XL cartridge filter and 0.22 µm with an Isopore Membrane). The MOPS buffer was prepared in MilliQ water, its pH adjusted with 65% HNO3 and sterilized by autoclave and filtration (0.22 µm Isopore Membrane, polycarbonate, Hydrophilic). The sterile exposure media were stored in the dark at 4°C.

TiO2 handling and characterization.

A stock suspension of 2 mg L -1 nano-TiO2 (Degussa P25: 80% anatase, 20% rutile) was prepared in ultrapure water, sonicated in an ultrasonication bath for 10 min (Telsonic 150/300 W) and then stored at 4°C in the dark for the duration of the experiments. Primary particle properties are provided in Fig. S1. For exposure experiments, an intermediate working stock suspension was prepared by sonicating the initial stock suspension for 15 min in an ultrasonic water bath (Branson 2510) and then sampling an aliquot of 1 mL into an Eppendorf tube, which was stored in the dark at 4°C for use within 1 d. Before every experiment, this intermediate stock suspension was retrieved, sonicated in an ultrasonic bath for 5min directly before the preparation of the final, nominal exposure concentrations of 10, 50, 100 and 200 mg L -1 nano-TiO2 within no more than 1 h. For all exposure conditions the number-/volume-/intensity-weighted hydrodynamic particle diameter distributions and zeta potentials were measured by dynamic light scattering (DLS) and electrophoresis with a Zetasizer Nano ZS (Malvern Instruments) at the beginning and end of exposure to cells. Hence, non-UV suspensions were measured at t = 0 and t = 60 min after suspension preparation while UVtreated suspensions at t = 20 min (initial contact with cells) and t = 80 min after suspension preparation. Samples were measured in triplicates, consisting of approximately 10 runs each. The z-average particle diameters were derived by the method of cumulants and the zeta potential was derived from the electrophoretic mobility using the Smoluchowski approximation.

UV treatment. Nano-TiO2 suspensions received a 20 min illumination with long wave UV (300 -420 nm = UVA) in the absorption range of nano-TiO2 (λ < 385 nm) [START_REF] Senna | [END_REF] before contact with algae.

The intensity of the UV lamp (Waldmann Typ 602352 230V 50Hz 2x4W) at the sample was 60 µW cm -2 nm -1 , at the wavelength 350 nm (Fig. S2), which corresponds to an integrated intensity of 2700 µW cm -2 in the wavelength range 300 -420 nm. This dose is in the range of intensities commonly occurring in natural aquatic environments 40 .

Extracellular pro-oxidant processes.

Extracellular abiotic ROS. ROS in abiotic exposure conditions was qualitatively detected with the common 2′,7′-Dichlorofluorescin diacetate (H2DCF-DA, D6883-250 MG, Sigma Aldrich) assay 41,42 . Before staining samples, the non-fluorescent dye was first dissolved in ethanol and then deacetylated by the addition of 0.01 M NaOH (pH = 7.2) in the dark, which yielded the H2DCF molecule sensitive to oxidation. The deacetylation reaction was halted after 30 min by the addition of 0.1 M sodium phosphate (pH = 7.2) on ice in the dark. Finally, samples were aliquoted into 96-well plates and incubated with a final concentration of 26 µM H2DCF for 30 min in the dark, after which fluorescence was measured in a plate reader (Tecan, Infinite M200) at 485/528 ± 20 nm. For positive controls, samples were spiked with 1 mM FeSO4 (Sigma Aldrich) and 13 mM H2O2 (Sigma Aldrich). 3*3 replicates were prepared for every exposure condition and measured at the beginning (t = 0 h) and end of the exposure time (t = 1 h).

Quantification of extracellular abiotic and biotic peroxide. Extracellular H2O2 was measured

with an optical, portable oxidative stress sensor (POSS), a non-invasive method recently described by Koman et al. 43 for the continuous quantification of H2O2 with an unprecedented limit of detection (LOD) in the nanomolar range (40 nM). The principal sensing element of the POSS consists of a ferrous heme group (Fe II ) embedded in the hemeprotein cytochrome c (cyt c), whose transmission spectrum at a wavelength of λ = 550 nm conspicuously evolves from a sharp peak to a broad flat dip upon oxidation to ferric iron (Fe III ) and the simultaneous reduction of H2O2 to water. This transformation can be related to the concentration of oxidizing agents, such as H2O2, present in in the sample. Optical measurements (in transmission mode) were performed in the reaction chamber, the core component of the POSS, consisting of an Oring (8.0 mm * 1.0 mm, NBR Nitril, BRW) imperviously mounted onto a glass slide with grease (Dow Corning® high-vacuum silicone), forming a chamber with a volume of 60 µL to contain the sample and a cyt c spot, which is sealed with a cover slip. For every replicate, a new reaction chamber was prepared, filled with 80 µL of a freshly prepared sample, equipped with a freshly defrosted (fully reduced) cyt c spot, covered with a cover slip and excess liquid removed. Extracellular H2O2 was then continuously measured for 1 h, immediately after the initial contact of algae with TiO2 (max. lag time 5 min.). At the end of every 1 h measurement, reference measurements of the background scattering were performed for every sample. Cyt c sensing spots were previously printed onto filtration membranes, as described in Koman et al. 43 and stored in in a freezer at -20 °C until use. Nano-TiO2 suspensions did not affect the signal of the optical sensor (Fig. S3). Calibration curves for H2O2 were prepared for both exposure media (Fig. S4), yielding the required values of the interaction constant k for the derivation of H2O2 concentrations according to Koman et al. 43 .

Cellular pro-oxidant processes. The cellular responses oxidative stress and membrane integrity

were qualitatively assessed by flow cytometry (FCM, BD Accuri C6, Accuri cytometers Inc., Michigan) equipped with a multisampler (Accuri CSampler), a 488 nm argon excitation laser, three fluorescence detectors (FL 1-3) and respective software (BD Accuri C6 1.0.264.15) for data acquisition and analysis. The fluidics rate for sampling was set to slow (14 μL min -1 , core size 10 μm) and run limits were set to 20,000 events per sample (gated on algae in FL3). Details on the FCM gating strategy applied for data analysis are supplied in Fig. S5 and corresponds to that previously described 44 . For every exposure condition two aliquots of 250 µL were sampled at t = 0 h and t = 1 , which were stained with fluorescent probes (Invitrogen, Life Technologies) and incubated for 30 min prior measurements. Intracellular oxidative stress was assessed with the fluorescent probe CellROX Green reagent (CRG), which was added at a final concentration of 5 µM and analyzed with the green fluorescence detector FL1 (530 ± 15 nm). Positive controls for oxidative stress were obtained by exposing algae to 0.8 mM cumene hydroperoxide for 30 min before staining with CRG. Membrane integrity was evaluated with propidium iodide (PI), added at a final concentration of 7 µM and analyzed in the orange fluorescence detector FL2 (585 ± 20 nm). Positive controls for membrane damage were obtained by exposing cells to 1 M CH2O for 30 min before adding PI.

Statistical analysis. Graphs were prepared with Origin Pro 8 and R version 3.1.3 "Smooth Sidewalk". For statistical analysis, FCM data was log-transformed and two obviously aberrant outliers were removed. To analyze the underlying data generating factors of the discrete data sets obtained for abiotic ROS (measured by H2DCF-DA), oxidative stress (CellRox Green) and membrane integrity (propidium iodide) a linear regression model was fit with R, containing the main predictors "exposure medium", "exposure time", "exposure concentration" and "UV treatment" (medium + time + concentration + UV) and all their interactions (medium:time, medium:concentration etc.). Model selection was performed by the BIC (simple model) and AIC (complex, more predictive model, see ESI 2.1). Since abiotic ROS measurements and cellular endpoints have different units (fluorescence in a.u. and % affected cells, respectively), statistical analyses were performed separately. All R output tables and residual analyses are provided in the Electronic Supplementary Information (ESI, Tables S2 -S11, Figs. S10 andS11).

Hydrodynamic diameter (dh).

Nano-TiO2 suspensions are heavily agglomerated in both exposure media, with number-weighted dh roughly ranging between 200 nm and 2500 nm (Fig. 1, A andC). Volume-and intensity-weighted dh are provided in the ESI (Fig. S6) and showed similar trends (Fig. S7, A andB). The hydrodynamic diameters detected are at the extreme limit of the DLS method and the measured sizes will be unreliable but the trends seem to be consistent. General trends of untreated particles showed that dh increased with nano-TiO2 concentration (dh (10 mg L -1 ) < dh (200 mg L -1 )) and time (dh (tinitial) < dh (tfinal)) in both media (Fig. 1, Fig. S7, C -F). Dh at tinitial were largely slightly higher in MOPS than in lake water.

These findings are in agreement with earlier observations of nano-TiO2 forming agglomerates of several hundred nanometers to several micrometers in diameter within minutes at environmentally relevant pH, ionic strengths and DOM 45,46 . A comprehensive study investigating the behavior of nano-TiO2 in natural matrices at the same concentrations employed here found very low sedimentation rates in freshwater suggesting ecotoxicologically relevant residence times of agglomerated nano-TiO2 for aquatic organisms in the water column 47 .

The trends for UV pre-illuminated suspensions in lake water showed dh (tinitial UV) > dh (tfinal UV) in 50 mg L -1 , 100 mg L -1 and 200 mg L -1 suspensions. This relation was inverse in 10 mg L -1 suspensions. In the MOPS buffer dh (tinitial UV) < dh (tfinal UV), except in the 200 mg L -1 suspension (Fig. 1, A andC).

All in all, particles in the present exposure system were no longer in the nanometer range but much ranger formed colloid-sized agglomerates in the micrometer size range. For comparison, C. reinhardtii cells have diameters around 5 -10 µm, and are thus roughly equal to one order of magnitude larger than agglomerates observed here.

Zeta potential. The zeta potential values of untreated particles in lake water lay between -20 mV and -15 mV and decreased slightly with increasing particle concentrations (Fig. S7, E andF), reaching values around -10 mV in the 200 mg L -1 suspension. This could contribute to the comparably smaller dh (i.e. greater suspension stabilities) observed in the lower concentrations.

The values were largely comparable at tinitial and tfinal. UV treated particles in lake water were generally less negative with values ranging from -3 mV to -15 mV and values at tfinal UV were either equal or more negative than at tinitial UV, which matches the elevated dh observed at tinitial UV and also the decrease in dh observed at tfinal UV (Fig. 1,A). Likewise, the zeta potential values in the MOPS buffer roughly varied between -20 mV and -10 mV but were overall nearer -10 mV and thus slightly less negative than particles in lake water, likely due to the absence of DOM. Previous findings have also shown that charge repulsion between particles and HS occurred in conditions where pH > isoelectric point 48 , which may explain the overall similarity of dh in the two media observed here. In the MOPS buffer zeta potentials of untreated particles remained more or less the same at the beginning and end of exposure. The zeta potential of UV-treated particles at tinitial UV were generally either equal or more negative than at tfinal UV (Fig. 1D) but, rather puzzlingly, dh( tfinal UV) were greater than dh(tinitial UV). Abiotic ROS. In both media elevated, above control median levels of ROS, as measured by H2DCF-DA, were only observed at 200 mg L -1 nano-TiO2 (Fig. 2A and2B). UV treatment of nano-TiO2 suspensions did not significantly affect results in lake water, but in the MOPS buffer produced higher median levels of abiotic ROS, most pronounced at 100 and 200 mg L -1 nano-TiO2 (t = 0 h and t = 1 h, Fig. 2B).

Extracellular pro-oxidant processes

Abiotic and extracellular biotic peroxide in lake water. In contrast to the measured abiotic ROS, the peroxide measurements obtained by POSS indicate a nano-TiO2 concentration dependent level of abiotic peroxide in lake water (Fig. 3A) and overall increased abiotic peroxide levels in UV treated nano-TiO2 suspensions (Fig. 3B). Much like abiotic ROS measured by H2DCF-DA, the highest level of peroxide was observed at 200 mg L -1 nano-TiO2 (Fig. 3A). UV treatments produced cH2O2 peaks immediately at t = 0, which exponentially decreased within the first 10 to 20 min (Fig. 3B) but did not show any concentration dependent trends. The respective decay rate constants are provided in Table 1 and plotted in Figure S9.

In biotic settings, the algae only treatment as well as algae exposed to 10 mg L -1 nano-TiO2 yielded the highest cH2O2 values around 50 -60 nM, all other conditions produced lower concentrations of peroxide (Fig. 3C). The combination of UV pre-treated nano-TiO2 suspensions with algae (Fig. 3D) did not reproduce the characteristic peroxide peak previously observed in the abiotic UV treatment (Fig. 3B) and consistently yielded lower peroxide levels than in biotic conditions (Fig. 3C).

H2O2 has a lifetime of up to 10 h (or longer) in the environment (pH = 7) 49,50 that may explain the steady concentrations observed during 1 h, which are well in the range of those typically occurring in natural surface waters ranging from 10 pM to 100 nM 49 .

These observations lead us to the following interim conclusions: i) In lake water the POSS method is more sensitive to peroxide than the H2DCF-DA stain to peroxide and/or ROS, ii) in lake water the presence of algae can either increase (10 and 50 mg L -1 treatments) or decrease extracellular peroxide levels (100 and 200 mg L -1 treatments) observed in abiotic conditions, iii) C. reinhardtii do not release excess peroxide above the baseline of negative controls when exposed to nano-TiO2 (Fig. 3C) and finally iv) algae exposed to UV activated nano-TiO2 (Fig. 3D) quench the excess peroxide peak observed in abiotic UV treatments (Fig. 3B).

Abiotic and extracellular biotic peroxide in MOPS buffer. Unlike in the lake water system, cH2O2 in the MOPS buffer peaked at t = 0 h in all concentrations, in both treatments, and exponentially decreased within the first 10 -20 minutes to relatively stable lower values hovering around or below the LOD (Fig. 4A-D). The corresponding exponential decay rates are provided in Table 1 and plotted in Figure S9.

Overall, peroxide levels increased in the presence of C. reinhardtii and even surpassed the baseline level of the negative control in exposures to 50, 100 and 200 mg L -1 nano-TiO2 (Fig. 4C).

This was most evident in the 200 mg L -1 nano-TiO2 exposure where an additional surge in cH2O2 was observed beginning at t = 10 min and peaking at t = 60 min, which did not occur in any other treatment (Fig. 4C).

In MOPS the UV pre-irradiation of nano-TiO2 actually increased initial biotic cH2O2 peak values but on the long run lead to lower biotic peroxide levels compared to non-UV treatments, except in the 10 mg L -1 treatment showing higher cH2O2 values than the negative control (Fig. 4D).

The initial peak in cH2O2 was observed in all conditions without exception and presumably resulted from the generation of hVB + /eCB -on particle surfaces through sonication prior exposure and photocatalytic reactions with H2O and O2 51 . The generation of ROS in aqueous solutions through sonication and acoustic cavitation effects is also a well-known phenomenon [START_REF]Free radicals in biology and medecine[END_REF] . There are several possible pathways for the subsequent extinction of H2O2. For example, measured cH2O2 levels and the respective decay rate constants obtained in abiotic conditions were inversely related to particle concentrations in both media (Table 1, Fig. S9), indicating that total particle surface area is an important factor governing ROS generation and quenching.

Therefore, nano-TiO2 probably behaved as both a source and sink of cH2O2 by sonication/UVgenerated reactive hVB + /eCB -at the particle surface reacting both with H2O/O2 and ensuing ROS.

Since the dynamics of cH2O2 decay were identical in distilled water it can be inferred that the reaction of H2O2 with nano-TiO2 was presumably the dominant decay mechanism.

These observations lead us to the following tentative conclusions i) in the MOPS buffer sonication of nano-TiO2 may explain the initial peroxide peak observed in nearly all MOPS treatments, which unlike lake water does not possess ROS quenching species such as DOM, ii) C. reinhardtii slightly elevate peroxide levels when exposed to nano-TiO2 in MOPS, suggesting a biological contribution either from redox reactions at the cell surface or from intracellular ROS/peroxide diffusing out.

Cellular pro-oxidant processes: oxidative stress and damage

Oxidative stress in lake water. In lake water exposures the proportion of cells affected by oxidative stress did not exceed 5 % (Fig. 5A, Table S2) but intracellular ROS levels showed a concentration dependent increase after 1 h of exposure in both treatments, with highest responses obtained for algae exposed to 100 and 200 mg L -1 nano-TiO2. UV treatments only minutely enhanced median intracellular ROS levels in lake water, leading to slightly higher values at t = 1 h in [50 mg L -1 ]UV and [100 mg L -1 ]UV treatments (Fig. 5A, Table S2). Controls and 10 mg L -1 nano-TiO2 exposures produced comparable responses in both treatments (Fig. 5A).

Oxidative stress in MOPS.

In the MOPS buffer, although high extracellular cH2O2 were measured in biotic conditions, no effects on intracellular ROS levels were observed in both treatments (Fig. 5A, Table S3). A marked increase in the proportion of cells with elevated intracellular ROS was observed in all conditions after 1 h, including the controls, suggesting that the MOPS medium may have acted as a stressor itself (Fig. 5A). The pre-irradiation of nano-TiO2 with UV slightly reduced median intracellular ROS responses at all concentrations in the MOPS buffer.

Membrane integrity in lake water. Membrane damage predominantly occurred in lake water (Fig. 5B, Table S8 andS9) and the proportion of affected cells was more than one order of magnitude higher than in MOPS. In lake water, membrane impairment was considerably elevated in cells exposed to 100 and 200 mgL -1 untreated nano-TiO2 for 1 h reaching 12 % and 19 % affected cells, respectively, compared to ca. 8 % in controls. There was no difference in membrane damage between controls and cells exposed to 10 mg L -1 and 50 mg L -1 nano-TiO2 (Fig. 5B, Table S8). UV pre-treated nano-TiO2 did not greatly affect responses, but rather even decreased the proportion of affected cells (Fig. 5B).

Membrane integrity in MOPS.

In MOPS the effects of nano-TiO2 on the membrane integrity of C. reinhardtii were altogether negligible (< 1 %, Fig. 5B, Table S7). No differences in membrane damage were observed between control and exposed cells but opposed to results obtained for intracellular ROS, the membrane integrity of controls was not affected by the MOPS medium itself.

Overall, cellular responses were higher in lake water exposures, which is in agreement with earlier results showing a heightened toxicity of nano-TiO2 on developing zebrafish in the presence of humic acid 52 but contradicts others showing a mitigation of nano-TiO2-induced pro-oxidant effects on the alga Chlorella sp. through increased electrosteric repulsion 53 .

Main predictors of pro-oxidant processes.

For all three endpoints, all main effects including several interaction terms were retained in the more complex AIC selected models. In the simpler BIC models (Tables S2, S6 and S10) the main effect "exposure concentration" was not retained in the models fitted to the cellular endpoints (Table 2).

For the ROS related endpoints (H2DCF-DA and CellRox Green) the AIC models suggest a significant effect of exposure concentration, as well as interaction between exposure concentration and exposure medium. More generalized, this finding implies that pro-oxidant processes of varying nano-TiO2 concentrations will differ as a function of the ambient medium.

On the other hand, the effect of varying nano-TiO2 concentrations on membrane damage is simply an additive factor, independent of the respective levels of all the other factors while the impact of the medium will be influenced by the exposure time and UV treatment and vice versa. More generalized, this implies that the medium itself plays a significant role in membrane integrity.

Furthermore, we can infer from the fitted models that the effect of UV treatment on all endpoints considered will significantly depend on the exposure medium.

These relationships can also be visually related to the data presented above (Figs. 2, 5 and6).

Linking extracellular processes to cellular pro-oxidant processes.

Overall, the cH2O2 results obtained by POSS provide evidence that (i) irrespective of the medium, agglomerated nano-TiO2 in the micrometer size range produced abiotic H2O2 in biologically relevant media, which is enhanced by UV irradiation, (ii) the generation of H2O2 and/or the measured cH2O2 is a dynamic process and is modified by the ambient medium as well as the nano-TiO2 concentration and lastly (iii) the presence of cells either decreased or increased total cH2O2 measured in equivalent cell-free conditions. In view of our results showing both an additive and quenching effect of cH2O2 in the presence of algae, we will now examine cellular responses in an attempt to link extracellular processes with pro-oxidant processes at the cellular level.

The exposure of algae to nano-TiO2 in lake water did not produce cH2O2 above the level of unexposed controls. Nonetheless, the presence of cells reduced cH2O2 levels obtained for 100 and 200 mg L -1 exposures but increased those of 10 and 50 mg L -1 exposures with respect to the corresponding abiotic values (Fig. 3). On the other hand, exposure of algae to 50, 100 and 200 mg L -1 nano-TiO2 in MOPS produced higher cH2O2 levels than in the respective controls. After t = 1 h the 10 mg L -1 exposure also surpassed the negative control. Similarly, exposure to UV treated nano-TiO2 surpassed the baseline cH2O2 levels of controls in the initial 10 -20 min of exposure with the 10 mg L -1 exposure remaining high until the end of exposure at t = 1 h (Fig. 4). Based on these observations and assuming the initially stated working hypothesis whereby peroxide may serve as a marker for oxidative stress in cells as true, we would thus anticipate i)

no intracellular oxidative stress and damage in lake water treatments (Fig. 5) and ii) elevated intracellular ROS levels and membrane damage in cells exposed in MOPS (Fig. 6).

In fact, quite the opposite was observed. Oxidative stress (albeit low values) and membrane damage (up to 15 % cells affected) were primarily observed in lake water exposures while there was no evidence of elevated cellular pro-oxidant stress in either controls or treatments conducted in MOPS. From the point of view of the original working hypothesis from which we departed these observations are not immediately obvious but there are several plausible explanatory hypotheses we can infer. However, the present experimental setup does not provide sufficient resolution as to support one hypothesis over the other.

Lake water. Oxidative stress and membrane damage in lake water treatments were highest in exposures to 100 and 200 mg L -1 nano-TiO2. At the same time biotic cH2O2 in 100 and 200 mg L -1 exposures decreased compared to identical abiotic conditions while biotic cH2O2 in exposures to 10 and 50 mg L -1 increased. UV treatment produced abiotic cH2O2 peaks at the beginning of exposure time which disappeared in the identical biotic settings. Oxidative stress only marginally increased in UV treatments and biotic cH2O2 remained markedly below the level in unexposed controls.

By inference we may conclude that oxidative stress and membrane damage was likely mediated by free extracellular H2O2 reacting at the cell surface. H2O2 can directly oxidize DNA, lipids, -SH groups and can inactivate enzymes 54 . Although H2O2 is a rather weak oxidizing agent for biomolecules and particularly lipids 20 , it readily reacts with e -CB, O2 -or, in the presence of transition metal ions, engages in Fenton reactions leading to the formation of the highly reactive, short-lived, indiscriminate oxidant OH  49,51 . DOM is another source of hydroxyl radical formation 49,54 . The likely presence of trace amounts of metals in lake water samples in combination with DOM/nano-TiO2-generated H2O2 may have facilitated the generation of the more reactive OH  that more rapidly and readily oxidizes biomolecules such as lipids and thereby escaped detection by cyt c, leading to relatively low measured cH2O2 and higher levels of oxidative stress and membrane damage in exposed cells. In extension, UV treatment induced slightly elevated oxidative stress levels (except at 200 mg L -1 ) as a consequence of extra OH  emerging from UV generated cH2O2. The ecotoxicological implications of Haber-Weiss reactions have previously been evoked 55 and demonstrated that environmentally relevant concentrations of redox and nonredox active metals enhanced intracellular ROS in C. reinhardtii, without affecting algal photosynthesis.

Alternatively, DOM is a known ROS scavenger. It is quite feasible that DOM competed with the cyt c for H2O2 possibly emanating from stressed cells and thereby buffered extracellular cH2O2.

Finally, the observed responses may have also ensued from direct physical interactions between nano-TiO2 and cells. It is well-known that increased ROS and oxidative damage may not only result from a contaminant's direct pro-oxidant effects. Rather, interactions with a contaminant can lead to some physical injury, which in turn can lead to excess ROS or ROSgenerating species 16 .

MOPS.

Much in contrast, in the MOPS buffer significant concentrations of cH2O2 were measured in the presence of nano-TiO2 both in abiotic and biotic exposure conditions, with no significant cellular effects whatsoever. Hence, results suggest that neither direct particle -cell interactions nor freely diffusing extracellular H2O2 adversely affected C. reinhardtii. In the biotic setting, initial and final cH2O2 were higher compared to equivalent abiotic conditions for all concentrations but the 10 mg L -1 nano-TiO2 treatment. This implies that algae contributed to the net measured cH2O2, either through the leaching of intracellular ROS or through reactions of H2O2 with the cell surface. Both these hypotheses seem feasible in view of the above observations:

If we consider that H2O2 is a relatively weak oxidizer 20,49 (particularly in absence of transition metal ions that would enable the formation of the more reactive hydroxyl radical) and both intracellular ROS and membrane integrity were not significantly elevated in exposed cells at the beginning of exposure, the excretion of H2O2 by cells is a plausible explanation for the net increase in cH2O2 at the beginning of exposure. It is known from plants for example that extracellular H2O2 concentrations can increase in response to abiotic stressors and environmental pollutants such as metals, pesticides and salt during what is known as the oxidative burst.

On the other hand, previous findings showed an accumulation of nano-TiO2 on the cell surface of microalgae exposed to similar nano-TiO2 concentrations 37,[56][57][58] and postulated that oxidation occurred through surface-bound ROS which are not free to diffuse into the cell 59 , which is the second possible explanation for the observed increase in cH2O2 in biotic conditions. It is widely acknowledged that proximity or direct contact is a prerequisite for ENP toxicity, without which direct oxidation of cellular components or physical disruption of cell walls and membranes would not occur 30,59,60 . However, assuming this scenario, one would expect oxidative stress or oxidative damage in exposed cells, which was not the case. The absence of transition metals in the MOPS buffer may explain why membrane integrity did not degenerate as fast, despite the elevated levels of cH2O2 both in UV pre-treated and untreated nano-TiO2 suspensions.

Therefore, cH2O2 excretion by cells seems more plausible.

UV treatment. The pre-illumination of nano-TiO2 suspensions considerably affected abiotic cH2O2 in both media and rather unexpectedly only marginally enhanced oxidative stress in lake water exposures, which is in agreement with earlier findings showing no UV enhanced toxicity of nano-TiO2 on microalgae 34 . The pre-illumination of lake water itself generated the highest concentration of ROS, which can result from the absorption of UV by DOM and its subsequent photolysis 50,61 . An initial peak in cH2O2 was observed in all conditions in MOPS but only in the abiotic UV treatments in lake. DOM is also known to scavenge cH2O2, which may explain the fairly steady cH2O2 over time compared to concentrations measured in MOPS. Thus, the additional energy input by UV may have provided an additional source of H2O2 generation through DOM photolysis, which could no longer be scavenged by DOM and thus produced the initial cH2O2 peaks in the UV treatments. The elevated cH2O2 measured in cell-free UV pre-treated nano-TiO2 suspensions in lake water disappeared in the presence of cells, which supports the above hypothesis by which free extracellular H2O2 react with exposed algae (and/or possibly transform into highly reactive hydroxyl radicals via Fenton reactions). Plant cells are actually known to consume extracellular H2O2 concentrations as high as 10 mM in less than 10 min, as demonstrated by its rapid depletion by Arabidopsis thaliana within 8 -10 min 62,63 . The proportion of cells with oxidative stress was higher than in non-UV treatments and increased in a concentration dependent manner, except in the [200 mg L -1 ]UV exposure. Bearing the biotic, non-UV responses in mind, this suggests an additive pro-oxidant effect of H2O2 and thus supports the hypothesis of extracellular H2O2 reacting with cells in lake water as primary cause over direct cell-particle interactions. This contradicts previous findings reporting that artificial UV irradiance of two temperate German lakes with moderate DOM levels (dissolved organic carbon (DOC) = 15 mg L -1 ) generated H2O2 concentrations in the range of 150 -360 nM, which did not adversely affect the three microalgae Chlorella sp., Desmodesmus subspicatus and Scenedesmus obliquus (measured by delayed fluorescence kinetics) 50 . Despite lower DOM levels in Lake Geneva water samples (DOC = ca. 1 mg L -1 , Table S1) the cH2O2 values obtained for

[Lake]UV were one order of magnitude above those previously reported 50 , which may account for the additive effect of cH2O2 on oxidative stress observed in C. reinhardtii in our experimental setup.

In MOPS, abiotic cH2O2 were higher in UV treatments but all other trends by and large remained the same as in the non-UV treatment.

CONCLUSION

This is the first in-depth nano-ecotoxicological study to continuously quantify abiotic and biotic nano-TiO2 -stimulated extracellular H2O2 during 1 h exposure of C. reinhardtii. It is also the first attempt to link extracellular H2O2 to standard nano-ecotoxicological endpoints of cellular prooxidant processes.

It was found that agglomerated nano-TiO2 generated cellular pro-oxidant responses, which are significantly modified by the parameters exposure medium, exposure time, UV pre-illumination as well as exposure concentrations. Furthermore, extra-and intracellular pro-oxidant processes differed significantly: intracellular oxidative stress increased in conditions where no significant increase in extracellular H2O2 was measured and elevated extracellular levels of abiotic H2O2 did not point to intracellular oxidative stress.

Thus, nano-TiO2 toxicity is not mediated by pro-oxidant processes alone and extracellular H2O2 cannot serve as a predictor of cellular oxidative stress and damage. These findings are important with respect to future ecotoxicological endeavors seeking to link physicochemical particle properties with exposure scenarios and biological responses for ENM hazard assessment and prediction.

Figure 1 .

 1 Figure 1. Mean and standard deviation of number-weighted hydrodynamic diameters (dh -Nb.) in [nm] (A, C) and zeta potentials in [mV] (B, D) of different nano-TiO2 concentrations in lake water (A, B) and MOPS (C, D) at the beginning (tinitial) and end (tfinal) of the 1 h exposure. Diameters for untreated samples correspond to times tinitial = 0 min and tfinal = 60 min (after the preparation of the suspension). Diameters of UV pre-treated samples correspond to times tinitial UV = 20 min and tfinal UV = 80 min after suspension preparation.

Figure 2 .

 2 Figure 2. Abiotic ROS measured as DCF fluorescence after t = 0 h and t = 1 h in lake (A) and MOPS (B) at different nano-TiO2 concentrations with and without 20 min UVA pre-illumination of nano-TiO2 suspensions. Box-whisker plots show median of at least triplicate measurements and the respective quantiles with whiskers indicating minima and maxima within 1.5 of the interquantile range.

Figure 3 .

 3 Figure 3. Average extracellular H2O2 [nM] (cH2O2) (of at least triplicate measurements) produced during 60 min by four nano-TiO2 concentrations with (B, D) and without UV pretreatment (A, C) in abiotic (A, B) and biotic (C, D) conditions in lake water: nano-TiO2 only (A), nano-TiO2 after 20 min UV pre-treatment (B), algae exposed to nano-TiO2 (C) and algae exposed to UV pre-treated nano-TiO2 (D). The horizontal red line represents the LOD and the inset in (B) shows a close-up of the 0 -100 nM concentration range.

Figure 4 .

 4 Figure 4. Average extracellular H2O2 [nM] (cH2O2) (of at least triplicate measurements) produced during 60 min by four nano-TiO2 concentrations with (B, D) and without UV pre-treatment (A, C) in abiotic (A, B) and biotic (C, D) conditions in the MOPS buffer: nano-TiO2 only (A), nano-TiO2 after 20 min UV pre-treatment (B), algae exposed to nano-TiO2 (C) and algae exposed to UV pre-treated nano-TiO2 (D). The horizontal red line represents the LOD and insets depict enlargements of the respective 0 -1000 nM concentration range.

Figure 5 .

 5 Figure 5. Box-whisker plots showing the median, quantiles and S.D. (of at least triplicates) of intracellular ROS (A) and membrane damage (B) in [% affected cells] of the total number of cells exposed to 0, 10, 50, 100 and 200 mg L -1 nano-TiO2 concentrations without (left plot) and with UV pre-treatment (right plot) in lake water at the beginning (t = 0 h) and end of exposure (t = 1 h).

Figure 6 .

 6 Figure 6. Box-whisker plots showing the median, quantiles and S.D. (of at least triplicates) of intracellular ROS (A) and membrane damage (B) in [% affected cells] of total number of exposed cells to different TiO2 concentrations without (left plot) and with UV pre-treatment (right plot) in the MOPS buffer at the beginning (t = 0 h) and end of exposure (t = 1 h).

Table 1 .

 1 Decay rate constants (b [min-1 ] extracted from fitting the function Ae -bt to cH2O2 (t)) in the initial exponential phase to extracellular H2O2 measured by POSS, as plotted in Fig.3and Fig.4. Also plotted in FigureS9. NA = not available.

		without TiO2	10 mgL -1 TiO2	50 mgL -1 TiO2	100 mgL -1 TiO2	200 mgL -1 TiO2
	abiotic ROS	NA	0.203	NA	0.206	NA	0.173	NA	NA	NA	0.167
	biotic ROS	NA	NA	NA	NA	NA	NA	NA	NA	NA	NA
	MOPS buffer
	abiotic ROS	NA	0.06	0.259	NA	0.143	0.420	0.094	0.222	0.076	0.275
	biotic ROS	0.042	NA	NA	0.261	0.108	0.175	0.139	0.189	0.234	0.275

Table 2 .

 2 Summary of linear models fitted to data for abiotic ROS and cellular endpoints of stress (Tables S2-S11, Figs.S10 and S11).

	Endpoint: abiotic ROS
	BIC model: medium + time + UV + conc + medium:UV +	medium:conc + UV:conc + medium:UV:conc
	AIC model: medium + time + UV + conc + medium:UV + time:UV + medium:conc + UV:conc + medium:UV:conc
	Endpoint: oxidative stress
	BIC model: medium + time + UV +	medium:UV
	AIC model: medium + time + UV + conc + medium:UV +	medium:conc
	Endpoint: membrane damage
	BIC model: medium + time + UV
	AIC model: medium + time + UV + conc + medium:UV + time:UV + medium:time + medium:time:UV

ACKNOWLEDGEMENTS

This work was part of the National Research Program 64 on the Opportunities and Risk of Nanomaterials with the project number 406440-131280, funded by the Swiss National Science Foundation. Many thanks are extended to Maurus Thurneysen for support in statistical analyses with R.

DECLARATION OF INTEREST

All authors declare that there are no conflicts of interest.


Jay C Bullen
email: j.bullen16@imperial.ac.uk

Sarawud Saleesongsom

Kerry Gallagher

Dominik J Weiss
email: d.weiss@imperial.ac.uk

A revised pseudo-second order kinetic model for adsorption, sensitive to changes in adsorbate and adsorbent concentrations

Keywords: Adsorption kinetics, kinetic model, pseudo-second order, Lagergren, water treatment, particle size

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Abstract

The development of new adsorbent materials for the removal of toxic contaminants from drinking water is crucial to achieving the United Nations Sustainable Development Goal 6 (clean water and sanitation). The characterisation of these materials includes fitting models of adsorption kinetics to experimental data, most commonly the pseudo-second order (PSO) model. The PSO model, however, provides no sensitivity to changes in experimental conditions such as adsorbate and adsorbent concentrations (C 0 and C s ) and consequently is not able to predict changes in performance as a function of operating conditions. Furthermore, the experimental conditionality of the PSO rate constant, k 2 , can lead to erroneous conclusions when comparing literature results. In this study, we analyse 108 kinetic experiments from 47 literature sources to develop a relatively simple modification of the PSO rate equation, yielding:

Unlike the original PSO model, this revised rate equation (rPSO) demonstrates the first-order and zero-order dependencies upon C 0 and C s that we observe empirically. Our new model reduces the residual sum of squares by 66% when using a single rate constant to model multiple adsorption experiments with varying initial conditions. Furthermore, we highlight how the rPSO rate constant k' is more appropriate for literature comparison, highlighting faster kinetics in the adsorption of arsenic onto alumina versus iron oxides. This revised rate equation should find applications in engineering studies, especially since unlike the PSO rate constant k 2 , the rPSO rate constant k' does not show a counter-intuitive inverse relationship with the increasing reaction rate when C 0 is increased.

Introduction

There is a wealth of recent literature concerning the development of novel adsorbent materials for the remediation of contaminated water, such as composite materials offering superior stability [START_REF] Boddu | Removal of Copper(II) and Nickel(II) Ions from Aqueous Solutions by a Composite Chitosan Biosorbent[END_REF] , ease of separation from the effluent 2 3 , or multifunctional capabilities such as photocatalytic activity 4 5 . Energy input typically forms about one third of water treatment plant operation costs [START_REF] Smith | Energy for Conventional Water Supply and Wastewater Treatment in Urban China: A Review[END_REF] , and if energy efficiencies are to be improved then accurate models of adsorption kinetics (including rate constants) are needed to (a) identify the minimum duration needed for batch treatments and (b) estimate maximum flow rates for column or continuous-flow treatments [START_REF] Dichiara | On the Choice of Batch or Fixed Bed Adsorption Processes for Wastewater Treatment[END_REF] . Laboratory experiments can only partially capture the environments in which new adsorbents will operate, and in practice different concentrations of adsorbent (C s ) will be needed to treat different concentrations of contaminant in the influent (C 0 ). It is thus important that adsorption models are made sensitive to operating conditions, providing predictive capabilities.

The pseudo-second order (PSO) rate equation [START_REF] Blanchard | Removal of Heavy Metals from Waters by Means of Natural Zeolites[END_REF] , popularised by Ho and McKay (1999) [START_REF] Ho | Pseudo-Second Order Model for Sorption Processes[END_REF] , is probably the most popular model currently used to describe adsorption kinetics [START_REF] Simonin | On the Comparison of Pseudo-First Order and Pseudo-Second Order Rate Laws in the Modeling of Adsorption Kinetics[END_REF] . The PSO rate equation takes the form: Equation 1where t is time (minutes), q t is the amount of adsorbate adsorbed per mass of adsorbent at time t (mg g -1 ), k 2 is the pseudo-second order rate constant (g mg -1 min -1 ), and q e is the amount of adsorbate adsorbed at equilibrium (mg g -1 ) [START_REF] Ho | Pseudo-Second Order Model for Sorption Processes[END_REF] .

The decrease in the concentration of aqueous adsorbate with time is given by the equation: Equation 2where C t is the concentration of aqueous adsorbate at time t (mg L -1 ), C 0 is the initial adsorbate concentration at t=0 (mg L -1 ) and C s is the concentration of adsorbent (g L -1 ).

The PSO model is popular for several reasons. Firstly, it has a simple mathematical form. Secondly, despite being applied by Ho and McKay as a mechanistic model for the bidentate adsorption of copper onto peat [START_REF] Ho | The Kinetics of Sorption of Divalent Metal Ions onto Sphagnum Moss Peat[END_REF] , the PSO model is able to fit kinetic data for a wide range of systems with different reaction mechanisms 9 12 13 (including where diffusion control is to be expected [START_REF] Xiao | Erroneous Application of Pseudo-Second-Order Adsorption Kinetics Model: Ignored Assumptions and Spurious Correlations[END_REF] ). Thirdly, Equation 1 can be integrated and rearranged to provide linear equations (of the form y=mx+c) from which the model parameters k 2 and q e can be easily obtained by linear regression [START_REF] Canzano | Use and Misuse of Sorption Kinetic Data: A Common Mistake That Should Be Avoided[END_REF] .

However, the PSO model has several important limitations due to the absence of adsorbate and adsorbent parameters within the rate equation, with k 2 and q e parameters being valid only under the specific experimental conditions under which the PSO model was fitted. The first limitation is that the model cannot predict how adsorption kinetics will change as a function of C 0 and C s , limiting its application in engineering or optimisation studies. Furthermore, rate constants from different literature sources with different experimental conditions cannot be meaningfully compared: greater values of k 2 do not necessarily indicate adsorbents possessing superior adsorption kinetics.

The aim of the present study was to modify the popular PSO equation, introducing sensitivity towards changes in C 0 and C s , with the objective of both improving predictive capabilities for engineering studies, and normalising rate constants for easier comparison between literature sources. The PSO model does not necessarily reveal insights into the adsorption mechanisms (i.e. whether intraparticle diffusion or chemisorption is the rate determining step) 13 16 , and our aim was similarly to develop an empirical model, rather than a mechanistic model. We thus conducted an empirical analysis of the adsorption kinetics reported by the literature to assess the influence of C 0 and C s on adsorption rates, and to modify the PSO rate equation accordingly. We used the method of initial rates to determine the order of reaction with respect to both C 0 and C s (given the possibility for data at later times to disguise the true reaction order [START_REF] Ollis | Kinetics of Photocatalyzed Reactions: Five Lessons Learned[END_REF] , such as when slower surface precipitation processes coincide with monolayer adsorption [START_REF] Lützenkirchen | On the Surface Precipitation Model for Cation Sorption at the (Hydr)Oxide Water Interface[END_REF] ). We first performed quality control experiments, investigating various methods for calculating initial rates when the availability of early kinetic data is limited (as per many adsorption experiments). We then compiled a wide range of literature data sets wherein multiple adsorption kinetic experiments with different values of C 0 and C s are reported (with each data set being a specific adsorbate-adsorbent system) and determined the order of reaction with respect to each variable. We used mineral and organic adsorbents, and metal, inorganic and organic adsorbates, to achieve a model that is generally applicable to a wide range of systems, as per the original PSO model. We built the observed C 0 and C s dependence into a revised form of the PSO rate equation (which we refer to as rPSO) and verified that the rate constants given by this new model are more stable with respect to changes in experimental conditions than the PSO rate constant k 2 . Finally, we used two application studies to assess the potential of this revised PSO model to overcome current limitations: (1) describing multiple experiments with varying values of C 0 and C s using a single rate constant, and (2) achieving a more meaningful comparison of the adsorption kinetics reported across the literature.

Experimental Data sets

Literature sources that experimentally investigated the influence of C 0 , C s or particle size upon adsorption kinetics were compiled and the experimental data tabulated (referenced in the Supplementary Information: SI Table S1). Both mineral adsorbents and organic adsorbents (activated carbon and chitosan) were included, however zeolites and metal-organic frameworks (MOFs) were not since the sorption mechanism of adsorbate trapping within cages might produce contrary results. None of the literature sources found gave any mechanistic account or mathematical explanation for observed differences in adsorption kinetics due to varying values of C 0 , C s or particle size. The majority of the compiled literature used the PSO model to describe adsorption kinetics.

In total 47 (25 experiments) for the adsorption of inorganic arsenic onto iron oxide and alumina adsorbents. A data set is considered to be all kinetic experiments using the same adsorbate-adsorbent system within a single literature source. The compiled data sets are available elsewhere [START_REF] Bullen | Adsorption Kinetics Data Sets, Compiled from the Literature. As Used in the Research Article "A Revised Pseudo-Second Order Kinetic Model for Adsorption, Sensitive to Changes in Adsorbate and Adsorbent Concentrations[END_REF] .

Mathematical approaches for the determination of initial rates

Three approaches towards the calculation of initial rates were compared: (1) the initial slope, (2) linearised PSO kinetics, and (3) non-linear PSO kinetics.

In the initial slope approach, the initial rate ( at t=0) was calculated as the slope between the origin at (0,0) and the earliest available data point at t>0 [START_REF] Atkins | Physical Chemistry[END_REF] .

Initial rates were also calculated using the following linearised form of the integrated PSO rate equation:

Equation 3
Kinetic profiles were plotted as as a function of t and the linear regression was obtained using the LINEST function in Excel (where the residual sum of squares between the data points and the linear regression is minimised). The equilibrium adsorption parameter q e was obtained via the relationship where m is the slope of the linear regression, and k 2 via where c is the y-intercept.

The initial rate of adsorption was then calculated through the simplification of Equation 1:

Initial rate = Equation 4
Uncertainties in k 2 , q e and the initial rate were calculated by linear propagation of the standard errors in m and c given by the LINEST function.

Finally, initial rates were calculated using non-linear PSO kinetics. Non-linear fitting of the PSO model to experimental data was achieved by using Microsoft Excel's Solver function to optimise k 2 and q e values, minimising the sum of squared residuals between the model and experiment. Uncertainties in k 2 and q e were calculated using a Monte-Carlo approach with 200 simulations as described by Hu et al. [START_REF] Hu | Evaluation of Parameter Uncertainties in Nonlinear Regression Using Microsoft Excel Spreadsheet[END_REF] .

For all literature sources, parameters were recalculated to the same units for ease of comparison: k 2 (g mg -1 min -1 ); q e (mg g -1 ); and the initial rate (mg g -1 min -1 ).

The influence of the availability of early kinetic data on the accuracy and precision of initial rates calculated using these three approaches was evaluated as follows. Fourteen data sets containing early kinetic data were collected (defined as adsorption experiments containing data within the range ). Initial rates were then re-calculated as data points were consecutively removed from the earliest to the latest. A linear regression between the calculated initial rates and the value of at the first available kinetic data was determined and extrapolated to to provide a theoretical 'true' initial rate (representing if kinetic data were to be collected within an infinitesimally small time period). This theoretical 'true' initial rate was used as a reference value to determine variation in the calculated initial rate as a function of the availability of early kinetic data.

A systematic error was calculated using the average error across all data sets and a random error was calculated as the standard deviation in the error across all data sets. The significance of the differences observed in the initial rate errors given by the three mathematical approaches was determined using the paired samples t-test.

After evaluating the three approaches (see Results and Discussion), non-linear PSO kinetics were used to determine initial rates in all subsequent calculations.

Determining the order of reaction

The order of reaction with respect to the independent variables C 0 and C s was calculated as the slope of log(initial rate) versus log(independent variable) [START_REF] Kirby | Determining the Effect of PH on Iron Oxidation Kinetics in Aquatic Environments: Exploring a Fundamental Chemical Reaction to Grasp the Significant Ecosystem Implications of Iron Bioavailability[END_REF] . The order of reaction was calculated for each data set (kinetic experiments using the same adsorbate-adsorbent system within a single literature source), with each data point used within the linear regression representing a single kinetic experiment at a unique value of C 0 or C s . This is represented by the equation:

Equation 5
For each data set, the order of reaction with respect to the independent variable was calculated using the LINEST function in Microsoft Excel. The error for each data set was specified as the standard error of the slope. An average (mean, ) order of reaction representing all data sets was calculated, with errors reported as the standard deviation. The dependencies of k 2 and k' upon C 0 and C s were determined using the same method, only substituting the initial rate with k 2 or k'.

Modelling the revised rate equation

Our final rate equation, developed and derived in the Results and Discussion (Equation 11, referred to as the rPSO), is:

Equation 6
where the rate constant k' takes the units L g -1 min -1 . As the rPSO rate equation is not easily integrated, experiments were simulated using Microsoft Excel. The quantity of adsorbate adsorbed at the n th data point was calculated using the following formula:

Equation 7
The time interval between data points, or Δt, was reduced until the magnitude of Δt had no significant effect on the results. The rPSO parameters k' and q e were obtained by non-linear fitting, using Microsoft E cel's Solver function to minimise the residual sum of squared residuals between the model and experiment. For all literature sources, parameters were recalculated to the same units for ease of comparison: k 2 (g mg -1 min -1 ); q e (mg g -1 ); and the initial rate (mg g -1 min -1 ). KERRR Equation 8where t is time (minutes), q t is the amount of adsorbate adsorbed per mass of adsorbent at time t (mg g -1 ), k 2 is the pseudo-second order rate constant (g mg -1 min -1 ), and q e is the amount of adsorbate adsorbed at equilibrium (mg g -1 ) [START_REF] Ho | Pseudo-Second Order Model for Sorption Processes[END_REF] .

The decrease in the concentration of aqueous adsorbate with time is given by the equation: Equation 9where C t is the concentration of aqueous adsorbate at time t (mg L -1 ), C 0 is the initial adsorbate concentration at t=0 (mg L -1 ) and C s is the concentration of adsorbent (g L -1 ).

KERRR

Application studies

In the first application study (evaluating the potential of the rPSO model for predictive applications), 6 data sets were used: 3 where C 0 was varied and 3 where C s was varied. In the first step, values of the equilibrium adsorption capacity (q e ) to constrain the model were obtained by fitting kinetic experiments individually using the PSO and rPSO models (optimising the rate constant and q e simultaneously to minimise the sum of squared residuals). After constraining q e to these values, all experiments within the given data set (i.e. experiments with the same adsorbate-adsorbent system but different C 0 or C s values) were simultaneously fit with the PSO and rPSO models, using a single rate constant (k 2 or k', depending upon the model) to describe all experiments.

In the second application study (evaluating the potential of the rPSO rate constant k' for more meaningful comparisons across the literature), 18 experiments reporting the adsorption kinetics of inorganic arsenic onto iron oxide minerals were collated. Both As(V) and As(III) experiments were included, since no significant difference between the adsorption kinetics was observed. A further 7 experiments reporting the adsorption of inorganic arsenic onto alumina (Al 2 O 3 ) were collected for comparison. The particle radius (r) was taken as reported by each study.

Results and Discussion

Determination of the influence of experimental conditions (C 0 and C s ) on the initial rate of adsorption

Quality control: The calculation of initial rates and setting criteria for the selection of literature data sets Many adsorption experiments reported by the literature begin their collection of kinetic data at high values of , where a significant proportion of the reaction has already been completed. This may be due to challenges in collecting samples quickly (especially when filtering is required) given that many adsorption reactions reach equilibrium in the minutes timescale. Other possible reasons include a lack of appreciation over the time-scale at which adsorption kinetics are best measured, with many papers fitting kinetic models on timescales when the reaction has already plateaued and reached equilibrium.

Whilst we chose to investigate the influence of C 0 and C s using the method of initial rates, the lack of literature reporting early stage kinetic data (i.e. low values of ) is a challenge. A popular approach to calculate initial rates is to determine the slope of a line that is tangent to the experimental data curve and passes through the origin at (0,0) [START_REF] Atkins | Physical Chemistry[END_REF] . However, the later the first kinetic data is collected, the shallower the slope will be, creating a systematic underestimation of the initial rate. In the most extreme case, an infinite delay before collection of kinetic data will yield a slope of zero and subsequently an initial rate of zero. Application of a kinetic adsorption model allows for the extrapolation of adsorption rates to t=0, however the accuracy of the calculated initial rates depends upon how closely the experimental data obeys the applied model.

We therefore conducted a preliminary experiment to investigate how a limited availability of early kinetic data would influence the accuracy and precision of initial rates calculated using the initial slope or using the original PSO model (given that the PSO model is known to approximately describe a wide range of adsorbate-adsorbent systems [START_REF] Ho | Review of Second-Order Models for Adsorption Systems[END_REF] ). These results were used to set quality control criteria for which data sets would be included within the investigation of the influence of C 0 and C s on adsorption kinetics.

The literature search identified fourteen data sets satisfying the criteria that , a relatively small number, highlighting how most authors fail to collect early kinetic data (SI Figure S1). Initial rates calculated from the slope between the origin and the earliest available data point show the typical systematic underestimation of initial rates when early kinetic data is missing (Figure 1a, with the solid line representing the average of all data sets). In contrast, when calculated using the PSO model, there is no systematic error in the calculation of initial rates when the first kinetic data is collected in the range (Figure 1b,c). In all three approaches, significant variation between data sets is observed, i.e. an error in the initial rate unique to each data set. This error (with dashed lines representing one standard deviation) is approximately constant in the initial slope approach, being significant even when early kinetic data is available. In contrast, this error is insignificant using early kinetic data and either of the PSO approaches, however this error increases in magnitude as early data is sequentially removed.

Additionally, the absolute values of the initial rates calculated using the earliest possible kinetic data were compared (i.e. 14 initial rate calculations), with Student's t-test indicating that both linearised PSO kinetics and non-linear PSO kinetics tend to return an initial rate greater than that calculated using the initial slope (with p=0.87 and 0.97 respectively). The increase in initial rates calculated using non-linear PSO kinetics versus linearised PSO kinetics is not significant (p=0.33 using the 14 initial rate calculations). However, when comparing the calculated initial rates with all possible data cut-offs (i.e. 115 initial rate calculations, Figure 1d) the differences are more significant: non-linear PSO kinetics return greater initial rates than linearised PSO kinetics (p=1.00). This is due to the biased weighting of linearised PSO kinetics towards data at later times, with the slope of versus t increasing as equilibrium adsorption is approached, returning smaller values of q e and thus giving a smaller initial rate. The difference between initial rates calculated using linearised and non-linear PSO kinetics was observed despite equilibrium adsorption kinetic data from the original literature being excluded in our calculations.

Finally, we considered the uncertainties in the initial rates calculated using the two PSO approaches with varying availability of early kinetic data (Figure 1e). The uncertainties in the initial rates calculated using linearised PSO kinetics were calculated from the standard error in the linear regression fit to versus t, whilst the uncertainties in non-linear PSO kinetics were calculated using synthetic data and 200 Monte-Carlo simulations [START_REF] Hu | Evaluation of Parameter Uncertainties in Nonlinear Regression Using Microsoft Excel Spreadsheet[END_REF] . Using linearised PSO kinetics, the propagated uncertainty in the initial rate increases exponentially as early kinetic data is removed. This is explained by how late stage adsorption kinetic data is weighted too heavily when fitting linearised PSO kinetics [START_REF] Simonin | On the Comparison of Pseudo-First Order and Pseudo-Second Order Rate Laws in the Modeling of Adsorption Kinetics[END_REF] , where the differences in q t are smaller relative to the measurement uncertainty: this increases the uncertainty of the linear regression. At , the average uncertainty in the initial rates calculated using linearised PSO kinetics is 9.5%. In contrast, the uncertainty in the initial rates calculated using non-linear PSO kinetics is essentially independent of the availability of early kinetic data, and at the average uncertainty is only 5.7%.

As highlighted here, ideally the data sets used to explore the influence of experimental parameters such as the adsorbate concentration (C 0 ) and the adsorbent concentration (C s ) should include early kinetic data to reduce the uncertainty in the calculated initial rates. However, literature data reporting early adsorption kinetics is limited. To provide a balance between the accuracy of our initial rate calculations and the collection of a sufficient quantity of data sets for statistical analysis, we set the criterion that data sets must include kinetic data with . This boundary condition gives an average error in the calculated initial rate of -44 ±22 % for the initial slope approach, +17 ±38 % using linearised PSO kinetics, and only +3 ±27% for the non-linear PSO kinetics. The results indicate that non-linear PSO kinetics are most appropriate for calculating initial rates, with an insignificant systematic error. A ~30% uncertainty remains, associated with how closely the adsorption kinetics fit to the PSO form (with deviations being both due to inaccurate measurements and real chemical mechanisms).

By considering a literature source reporting two kinetic experiments only, with a different value of C 0 or C s in each, a 30% error in the initial rate of the first experiment (as per the boundary condition of ) will confer an error of <0.5 in the calculated reaction order. In this work, the average value of in the first available kinetic data was 0.37 for experiments where C 0 is varied, and 0.27 for experiments were C s is varied. Furthermore, the number of kinetic experiments in each literature data set was 3-6, indicating that the uncertainties in the calculated initial rates will be <30%. Under these conditions, the calculated orders of reaction will be accurate to the nearest integer value. This was deemed appropriate for the purposes of developing the revised PSO model, given that it is common practice for kinetic adsorption models to use integer values for reaction orders (e.g. the original PSO equation, and the pseudo-first order kinetic model [START_REF] Ho | Pseudo-Second Order Model for Sorption Processes[END_REF] ). In principle, however a similar analysis could be made relaxing this condition.

Consequently, non-linear fitting of the PSO model was used to determine the initial rate of adsorption in each kinetic experiment in all subsequent work. The influence of C 0 and C s upon the initial rate of adsorption (calculated as the rate at t=0 using nonlinear PSO kinetics) is presented in Figure 2. For each data set, the order of reaction was determined from the slope of log(initial rate) versus log(C 0 ) or log(C s ) (Figure 2a,b). Based upon the evaluation of uncertainties in the initial rates calculated using non-linear PSO kinetics (in the previous section), the reaction orders are accurate to the nearest integer value. The data sets tend to show a first-order dependency of initial rate upon the initial adsorbate concentration (C 0 ) (Figure 2c), with an average dependency and standard deviation of 0.80 ±0.38, and a median value of 0.67. Of the 9 data sets, 7 were closest to first-order dependency, and 2 were closer to zero-order dependency. The sum of normal distributions representing the reaction orders and uncertainties calculated for each data set was approximated by a single normal distribution. (However, a larger number of data sets are needed to verify this). Based upon the standard deviation and assuming a normal distribution in the results, the relationship between the initial rate and C 0 is first-order with a 90% confidence interval (1.65 standard deviations). This is intuitive for both diffusion and adsorption-controlled mechanisms, as twice as much adsorbate should lead to twice as much adsorbate flux from adsorbent surface into pores, and collisions with the adsorbent surface should be twice as frequent 36 9 .

(a) (b) (c) (d) (e)
A first-order dependency of initial rates (normalised to mg L -1 min -1 ) with respect to the adsorbent concentration (C s ) is also observed (Figure 2d), albeit with a wider distribution of results: an average value of 1.11 ±0.33 and a median of 1.07. Of the 8 data sets, 6 were closest to first-order dependency, with 1 data set closer to zero-order and another closer to second-order. Based upon the standard deviation and assuming a normal distribution in the results, the relationship between the initial rate and C s is also first-order with a 90% confidence interval (1.65 standard deviations). This is again intuitive for both diffusion and adsorption-controlled mechanisms, as when C s is doubled the total surface area available to solution (m 2 L -1 ) is doubled, the overall flux of adsorbate entering adsorbent pores is doubled, and the rate of collisions between adsorbate and total adsorbent surface is also doubled. When normalised to mass (mg g -1 min -1 ) the initial rate is zeroorder with respect to C s , as expected. For both predictive modelling and the comparison of adsorption kinetics between literature sources, it is necessary that rate constants are not affected by the experimental conditions. Whilst adsorption kinetics are typically first-order with respect to C 0 , the PSO rate constant k 2 is inversely proportional to C 0 (Figure 3a). Consequently, whilst doubling the initial adsorbate concentration typically increases the initial rate of adsorption by a factor of two, counter-intuitively the PSO rate constant k 2 will decrease by a factor of two. The average slope of log(k 2 ) versus log(C 0 ) is -0.73±0.46. The inverse relationship between k 2 and C 0 is explained by the second-order dependence of the PSO model upon the absolute concentration of available adsorption capacity remaining through the term (q e -q t ) 2 . In cases where the adsorbent is unsaturated, increasing C 0 by a factor of two will approximately double q e . The parameter (q e -q t ) 2 at t=0 will increase by a factor of four, and consequently k 2 must decrease by a factor of two to achieve the observed doubling of initial rates.

(a) (b) In contrast, a positive relationship between C s and k 2 is observed, with an average dependency of 1.57±0.85 (Figure 3b). (First and second-order dependencies between k 2 and C s are both included within the standard deviation). This is explained by how when C s is doubled, q e will decrease by a factor of between 0 and 2: zero when the adsorbate is in excess and q e is insignificant compared to C 0 , and two when the adsorbent is in excess and q e is large relative to C 0 . Consequently, as C s increases, (q e -q t ) 2 decreases with a zero-to-second-order dependency, and to achieve the zero-order relationship between C 0 and initial rates (mg g -1 min -1 ), k 2 must also increase with a dependency that is between zero and second-order.

Revision of the pseudo-second order (PSO) rate equation to account for changes in adsorbate (C 0 ) and adsorbent (C s ) concentrations

Modification of the PSO rate equation

In this section, we modify the original PSO rate equation to include the appropriate sensitivity towards C 0 and C s , meeting the aims of (a) improving the predictive capacity of this model, and (b) normalising rate constants for better comparison across the literature.

Firstly, for a given concentration of adsorbent, the total concentration of adsorption surface sites is constant regardless of the value of C 0 . The term within the rate equation used to represent the contribution of adsorption surface site availability towards the rate of reaction should therefore be independent of C 0 . The original PSO rate equation contains a second-order dependence upon the absolute amount of adsorption capacity remaining, (q e -q t ) 2 , which gives the undesirable inverse relationship between the rate constant and C 0 demonstrated in Figure 3a. This term can be replaced with a second-order dependence upon the relative amount of adsorption capacity remaining, , which will always return a value of 1 at time t=0, independent of C 0 . This term therefore describes the contribution of adsorption surface site availability towards the rate of adsorption more appropriately than the original PSO term (q e -q t ) 2 . Here, is the same as the parameter θ used in the Langmuir adsorption isotherm model [START_REF] Atkins | Physical Chemistry[END_REF] . This modification of Equation 1gives the following:

Equation 10
where .

The first-order dependence of the reaction rate upon the adsorbate concentration observed experimentally (Figure 2c) is then defined within the rate equation, giving:

Equation 11
where . The rate constant k' takes the units L g -1 min -1 .

The initial rate of adsorption tends to be zero-order with respect to C s (when normalised to adsorbent mass with the units mg g -1 min -1 ). The original PSO model gives an initial rate that varies with changes in C s , due to its second-order dependence upon the absolute adsorption capacity remaining (q e -q t ) 2 and the decrease in q e with increasing C s . In contrast, since the rPSO depends on the relative adsorption capacity remaining through the term , this rate equation displays the zero-order dependency of C s identified from analysis of the literature. The rPSO rate constant k' is therefore theoretically independent of changes in C s , unlike the original PSO rate constant k 2 . The rPSO model is similar to the adsorption-only form of the kinetic Langmuir model (kLm), which at high surface coverage is first order with respect to C t and second order to ) 37 38 .

Validation of the rPSO rate equation

The removal of experimental conditionality (i.e. the dependency upon C 0 and C s ) from the revised model was verified using experimental data from the literature. The ideal rate constant is unaffected by the experimental conditions, and subsequently the dependency of the rate constant with respect to C 0 and C s should be zero. These dependencies were calculated from the slope of log(rate constant) versus log(C 0 ) or log(C s ). As highlighted by Figure 4a, the original PSO rate constant k 2 is strongly dependent upon the experimental conditions, being inversely proportional to C 0 and second-order with respect to C s . The average C 0 dependency is -0.73±0.46, and the average C s dependency is 1.57±0.79. In contrast, k' is appro imately zero-order with respect to both e perimental variables. Furthermore, the dependencies of k' upon C 0 and C s vary less (there is less scattering) than k 2 . The average C 0 dependency is -0.20±0.38 and the average C s dependency is 0.10±0.45. These results demonstrate that the new rate constant k' is less conditional than k 2 , and that the dependency of adsorption kinetics upon C 0 and C s is captured by the new model.

When the adsorbate is in excess (q e <C 0 ) the rPSO model approximates the form of the original PSO model. At higher values of q e relative to C 0 , the graphical form of the two models deviates, due to the rPSO kinetics decreasing more rapidly than PSO kinetics due to the consumption of the adsorption, which decreases the parameter C t . (However, this effect is logical, given that at low C t values, the rate of adsorption will be limited by the availability of adsorbate). Therefore, rate constants for the rPSO (k') can be readily calculated from PSO parameters k 2 and q e when the adsorbate is in excess, using the formula . When the adsorbent removes the majority of the adsorbate, however, the rPSO model will require re-fitting due to the increasing difference in the graphical form of the PSO and rPSO models. ). The original PSO model is presented in blue and the rPSO model in red, with equal to 1 (dotted lines), 2 (dashed lines), and 5 (solid lines).

Example applications Application 1: Evaluating the predictive capability of the revised PSO model

The first objective of this work was to provide a simple modification of the popular PSO model to introduce predictive capabilities, for the purpose of engineering studies [START_REF] Bullen | On the Application of Photocatalyst-Sorbent Composite Materials for Arsenic(III) Remediation: Insights from Kinetic Adsorption Modelling[END_REF] . If the kinetic model is to be used to predict adsorption performance under different conditions, then it is essential that the model parameters obtained experimentally are valid in a range of scenarios. Consequently, we evaluated whether the rPSO model would provide a better fit to experimental data compared against the PSO model, if a single rate constant is used to model multiple experiments with different values of C 0 and C s .

The original PSO model tends to systematically overestimate q t for experiments with high C 0 or low C s values and systematically underestimate q t for experiments with low C 0 or high C s values (Figure 5). This is due to the negative and positive relationships between k 2 and C 0 and C s respectively as previously discussed, with this relationship being denied when a single value of k 2 is used to model all experiments. The rPSO model gave a better fit to experimental data (i.e. a smaller sum of squared residuals) in 5 of the 6 data sets tested (all panels in Figure 5 except d) and the median decrease in the sum of squared residuals when changing from the PSO to the rPSO model was 66%. The average relative error of q t calculated by the rPSO is just 1±17%, versus -11±27% for the original PSO model (Figure 6a), indicating that the rPSO model provides greater accuracy when modelling adsorption kinetics with changing values of C 0 and C s using a single rate constant. Furthermore, the calibration curve of q t (model) against q t (experiment) is closer to the ideal one-to-one line in the rPSO model, with an R 2 value of 0.999 versus just 0.9217 for the original PSO model (Figure 6b). Considering each experimental data series in turn, the typical slope of the calibration curve was closer to one, with less scattering, in the rPSO model (a slope of 0.99±0.11) compared against the original PSO model (with a slope of 1.23±0.35) (Figure 6c). S3.

Whilst the rPSO rate constant k' appears to be more stable to changes in experimental conditions than the PSO rate constant k 2 , the parameter q e is conditional, depending upon C 0 and C s . For predictive modelling, this limitation can be rectified by using an adsorption isotherm to predict q e (such as the Langmuir or Freundlich model [START_REF] Ayawei | Modelling and Interpretation of Adsorption Isotherms[END_REF] ). Though a single value of q e can be determined for the entirety of each experiment, in scenarios such as a column reactor the equilibrium adsorbate concentration parameter C e has diminished physical significance, and it may be better to replace this term with C t , recalculating the hypothetical value of q e at each point in time. Huang et al. previously demonstrated that this approach can give a better account of the true driving force of the reaction during the initial stages of adsorption [START_REF] Huang | Model Fitting of Sorption Kinetics Data: Misapplications Overlooked and Their Rectifications[END_REF] .

Application 2: Comparison of rate constants between different experimental studies

Comparison of the rPSO rate constant k' is expected to be more meaningful and more appropriate than k 2 when comparing the adsorption kinetics reported in the literature using different experimental conditions, since some of the experimental conditionality (towards changes in C 0 and C s ) is accounted for. To demonstrate the potential application of normalised rate constants towards achieving a meaningful comparison of the adsorption kinetics reported across the literature, we collected 14 and 7 literature sources reporting the kinetics of inorganic arsenic As(V) and As(III) adsorption onto iron oxide and alumina adsorbents respectively.

The average value of log(k 2 ) for all iron oxide studies is -0.93±1.50, whilst the average of log(k') is -1.05±1.08. In both cases, the standard deviation is large, with more than an order of magnitude variation in k 2 and k' values: neither model provides a rate constant that is generally valid for iron oxide adsorbents used by different studies. Similarly, the adsorption of inorganic arsenic onto alumina gives average values of log(k 2 ) = 0.98±1.84 and log(k') = 0.35±1.75.

Whilst the variation in both k 2 and k' across the literature is large, the influence of adsorbent morphology on adsorption kinetics has not been incorporated into either of the PSO and rPSO models. Adsorption kinetics are often faster for adsorbent materials with smaller particles, due to the improved rate of mass transport of the adsorbate to adsorbent surface sites [START_REF] Cooney | Effect of Particle Size Distribution on Adsorption Kinetics in Stirred Batch Systems[END_REF] . For instance, in intraparticle diffusion model gives a rate of adsorption that is proportional to r -1 (where r is particle size) 49 .

Whilst a faster reaction is anticipated for small iron oxide particle sizes, plotting log(k 2 ) as a function of log(r -1 ) shows a weak inverse relationship (linear regression gives a slope of -0.49±0.20 and R 2 =0.3234) (Figure 7a). This is due to the significant increase in q e as the particle size decreases (SI Figure S4) and the inverse relationship between k 2 and q e through the term (q e -q t ) 2 as previously discussed. In contrast, alumina shows the anticipated positive relationship (with a slope of 0.61±0.28 and R 2 =0.4783). Since adsorption onto smaller particles is typically faster than onto larger particles, the comparison of k 2 values for different adsorbent sizes and different adsorbent materials (i.e. iron oxides versus alumina) is not useful and is likely to lead to false conclusions.

In contrast, we see the anticipated positive relationship for log(k') as a function of log(r -1 ) for both iron oxide and alumina adsorbents (Figure 7b). For the iron oxides we see a slope of 0.44±0.1 with R 2 =0.5211, and for alumina we see a slope of 0.62±0.25 and R 2 =0.5547. The limited goodness of fit in the linear regression is controlled by factors including poor characterisation of the particle size and differences in experimental conditions beyond C 0 and C s , such as the pH. The accurate characterisation of particle size is especially challenging, given the range of techniques used by the literature including transmission electron microscopy (TEM), dynamic light scattering (DLS) and sieve fractionation.

The results from this literature survey show greater values of log(k') at each specific particle size for alumina versus iron oxides, suggesting that the adsorption of inorganic arsenic is faster onto alumina than onto iron oxides. With more rigorous analysis, such a study would have important implications in the design of engineered solutions for arsenic remediation, i.e. opting to use alumina in place of iron oxides when high flow rates in a column filter are required, or when larger particle sizes are necessary to achieve the desired flow rate and porosity. This highlights that normalisation of the PSO rate constant k 2 to the rPSO rate constant k', or some other constant, can achieve a more meaningful comparison of the adsorption kinetics reported by the literature.

(a) (b) The new rate equation is similar to an adsorption-only form of the kinetic Langmuir model (kLm), which at high surface coverage is first order with respect to C t and second order with respect to ) 37 38 . However, the rPSO equation is simpler, with fewer fitting parameters needed, and may thus be more useful for the non-expert. The rPSO equation may prove more useful than the original PSO model in engineering studies where operating conditions are likely to vary.

One of the reasons for the popularity of the original PSO model is its linearised forms, from which k 2 and q e parameters can be readily obtained from experimental data. Whilst, we have unfortunately not yet found a way to linearise the rPSO rate equation, it is worth noting that linearisation of the original PSO model often results in a poorer quality of fit versus when using non-linear fitting [START_REF] Xiao | Erroneous Application of Pseudo-Second-Order Adsorption Kinetics Model: Ignored Assumptions and Spurious Correlations[END_REF] .

Where necessary, the rate constant for this revised model can be quickly obtained from linearised PSO kinetics using the expression .

In our analytical approach, we demonstrate that non-linear fitting of PSO kinetics is more appropriate than linearised PSO kinetics and the initial slope approach when early kinetic data is limited, as is often the case in adsorption experiments. Our literature survey only yielded 9 data sets where C 0 was varied and 8 where C s was varied, satisfying our requirements for early kinetic data with . This highlights a need for experimental work to investigate the influence of these independent variables more systematically, and to ensure that initial adsorption kinetics are adequately captured, given that the method of initial rates is typically considered the superior approach towards determining reaction orders [START_REF] Ollis | Kinetics of Photocatalyzed Reactions: Five Lessons Learned[END_REF] .

Figure 2 :

 2 Figure 2: Determining the influence of initial adsorbate concentration (C 0 ) and adsorbent concentration (C s ) on the initial rate of adsorption using literature data sets. The order of reaction was determined from the slope of log(initial rate) as a function of (a) log(C 0 ) and (b) log(C s ), with each data point representing a single kinetic experiment (with unique values of C 0 and C s ). All experiments in a given data set (one literature paper, where all experimental conditions except for either C 0 or C s are constant) are grouped by colour and symbol, with oxyanions in red, metal cations in blue, and organic dyes in yellow (a legend referencing the literature sources is presented in SI Figure S2). Uncertainties were calculated from the standard error in the slope. The results are alternatively presented using normal distributions with the mean given by the slope and the standard deviation given by the standard error of the slope (c and d). Here, each data set is represented by a dotted line, and the sum of all data sets given by the dashed line. Solid black lines represent the normal distribution obtained by the average of all data sets.

Figure 3 :

 3 Figure 3: Dependence of the pseudo-second order (PSO) rate constant k 2 upon (a) initial adsorbate concentration (C 0 ), and (b) adsorbent concentration (C s ). The data is presented as described in Figure 2, with a legend referencing the literature sources presented in SI Figure S2.

Figure 4 :

 4 Figure 4: Verifying that experimental conditionality (with respect to C 0 and C s ) is decreased in the rPSO model versus the original PSO model. (a) The dependencies of the original PSO rate constant k 2 and the rPSO rate constant k' upon C 0 and C s were calculated from the slope of log plots, as per Figure 3, with the ideal rate constant giving a reaction order or 'dependency' of zero. The data is shown as normal distributions with the mean and standard deviation set equal to the slope and standard error of the slope in the log plots (with 9 data sets for C 0 and 8 for C s ). These values are: -0.73±0.46 for k 2 and C 0 ; 1.57±0.79 for k 2 and C s ; -0.20±0.38 for k' and C 0 ; 0.10±0.45 for k' and C s . (b) A comparison of the form of the original PSO equation with the rPSO equation, with different ratios (with C s =1 g L -1

Figure 6 :

 6 Figure 6: Application study 1: Cross-calibration of the rPSO model against experimental data (6 literature sources, 22 experiments and 198 data points). (a) Box and whisker plot presenting the relative error of q t calculated via the original PSO model and the rPSO model. The boxes highlight the 25%, 50% (median) and 75% percentiles, whilst the whiskers represent

Figure 7 :

 7 Figure 7: Application study 2: Use of the rPSO rate constant k' to compare literature sources with adsorption kinetics determined under different experimental conditions. Particle size decreases from left to right. Presented are As(V) (red filled shapes) and As(III) (orange filled shapes) adsorption onto iron oxides, and As(V) (dark blue open squares) and As(III) (light blue open circles) adsorption onto alumina. Each data point indicates a different literature source, with a legend given in SI Figure S5.

  literature sources with approximately 100 kinetic experiments were collected. This

	includes: 14 literature sources with early kinetic data (where	) to investigate the
	influence of early data availability on calculations of the initial rate; 8 literature sources (9 data sets)
	with a total 37 experiments where C 0 is varied; 6 literature sources (8 data sets) with a total 27
	experiments where C s is varied; and 21 literature sources

0 and C s upon the rate of adsorption and the PSO rate constant k 2

  Figure 1: The influence of the limited availability of early kinetic data on the calculation of initial rates of adsorption, assessed by the analysis of 14 literature sources. The influence of removing early kinetic data on the calculated initial rates was assessed using (a) the initial slope, (b) linearised PSO kinetics, and (c) non-linear PSO kinetics. The theoretical 'true'initial rate is given by initial rate 0 and the initial rate calculated with the available data given by initial rate i . The solid black line represents the average of the 14 data sets, whilst dotted lines indicate one standard deviation. The three approaches are compared, assessing the influence of data availability on (d) the relative error in the initial rate, and (e) the uncertainty in the calculated initial rate. The horizontal error bars in (d) and (e) represent the size of the bins used for grouping data,

	whilst the vertical error bars indicate the standard deviation calculated between the 14 unique data sets. The data sets listed are in order: (1) Yang et al., 2019 23 , (2) Yang et al., 2001 24 , (3 and 4) Liu and Shen, 2008 25 , (5) Zhu et al. 2016 26 , (6) Yang et al. 2019 27 , (7) Yang et al. 2019 28 , (8) Mohamed et al. 2007 29 , (9) Drenkova-tuhtan et al. 2015 30 , (10) Liu et al. 2016 31 , (11) Ornek et al. 2007 32 , (12) Zhan et al. 2018 33 , (13) Ai et al. 2020 34 , (14) Nadiye-tabbiruka and Sejie 2019 35 .
	Determining the influence of C

  Figure 5: Application study 1: Application of the revised PSO (rPSO) model (solid lines) to describe multiple experiments with a single rate constant, compared against the original PSO model (dotted lines). (a,b,c) present 3 experiments where C 0 is varied,and (d,e,f) present 3 experiments where C s is varied. Experimental data was sourced fromManna et al. (2003)

	Singh et al. (1996)	41 , Mezenner and Bensmaili (2009)	42 , Debnath et al. (2017)	43 , Lazaridis et al. (2004)	40 , , and Shipley et al. 44
	(2013)

45

.

  The boxes highlight the 25%, 50% (median) and 75% percentiles, whilst the whiskers represent the minima and maxima excluding 'outlier' data points, defined as those greater than the top of the box plus 1.5 times the interquartile range, or less than the bottom of the box minus 1.5 times the interquartile range. (b) Cross-calibration plot highlighting the goodness of fit against the one-to-one line. Open shapes indicate q t values calculated using the originalPSO model, whilst filled shapes indicate the rPSO model. Values of R 2 indicate the goodness of fit against the ideal one-to-one line. (c) Comparison of the cross-calibration slopes with each model and each data set. Literature sources are denoted as

	As(III)/HFO Cr(VI)/Fe 2 O 3 40 (dark blue squares), As(V)/Fe 2 O 3 43 (light green triangles), Cr(VI)/Mg-Al-CO 3 41 (light blue circles), HPO 4 2-/iron hydroxide 44 (orange squares), and Cd(II)/Fe 2 O 3 45 (red circles). Further results 42 (dark green diamonds),
	are presented in SI Figure

Acknowledgements

The authors acknowledge support from the Engineering Physical Sciences Research Council (EPSRC) [grant number EP/N509486/1].

Supporting information

The supplementary information provides references to all data sets used in this work; figures showing the influence of adsorbent morphology on k 2 ; and tabulates all parameters (q e , k 2 , k' and initial rate) calculated in this work.