1 Introduction

Over the past several years, water pollution has been a global problem with rapid industrial development [1, 2]. Mercury pollution is one of the most severe problems because of its high toxicity and non-biocompatibility. Various industrial activities have discharged a large amount of mercury into wastewater, such as batteries, metal plating, mining, paint manufacturing and photographic industry [3]. Besides, mercury enters the food chain through various pathways during the circulation of water, leading to many diseases, such as Alzheimer’s disease, loss of vision, deafness, mental disorder, congenital malformation of children and memory impairment [4,5,6]. Therefore, it is a very important to remove mercury ions from aqueous solutions.

Generally, the removal methods of mercury ions from aqueous solutions include chemical precipitation, ion exchange, liquid extraction, photo reduction, coagulation, membrane filtration, solvent extraction and adsorption [7,8,9]. Among these removal methods, adsorption was considered as a promising method to remove mercury ions from aqueous solutions because of its high efficiency, good reutilization, low cost and environmental friendliness [10, 11]. Many adsorbents were reported in the literatures, such as rice straw, activated carbon, carbon nanotube, polystyrene, chitosan, silica gel, zeolite, coconut husk, cellulose, polymer materials, nanosilica, and so on [12,13,14,15]. Nanosilica is widely concerned because of their unique physical and chemical properties, such as low cost, large specific surface area and easy regeneration [16]. Besides, nanosilica is easy to be functionalized by chemical or physical methods than other adsorbents [17]. In general, the adsorption capacity of adsorbent is mainly determined by surface area and functional group density. The functionalized nanosilica has the intrinsic characteristics of nanosilica and the inherent characteristic of the functional group [18]. So, surface functionalized nanosilica is a promising sorbents for metal ions removal.

In recent years, hypophosphate has attracted great interests because phosphorus compounds with P–O or P=O groups are very effective ligands for heavy metal ions based on Lewis acid base theory [19]. Wang, et al. reported that phosphinic acid groups coordinated with a variety of transition-metal ions [20]. Therefore, many phosphated materials are used for metal ion removal. The phosphate nanocelluloses selectively adsorbed Ag+, Cu2+, and Fe3+ from industrial effluents [21]. The phosphorylated nanofibrils improved the sorption rate and capacity of copper from contaminated water [22]. Therefore, it is a potential solution for water treatment to modify phosphate groups on the surface of materials.

In this article, a novel adsorbent was synthesized by modifying nanosilica with phosphinic acid for mercury ions removal from aqueous solutions. This adsorbent was characterized by Fourier transform infrared spectroscopy (FT-IR), X-ray photoelectron spectroscopy (XPS), transmission electron microscope (TEM) and thermogravimetric analysis (TGA). Simultaneously, the effect of the solution pH, initial mercury ions concentration, contact time and coexisting ions was systematically investigated. Besides, selectivity, reusability (five cycles), adsorption mechanism, adsorption isotherms (Langmuir and Freundlich) and kinetics (Pseudo-first-order kinetic and Pseudo-second-order kinetic) were also studied.

2 Materials and methods

2.1 Materials

Polyethyleneimine (PEI), (3-glycidyloxypropyl)trimethoxy-silane and nanosilica (SNPs, Hydrophilic-380, 99.8%) were purchased from the Aladdin Chemistry Co. Ltd., Shanghai, china. Ammonia (25%), phosphinic acid, thiourea (99%), ethanol (99.8%) and formaldehyde were purchased from Tianjin Chemical Reagent Co. Ltd. HNO3 (67%) and NaOH (99%) were purchased from Nanjing Chemical Reagent Co. Ltd. Hg(NO3)2, Zn(NO3)2, Ni(NO3)2, Mn(NO3)2, and Ge(NO3)4 were purchased from National Nonferrous Metals Research Institute, Peking, China. All of the chemical reagents were analytical grade and without further treatment. The effects of coexisting ions on mercury adsorption were detected by adding the salt of the coexistence ion into the mercury solutions. The pH value of solutions was adjusted by 0.1 mol/L HNO3 and 0.1 mol/L NaOH solutions.

2.2 Preparation of phosphinic acid functionalized nanosilica

The preparation process of the phosphinic acid functionalized nanosilica was presented in Scheme 1. Firstly, the mixture of 3-glycidoxypropyltrimethoxysilane (10 mL), SNPs (5 g), ethanol (100 mL) and ammonia (2 mL) was stirred and refluxed at 65 °C for 48 h. After centrifugation, the solid was washed with ethanol five times, and dried at 60 °C for 12 h in vacuum drying oven. The obtained sample was denoted as GTS-SNPs. Secondly, the mixture of GTS-SNPs (4.5 g), PEI (100 mg/mL) and ethanol (50 mL) was stirred and refluxed at 65 °C for 24 h. After centrifuged, the solid was washed with distilled water and then dried at 60 °C for 12 h. The obtained sample was denoted as PEI-SNPs. Finally, the mixture of PEI-SNPs (5 g), phosphinic acid (15 mL), formaldehyde (40%, 15 mL) and ethanol (30 mL) was stirred and refluxed at 65 °C for 24 h under N2. After centrifuged, the solid was washed with distilled water and dried at 60 °C for 12 h. The obtained white powder was denoted as PA-SNPs.

Scheme 1
scheme 1

The synthesis process of PA-SNPs

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2.3 Characterization

Fourier transform infrared spectroscopy (FT-IR, Nicolet iS50, USA) was used to characterize the surface functional groups of adsorbents. Thermogravimetric analysis (TGA) were measured by PerkinElmer TGA-8000 (US) thermogravimetric analyzer at a heating rate of 10 °C/min. X-ray photoelectron spectroscopy (XPS, Physical Electronics, Inc., Chanhassen, MN, 200 W Mg radiations, USA) was used to investigate the surface chemical states of samples. Transmission electron microscope (TEM, JEM-3200) was used to evaluate the morphology of the nanoparticles. Inductively coupled plasma optical emission spectrometer (ICP-OES, Leeman prodigy 7, America) was used to detect the concentration of metal ions.

2.4 Mercury adsorption experiments

The adsorption experiments of mercury ion were carried out by adding 20 mg PA-SNPs into a series of centrifuge tube containing mercury ion solution (10 mL) with desired initial concentration and pH. The mixtures were shaken in a ZD-85 thermostat oscillator at 25 °C with constant rate of 300 rpm for a given time. After centrifuged at 8000 rpm for 15 min, the concentration of mercury ion was determined by ICP-OES.

The effects of pH, initial mercury ion concentration and contact time were investigated. The desired pH (ranging from 1.0 to 6.0) was adjusted by HNO3 (0.1 mol/L) and NaOH (0.1 mol/L) solutions. The selective separation of mercury ion from coexisting ions (Ge(IIII), Ni(II), Zn(II) and Mn(II)) was carried out when all ions concentrations were 100 mg/L. The concentration of mercury ion was 100 mg/L for kinetic study and 100–800 mg/L for isotherm study. In addition, the contact time of mercury ion with PA-SNPs was 5–720 min. The equilibrium adsorption capacity (q, mg/g) and removal rate (R, %) of mercury ion by PA-SNPs were calculated by following equation [23, 24].

$$q = \left( {C_0 - C_e} \right)\frac{V}{m}$$
(1)
$$R = \left( {{\mathrm{1 - }}\frac{{C_{\mathrm{e}}}}{{C_0}}} \right) \times 100\%$$
(2)

Where C0 (mg/L) and Ct (mg/L) are the initial and final concentration of mercury ion, respectively. V (mL) represents the volume of the mercury ion solution and m (mg) represents the mass of PA-SNPs sorbent.

2.5 Mercury desorption

The desorbing agent is made up of nitric acid (3 mol/L) and thiourea (1.5 mol/L). In order to better examine the reusability of PA-SNPs, the mixture of PA-SNPs (80 mg) and mercury ions solution (40 mL, 100 mg/L) was oscillated in thermostatic oscillator for 2 h at 25 °C. The solid was filtrated and the solution was retained for detection. The remaining solid was treated with the desorbing agent for 24 h. Then, the solid was washed with distilled water for five times and conduct the second round of adsorption. The above procedures were repeated for five times.

3 Results and discussion

3.1 Characterization of the phosphinic acid functionalized nanosilica

Figure 1 showed the TEM image of SNPs and PA-SNPs. The particles sizes of SNPs and PA-SNPs were 12.5 ± 5.0 nm. There was no significant size change between SNPs and PA-SNPs.

Fig. 1
figure 1

TEM images of SNPs and PA-SNPs

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Figure 2 has given FT-IR spectra of GTS-SNPs, PEI-SNPs and PA-SNPs. The peaks at 470, 803, and 1101 cm−1 were assigned to Si–O–Si bending vibration, symmetric stretching and stretching vibration, respectively [25,26,27]. The peak at 950 cm−1 was attributed to Si–OH group vibration [28]. For the effects of adsorbed water and silanol groups, the peaks at 3420 and 1637 cm−1 were attributed to O–H stretching vibration and deformation vibration, respectively. In the spectra of GTS-SNPs, the peaks at 2920 and 2850 cm−1 were attributed to the C-H stretching vibration, indicating that organic silane was grafted onto the surface of silica nanoparticles successfully. In the spectrum of PEI-SNPs, The band at 1384 cm−1 was attributed to the –NH2 and –NH bending vibration. For PA-SNPs, the peaks at 1384 cm−1 for the –NH2 and –NH bending vibration were disappeared, due to the reaction between phosphonic acid groups and the –NH2 and –NH [19]. Besides, the peaks for phosphinic acid groups (including P=O bond adsorption peak at 1185 cm−1, PO2 stretching vibrations at 1054 and 1185 cm−1) were no obvious indication in FT-IR spectra of PA-SNPs because they are overlapped by the broad bond between 1101 cm−1[19, 24, 29].

Fig. 2
figure 2

FT-IR spectra of GTS-SNPs, PEI-SNPs, and PA-SNPs

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Figure 3 presented the TGA analysis of SNPs, GTS-SNPs, PEI-SNPs and PA-SNPs. The mass loss of SNPs, GTS-SNPs, PEI-SNPs and PA-SNPs were 6.1, 8.5, 12.4, and 15.7%, respectively. The mass loss was due to the decomposition of organic matter and evaporation of water. The calculated amounts of 3-glycidoxypropyltrimethoxysilane, polyethyleneimine and phosphinic acid on the surface of SNPs were 2.66, 3.39, and 6.44%, respectively.

Fig. 3
figure 3

TGA analysis of SNPs, GTS-SNPs, PEI-SNPs, and PA-SNPs

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Figure 4 (a) showed the wide-scan spectra of GTS-SNPs, PEI-SNPs and PA-SNPs. The nitrogen peak around 400 eV was appeared after the GTS-SNPs were modified by PEI, indicating PEI groups were grafted onto the surface of silica nanoparticles successfully. A signal spectra at 168.2 eV presented in XPS wide-scan spectra of PA-SNPs was attributed to P2p, arising from the phosphinic acid group. This indicated phosphinic acid groups were grafted onto the surface of silica nanoparticles successfully. The C1s spectra of GTS-SNPs, PEI-SNPs and PA-SNPs were showed in Figs. 4b–d. Two peaks at 284.6 and 285.8 eV (Fig. 4b) in the C1s spectra of GTS-SNPs were corresponded to C-C (C-H) and C-O bonds, respectively. The new peak at 286.4 eV was corresponded to C-N bond (Fig. 4c). After modification by phosphinic acid, the new peak at 285.4 eV was attributed to the C-O(PO) of the phosphinic acid groups(Fig. 4d). Therefore, The XPS analyses further demonstrated that the phosphinic acid functionalized nanosilica was successfully synthesized. Besides, the isotherm linear of PA-SNPs was shown in Fig. 5. The BET results showed that the specific surface area, pore volume and pore size of PA-SNPs were 116.459 m2/g, 0.328 cm3/g, and 3.253 nm, respectively.

Fig. 4
figure 4

XPS wide-scan spectra and C1s spectra of GTS-SNPs, PEI-SNPs, and PA-SNPs

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Fig. 5
figure 5

The isotherm linear of BET surface area

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3.2 Adsorption performance

3.2.1 Effect of pH

The pH of the solution is a crucial factor in the adsorption process. Because pH affects the surface properties of adsorbents in terms of dissociation of functional groups and surface charge. In order to prevent the hydrolysis of Hg2+ when pH value is greater than 6, the adsorption experiments of mercury ion were carried out in the pH range of 1.0 to 6.0. PA-SNPs (20 mg) and mercury ion solution (10 mL, 100 mg/L) were stirred in a ZD-85 thermostat oscillator at 25 °C for 12 h. Figure 6 showed the effect of pH on the removal rate of mercury ions. The result showed that maximum removal rate of mercury ion was obtained at pH 2. At low pH (<2.0), the removal rate of mercury ions by PA-SNPs increases with the increase of pH. Because H+ will neutralize the surface charge, the H3O+ ions was preferably adsorbed on PA-SNPs and prevented the sorption of mercury ions at low pH [19, 30]. At higher pH, the concentration of H+ in solution decrease with the increase of pH. Moreover, Hg2+ converted into the metal hydroxide species, such as Hg(OH)2 or Hg(OH)+. Therefore, the removal rate of mercury ions by PA-SNPs decreases with the increase of pH from 2 to 6. So, the following experiments were carried out at pH 2.

Fig. 6
figure 6

Effect of initial pH on Hg(II) adsorption

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3.2.2 Effect of contact time on the removal rate and adsorption kinetics

20 mg of PA-SNPs was added into 10 mL of the mercury solution with the concentration of 100 mg/L at pH 2. The influence of the contact time on removal rate of Hg2+ was shown in Fig. 7. It can be found that the removal rate was extremely rapid within the initial few minutes and reached the adsorption equilibrium at 60 min. In the initial stage, the fast adsorption should attribute to the small mass transfer resistance, plenty of functional groups and large specific surface area on the PA-SNPs [31, 32]. In the equilibrium stage, the adsorption was slowed down as the active sites of PA-SNPs were filled up.

Fig. 7
figure 7

Effect of contact time on the removal rate of Hg(II)

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In order to evaluate the kinetic mechanism, two models were employed to interpret the experimental data. The pseudo-first-order and pseudo-second-order models are represented as Eq. (3) and Eq. (4) [33, 34]:

$$\ln \left( {q_{\mathrm{e}} - q_{\it{t}}} \right) = \ln \,q_{\mathrm{e}} - k_1t$$
(3)
$$\frac{{\mathrm{t}}}{{q_t}} = \frac{1}{{k_{\mathrm{2}}q_e^2}} + \frac{{\it{t}}}{{q_e}}$$
(4)

Where, qe and qt were the adsorption capacity (mg/g) at equilibrium time and at time t (min), respectively. k1 and k2 referred to the pseudo-first-order rate constant (min−1) and pseudo-second-order rate constant (g/mg/min), respectively.

The experimental data of adsorption was fitted by linear plot of pseudo-first-order and pseudo-second-order models as ln(qe−qt) vs. t and t/qt vs. t, respectively. Figure 8 has given the results of the two models. The correlation coefficients (R2) and kinetic parameters were listed in the Table 1. The pseudo-second-order model showed good fit for the adsorption of mercury on PA-SNPs. Because the correlation coefficient (R2) of pseudo-second order model (0.9999) was much higher than that of the pseudo-first order model (0.7741). The pseudo-second-order model assumed that the rate determining step was chemisorption. This suggested that the adsorption of mercury ions on PA-SNPs was mainly the chemical reactive adsorption [35].

Fig. 8
figure 8

The kinetic models of Hg(II) adsorption on PA-SNPs

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Table 1 Kinetics parameters for mercury ions adsorption on PA-SNPs
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3.2.3 Effect of initial concentration on Hg(II) adsorption and adsorption isotherms

Figure 9 presented the relationship between adsorption capacity and the initial Hg(II) concentration. The adsorption capacity increased before the concentration of 600 mg/L and then attained a platform with the increase of mercury ions concentration. At low mercury ions concentration (<600 mg/L), there are enough functional groups and active sites on the surface of PA-SNPs. The interaction between the metal ions and active sites can generate a potent driving force to defeat the resistances of the mercury ions mass transfer between the bulk solution and adsorbent [36]. So, the adsorption capacity increased with the increase of the mercury ions concentration. The maximum absorbability of mercury ions on PA-SNPs is 274.32 mg/g at the adsorption equilibrium.

Fig. 9
figure 9

Effect of initial concentration on Hg(II) adsorption

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In this study, Langmuir and Freundlich isotherm models were employed to describe the interaction between mercury ion and PA-SNPs. The linear form of the Langmuir and Freundlich are expressed by Eq. (7) and Eq. (8), respectively [37, 38]:

$$\frac{{C_{\mathrm{e}}}}{{q_e}} = \frac{1}{{q_{{\mathrm{max}}}K_L}} + \frac{{C_e}}{{q_{\max }}}$$
(7)
$$\ln q_e = \ln K_F + \frac{1}{n}\ln C_e$$
(8)

Freundlich isotherm assumes that the adsorption sites with stronger affinity are first occupied and is applied to multilayer adsorption on heterogeneous surfaces with interactions between adsorbed molecules [39, 40] Langmuir isotherm assumes a monolayer adsorption on a homogenous surface where the binding sites have the same adsorption affinity and no interactions between adsorbates are considered [41, 42].

The experimental data of adsorption was fitted by linear plot of Langmuir and Freundlich models as Ce/qe vs. Ce and lnqe vs. lnCe, respectively. Figure 10 presented the fitting results of the two isotherm models. The correlation coefficients (R2) and fitting model parameters of the two models have been listed in Table 2. The Langmuir model better described the mercury ion adsorption process on PA-SNPs because the correlation coefficient (R2, 0.9996) of Langmuir model is higher than that of Freundlich model (0.6976). Simultaneously, the theoretical maximum adsorption capacity of Langmuir model (277.78 mg/g) was closer to the experimental value (274.32 mg/g). Therefore, Langmuir model was more suitable for the novel adsorbent. Table 3 compares the adsorption capacity of the present work with others adsorbents reported in the literature. It can be clearly seen that PA-SNPs has the highest adsorption capacity for mercury ions among the listed absorbents.

Fig. 10
figure 10

The isotherm models of Hg(II) adsorption on PA-SNPs

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Table 2 Isotherm parameters of mercury (II) adsorption
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Table 3 Comparison of adsorption capacity of GD-SNPs with other adsorbents reported in literature
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3.2.4 Selectivity

Selectivity is an indispensable factor in evaluating the adsorption properties of the absorbent. In our work, the selectivity of PA-SNPs was investigated in existing four coexisting ions (Zn(II), Ni(II), Mn(II) and Ge(IIII))[32, 43]. 20 mg of PA-SNPs was immersed into 10 mL of the mixed-ion solution (Hg(II), Zn(II), Ni(II), Mn(II), and Ge(IIII)) at pH 2.0. The initial concentration of all ions was 100 mg/L. The mixed-ion solution was vibrated for 6 h. Figure 11 has given the results of the selective adsorption of Hg(II) on PA-SNPs. The removal rate of mercury ions reached 98.94%, which is remarkably higher than that of the coexisting metal ions. Therefore, the mercury ion could be selectively separated from the mixed metal ions solution by PA-SNPs. The high selectivity for mercury ions was resulted from the phosphonic acid groups of PA-SNPs. According to the theory of hard and soft acids-base [44, 45], the mercury ions was the soft nature metal ion and P atoms of PA-SNPs was soft bases which have a superior affinity with mercury ions. Therefore, mercury ions can be easily adsorbed by PA-SNPs and high selectivity was obtained.

Fig. 11
figure 11

Selective adsorption of Hg(II) on PA-SNPs

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3.2.5 Cyclic utilization and desorption experiments

To reduce the cost of removal process during practical application, it is very important to examine the desorption of Hg(II) from TPA-SNPs and regeneration of the spent adsorbent. Figure 12 has given the removal rate in each cycle. The adsorbent still remained good adsorption efficiency. The removal rate of mercury ions decreased slightly from 96.29 to 94.01%. The decrease might be caused by the loss of absorbent or irreversible occupation of part adsorption sites [31]. So, PA-SNPs shows good reusability for the mercury recovery from aqueous solution.

Fig. 12
figure 12

Reusability of Hg(II) adsorption on PA-SNPs

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Figure 13 has shown the desorption isotherm of PA-SNPs. Compared with Fig. 10a, the y-intercept values of the linearized Langmuir model of adsorption and desorption were the same. It is proved that mercury ions competed with the desorbing agent on the active adsorption sites [46]. Under the condition of nitric acid (3 mol/L), the surface of PA-SNPs is protonated, leading to the desorption of positively mercury ions [47]. The stable complexes and the electrostatic interactions between the charged thiourea species and mercury ions would weaken the interaction between the PA-SNPs and mercury ions and promoted desorption[48].

Fig. 13
figure 13

Linearized Langmuir plots (Ce/qe Vs. Ce) for desorption of PA-SNPs (desorbing agent of nitric acid (3 mol/L) and thiourea (1.5 mol/L))

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3.2.6 Adsorption mechanism

Based on some similar works, six adsorption mechanisms of mercury ions adsorption might be supposed (electrostatic interaction, ion exchange, ion-dipole interactions, coordination by surface metal cations, hydrogen bonding, and hydrophobic interaction) [49, 50]. In order to better understand the adsorption mechanism of mercury ions on the surface of PA-SNPs. The IR and XPS analysis were carried out in this study.

We defined Hg2+ loaded PA-SNPs as PA-SNPs-Hg. Figure 14 showed the FT-IR spectra of PA-SNPs and PA-SNPs-Hg. Compared with PA-SNPs, PA-SNPs-Hg presented a strong absorption peak at 1245 cm−1, due to that the bonds of P=O and P-OH in phosphonic acid groups interacted with Hg2+. This demonstrated that mercury ion was adsorbed through the chelating binding between Hg2+ and phosphorus groups on the surface of PA-SNPs.

Fig. 14
figure 14

FT-IR spectra of PA-SNPs and PA-SNPs-Hg

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To further verify the obtained results of adsorption mechanism from IR analysis, XPS analysis was carried out to elucidate the adsorption mechanism. Figure 15a showed the wide-scan spectrum of PA-SNPs and PA-SNPs-Hg. Compared to the spectra of PA-SNPs, it can be clearly see that the Hg4f and Hg4d were appeared in PA-SNPs-Hg, suggesting the mercury ions were adsorbed by PA-SNPs. The peaks at 358.3 and 377.8 eV were attributed to Hg4d5 and Hg4d3, respectively (Fig. 15 b). The peaks at 100.7 and 103.9 eV in Fig. 15c were attributed to Hg-P bond and Hg4f5, respectively. The XPS analyses of mercury indicated that there existed stronger electron transfer between the adsorbed Hg2+ and PA-SNPs. So, the phosphorus atom groups on the surface of PA-SNPs play an important role in the adsorption process of mercury ion.

Fig. 15
figure 15

XPS analyses of PA-SNPs and PA-SNPs-Hg a, XPS analyses of mercury b, c and P2p analyses d, e

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The P2p spectra of PA-SNPs and PA-SNPs-Hg were showed in Figs. 15d, e. The P2p spectra of PA-SNPs can be split into two peaks (131.28 and 132.75 eV), which correspond to P-C and P-O/P=O, respectively [51]. After adsorbed mercury ion, the two peaks were shifted to 132.0 and 133.74 eV. Because the mercury ions interacted with the P=O/P-OH groups on the surface of PA-SNPs [52]. Besides, Hg2+ shared the lone pairs of electrons from P atoms, resulting in the higher binding energies [19]. Therefore, chelation interaction was main mechanisms for the adsorption of mercury ions on the surface of PA-SNPs. The possible mechanism was presented in Scheme 2.

Scheme 2
scheme 2

Adsorption mechanism of mercury ions by PA-SNPs

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4 Conclusions

In this work, a novel adsorbent was prepared by modifying silica nanoparticles with phosphinic acid for mercury removal from aqueous solutions. The prepared adsorbent was characterized by TEM, FT-IR, TGA, and XPS. The adsorption properties of the adsorbent for mercury ions were investigated by batch adsorption experiments. Results showed that the optimum pH and the maximum adsorption capacity were 2.0 and 217.17 mg/g, respectively. PA-SNPs presented excellent selectivity for mercury ions among Zn(II), Ni(II), Mn(II), and Ge(IIII). Meanwhile, the reusability experiment showed that the novel adsorbent can be reused at least five times for the recovery of mercury ion. Besides, the adsorption process of mercury ions via PA-SNPs followed better with the pseudo-second-order kinetic model, which indicated that the interaction between Hg2+ and active groups on the PA-SNPs was chemical adsorption. Moreover, the equilibrium data fitted the Langmuir isotherm model very well, with a monolayer adsorption capacity of 277.78 mg/g. The main adsorption mechanism of mercury ion on PA-SNPs was chelation interaction between Hg2+ and PA-SNPs. Therefore, the novel nanosilica adsorbent has a potential application in the field of recovery of mercury ion from aqueous solution.