Introduction

Adsorption of surfactants at solid–liquid interfaces has been extensively investigated owing to its fundamental and practical importance [1,2,3,4,5,6,7]. For instance, it plays an important role in many technological and industrial applications, such as detergency, mineral flotation, dispersion of solids, and oil recovery [3]. Surface modification of solid adsorbents through surfactant adsorption can significantly enhance their removal efficiency of contaminants from aqueous solutions, showing a good potential for application in wastewater treatment [8,9,10]. It is well known that the behavior of surfactant adsorption is generally governed by a number of forces, including electrostatic interaction, covalent bonding, hydrogen bonding, hydrophobic effects, and solvation/desolvation [3, 5]. A total adsorption is usually the cumulative result of some or all of these forces, depending on the natures of both surfactants and solid surfaces as well as on the environmental conditions. Therefore, surfactant adsorption is a complex interfacial phenomenon. Three types of adsorption isotherms have been identified, namely, Langmuir type (L-type), S-type, and “double plateau” type (LS-type) isotherms [1]. The L-type isotherm can be described using the classical Langmuir model, while the mechanisms of S-type and LS-type adsorption are complex [1, 5]. To explore the adsorption mechanism and describe the adsorption isotherms, several surface micellization models, such as “one-step” [11], “two-step” [12,13,14], and “four-region” [15,16,17], were developed [1,2,3, 5]. The one-step and two-step models can describe the S-type and LS-type isotherms, respectively, and, more importantly, can provide information of surfactant micellization at solid–liquid interfaces [1, 5, 11,12,13,14].

A wide variety of solid adsorbents, such as metal/nonmetal oxides (alumina [9, 15, 18,19,20,21], titania [16, 17, 22], iron oxide [23, 24], and silica [4, 25,26,27,28]), mineral clays (kaolinite [15, 29] and montmorillonite [29, 30]), polymer beads or films [31,32,33], carbon materials (activated carbon [34], carbon nanotubes [35, 36], and graphene nanosheets [36]), and nanocellulose [7], have been used to study the adsorption behavior of surfactants at solid–liquid interfaces. Amorphous silica (SiO2) is one of the most widely used adsorbents [4, 25,26,27,28], which has a lowest point of zero charge (PZC, pH ~ 2) and unusual surface properties compared to other well-characterized oxide surfaces [2, 3]. There has been a well understanding for the effects of the particle surface features, such as the pore structure and the chemical state of surface hydroxyl groups (or silanol groups, Si-OH), and the environmental conditions, such as pH, temperature, and ionic strength, on the adsorption [11, 27, 28, 37,38,39,40,41,42,43]. However, understanding on the effect of particle sizes is still limited, owing to the fact that little attention has been focused on this aspect. To our knowledge, only Penfold’s group [44, 45] has investigated the effect of silica sol sizes (~ 7.8 and 13.8 nm) on the adsorbed layer thickness of the nonionic surfactant alkyl polyoxyethylene ether (CnEOm) by using the small-angle neutron scattering (SANS) technique. A difference in the adsorbed layer thickness between the two silica sols was observed, but no information on the adsorption amount and, especially, the adsorption thermodynamics were provided in the studies [44, 45]. In principle, the size (or surface curvature) of adsorbent particles can affect the physicochemical state of their surface molecules or groups, which may affect their adsorption for surfactants from solutions. In addition, there have been some reports focused on the effect of particle sizes on adsorption of polymers and biomolecules at solid–liquid interfaces [46,47,48,49]. Greenwood et al. [46] found that the adsorbed layer thickness of the ABA block copolymer Synperonic F127 (PEO-PPO-PEO) on poly(methyl methacrylate) (PMMA) and polystyrene (PS) latexes increased with an increase in the latex particle sizes. Vertegel et al. [47] reported that the adsorbed amount of chicken egg lysozyme on silica nanoparticles increased with the increase of particle size, demonstrating that stronger protein–particle interactions exist in the case of larger nanoparticles. Lindman et al. [48] determined the surface coverage degree of human serum albumin (HSA) on N-iso-propylacrylamide/N-tert-butylacrylamide (NIPAM/BAM) copolymer nanoparticles with a size range of 70–700 nm; they found that the surface coverage degree on the smaller particles (70 nm) was lower than that on the larger particles (120–700 nm), suggesting that a higher degree of surface curvature interferes with binding of HSA. Recently, Walkey et al. [49] found that the adsorbed surface density of serum protein on gold nanoparticle decreased with an increase in particle sizes. These previous studies clearly demonstrate that adsorbent particle sizes may obviously impact adsorption and the size-effect is complex. Up to now, the size-effect mechanism has been less understood. Therefore, research on particle size dependence of surfactant adsorption at solid–liquid interfaces are very essential, which can deepen our understanding of the adsorption phenomena.

Benzyldimethyldodecylammonium halide (BDDA+X−, commonly X− = Cl− and Br−) is one kind of quaternary ammonium surfactants and widely applied in clinical, cosmetic, and industrial areas [31, 50,51,52]. However, research on the aggregation features of BDDA+X− in solutions [51, 53, 54] and those at solid–liquid interfaces [31, 37, 39] are scarce. Harkot and Jańczuk [31] studied the effect of BDDABr adsorption on wetting of polytetrafluoroethylene and poly(methyl methacrylate) surfaces. Partyka’s group [37, 39] investigated the adsorption of BDDABr on silica particles with a mean size of 130 nm in water at free pH. It is interesting to understand the influence of silica particle sizes on the BDDABr adsorption from aqueous solutions at various pH values.

In the current work, the adsorption of BDDABr on amorphous silica particles with different sizes (~ 12 and 31 nm, denoted as S-SiO2 and L-SiO2, respectively) in water was investigated at various solid dosages (Cs, 10–40 g/L), pH (3–10), and temperature (T, 298–308 K). The original aim of this study is to examine whether the “sorbent concentration effect” (Cs-effect) exists in the adsorption process. The so-called Cs-effect (or “solid effect”) is a phenomenon that the adsorption isotherms (or saturation adsorption capacity) decline with increasing Cs under given conditions (such as temperature, pressure, and medium composition) [55, 56]. No Cs-effect was observed in the adsorption systems. However, interestingly, it is found that, besides pH and T, the size of the silica particles has an obvious influence on the adsorption. The affinity of the S-SiO2 toward BDDABr is lower than that of the L-SiO2. In addition, we find that the adsorption may show the S-type, LS-type, and L-type isotherms, depending on silica particle sizes and pH. The obtained S-type and LS-type isotherms are analyzed using the one-step and two-step models, and the effect of silica particle sizes on the surface micellization of BDDABr is observed. To the best of our knowledge, this is the first report on the effect of adsorbent particle sizes on adsorption thermodynamic features. We think that this work provides a better understanding of cationic surfactant adsorption at silica–liquid interfaces.

Experimental section

Materials

Two high-purity (≥ 99.5%) silica nanoparticle samples, with average diameters of ~12 and 31 nm, respectively, were purchased from Macklin, China. Benzyldimethyldodecylammonium bromide (BDDABr, ≥ 97% for purity) was purchased from TCI, China, and used as received. Its molecular structure is shown in Fig. S1 in the Electronic Supplementary Material (ESM). All other chemicals used in this work were of analytical reagent grade. Ultrapure water with a resistivity of 18.25 MΩ cm was obtained using a Hitech-Kflow water purification system (Hitech, China).

Acid-base titration

Acid-base titration was performed at 25 °C to determine the surface density (Ns) and apparent dissociation constant (Ka) of surface hydroxyl groups (or silanol groups) of the SiO2 adsorbents.

Silica particles (0.75 g) were dispersed in 75 mL water under magnetic stirring, and the pH of the suspension was adjusted to be ~ 2.8 using 1.00 M HCl. Prior to acid-base titration, the resultant suspension was magnetically stirred for 24 h at 25 ± 0.5 °C in an atmosphere of N2, and its pH was adjusted again if necessary. Afterwards, the suspension was back-titrated using 0.20 mM NaOH in 0.1 mL increments to pH ~ 11.0. During the titration period, N2 was bubbled to exclude CO2, the mixture was magnetically stirred, and the temperature was kept at 25 ± 0.5 °C using a HK-2A thermostatic bath (Nanjing Nanda Wanhe Science & Technology Co., Ltd., China). After each addition of the NaOH, the pH value of the suspension and the cumulative volume of the NaOH added were recorded when the change in pH value did not exceed 0.02 pH unit in 5 min. The pH values of the suspension were measured using a FE28 pH meter (Mettler-Toledo Instruments (Shanghai) Co., Ltd., China), calibrated by three buffer solutions with pH 4.00, 6.86, and 9.18. The titration was also performed on the blank system (water) as control. The tests were performed in triplicate, and the final values are the average of the three measurements.

The titration data were analyzed using Gran functions [57, 58] to estimate the Ns value. Two Gran functions, Ga and Gb, were used:

$$ {G}_{\mathrm{a}}=\left({V}_0+V\right)\ 1{0}^{\hbox{--} \mathrm{pH}}\kern1em \left(\mathrm{for}\ \mathrm{the}\ \mathrm{acidic}\ \mathrm{side}\right) $$
(1)
$$ {G}_{\mathrm{b}}=\left({V}_0+V\right)\ 1{0}^{\mathrm{pH}\hbox{--} 14}\kern0.75em \left(\mathrm{for}\ \mathrm{the}\ \mathrm{alkaline}\ \mathrm{side}\right) $$
(2)

where V0 (mL) is the start volume of suspension, and V (mL) is the cumulative volume of added NaOH solution. Plots of the Gran functions Ga and Gb versus V can form two straight lines, which intersect the V-axis at Veq1 and Veq2, respectively. The difference (Veq2 − Veq1) is the volume of added NaOH needed to achieve a complete neutralization reaction for the hydroxyl groups of silica surfaces. Notably, the hydroxyl groups of silica surfaces (Sur-OH, here “Sur” represents the surface) cannot be protonated to form protonated hydroxyl groups (Sur-OH2+), owing to the PZC of SiO2 being about pH 2 [3, 38, 39]. Therefore, the Ns (sites/nm2) of the SiO2 adsorbents can be estimated by:

$$ {N}_{\boldsymbol{s}}=\frac{\left[{\left({V}_{\boldsymbol{eq2}}-{V}_{\boldsymbol{eq1}}\right)}_{\boldsymbol{s}\boldsymbol{ample}}-{\left({V}_{\boldsymbol{eq2}}-{V}_{\boldsymbol{eq1}}\right)}_{\boldsymbol{control}}\right]{C}_{\boldsymbol{NaOH}}{N}_{\boldsymbol{A}}}{mA_{\boldsymbol{s}}}\times {10}^{-24} $$
(3)

where CNaOH (mM) is the concentration of NaOH solution, NA is the Avogadro’s number (6.02 × 1023), m (g) is the used amount of adsorbent, As (m2/g) is the specific surface area of adsorbent, and 10−24 is a conversion factor for units.

The pKa of the SiO2 was estimated from the titration data via the estimation of the dissociation degree (αSH) of surface hydroxyl groups at various pH. The dissociation reaction of surface hydroxyl groups for SiO2 can be expressed as:

$$ Sur- OH\rightleftharpoons Sur-{O}^{-}+{H}^{+} $$
(4)

The apparent equilibrium constant (Ka) of the dissociation reaction is given by:

$$ {K}_{\boldsymbol{a}}=\frac{\left[\boldsymbol{Sur}-{\boldsymbol{O}}^{-}\right]\left[{\boldsymbol{H}}^{+}\right]}{\left[\boldsymbol{Sur}-\boldsymbol{OH}\right]} $$
(5)

where [Sur-OH] and [Sur-O−] are the surface densities of Sur-OH and Sur-O− groups, respectively. The αSH of surface hydroxyl groups is defined as:

$$ {\alpha}_{\boldsymbol{SH}}=\frac{\left[\boldsymbol{Sur}-{\boldsymbol{O}}^{-}\right]}{N_{\boldsymbol{s}}} $$
(6)

The [Sur-O−] value at a given pH was calculated by the following:

$$ \left[\boldsymbol{Sur}-{\boldsymbol{O}}^{-}\right]=\frac{\left({\left[{\boldsymbol{O}\boldsymbol{H}}^{-}\right]}_{\boldsymbol{control}}-{\left[{\boldsymbol{O}\boldsymbol{H}}^{-}\right]}_{\boldsymbol{s}\boldsymbol{ample}}\right)\left({V}_0+V\right)}{A_{\boldsymbol{s}}m} $$
(7)

where [OH−]control and [OH−]sample are the OH− concentrations of the control and test sample systems, respectively, at the V of added NaOH solution. The pKa value of surface hydroxyl groups for the test samples was given by the pH value at which αSH = 0.5 (i.e., [Sur-OH] = [Sur-O−]).

Conductivity measurement

Conductivity (κ) measurements were performed on a DSJ-308A digital conductivity meter (Shanghai REX Instrument Factory, China) with a DJS-1C glass electrode. The temperature of the measurement cell was controlled at 25.0 ± 0.1 °C using the HK-2A thermostatic bath. The tests were performed in triplicate, and the final values are the average of the three measurements.

Characterization of adsorbent

The morphology and size of solid samples were analyzed using a JEM-2100 transmission electron microscopy (TEM, JEOL, Japan) and a JSM-6700F scanning electron microscopy (SEM, JEOL, Japan). The elemental composition of the samples was determined using energy dispersive spectrometer (EDS) equipped in the SEM instrument. Specific surface area (As) of the samples was determined by measuring volumetric N2 adsorption–desorption isotherms at liquid nitrogen temperature using an Autosorb IQ-MP instrument (Quantachrome Instruments, USA). Samples were degassed at 200 °C for 3 h under vacuum before measurement.

Adsorption experiment

The adsorption experiments were performed using a batch technique at different adsorbent dosages (Cs, 10–40 g/L), BDDABr concentrations (Ci, 0–30 mmol/L), pH (3–10), and temperatures (T, 298, 303, and 308 K). Known masses (0.25–1.00 g) of the adsorbents were added to 25 mL of test BDDABr solutions in polyethylene centrifuge tubes. The centrifuge tubes were shaken using a thermostatic water bath shaker (Jiangsu Medical Instrument Factory, China) for 24 h at a given T. The adsorption kinetic tests showed that the contact time of 24 h was sufficient to reach adsorption equilibrium. The adsorbent particles were then separated from the adsorption systems by centrifugation (GT16-3, Beijing Shidai Beili Centrifuge Co., Ltd., China) at 12,000 rpm for 30 min. The concentrations of BDDABr remaining in the resultant supernatants were determined using UV–vis spectrometry (SP-4100, Shanghai Spectrum Instruments Co., Ltd., China) at 262 nm. The equilibrium adsorption amounts were calculated from the difference between the initial and remaining (or equilibrium) concentrations, using the following equation:

$$ {\varGamma}_{\boldsymbol{e}}=\frac{C_{\boldsymbol{i}}-{C}_{\boldsymbol{e}}}{C_{\boldsymbol{s}}} $$
(8)

where Γe (mmol/g) is the equilibrium adsorption amount, Ci (mM) and Ce (mM) are the initial and remaining (equilibrium) concentrations, respectively, and Cs (g/L) is the adsorbent dosage.

For pH effect tests, the pH values of the adsorption systems were adjusted using NaOH and HCl solutions during the adsorption procedures. Other tests were performed at free pH.

Each test run was performed in triplicate, and the final values were presented as an average of the three measurements. The relative error was less than 5%.

Results and discussion

Characterization of silica samples

The S-SiO2 and L-SiO2 samples show spherical particles, as observed using TEM and SEM (Fig. 1). Their mean diameters (Dm), measured from the TEM and SEM images, are ~ 12 and 31 nm, respectively. The As values of the S-SiO2 and L-SiO2 were determined using the BET method to be 235 and 164 m2/g (Fig. S2, ESM), respectively. The Si/O molar ratios of the two SiO2 samples were determined by EDS all to be 0.49 ± 0.01, and no metallic elements are detectable, suggesting that the two samples have an extremely high purity and their surface hydroxyl groups exist in the neutral state (Sur-OH).

Fig. 1
figure 1

a,b TEM and c,d SEM images of a,c S-SiO2 and b,d L-SiO2

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The acid-base titration was performed at 298 K for the silica samples, and the data were analyzed using the Gran function method (Fig. S3, ESM). The so-obtained Ns values of the S-SiO2 and L-SiO2 are 0.72 and 0.48 mmol/g (or 3.06 and 2.93 μmol/m2, or 1.83 and 1.82 sites/nm2), respectively, which are consistent with the literature values (2.3–10 μmol/m2) [38, 59,60,61]. The two silica samples have the same surface hydroxyl density (~ 1.8 sites/nm2). In addition, the Ns values suggest that the maximum surface charge densities (σm, those at αSH = 1) of the S-SiO2 and L-SiO2 are about − 0.29 and − 0.28 C/m2, respectively.

The change in αSH with pH for the S-SiO2 and L-SiO2 was estimated (Fig. S4, ESM). The results show that the dissociation of surface hydroxyl groups for the silica samples obviously occurs at pH higher than ~ 6, similar to the literature reports [3, 38, 60]. It is worth to note that the pH at which the αSH begins to increase for the S-SiO2 (pH ~ 6.0) is slightly lower than that for the L-SiO2 (pH ~ 6.5), suggesting that the dissociation tendency of surface hydroxyl groups for the S-SiO2 is stronger than that for the L-SiO2. Based on the αSH data, the pKa values of the S-SiO2 and L-SiO2 are obtained at αSH = 0.5 to be 9.43 and 9.57, respectively, which are similar to the literature values (~ 7–10) [60]. The pKa value of the S-SiO2 is slightly lower than that of the L-SiO2. A low pKa value represents a strong dissociation tendency of the surface hydroxyl groups. Therefore, the pKa data, along with the αSH data, suggest that the surface hydroxyl groups of the S-SiO2 exhibit a higher dissociation tendency than those of the L-SiO2. Furthermore, these results also suggest that the affinity of the S-SiO2 for cationic species may be weaker than that of the L-SiO2, which is confirmed by the results of BDDABr adsorption tests. For clarity, the characterization results of the two silica samples are summarized in Table 1.

Table 1 Basic properties of the silica samples
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Micellization behavior of BDDABr in water

To understand the difference between the micellization of surfactant at solid–liquid interfaces and that in bulk solutions, the micellization behavior of BDDABr in bulk water was examined through conductivity (κ) measurements at three T (298, 303, and 308 K). Each plot of κ versus BDDABr concentration (C0) at a given T exhibits two straight lines with different slopes (Fig. S5, ESM). The slope change arises from the formation of micelles, thereby the concentration corresponding to the intersection point between the two straight lines is assigned to the critical micelle concentration (cmc) [53]. In addition, the degree of micellar counterion dissociation (αm) can be estimated from the ratio between the slopes of the κ–C0 lines above and below cmc [53]. The cmc and αm values of BDDABr in water are listed in Table 2 (and Fig. S6, ESM), which are close to the literature values [37, 53]. An increase in T leads to a rise of both cmc and αm values, which is consistent with the previous report [53].

Table 2 Micellization parameters of BDDABr in water at different T
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The thermodynamic parameters, including the standard Gibbs free energy (\( \Delta {\tilde{G}}_{\boldsymbol{m}}^{{}^{\circ}} \)), enthalpy (\( \Delta {\tilde{H}}_{\boldsymbol{m}}^{{}^{\circ}} \)), and entropy (\( \Delta {\tilde{S}}_{\boldsymbol{m}}^{{}^{\circ}} \)), for the micelle formation per mole of BDDABr can be calculated from the cmc and αm values using the following equations:

$$ \Delta {\tilde{G}}_{\boldsymbol{m}}^{{}^{\circ}}=\left(2-{\alpha}_{\boldsymbol{m}}\right) RT\ln {X}_{\mathrm{cmc}} $$
(9)
$$ \Delta {\tilde{H}}_{\boldsymbol{m}}^{{}^{\circ}}=-\left(2-{\alpha}_{\boldsymbol{m}}\right){RT}^2\frac{d\ln {X}_{cmc}}{dT} $$
(10)
$$ \Delta {\tilde{S}}_{\boldsymbol{m}}^{{}^{\circ}}=\frac{\Delta {\tilde{H}}_{\boldsymbol{m}}^{{}^{\circ}}-\Delta {\tilde{G}}_{\boldsymbol{m}}^{{}^{\circ}}}{T} $$
(11)

where X cmc is the cmc in terms of mole fraction, R is the gas constant (8.314 J/(mol K)), and T (K) is the absolute temperature. The values of the micellization thermodynamic parameters are also listed in Table 2. The negative \( \Delta {\tilde{G}}_{\boldsymbol{m}}^{{}^{\circ}} \) values indicate the thermodynamically spontaneous nature of the micellization, and the slight decrease in the absolute values of \( \Delta {\tilde{G}}_{\boldsymbol{m}}^{{}^{\circ}} \) with increasing T indicates that a high T is unfavorable for the micellization. The negative \( \Delta {\tilde{H}}_{\boldsymbol{m}}^{{}^{\circ}} \) value indicates that the micellization process is exothermic in nature. The positive \( \Delta {\tilde{S}}_{\boldsymbol{m}}^{{}^{\circ}} \) value arises from the release of solvated water molecules during the micellization process. The negative \( \Delta {\tilde{H}}_{\boldsymbol{m}}^{{}^{\circ}} \) and positive \( \Delta {\tilde{S}}_{\boldsymbol{m}}^{{}^{\circ}} \) values together reveal that the micellization is driven by both enthalpy and entropy. Owing to the absolute value of \( \Delta {\tilde{H}}_{\boldsymbol{m}}^{{}^{\circ}} \) being lower than that of \( -T\Delta {\tilde{S}}_{\boldsymbol{m}}^{{}^{\circ}} \), the micellization can be considered to be mainly entropy-driven. These results are similar to the previous report [53].

Adsorption isotherms

Effect of adsorbent dosage and particle size

The effect of silica dosages (Cs) on the BDDABr adsorption was examined at free pH and 298 K, as shown in Fig. 2. Obviously, the adsorption isotherms for the two silica samples are all independent of Cs, demonstrating that no Cs-effect exists in the adsorption under the studied conditions. Actually, previous reports on the Cs-effect are all associated with the adsorption of heavy metal ions or non-active organic substances at solid–liquid interfaces [55, 56], and there have been no reports about the Cs-effect existing in surfactant adsorption. Whether this is a feature of surfactant adsorption remains to be examined.

Fig. 2
figure 2

Adsorption isotherms of BDDABr onto a S-SiO2 and b L-SiO2 at different Cs (T = 298 K, free pH)

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Interestingly, it can be seen from Fig. 2 that the S-SiO2 and L-SiO2 exhibit different types of isotherms, namely, S-type and LS-type, respectively. An obvious double-plateau adsorption is observed for the L-SiO2, while for the S-SiO2, only at Ci higher than a given value (~ 3.5 mM), the adsorption obviously occurs. This suggests that the affinity of the S-SiO2 toward BDDABr is lower than that of the L-SiO2, which is consistent with the affinity of the two solids toward H+ ions, as suggested by their pKa values. In addition, the particle size-dependence of adsorption affinity observed here seems to be similar to those reported in the literature [47, 48] for adsorption of chicken egg lysozyme on silica nanoparticles [47] and HSA on NIPAM/BAM copolymer nanoparticles [48].

Notably, the above results are obtained at free pH. We noted that the free pH values of the S-SiO2 and L-SiO2 dispersions in equilibrium are different, being ~ 4.7 and 7.2, respectively, at a Cs of 20 g/L, which can be mainly attributed to the difference in the dissociation tendency of their surface hydroxyl groups. In addition, we also noted that, with the BDDABr adsorption, the free pH values of the S-SiO2 and L-SiO2 dispersions decrease to ~ 3.8 and 5.4 (Fig. S7, ESM), respectively, showing the adsorption is accompanied by the dissociation (or H+ release) of solid surface hydroxyl groups. Similar results were reported in the literature for the adsorption of cationic surfactants on silica in water [37, 39, 41, 42]. Therefore, the effect of particle size on the adsorption observed at free pH includes the effect of pH changes caused by the change in particle sizes. To further understand the difference in adsorption for the two silica samples, the adsorption was determined under controlled pH conditions.

Effect of pH

The change in the Γe with pH was first determined for the two silica samples at Ci = 30 mM, Cs = 20 mg/L, and T = 298 K, as shown in Fig. 3. The Ci of 30 mM corresponds to the maximum adsorption for the adsorption systems at various pH. It can be seen from Fig. 3 that, with increasing pH from ~ 3 to 10, the Γe values gradually increase, but which shows three successive stages. Initially, at pH < 4, the change of Γe with pH is very small (or the Γe keeps almost constant). Subsequently, at pH higher than ~ 4, the Γe shows a sharp increase. Finally, at pH higher than ~ 7, the Γe reaches a plateau. Notably, at pH higher than ~ 9.5, the Γe shows a slight increase, probably arising from the screening effect of electrolyte (NaOH) for the adsorption force [20] and the competitive adsorption of Na+ with BDDA+. The adsorption force of BDDABr on silica arises mainly from the electrostatic interaction between the cationic species BDDA+ and the negatively charged surface sites (Sur-O−). A high pH results in a high αSH value, thereby in a high Γe value. Actually, the change in the Γe with pH is similar to that in the αSH (Fig. S4, ESM), but the BDDABr adsorption facilitates the dissociation of surface hydroxyl groups. The plateau adsorption appearing at pH > 7 corresponds most likely to the complete dissociation of surface hydroxyl groups (i.e., αSH = 1).

Fig. 3
figure 3

Adsorption amounts of BDDABr onto S-SiO2 and L-SiO2 as a function of pH at Ci = 30 mM, Cs = 20 mg/L, and T = 298 K

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The Γe values of the S-SiO2 and L-SiO2 at low pH (< 4) are ~ 494 and 425 μmol/g (or ~ 2.13 and 2.44 μmol/m2), respectively, and those at high pH (> 7) are ~ 959 and 578 μmol/g (or ~ 4.08 and 3.52 μmol/m2), respectively. Partyka et al. [37, 39] reported that, at pH ~ 4.1 and 6.7, the maximum Γe values of BDDABr on a silica with 130 nm in size were ~ 2.08 and 4.32 μmol/m2, respectively. Our Γe values are close to the literature values. In addition, for our two silica samples, the adsorption amounts per unit area (μmol/m2) are close to each other, indicating that the higher adsorption amounts per unit mass (μmol/g) of the S-SiO2 result mainly from its larger As. The area occupied per BDDABr molecule at water–gas interfaces (aL-G) was reported to be ~ 0.71 nm2 [37], where the BDDABr molecule can be regarded in a “fully” hydrated state. The intrinsic cross-sectional area of a BDDABr molecule (a0, without hydration) is estimated using the molecular mechanic method to be ~ 0.54 nm2. The aL-G and a0 values suggest that the maximum adsorption capacity of a saturated adsorption monolayer should be in the range of ~ 2.34–3.08 μmol/m2. Therefore, our Γe data suggest that the adsorption of BDDABr on the two adsorbents form a monolayer (or small, isolated surface micelle) structure at low pH (< 4) and a bilayer structure at high pH (> 7). If this is true, the areas occupied per BDDABr molecule (aS-L) in the monolayers at low pH and in the bilayers at high pH are estimated all to be ~ 0.77 nm2, which is close to its aL-G value (~ 0.71 nm2). A similar aS-L value (~ 0.8 nm2) was reported by Partyka et al. [39] for BDDABr adsorption on 130 nm silica particles at free pH. Such high aS-L value (close to the aL-G value) suggests that the adsorbed layers are less compact. It has been revealed that, at Ce > cmc, the adsorption of surfactant molecules at solid–liquid interfaces commonly forms a bilayer structure [5, 62, 63] and that the adsorption generally leads to a strong dehydration of the surfactant molecules (or a low aS-L value) [62, 63]. Therefore, it might be more reasonable to consider that all the adsorption layers at both low and high pH consist of isolated surface micelles, namely, isolated admicelles (or noncomplete bilayers [39]).

To understand the effect of pH on the isotherm types, the adsorption of BDDABr on the S-SiO2 and L-SiO2 at controlled pH of 4.0, 5.4, and 9.0 was determined at Cs = 20 mg/L and T = 298 K, as shown in Fig. 4. The isotherms at pH 9.0 for both the two silica samples can be recognized as LS-type or as L-type owing to the fact that the first-step adsorption is not very apparent. Therefore, a changing trend in isotherm types, namely, from S-type through LS-type to L-type, is observed with increasing pH under the studied conditions. This indicates that the affinity of the adsorbents toward BDDABr increases with a raise of pH, which arises from the increase in the negative charges of solid surfaces. In addition, it can be seen from Fig. 4 that the difference of isotherm types between the S-SiO2 and L-SiO2 appears only at low pH (especially at pH = 4.0). A possible reason for this result is that, at high pH, the large surface density of ionized surface hydroxyl groups (Sur-O−) masks the effect of particle sizes on the adsorption.

Fig. 4
figure 4

Adsorption isotherms of BDDABr onto a S-SiO2 and b L-SiO2 at various controlled pH. T = 298 K, Cs = 20 g/L. The dots represent the experimental data, and the solid lines represent model fitting

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Effect of temperature

The effect of temperature on the adsorption of BDDABr onto the S-SiO2 (pH 4.0) and L-SiO2 (pH 5.4) was determined, as shown in Fig. 5. At the three T (298, 303, and 308 K) studied, the isotherms of the S-SiO2 all are S-type, while those of the L-SiO2 all are LS-type. That is, the change in T in the studied range has no influence on the isotherm type for each adsorbent. Interestingly, with increasing T from 298 to 308 K, the maximum Γe value of the S-SiO2 exhibits an obvious decrease from ~ 361 to 231 μmol/g, while that of the L-SiO2 exhibits a very slight decrease from ~ 410 to 375 μmol/g. Partyka et al. [37] reported that the maximum Γe values of BDDABr on large silica particles (130 nm in size) at free pH (~ 6.7) decreased slightly from ~ 173 to 160 μmol/g with an increase in T from 298 to 308 K. Our results are similar to the previous report [37]. The reason for the difference in the T-dependence of adsorption amounts for the S-SiO2 and L-SiO2 is not clear. A possible explanation is that the stronger affinity of the L-SiO2 for BDDABr results in the weaker effect of T on the adsorption.

Fig. 5
figure 5

Adsorption isotherms of BDDABr onto a S-SiO2 (pH 4.0) and b L-SiO2 (pH 5.4) at various T and Cs = 20 g/L. The dots represent the experimental data, and the solid lines represent model fitting

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Surface micellization model analysis

Theoretical basis

To understand the surface micellization behavior, the obtained S-type and LS-type isotherms were analyzed using the one-step and two-step models, respectively. For clarity, the theoretical bases of the two models are summarized as follows:

One-step model

The one-step model [1, 11] supposes that the adsorption of surfactants at solid–liquid interfaces occurs in one step. That is, surfactant monomers adsorb on a surface site to directly form a surface micelle. The adsorption equilibrium can be represented as:

$$ \mathrm{Site}+n\;\mathrm{Monomer}\rightleftharpoons \mathrm{Surface}\ \mathrm{micelle} $$
(12)

and its equilibrium constant, Kos, is:

$$ {K}_{\boldsymbol{os}}=\frac{a_{\boldsymbol{s}\boldsymbol{m}}}{a_{\boldsymbol{s}}{a}_{\boldsymbol{m}}^n} $$
(13)

where n is the average aggregation number of surface micelles, and as, am, and asm are the activities of unoccupied surface sites, free monomers, and surface micelles, respectively. For dilute adsorption systems, a = C. An isotherm equation of the one-step model can be derived as:

$$ {\varGamma}_{\boldsymbol{e}}=\frac{\varGamma_{\infty }{K}_{\boldsymbol{os}}{C}_{\boldsymbol{e}}^n}{1+{K}_{\boldsymbol{os}}{C}_{\boldsymbol{e}}^n} $$
(14)

where Γ∞ is the limiting adsorption amount.

Two-step model

The two-step model [1, 12–14] supposes that the adsorption of surfactants at solid–liquid interfaces occurs in two steps. In the first step, surfactant monomers adsorb on solid surface sites as individuals through electrostatic attraction and/or specific (i.e., van der Waals) attraction, and no aggregates form. The first-step adsorption equilibrium can be represented as:

$$ \mathrm{Site}+\mathrm{Monomer}\rightleftharpoons \mathrm{Adsorbed}\ \mathrm{monomer} $$
(15)

and its equilibrium constant, k1, is:

$$ {k}_{\boldsymbol{1}}=\frac{a_{\boldsymbol{am}}}{a_{\boldsymbol{s}}{a}_{\boldsymbol{m}}} $$
(16)

where aam is the activity of adsorbed monomers.

In the second step, free surfactant monomers adsorb on the previously adsorbed monomers through hydrophobic interaction between tails of surfactants, to form surface micelles. The second-step adsorption equilibrium can be represented as:

$$ \left(n\hbox{--} 1\right)\ \mathrm{Monomers}+\mathrm{Adsorbed}\ \mathrm{monomer}\rightleftharpoons \mathrm{Surface}\ \mathrm{micelle} $$
(17)

and its equilibrium constant, k2, is:

$$ {k}_{\boldsymbol{2}}=\frac{a_{\boldsymbol{sm}}}{a_{\boldsymbol{am}}{a}_{\boldsymbol{m}}^{n-1}} $$
(18)

A general isotherm equation of the two-step model can be derived as:

$$ {\varGamma}_{\boldsymbol{e}}=\frac{\varGamma_{\infty }{k}_1{C}_{\boldsymbol{e}}\left(\frac{1}{n}+{k}_2{C}_{\boldsymbol{e}}^{n- 1}\right)}{1+{k}_1{C}_{\boldsymbol{e}}\left( 1+{k}_2{C}_{\boldsymbol{e}}^{n- 1}\right)} $$
(19)

Notably, the two-step isotherm has two important limiting cases [1, 14]. If k2 → 0 and n → 1, it reduces to the Langmuir isotherm. If n > 1 and klCe ≪ 1, it reduces to the one-step isotherm. Therefore, the Langmuir and one-step models can be regarded as two limiting cases of the more general two-step model.

Critical surface micelle concentration

The critical surface micelle concentration (csmc) is defined as the concentration at which surface micelles begin to form on the solid surface and can be estimated from model parameter values using the following equations [1, 14]:

$$ \mathrm{for}\ \mathrm{S}\hbox{-} \mathrm{type}\ \mathrm{isotherm}, csmc={\left(\frac{n-1}{n+1}\right)}^{\left(n+1\right)/n}{K}_{\boldsymbol{os}}^{-1/n} $$
(20)
$$ \mathrm{for}\ \mathrm{LS}\hbox{-} \mathrm{type}\ \mathrm{isotherm}, csmc={\left(\frac{n-2}{n}\right)}^{n/\left(n-1\right)}{k}_{\boldsymbol{2}}^{1/\left(1-n\right)} $$
(21)

Thermodynamics of surface micellization

From the equilibrium constants of adsorption, the standard Gibbs free energy of surface micellization per mole of surfactant (\( \Delta {\tilde{G}}_{\boldsymbol{sm}}^{{}^{\circ}} \)) can be calculated using the following equations [1, 14]:

for S-type isotherm,

$$ \Delta {\tilde{G}}_{\boldsymbol{sm}}^{{}^{\circ}}=-\frac{1}{n} RT\ln {K}_{\boldsymbol{os}} $$
(22)

for LS-type isotherm,

$$ \Delta {\tilde{G}}_{\boldsymbol{sm}}^{{}^{\circ}}=-\frac{1}{n} RT\ln {k}_{\boldsymbol{2}} $$
(23)

The standard entropy (\( \Delta {\tilde{S}}_{\boldsymbol{sm}}^{{}^{\circ}} \)) and enthalpy (\( \Delta {\tilde{H}}_{\boldsymbol{sm}}^{{}^{\circ}} \)) of surface micellization per mole of surfactant can be calculated using the following equations [1, 14]:

$$ \Delta {\tilde{S}}_{\boldsymbol{sm}}^{{}^{\circ}}=-\frac{d\Delta {\tilde{G}}_{\boldsymbol{sm}}^{{}^{\circ}}}{dT} $$
(24)
$$ \Delta {\tilde{H}}_{\boldsymbol{sm}}^{{}^{\circ}}=\Delta {\tilde{G}}_{\boldsymbol{sm}}^{{}^{\circ}}+T\Delta {\tilde{S}}_{\boldsymbol{sm}}^{{}^{\circ}} $$
(25)

Modeling for adsorption data

The fitting procedure of surface micellization models for adsorption data was performed using the 1stOpt software with the Levenberg-Marquardt algorithm. The best-fit values of model parameters for a set of Γe − Ce data (or an isotherm) were automatically obtained from the software, which correspond to the minimum sum of square of residuals, SSresiduals = Σ(Γcal − Γexp)2, where Γcal and Γexp are the calculated and experimental adsorption amounts, respectively.

The fitting plots of the one-step and two-step models for the adsorption data are shown in Figs. 2, 4, and 5. All of the model plots coincide well with the experimental data, demonstrating that the one-step and two-step models can adequately describe the S-type and LS-type isotherms, respectively, obtained for the adsorption systems. The best-fit values of the model parameters and the correlation coefficient (R2) are listed in Tables 3 and 4 (and Tables S1 and S2, ESM). Notably, the expressed unit of concentration (mmol/L) in the Kos ((mmol/L)–n), k1 ((mmol/L)−1), and k2 ((mmol/L)–(n–1)) is converted to mole fraction (mol/mol) for calculation of \( \Delta {\tilde{G}}_{\boldsymbol{sm}}^{{}^{\circ}} \) using Eqs. (22) and (23). In addition, the isotherms at pH 9.0 for the S-SiO2 and L-SiO2 were also fitted using the Langmuir model (see Section S1, Fig. S8, and Table S3, ESM), showing that they can be described also using the Langmuir model, namely, can also be recognized as L-type. However, it is better to recognize the two isotherms as LS-type owing to the maximum adsorption amounts (Γm) obtained from the Langmuir model (1230 and 723 μmol/g) being obviously higher than the experimental (Γ∞) values (~ 1040 and 580 μmol/g).

Table 3 Best-fit values of model parameters for BDDABr adsorption on S-SiO2 and L-SiO2 at different pH (T = 298 K)
Full size table
Table 4 Best-fit values of model parameters for BDDABr adsorption on S-SiO2 (pH 4.0) and L-SiO2 (pH 5.4) at different T
Full size table

From Tables 3 and 4, it can be seen that the n values obtained at different conditions are in the range of 3.16–16.1, which is consistent with the reported n values (3–20) for surface micelles in the literature [1, 14, 21]. With increasing pH, the n values for the two silica samples all decrease obviously (Table 3). Similar results were reported in the literature [1]. It can be seen from Table 3 that, for the LS-type adsorption, an increase in pH results in a rise of the first-step adsorption (Γ1, see Table S1, ESM). Enough high Γ1 may result in one surface micelle developing around more than one anchor chain [1], thereby resulting in a decrease in n. With increasing T, the n values for the two silica samples also decrease (Table 4), which is consistent with the previous report on BDDABr adsorption on silica [37] but contrary to the previous report on adsorption of the nonionic surfactant decylmethylsulphoxide (DEMS) on carbon black [13]. We noted that the Γ∞ values of DEMS on carbon black increase with increasing T [13]. However, for the adsorption of BDDABr on silica, an opposite change in Γ∞ with T occurs. Therefore, our result is reasonable. In addition, the T-dependence of n values for BDDABr observed here is similar to the reports for ionic surfactants in bulk solutions [64, 65]. Notably, the change of the n value for the L-SiO2 with T is very small, which is probably related to the small change in its Γ1 and Γ∞ values.

The csmc values obtained at different conditions are in the range of 0.47–4.69 mM (Tables 3 and 4), which is obviously lower than the cmc of BDDABr in bulk water (5.56–6.18 mM). This is a feature of surface micellization of surfactants [1, 14, 21, 28]. With increasing pH, the csmc values on the two silica samples obviously decrease (Table 3). Pham et al. [21] reported that the csmc values for adsorption of the anionic surfactant sodium dodecyl sulfate (SDS) on α-Al2O3 increase with increasing pH. It should be noted that, with increasing pH, the affinity of SDS toward α-Al2O3 decreases while that of BDDABr toward silica increases. Therefore, the result of the csmc values decreasing with pH obtained here can be attributed to the enhancement of the affinity between BDDABr and silica. With increasing T, the csmc values on the two silica samples slightly increase (Table 4), which is similar to the change of cmc of BDDABr in bulk water.

The \( \Delta {\tilde{G}}_{\boldsymbol{sm}}^{{}^{\circ}} \) and \( \Delta {\tilde{H}}_{\boldsymbol{sm}}^{{}^{\circ}} \) values obtained at different conditions are negative and the obtained \( \Delta {\tilde{S}}_{\boldsymbol{sm}}^{{}^{\circ}} \) values are positive (Tables 3 and 4), which are the same as the case of micellization of BDDABr in bulk water. Notably, the absolute values of \( \Delta {\tilde{G}}_{\boldsymbol{sm}}^{{}^{\circ}} \) (17.28–23.49 kJ/mol) are lower than those of \( \Delta {\tilde{G}}_{\boldsymbol{m}}^{{}^{\circ}} \) of BDDABr in bulk water (37.75–38.32 kJ/mol). One possible reason is that the degree of counterion dissociation of surface micelles is not taken into account in the one-step and two-step models. In addition, the absolute values of \( \Delta {\tilde{H}}_{\boldsymbol{sm}}^{{}^{\circ}} \) are obviously lower than that of \( -T\Delta {\tilde{S}}_{\boldsymbol{sm}}^{{}^{\circ}} \), demonstrating that the surface micellization is mainly entropy-driven, which is similar to the bulk micellization. With increasing pH, the absolute values of \( \Delta {\tilde{G}}_{\boldsymbol{sm}}^{{}^{\circ}} \) for the two silica samples all decrease (Table 3), suggesting the surface micellization tendency becomes weaker. With increasing T, the absolute values of \( \Delta {\tilde{G}}_{\boldsymbol{sm}}^{{}^{\circ}} \) for the two silica samples all slightly increase (Table 4), suggesting the surface micellization tendency becomes stronger, which is contrary to the bulk micellization. Furthermore, we can see from Table 3 that, at given pH and T, the n and csmc values of the S-SiO2 are higher than those of the L-SiO2, demonstrating that the affinity of the S-SiO2 toward BDDABr is lower than that of the L-SiO2. The model fitting result is consistent with the results of isotherm type observations and pKa determinations.

Conclusions

The adsorption of BDDABr on two silica samples, S-SiO2 and L-SiO2, in water was investigated. No Cs-effect is observed for the adsorption, while the size of silica particles, besides pH and T, has an obvious impact on the adsorption. The S-SiO2 shows a S-type isotherm while the L-SiO2 shows a LS-type isotherm at low pH (~ 4.0). An increase in pH may lead to a change in the isotherm types from S-type through LS-type to L-type. The S-type and LS-type isotherms can be adequately described using the one-step and two-step models, respectively. With increasing pH, the n and csmc values all decrease. With increasing T, the n values decrease while the csmc values slightly increase. At given pH and T, the n and csmc values for the S-SiO2 are obviously higher than those for the L-SiO2. These results suggest that the affinity of the S-SiO2 toward BDDABr is lower than that of the L-SiO2, which is consistent with the dissociation tendency of their surface hydroxyl groups. This work provides a well understanding of the particle size-dependence of cationic surfactant adsorption at silica–liquid interfaces.