Theoretical Investigation of the Adsorption of Cadmium Iodide from Water Using Polyaniline Polymer Filled with TiO 2 and ZnO Nanoparticles

: The removal of heavy metals from drinking water has attracted great interest in water puriﬁcation technology. In this study, a biocompatible Polyaniline (PANI) polymer ﬁlled with TiO 2 and ZnO nanoparticles (NPs) is considered as an adsorbent of cadmium iodide from water. Theoretical investigation of the van der Waals (vdW) interactions deduced from the Hamaker constant calculated on the basis of Lifshitz theory was presented. It was found that the surface energy as well as the work of adhesion between water and PANI/NPs across air increases with an increasing volume fraction of the TiO 2 and ZnO nanoparticles. Consequently, an increase in the Laplace pressure around the cavities/porosities was found, which leads to the enhancement of the speciﬁc contact surface between water and PANI/NPs. On the other hand, for the interactions between CdI 2 particles and PANI/NPs surface across water, we show that the interactions are governed principally by the attractive London dispersion forces. The vdW energy and force increase proportionally with the augmentation of the volume fraction of nanoparticles and of the radius of the CdI 2 particle. Particularly, the PANI/TiO 2 has been proved to be a better candidate for adsorption of cadmium iodide from water than PANI/ZnO.


Introduction
Cadmium metal has become a material that is inescapable for a wide variety of industrial applications such as batteries, alloys, coatings, solar cells, plastic stabilizers, and pigments [1][2][3][4][5]. These successful industrial features mask many environmental and health compatibility problems. Cadmium is non-biodegradable and hence, once released to the environment, it stays in circulation, especially in potable water and soil [6][7][8]. Cadmium iodide is one of the cadmium compound wastes. It can get into drinking water (groundwater or surface water) from the wastewater of electroplating industries and the manufacturing of phosphors [9][10][11]. In fact, such industries use an aqueous solution containing cadmium iodide Cd-I complexes such as CdI + , CdI 2 , CdI 3 − , and CdI 4 2− as an electrolyte [12,13]. Humans get exposed to cadmium by ingestion (drinking or eating) or inhalation. Ailments such as bone disease, renal damage, and several forms of cancer are attributed to overexposure to cadmium [14]. For human health safety, the World Health Organization of the effect of the volume fraction Φ and the type of nanoparticles filler on the vdW interactions between water and PANI/NPs in air. In the fourth section, we evaluate the vdW interactions between a spherical cadmium particle waste CdI 2 and the surface PANI/NPs in a water medium as an adsorption case study. The effect of the volume fraction, the type of nanoparticles filler, and the particle waste of CdI 2 on the vdW interactions is discussed. Section 4 presents the conclusions of this work.

Computational Methods
The computational method used in this study has been validated by many research works for evaluating the vdW interactions for adsorption studies for the diversity of materials liquids, solids, and gas. It has presented good accordance with experimental results [34][35][36][37][38].

VdW Interactions between Two Flat Surfaces across Air
VdW interactions are the principal driving interactions responsible for the adsorption phenomenon. They are responsible for the attractions between any two bodies because, while they are not as strong as Coulombic or H-bonding interactions, they are always present and can be important both at small and large separations [38] (pp. 107-130). Two flat surfaces interacting across the third medium, like the one that exists between the surface of PANI/NPs nanocomposites and water (Figure 1), involve vdW energy as given by the following equation [38] (p. 254): where x is the separating distance between two interacting mediums and H is the famous Hamaker constant.
Water 2021, 13, x FOR PEER REVIEW 3 of 17 the refractive index of PANI/NPs as a function of volume fraction Φ of the nanoparticles and the study of the effect of the volume fraction Φ and the type of nanoparticles filler on the vdW interactions between water and PANI/NPs in air. In the fourth section, we evaluate the vdW interactions between a spherical cadmium particle waste CdI2 and the surface PANI/NPs in a water medium as an adsorption case study. The effect of the volume fraction, the type of nanoparticles filler, and the particle waste of CdI2 on the vdW interactions is discussed. Section 4 presents the conclusions of this work.

Computational Methods
The computational method used in this study has been validated by many research works for evaluating the vdW interactions for adsorption studies for the diversity of materials liquids, solids, and gas. It has presented good accordance with experimental results [34][35][36][37][38].

VdW Interactions between Two Flat Surfaces across Air
VdW interactions are the principal driving interactions responsible for the adsorption phenomenon. They are responsible for the attractions between any two bodies because, while they are not as strong as Coulombic or H-bonding interactions, they are always present and can be important both at small and large separations [38] (pp. 107-130). Two flat surfaces interacting across the third medium, like the one that exists between the surface of PANI/NPs nanocomposites and water (Figure 1), involve vdW energy as given by the following equation [38] (p. 254): where x is the separating distance between two interacting mediums and H is the famous Hamaker constant.
where x0 is the cut-off which defines as the interatomic distance between the water and the PANI/NPs surface. It is typically equal to 0.165 nm. The distance x0 yields values for where x 0 is the cut-off which defines as the interatomic distance between the water and the PANI/NPs surface. It is typically equal to 0.165 nm. The distance x 0 yields values Water 2021, 13, 2591 4 of 17 for surface energy in such good agreement with those measured, even for very different liquids and solids [38] (p. 257).

Van der Waals Interactions between Spherical Particle and Flat Surface across Water
As a case study of the adsorption, we consider a spherical cadmium iodide CdI 2 particle of a radius R c interacted with the flat surface of PANI/NPs over water medium at room temperature (298.15 K), as shown in Figure 2. The CdI 2 particle is subjected to the vdW interaction by the surface of PANI/NPs that can be attractive or repulsive. At a separating distance x<<R (in the range of 0.1-50 nm), the corresponding vdW energy is a function of the Hamaker constant H and the radius R c of the particle of CdI 2 and the separating distance x as [38] (p. 254): The corresponding adhesion force F resulting from the vdW interaction is defined as the first derivate of the vdW energy as [38] (p. 254): surface energy in such good agreement with those measured, even for very different liquids and solids [38] (p. 257).

Van der Waals Interactions between Spherical Particle and Flat Surface across Water
As a case study of the adsorption, we consider a spherical cadmium iodide CdI2 particle of a radius Rc interacted with the flat surface of PANI/NPs over water medium at room temperature (298.15 K), as shown in Figure 2. The CdI2 particle is subjected to the vdW interaction by the surface of PANI/NPs that can be attractive or repulsive. At a separating distance x<<R (in the range of 0.1-50 nm), the corresponding vdW energy is a function of the Hamaker constant H and the radius Rc of the particle of CdI2 and the separating distance x as [38] (p. 254): The corresponding adhesion force F resulting from the vdW interaction is defined as the first derivate of the vdW energy as [38] (p. 254): In the aim to study the effect of particle size, we used two size particles of CdI2 with radius Rc = 136 nm and Rc = 348 nm with known physical properties [11].

Method for Calculation of the Hamaker Constant
In the basis of the Lifshitz theory, the nonretarded Hamaker constant H between two mediums 1 and 2 interacting across medium 3 is expressed in terms of dielectric constant ε and the refractive index n [39]: where the H P is the polar component of the nonretarded Hamaker constant; it regroups the Keesom interaction that arises from permanent molecular dipoles and the Debye interaction that arises from permanent dipoles and induced dipoles [40][41][42]. It is expressed as function as dielectric permittivity by [38] (p. 260): In the aim to study the effect of particle size, we used two size particles of CdI 2 with radius R c = 136 nm and R c = 348 nm with known physical properties [11].

Method for Calculation of the Hamaker Constant
In the basis of the Lifshitz theory, the nonretarded Hamaker constant H between two mediums 1 and 2 interacting across medium 3 is expressed in terms of dielectric constant ε and the refractive index n [39]: where the H P is the polar component of the nonretarded Hamaker constant; it regroups the Keesom interaction that arises from permanent molecular dipoles and the Debye interaction that arises from permanent dipoles and induced dipoles [40][41][42]. It is expressed as function as dielectric permittivity by [38] (p. 260): where k B is the Boltzmann constant, T is the temperature, ε 1 , ε 2 , and ε 3 are the dielectric permittivity of the three interacting mediums. The H D regroups the London dispersion interactions between instantaneously induced dipoles. It is a function of the refractive index of the 3 interacting mediums and is given by [38] (p. 260): ν e is the main electronic absorption frequency in the UV, typically around 3 × 10 15 s −1 [38] (p. 260), h is the Planck constant, and (n 1 , n 2 , n 3 ) are respectively the refractive indexes which are visible (water, sample, air). For metals, the dielectric constant is very high as infinite, and then we may neglect the contribution of the term (ε 1 − ε 2 )/(ε 1 − ε 2 ) in Equation (3). Therefore, the polar component of the nonretarded Hamaker constant H P is function only of the dielectric constants and temperature, and is simplified as follows [43]:

Model for Calculation the Dielectric Constant of Nanocomposites
In this study, the pure PANI polymer and the PANI filled by the nanoparticles of TiO 2 and ZnO with volume fraction Φ = (20%, 40%, 60%, 80%) are used.
To calculate the dielectric constant of the nanocomposites, the power-law model was used, considering that the spherical particle filler is uniformly dispersed in a continuous PANI matrix [44].
where Φ is the filler volume fraction (%), ε c , ε m , and ε f are de dielectric constant respectively of the nanocomposite (PANI/NPs), the polymer matrix (PANI), and the nanoparticles filler (NPs of TiO 2 and ZnO).

Models for Calculation of the Refractive Index
The calculation of the refractive index for PANI/NPs was done referring to the pioneering work the most popular mixing theory of Maxwell garnet [45]: where Φ is the filler volume fraction (%), n c , n m and n f are the refractive indexes respectively of nanocomposites (PANI/NPs), matrix (PANI), and NPs filler (TiO 2 and ZnO).

Materials
The theoretical calculations of the nonretarded Hamaker constant on the basis of the Lifshitz theory provide a known dielectric constant and the refractive index of the three interacting materials. Table 1 summarizes the dielectric constants and the refractive indexes of the PANI, the NPs of TiO 2 and ZnO, water, and CdI 2 particle considered in this study. The values of the dielectric constants were taken at 1 MHz. Where, the refractive indexes were taken at visible wavelength 600 nm.

Results and Discussion
All calculations and discussions of the results using computational method are carried out at room temperature (298.15 K). Table 2 reports the value of the dielectric constant ε and the refractive index n of nanocomposites PANI/NPs with the correspondent volume fraction Φ of TiO 2 and ZnO nanoparticles using Equations (9) and (10). The ε of PANI/TiO 2 increases with the volume fraction Φ of TiO 2 NPs. However, the ε for PANI/ZnO decreases with the volume fraction Φ of ZnO NPs. This result is attributed to the high dielectric constant of TiO 2 NPs rather than the pure PANI and the less dielectric constant of the ZnO NPs. However, due to the higher refractive index of NPs compared with the pure PANI, we find that the refractive indexes n of PANI/TiO 2 and PANI/ZnO increases with the increasing the volume fraction Φ of NPs. The comparison of n values visualized in the table affirms that the PANI/TiO 2 exhibits a higher refractive index than those for PANI/ZnO.

VdW Interaction between PANI/NPs and Water across Air
This part of the study reports a prediction and evaluation of the type of the vdW interactions (attractive or repulsive) between water and PANI/NPs across air using the computational methods as described in Section 2.1.1. Figure 3 reports the values of the nonretarded Hamaker constants H, H P , and H D between PANI/NPs and water across air as function as the volume fraction Φ of the NPs calculated using Equations (5)- (7). It is remarked that all values of nonretarded Hamaker constants are positives, suggesting that the interactions of PANI/NPs and water across air are always attractive for the two types of NPs-TiO 2 and ZnO-and at all volume fraction Φ of NPs. However, for the same Φ, a higher dominance (>94%) of the dispersive nonretarded Hamaker constant H D in the total value of Hamaker constant H is observed. This result was particularly relevant because it affirms the dominance of vdW dispersive attractions between water and PANI/NPs across the air.   Figure 4 illustrates the vdW energy for PANI/TiO2 and PANI/ZnO at the interatomic distance x0 = 0.165 nm. We find that the vdW energy is much higher, with an increasing volume fraction Φ of NPs, and this fact is remarkable for PANI/TiO2. This result reflects that the TiO2 NPs is better than ZnO NPs for enhancing the vdW attractive energy between water and PANI/NPs.   [50,51], which shows the  Figure 4 illustrates the vdW energy for PANI/TiO 2 and PANI/ZnO at the interatomic distance x 0 = 0.165 nm. We find that the vdW energy is much higher, with an increasing volume fraction Φ of NPs, and this fact is remarkable for PANI/TiO 2 . This result reflects that the TiO 2 NPs is better than ZnO NPs for enhancing the vdW attractive energy between water and PANI/NPs.   Figure 4 illustrates the vdW energy for PANI/TiO2 and PANI/ZnO at the interatomic distance x0 = 0.165 nm. We find that the vdW energy is much higher, with an increasing volume fraction Φ of NPs, and this fact is remarkable for PANI/TiO2. This result reflects that the TiO2 NPs is better than ZnO NPs for enhancing the vdW attractive energy between water and PANI/NPs.     [50,51], which shows the accuracy of the calculation method. As already expected from the Hamaker constant evolution, the surface energy was increasing with the volume fraction Φ for two PANI/NPs except the polar part γ P for PANI/ZnO because of the decrease in their dielectric constant with the volume fraction Φ ( Table 1). accuracy of the calculation method. As already expected from the Hamaker constant evolution, the surface energy was increasing with the volume fraction Φ for two PANI/NPs except the polar part γ P for PANI/ZnO because of the decrease in their dielectric constant with the volume fraction Φ (Table 1). As expected from the Hamaker constant behavior, the PANI/TiO2 composites exhibit higher surface energy values for all volume fractions Φ of NPs. This is attributed to the higher dielectric constant and the refractive index of the PANI/TiO2.
It is noticeable that for the two PANI/TiO2 and PANI/ZnO, the surface energy is dominated principally by their dispersive γ D part, which represents >94% in the total surface energy γ. For such cases, the work of adhesion WA between PANI/NPs and water is approximated as [52]: where γ and γ represent respectively the polar part of the surface energy of PANI/NPs and water. γ is typically equal to 19.9 × 10 −3 N.m [52]. As already expected from Figure 6, the work of adhesion WA between PANI/NPs-Water across air increases with the volume fraction Φ of NPs and the highest adhesion occurs between PANI/TiO2-Water. Therefore, it is well known that the nanoparticles filler enhances the adhesion between PANI/NPs and water, and the adhesion is involved principally in the London dispersion forces. As expected from the Hamaker constant behavior, the PANI/TiO 2 composites exhibit higher surface energy values for all volume fractions Φ of NPs. This is attributed to the higher dielectric constant and the refractive index of the PANI/TiO 2 .
It is noticeable that for the two PANI/TiO 2 and PANI/ZnO, the surface energy is dominated principally by their dispersive γ D part, which represents >94% in the total surface energy γ. For such cases, the work of adhesion W A between PANI/NPs and water is approximated as [52]: where γ D 1 and γ D 2 represent respectively the polar part of the surface energy of PANI/NPs and water. is typically equal to 19.9 × 10 −3 N.m [52].
As already expected from Figure 6, the work of adhesion W A between PANI/NPs-Water across air increases with the volume fraction Φ of NPs and the highest adhesion occurs between PANI/TiO 2 -Water. Therefore, it is well known that the nanoparticles filler enhances the adhesion between PANI/NPs and water, and the adhesion is involved principally in the London dispersion forces. As mentioned previously in this section, we have shown that the PANI/NPs-water interactions at the interatomic distance x0 was enhanced with the increase in the rate of volume fraction Φ of NPs.  As mentioned previously in this section, we have shown that the PANI/NPs-water interactions at the interatomic distance x 0 was enhanced with the increase in the rate of volume fraction Φ of NPs.
Generally, the common adsorbent materials present much nano and micro porosity architecture; this porosity gives rise to a significant Laplace pressure that drives the interface in the concave direction, and then it is responsible for the driving water in the cavities as mentioned in Figure 7. This fact leads to an increase in the specific contact surface, and then the adsorption of waste from water in the cavities can be enhanced. As mentioned previously in this section, we have shown that the PANI/NPs-water interactions at the interatomic distance x0 was enhanced with the increase in the rate of volume fraction Φ of NPs.
Generally, the common adsorbent materials present much nano and micro porosity architecture; this porosity gives rise to a significant Laplace pressure that drives the interface in the concave direction, and then it is responsible for the driving water in the cavities as mentioned in Figure 7. This fact leads to an increase in the specific contact surface, and then the adsorption of waste from water in the cavities can be enhanced. The Laplace pressure is expressed as a function of surface energy γ and the cavity radius Rc as: The Laplace pressure is expressed as a function of surface energy γ and the cavity radius R c as: P L = γ R c (12) Figure 8 reports the evolution of the Laplace pressure for typical size cavity radius R c = 100 µm.  The attraction around the cavities/porosities caused by Laplace pressure is enhanced by incorporation of NPs. We compute that this finding enhances the specific contact surface and then the adsorption of particle waste from water in the contact by considering The attraction around the cavities/porosities caused by Laplace pressure is enhanced by incorporation of NPs. We compute that this finding enhances the specific contact surface and then the adsorption of particle waste from water in the contact by considering the contribution of the Laplace pressure.

The Nonretarded Hamaker Constants Behavior
We illustrate the interaction between cadmium iodide CdI 2 (medium 1) and PA-NI/NPs (medium 2) across water (medium 3), as shown in Figure 2 at room temperature (298.15 K).
Tables 3 and 4 summarize the nonretarded Hamaker constants H, H P , and H D , for the interactions between the spherical particle of CdI 2 and the flat surface of PANI/NPs nanocomposites across water. The calculation was done using Equations (5)-(8) for the two CdI 2 radius particle R c = 136 nm and R c = 348 nm and for various volume fractions (0, 20%, 40%, 60%, and 80%) of NPs of TiO 2 and ZnO.    The values of the nonretarded Hamaker constants are located in the range of most condensed phases (0.4-410 −20 J), which shows the accuracy of our calculation [38] (p. 255). The dispersive nonretarded Hamaker constant H D is positive and represents 99% of the total nonretarded Hamaker constant H. Therefore, we have led to the important ascertainment that the interaction between CdI 2 particle and PANI/NPs across water is always attractive (H > 0) and is governed essentially by the London disperse vdW forces. Whereas, the H P is negative and contributes about 1% in the total H value and then their repulsive effect on the vdW interactions can be neglected. Considering Equation (8), the negative value of H P is attributed to the lower dielectric constant of PANI/NPs to the dielectric constant of water (ε = 78.1), except for (ε(PANI/TiO 2 )) at (Φ = 0.6-0.8).
Consequently, these findings imply that the interactions between the spherical particle of CdI 2 and the flat surface of PANI/NPs nanocomposites across water are attractive. Note that the dispersive interactions depend only on the orbiting electron frequency, ν, and the refractive index, n (of the three interacting mediums), which are independent of temperature; we reveal that the attraction between CdI 2 and PANI/NPs is always admissible for the functional temperature around the ambient temperature.
Note that the dispersive nonretarded Hamaker constant H D is proportional to the refractive index of the mediums (Equation (7)), we found an increase of the nonretarded Hamaker constant H with increasing the volume filler fraction Φ and the refractive index of the CdI 2 particle. Interestingly, the increase in the vdW attraction was greater for TiO 2 NPs compared to that for ZnO.

VdW Energy
To evaluate the energy of the adsorption of the CdI 2 particle on the PANI/NPs surface from the water medium, we have computed the vdW energy given by relation (3) for the various volume fraction Φ of NPs and CdI 2 particle radius R c . We have scaled the vdW energy to the k B T room (T room = 298.15 K) to quantify the strength of the vdW energy along the separating distance x. As illustrated in Figure 9, the behavior of the vdW energy versus the separating distance x is typically for attractive energy curves. At the nonretarded regime (x < 20 nm), the vdW energy curve undergoes asymptotic growth at around U = −800 k B T room (for x = 1.4 nm), this behavior was found independently of the volume fraction Φ of NPs and of the radius R c of the CdI 2 particle. This finding supports that the vdW energy is sufficiently strong at a small distance from PANI/NPs. Compared to PANI/ZnO-CdI2 (Figure 9c,d), the PANI/TiO2-CdI2 (Figure 9a,b) exhibits a higher vdW energy throughout the separating distance x (0-50 nm), which is caused by the higher refractive index of TiO2 to that for ZnO NPs. Hence, the incorporation of the TiO2 NPs leads to an increasing attraction, and then we show that TiO2 NPs are better than ZnO NPs for enhancing the London attraction energy between CdI2 particles and Compared to PANI/ZnO-CdI 2 (Figure 9c,d), the PANI/TiO 2 -CdI 2 (Figure 9a,b) exhibits a higher vdW energy throughout the separating distance x (0-50 nm), which is caused by the higher refractive index of TiO 2 to that for ZnO NPs. Hence, the incorporation of the TiO 2 NPs leads to an increasing attraction, and then we show that TiO 2 NPs are better than ZnO NPs for enhancing the London attraction energy between CdI 2 particles and PANI/NPs nanocomposites across water medium.
Furthermore, as can be predicted from Equation (3), the vdW energy for the same type of NPs (Figure 9a-d) increased by increasing the radius R c of the CdI 2 particles.

VdW Force-Distance Curves
The vdW forces F between CdI 2 -PANI/NPs across water was calculated along the separating distance x (0.1-50 nm) using Equation (4). As illustrated in Figure 10, the vdW force-distance curves show a typically attractive behavior. For the nonretarded regime (approximately for x< 20 nm), we observe a dispersion in the force-distance curve (noted in Figure 10a increase of the vdW forces. The shape of this dispersion increases with the volume fraction Φ of the NPs and the particle radius R c of the CdI 2 . Water 2021, 13, x FOR PEER REVIEW 13 of 17 in Figure 10a increase of the vdW forces. The shape of this dispersion increases with the volume fraction Φ of the NPs and the particle radius Rc of the CdI2. Furthermore, the vdW force curves become stronger with the decreasing x, and this finding additionally supports that vdW attractions are the foremost driving force for particle adsorption at a long-range distance (x < 10 nm).
Thereby, taking account these above-mentioned findings, the vdW force and energy between CdI2 particle and PANI/NPs surfaces across water are enhanced by increasing the volume fraction Φ of NPs and the radius Rc of the CdI2 particle. Moreover, we can Furthermore, the vdW force curves become stronger with the decreasing x, and this finding additionally supports that vdW attractions are the foremost driving force for particle adsorption at a long-range distance (x < 10 nm).
Thereby, taking account these above-mentioned findings, the vdW force and energy between CdI 2 particle and PANI/NPs surfaces across water are enhanced by increasing the volume fraction Φ of NPs and the radius R c of the CdI 2 particle. Moreover, we can emphasize the advantageous effect of metal oxide nanoparticle filler (TiO 2 and ZnO) for improving the adsorption efficiency of the PANI matrix (unfilled).

Effect of the Nanoparticle Type
In the aim of evaluating the pertinent nanoparticles that mostly enhance the attractive vdW forces and energy, we have scaled the vdW force and energy of the PANI/NPs to those for pure PANI for the two-particle radius R c = 1.36 nm and R c = 348 nm. From relations (3) and (4) We have drawn the evolution of the scaled vdW and energy as a function of volume fraction Φ of NPs. As illustrated in Figure 11, the curves undergo a linear law of the form: where a and b present the intercept and slope of the curves that describe the magnitude of the regression of the scaled vdW energy and force. As clearly illustrated in Table 5, the slope for the PANI/TiO 2 is about two times higher than for PANI/ZnO. Thus, through this approach, we can demonstrate that theTiO 2 NP is the performer filler for enhancing the vdW energy and force that governed the adsorption of CdI 2 particle on PANI/TiO 2 surface from water. Table 5. Fitting results of curves in Figure 11 using Equation (14).

Material
Intercept Regarding the nonretarded Hamaker constant for the interaction between CdI 2 particles and PANI/NPs across water is mostly a function of the refractive index of the mediums, and therefore the enhancement on the magnitude of the nonretarded Hamaker constant is relative to the refractive index of the NP filler. Furthermore, the refractive index of NPs are higher and the magnitude of the attractive vdW interactions increase. Therefore, based on the theoretical analysis mentioned above, we can deduce that the adsorption of heavy metals on polymer composite from water can be improved by choosing the NP filler that exhibits the higher refractive index. stant is relative to the refractive index of the NP filler. Furthermore, the refractive index of NPs are higher and the magnitude of the attractive vdW interactions increase. Therefore, based on the theoretical analysis mentioned above, we can deduce that the adsorption of heavy metals on polymer composite from water can be improved by choosing the NP filler that exhibits the higher refractive index. In future research, we envisage the study of the effect of the temperature on the vdW interactions using the subjected model. Furthermore, these theoretical recommendations In future research, we envisage the study of the effect of the temperature on the vdW interactions using the subjected model. Furthermore, these theoretical recommendations can be exploited experimentally on the study of the removal of the heavy metals from water on the PANI/NPs adsorbent.

Conclusions
The vdW interaction that governed adsorption was investigated from the theoretical calculation of the nonretarded Hamaker constant on the basis of the Lifshitz macroscopic approach. For the interaction between water and PANI filled with TiO 2 and ZnO NPs across air, we demonstrate that the surface energy of PANI/NPs, as well as the work of adhesion between water and PANI/NPs across water, increase with the increasing volume fraction Φ of TiO 2 and ZnO NPs. Consequently, an increase in the Laplace pressure around the cavities/porosities is found. We suggest that this finding is responsible for an increase in the specific contact surface between water and PANI/NPs and then enhance the adsorption capacity of the PANI/NPs. However, the adsorption behavior of CdI 2 particles on the PANI/NP surface across water was explained in terms of the vdW energy and force. The results demonstrate that the vdW interactions are mostly governed by the London dispersion forces. At the nonretarded regime, the vdW energy and force increase strongly, indicating that vdW attraction is the most driving force for particle adsorption. We have demonstrated that an increase in the volume fraction Φ of TiO 2 and ZnO NPs and the radius of CdI 2 particles leads to an enhancement of the attractive vdW energy and forces between CdI 2 particles and PANI filled with TiO 2 and ZnO NPs across water. From scaling results, the TiO 2 NPs are better than ZnO NPs for enhancing the attractive vdW energy and force that governed adsorption of the CdI 2 particle on the PANI surface from the water. This fact is explained by the higher refractive index of the TiO 2 NPs that is relative to their London dispersion forces.
Collectively, considering that the vdW interaction is mostly governed by the dispersion London forces, the enhancement of the adsorption of nanocomposites can be done by choosing the NP filler with a higher refractive index rather than of the polymer matrix.
These results can be generalized for all other heavy metals that exhibit a refractive index in the vicinity of that for cadmium iodide.