Statistical Physics Quantification of Steric, Energetic and Thermodynamic Adsorption Attributes of Methylene Blue onto Super-Adsorbent Hydrogel (NaSS-DMA monomers) for water detoxification

doi:10.21203/rs.3.rs-5097565/v1

Through a statistical physics modeling approach, a detailed theoretical scrutiny was conducted utilizing four distinct models based on the grand canonical ensemble to fit the Methylene Blue adsorption isotherms onto NaSS-DMA hydrogel surface. Steriographic along with energetic-thermodynamic metrics have been inspected in response to combined effects of temperature and concentration. The uptake process was best described by a bimodal-energy linking monolayer scenario involving two sites and energies (𝜀₁ = 15.73 kJ/mol and 𝜀₂ = 17.85 kJ/mol) characterized by a multi-molecule adsorption process (n₁ = 8.383 and n₂ = 2.5967) at T = 295 K. Steriographic discussion revealed that the position of the adsorbate is non-parallel but a larger number of entities can be anchored in the same receptor site. The docking reaction is exothermic and when the concentration exceeds 95 mg/L, the adsorbed amount decreases significantly in response to incremented heat conditions. More importantly, the investigated linking process is primarily driven by weak van der Waals forces (energies below 45 kJ/mol) while the negative values of Gibbs free energy validated its spontaneity. These outcomes supported the development of a robust mathematical framework that accurately predicts removal efficiencies of Methylene Blue onto NaSS-DMA hydrogel surface providing a deeper understanding of the involved nanoscale surface linking.

Statistical physics

Methylene Blue

NaSS-DMA hydrogel

Thermodynamics and steriography

Due to rapid industrial growth and the widespread disregard for sanitary guidelines and ethical standards, water sources are increasingly contaminated by a variety of harmful effluents, dyes and radioactive substances [1]. Industries, particularly those involved in manufacturing, textiles, mining and energy production often discharge untreated or poorly treated wastewater directly into rivers, lakes and oceans [2, 3]. This wastewater contains hazardous chemicals, heavy metals and synthetic dyes that not only degrade water quality but also pose serious health risks to ecosystems and human populations. Additionally, the improper disposal of radioactive materials from certain industrial activities such as nuclear power generation and mining, exacerbates the contamination problem leading to long-lasting environmental damage [4]. The lack of enforcement of environmental regulations coupled with industrial negligence has thus resulted in widespread water pollution threatening biodiversity and public health on a global scale [5].

More concerning is the fact that several of these pollutants are carcinogenic and mutagenic in nature, meaning they have the potential to cause cancer and genetic mutations in humans and animals [6]. Exposure to such hazardous substances even at low concentrations can lead to severe health problems including respiratory issues, neurological damage and reproductive disorders [7]. Moreover, as these pollutants enter the food chain through contaminated water and aquatic life, they pose an even greater risk to public health by bioaccumulating in organisms and increasing in concentration at higher trophic levels. The unchecked release of these toxic effluents into water systems underscores the urgent need for stricter regulatory measures and ethical industrial practices to protect both the environment and human well-being [8, 9].

Methylene blue (MB) is widely used in various industries with its most prominent application being as a dye in the textile and fabric industry [10]. It is valued for its vibrant blue color which is often deployed in the dyeing of cotton, wool and silk fabrics. Beyond textiles, MB finds use in the paper and leather industries where it serves as a coloring agent [11, 12]. Additionally, it plays a crucial role in the chemical industry as a redox indicator in analytical chemistry and as a stain in biological research. Its antimicrobial properties make it useful in medical and pharmaceutical applications particularly for treating methemoglobinemia and as an antiseptic [13]. However, the extensive industrial use of MB particularly in textile dyeing often results in large amounts of this chemical being released into wastewater contributing to significant water pollution challenges due to its non-biodegradable and toxic nature. Direct contact with MB can cause irritation to the skin leading to redness, itching and sometimes burns [14]. If ingested, MB can cause nausea, vomiting, diarrhea, and abdominal pain. Ingesting large amounts may lead to more serious complications such as hemolytic anemia, especially in individuals with a deficiency of glucose-6-phosphate dehydrogenase (G6PD). Some individuals may experience allergic reactions including rashes, hives, swelling or difficulty breathing.

The removal of MB and other harmful dyes from contaminated water is a critical environmental concern. A variety of methods have been extensively employed to address this issue including adsorption, coagulation/flocculation, advanced oxidation processes (AOPs), membrane filtration and biological treatments [15–18]. From an experimental stand of view, coagulation/flocculation involves the addition of chemical agents to aggregate dye particles facilitating their removal [19]. AOPs such as photocatalysis and ozonation, utilize highly reactive species to break down dye molecules. Membrane filtration techniques including nanofiltration, ultrafiltration and reverse osmosis offer high removal efficiency but often come at a higher cost [20]. Biological treatments employing microorganisms or enzymes present a sustainable and eco-friendly alternative for dye degradation [21]. The selection of the most suitable method depends on various factors including dye concentration, wastewater characteristics and treatment costs. Each method has its own advantages and limitations and a combination of methods may be necessary to achieve optimal dye removal.

Scientists and researchers are encouraged to prioritize adsorption as an effective and versatile technique for removing organic compounds, dyes and toxins from contaminated water due to its proven efficiency, cost-effectiveness and simplicity. Adsorption offers several advantages including its ability to handle a wide range of pollutants and operate under varying environmental conditions [22–24]. By selecting optimal adsorbent materials such as activated carbon, clay minerals, biomass, Zeolites and novel nanomaterials researchers can achieve high extraction rates for even the most persistent contaminants [25]. However, to fully harness the potential of adsorption, rigorous experimental protocols must be followed. This includes optimizing parameters such as adsorbent dosage, pH, contact time, specific pressure and temperature to ensure maximum linking capacity and efficiency [26]. Furthermore, employing precise isotherm models and kinetic studies can aid in understanding the interaction mechanisms between adsorbents and pollutants, thus guiding the development of more effective systems [27, 28].

In recent years, experimental research in adsorption processes has increasingly been guided and governed by theoretical simulations and advanced computational algorithms [29–31]. These tools provide researchers with a multitude of possible and realistic scenarios based on precise adsorption inputs such as adsorbent material properties, pollutant characteristics and environmental conditions. By integrating molecular dynamics, density functional theory (DFT) and Monte Carlo simulations, scientists can predict adhesion behavior at the molecular level identifying optimal conditions and adsorbent configurations even before conducting experiments [32]. This computational approach not only saves time and resources but also allows for the exploration of a wide range of materials and operating conditions that might be difficult or expensive to test experimentally. Moreover, simulations enable the detailed examination of docking operation, isotherm behaviors and kinetic frameworks offering valuable insights that enhance experimental design and accuracy [33].

This examinative study seeks to unravel the microscopic mechanisms governing the retention of methylene blue (MB) onto super-adsorbent hydrogels based on sodium styrenesulfonate (NaSS) monomer through a rigorous statistical physics framework within the grand canonical ensemble. By leveraging this numerical approach, we will systematically inspect the steric, energetic and thermodynamic factors influencing the adsorption process. Key parameters including temperature and concentration will be meticulously examined to gain a comprehensive understanding of the molecular-level interactions and driving forces responsible for MB retention. This investigation aims to provide valuable insights into the fundamental mechanisms underlying the docking operation and facilitate the optimization of retention systems for MB extraction.

a. Materials and Reagents

Gelatin (type B, 100 bloom, derived from bovine skin), methacrylic anhydride, sodium carbonate, sodium bicarbonate, 2,2'-azobis(2-imidazolin-2-yl) propane dihydrochloride (VA-044), sodium styrene sulfonate (NaSS), methylene blue (MB), sodium hydroxide (NaOH), hydrochloric acid (HCl) and deuterium oxide (D₂O) were procured from Sigma-Aldrich and exploited without further purification. N,N-dimethylacrylamide (DMA) monomers were subjected to purification using a basic alumina column prior to use. Ultra-high purity argon gas (99.999%) was obtained from Airgas [34].

Gelatin methacryloyl (GelMA) was synthesized following the procedure outlined by M Zhu et al. [35]. Briefly, 20 g of gelatin type B was dissolved in 250 mL of carbonate-bicarbonate (CB) buffer (0.25 M) at 50°C under continuous stirring. The pH of the gelatin solution was adjusted to 9.4 using sodium hydroxide. Methacrylic anhydride (0.1 mL per gram of gelatin) was then slowly added to the gelatin solution while maintaining vigorous stirring. The reaction was allowed to proceed for 2 hours at 50°C. To terminate the reaction, the pH of the reaction mixture was adjusted to 7.4. The final product was purified through filtration and dialysis against ultrapure water using 12–14 kDa cutoff dialysis tubes. The purified GelMA was subsequently lyophilized and stored at -20°C for future use.

b. Super-adsorbent hydrogel synthesis

A free radical polymerization technique was employed to synthesize the super-adsorbent hydrogel within a closed kettle reactor assembly following previously established protocols [36]. A gelatin methacryloyl (GelMA) solution (1% w/v) was prepared in 75 mL of deionized water followed by the addition of sodium styrene sulfonate (NaSS, 13.5 g, 60 mol-% of total monomer) and N,N-dimethylacrylamide (DMA, 4.34 g, 40 mol-% of total monomer). The reaction mixture was purged with argon gas and stirred until a clear solution was obtained. Polymerization was initiated by the addition of VA-044 initiator (0.07 g, 0.2 mol-%) and the reaction was allowed to proceed for 24 hours at 30°C. The resulting hydrogel was subsequently cut into small cubes, dried at 60°C in an oven and pulverized prior to further characterization and evaluation. Figure 1 depicts the different reagents involved in this examinative study.

To gain a comprehensive understanding of the linking process, we will delve into the numerical framework of statistical physics. This rigorous approach will provide valuable insights into the underlying mechanisms governing the interactions between adsorbate entities and the adsorbent area [37]. By employing statistical physics principles, we can elucidate the complex interplay of forces and factors influencing the linking process such as steric effects, energetic considerations and thermodynamic properties. This theoretical foundation will serve as the basis for our analysis and interpretation of experimental data.

a. Preliminary assumptions

To lay the groundwork for a comprehensive understanding of the adsorption process within the framework of statistical physics, the following fundamental assumptions are introduced. These assumptions while simplifying the complexity of the system, provide a valuable foundation for theoretical analysis and interpretation. They offer a simplified yet informative perspective on the intricate interactions between adsorbate molecules and the adsorbent's cavities [38, 39].

Ideal gas behavior: MB entities are assumed to behave as an ideal gas neglecting intermolecular interactions. This simplification allows for easier theoretical treatment of the system.
Internal degrees of freedom: Each MB entity possesses multiple internal degrees of freedom, including electronic, translational, rotational and vibrational modes.
Negligibility of vibrational modes: The high energy barrier associated with most internal vibrational modes of MB atoms makes these modes negligible at the relevant temperature range allowing us to exclude them from the analysis.
Focus on translational motion: The translational degree of freedom is considered the most significant contributor to the overall behavior of MB entities within the system. Hence, the analysis primarily focuses on this aspect.

b. General adhesion formula

The proposed adhesion framework suggests that distinct amounts of MB entities can attach to specific available spots (interstitial sites) on the area of the NaSS-DMA hydrogel within a defined area. In this molecular scenario, each solute molecule (W) binds to a specific location (L). This interaction can be described mathematically using an equilibrium equation :

$$\:nW+L\rightleftarrows\:{W}_{n}L$$

1

This equilibrium expression illustrates the dynamic exchange between the attached state and the dissolved solute species in the bulk phase.

Take into account that Eq. (1) integrates a stoichiometric metric, n, which quantifies the mean occupancy of adsorbate entities per adsorption location.

Table 1

outlines the major possible n values, its physical interpretation and the main contributing reasons [40, 41].
Mathematical condition	n < 1	n > 1
Physical interpretation	This situation indicates a dispersed adsorption pattern where each adsorbate particle interacts with multiple receptor sites.	This situation suggests that multiple adsorbate species are occupying a single receptor site. It often depicts the formation of adsorbate clusters or multilayers within the receptor cavities.
Probable reasons	• An heterogeneous area with a wide range of adsorption site energies can lead to a dispersed adsorption pattern. • The adsorbate molecules are small and mobile: small, mobile adsorbate entities may readily diffuse across the adsorbent surface interacting with multiple sites. • The adsorbate-adsorbent interactions are weak: weak interactions between adsorbate molecules and the adsorbent surface can promote a dispersed adsorption pattern.	• Strong intermolecular forces : if the adsorbate entities exhibit strong attractive forces they may cluster together on the adsorbent area. • The geometry of the adsorbate entities and the receptor sites may promote the formation of multilayers. • With high adsorbate concentrations, the probability of multiple entities occupying a single site increases.

Table 1 : Mathematical condition on n, physical interpretation and probable reasons.

It’s worth mentioning that in the case of dispersed adsorption (n < 1), the reciprocal of 1/n, symbolizes the average number of receptor sites occupied by a single adsorbate entity. A high value of 1/n indicates that each adsorbate molecule is interacting with multiple receptor sites suggesting a dispersed adsorption pattern. Conversely, a low value of 1/n indicates a more localized adsorption where each adsorbate molecule occupies a smaller number of receptor sites.

Assuming independent and energetically equivalent binding positions, the grand canonical partition function for a system with N potential docking cavities per unit area can be outlined as follows [42]:

$$\:{Z}_{gc}={\left(\sum\:_{{N}_{i}}{e}^{-\beta\:\left({\epsilon\:}_{i}-\mu\:\right){N}_{i}}\right)}^{{N}_{M}}$$

2

The parenthetical term represents the grand canonical partition function for a single adsorption site, encapsulating all possible microscopic configurations at that specific location while $\:{\epsilon\:}_{i}$ portrays the adhesion energy of the interstitial site. The symbol µ characterizes the pore’s chemical potential, N_i determines the binding state and β is defined as 1/(k_BT) where k_B is the Boltzmann constant and T is the absolute temperature.

The average pore occupancy, N₀, was determined following the procedures described in [43] :

$$\:{N}_{0}={k}_{B}T\frac{\partial\:\:\text{l}\text{n}\left(\sum\:_{{N}_{i}}{e}^{-\beta\:\left({\epsilon\:}_{i}-\mu\:\right){N}_{i}}\right)}{\partial\:\mu\:}$$

3

At thermodynamic equilibrium, the chemical potentials reach a balanced state, expressed by the equation µ = m/n. Particularly, m is the chemical potential of the stacked molecule while n represent the molecular portion per site. The chemical potentiel $\:{\mu\:}_{m}$ assigned to the dissolved species is expressed [42, 44] :

$$\:{\mu\:}_{m}=\frac{1}{\beta\:}\text{ln}\left(\frac{N}{V{\left(\frac{2\pi\:m{k}_{B}\:T}{{h}^{2}}\right)}^{\raisebox{1ex}{$3$}\!\left/\:\!\raisebox{-1ex}{$2$}\right.}}\right)$$

4

In this context, V represents the volume populated by an adsorbate-adsorbent complex and N denotes the total number of adsorbed species.

The analytical expression for the adsorbed quantity Q is derived from the product of the adsorbate occupancy per site, n, and the average site occupation number, N₀, for each respective framework :

$$\:Q=n{N}_{0}$$

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c. Report on the tested scenarios

In this section, we will delve into the theoretical underpinnings of the suggested adhesion models grounded in the principles of statistical physics. Each scenario will be meticulously described with particular emphasis on its mathematical formulation and the physical significance of its incorporated parameters. By providing a clear and detailed exposition of these frameworks we aim to elucidate the underlying mechanisms governing the molecular retention and facilitate a comprehensive understanding of the system's behavior.

Monolayer linking scenario with single energy (MLSSE): it postulates a monolayer retention mechanism wherein MB entities form a single layer on the NaSS-DMA hydrogel surface. This idealized scenario provides a valuable starting point for understanding the fundamental principles of adsorption. W shall employ N_m to quantify the maximum number of available binding sites per unit surface area of the adsorbent. This parameter reflects the adsorbent's capacity for adsorbing molecules while n reflects the average number of MB entities occupying a single adsorption site. This parameter provides insights into the arrangement and distribution of adsorbate molecules on the surface. And finally C _1/2 is employed to quantify the half-saturation concentration representing the adsorbate concentration at which half of the available adsorption sites are occupied.
Bimodal-energy linking monolayer scenario (BELMS) : It suggests a monolayer retention reaction MB species form a single layer but involving two distinct types of binding sites with varying energetic properties. These binding sites have different densities (N_m1 and N_m2) and capacities (n₁ and n₂) for adsorbate binding. To fully describe this model, two half-saturation constants (C₁ and C₂) corresponding to the distinct binding sites must be incorporated.
Triple-energy linking monolayer scenario (TELMS) : It postulates a monolayer linking reaction wherein MB species form one layer NaSS-DMA hydrogel surface but involving three distinct energetic interactions with the substrate. These distinct binding energy states arise from the interplay of molecular geometry, intermolecular forces and electronic properties between the reagents.
Dual-energy linking scenario with bilayer (DELSL) : This interresting scenario posits a multilayered linking steps wherein adsorbate entities initially form a stable anchored monolayer followed by the creation of a second layer with lower adhesion energy. The occupancy state of each cavity, denoted by N_i, is quantified as follows: zero for vacant sites, one for sites occupied by a single adsorbate entity and 2n for sites hosting two adsorbate species.

Table 2

displays a comprehensive overview of the mathematical expressions describing the grand canonical partition function and the adsorbed quantities for each proposed adsorption model [39–41].
Model	Expression of Q_e	Expression of partition function z_gr
MLSSE	$\:{Q}_{e}=\frac{n{N}_{m}}{1+{\left(\frac{{c}_{1/2}}{c}\right)}^{n}}$	$\:{z}_{gr}=1+{e}^{\beta\:\left(\mu\:+\epsilon\:\right)}$
BELMS	$\:{Q}_{e}=\frac{{n}_{1}{N}_{{m}_{1}}}{1+{\left(\frac{{c}_{1}}{c}\right)}^{{n}_{1}}}+\frac{{n}_{2}{N}_{{m}_{2}}}{1+{\left(\frac{{c}_{2}}{c}\right)}^{{n}_{1}}}$	$\:{z}_{1gr}=1+{e}^{\beta\:\left(\mu\:+{\epsilon\:}_{1}\right)}$ $\:{z}_{2gr}=1+{e}^{\beta\:\left(\mu\:+{\epsilon\:}_{2}\right)}$
TELMS	$\:{Q}_{e}=n.{N}_{M}\frac{{\left(\frac{c}{{c}_{1/2}}\right)}^{n}+2{\left(\frac{c}{{c}_{1/2}}\right)}^{2n}+3{\left(\frac{c}{{c}_{1/2}}\right)}^{3n}}{1+{\left(\frac{c}{{c}_{1/2}}\right)}^{n}+{\left(\frac{c}{{c}_{1/2}}\right)}^{2n}+{\left(\frac{c}{{c}_{1/2}}\right)}^{3n}}$	$\:{z}_{gr}=1+{e}^{\beta\:\left(\mu\:+\epsilon\:\right)}+{e}^{2\beta\:\left(\mu\:+\epsilon\:\right)}+{e}^{3\beta\:\left(\mu\:+\epsilon\:\right)}$
DELSL	$\:{Q}_{e}=n{N}_{m}\frac{{\left(\frac{c}{{c}_{1}}\right)}^{n}+2{\left(\frac{c}{{c}_{2}}\right)}^{2n}}{1+{\left(\frac{c}{{c}_{1}}\right)}^{n}+{\left(\frac{c}{{c}_{2}}\right)}^{2n}}$	$\:{z}_{gr}=1+{e}^{\beta\:\left(\mu\:+{\epsilon\:}_{1}\right)}+{e}^{\beta\:\left(2\mu\:+{\epsilon\:}_{1}+{\epsilon\:}_{2}\right)}$

Table 2 : Analytical expressions of Q_eand z_grfor each tested model.

The optimal model will subsequently be utilized to extract valuable physicochemical parameters and calculate thermodynamic properties, providing invaluable insights into the fundamental mechanisms governing the docking reaction.

4.1. Consistency between empirical points and theory

A meticulous calibration of the model to experimental data is essential for validating theoretical frameworks and optimizing the removal of MB from water using NaSS-DMA hydrogel. By fine-tuning model variables to align with measured data, researchers can gain a deeper understanding of the underlying adsorption mechanisms, including the molecular-level interactions and the factors influencing the retention of MB. This knowledge can be leveraged to optimize the adsorption process, enhance removal efficiency, and develop more effective extraction strategies. Furthermore, isotherm modeling provides a valuable tool for comparing the performance of different adsorbent materials. By analyzing adsorption isotherms, researchers can identify materials with superior capacity and efficiency for MB removal. This comparative analysis aids in the selection of the most suitable adsorbent for specific applications and contributes to the development of more effective water purification technologies.

The experimental data points depicted in Fig. 2 illustrate a clear correlation between the adsorbed quantity and the intensity of thermal fluctuations.

We observe from the plot that the isotherm profile is divided into two parts based on the critical concentration of 95 mg/L. Indeed, for concentrations lower than the critical concentration, all the isotherms are almost identical. When the concentration exceeds 95 mg/L, we notice that as the temperature increases, the amount adsorbed decreases significantly. It is worth mentioning that this is an exothermic reaction. Here is the main explanation: at low concentrations, the available surface sites on the NaSS-DMA hydrogel are not fully occupied which leads to similar adsorption isotherms regardless of temperature. The system behaves in an equilibrium-like manner because the number of available sites is more than sufficient for the amount of MB in solution leading to minimal variation in retention. When the concentration exceeds 95 mg/L, the linking cavities on the NaSS-DMA hydrogel start to become saturated. Beyond this point, fewer linking points are available for additional MB entities and competition for available sites begins to intensify. This leads to a more pronounced difference in the adsorption behavior especially with temperature variations. As concentration increases, the system transitions to a state where adsorption becomes less efficient. At higher temperatures, the mobility of MB species increases, reducing their interaction with the surface due to thermal agitation. Thus, the adsorption capacity decreases at higher temperatures after reaching the critical concentration. Beyond a certain concentration, the movement of MB entities into deeper layers of the hydrogel becomes diffusion-limited. Higher concentrations would require more time for the MB molecules to access the inner adsorption sites. This limitation becomes more significant at elevated temperatures where kinetic energy leads to weaker interactions between the MB entities and the area.

4.2. Convergence criterion for accuracy evaluation

The convergence criterion hold a significant position in evaluating the accuracy of statistical physics frameworks deployed to inspect the MB anchoring onto NaSS-DMA hydrogel surfaces. It ensures that the optimization process used to fit empirical adsorption data reaches stable and reliable parameter values, allowing for precise estimation of key variables such as adsorption energy, number of active sites and thermodynamic properties. By confirming that the model has converged to a solution, the criterion helps prevent overfitting and ensures that the parameters are both mathematically valid and physically meaningful. By utilizing various convergence criteria, researchers can validate the stability and accuracy of the model from multiple angles, such as ensuring the minimization of energy, reducing fitting errors, and maintaining consistency across successive iterations. This multi-faceted approach helps confirm that the model has reached an optimal solution, ensuring that the results are both reliable and precise when evaluating complex systems like MB adsorption onto NaSS-DMA hydrogel surfaces.

Table 3 presents a comprehensive overview of the three convergence criteria employed in this study to select the most suitable model from among the various candidates. The specific criteria utilized in this analysis are detailed in [42, 43].

Convergence criteria	Analytical formula
Reduced Chi-Sqr	$\:{\chi\:}_{red}^{2}=\frac{\left(\sum\:_{i=1}^{n}\frac{{\left({Q}_{i,exp}-{Q}_{i,\:model}\right)}^{2}}{{\sigma\:}_{i}^{2}}\right)}{(n-p)}$ (6)
R²	$\:{R}^{2}=1-\left[1-\left(\frac{\sum\:i{\left({Q}_{i,\:exp}-\overline{{Q}_{i,\:exp}}\right)}^{2}-\sum\:i{\left({Q}_{i,\:exp}-{Q}_{i,\:model}\right)}^{2}}{\sum\:i{\left({Q}_{i,\:exp}-\overline{{Q}_{i,\:exp}}\right)}^{2}}\right)\right]\left[\frac{n-1}{n-p}\right]$ (7)

Table 3 : Analytical formula of the two deployed convergence criteria

Where Q_i,_exp is the measured value of the i-th data point while Q_i,_model denotes the predicted value of the iii-th data point from the model. $\:{\sigma\:}_{i}$ is the standard deviation of the i-th data point while n is the number of data points and p is the number of parameters estimated in the adopted model.

To ensure a well-organized and insightful error analysis, Table 4 exhibits the distinct statistical coefficients (R² and Reduced Chi-Sqr$\:)$ values for the tested scenarios.

Table 4

Values of the adjustment coefficients for the four tested models.
Model	MLSSE	BELMS	TELMS	DELSL
T = 295 K
$\:{\varvec{\chi\:}}_{\varvec{r}\varvec{e}\varvec{d}}^{2}$	5.71091	0.29552	4.60099	8.01409
$\:{\varvec{R}}_{\varvec{a}\varvec{d}\varvec{j}}^{2}$	0.98401	0.99987	0.98749	0.98401
		T = 305 K
$\:{\varvec{\chi\:}}_{\varvec{r}\varvec{e}\varvec{d}}^{2}$	9.65244	0.67803	2.25872	7.19764
$\:{\varvec{R}}_{\varvec{a}\varvec{d}\varvec{j}}^{2}$	0.98623	0.99965	0.97865	0.98656
		T = 315 K
$\:{\varvec{\chi\:}}_{\varvec{r}\varvec{e}\varvec{d}}^{2}$	8.45109	0.84679	5.94460	9.55823
$\:{\varvec{R}}_{\varvec{a}\varvec{d}\varvec{j}}^{2}$	0.98264	0.99777	0.98883	0.97757

Based on the values provided in Table 4, the second model (BELMS) consistently shows the most accurate performance across all tested temperaturs. This is evident from its significantly lower reduced Chi-Square values (0.29552, 0.67803, and 0.84679, respectively) compared to the other models. Additionally, BELMS demonstrates the highest R² values which are very close to 1 indicating an excellent fit to the experimental data. In contrast, the other models (MLSSE, TELMS and DELSL) show higher Chi-Square values implying less precise fits. Therefore, BELMS is clearly the most accurate scenario for describing the docking process as it exhibits the best balance of error minimization and data consistency across distinct temperatures.

Table 5

depicts the optimized numerical values of the model parameters obtained through the application of **BELMS** to the experimental data.
Temperature (K)	295	305	315
n₁	8.383±0.545	7.054±0.592	7.227±0.783
N_m2	55.059±1.098	68.745±3.908	66.498±2.443
C₁ (mg/L)	72.16±1.09	70.34±0.76	62.31±2.80
n₂	2.5967 ±0.2558	2.3401±0.0643	2.2873± 0.5390
N_m2	313.175 ±1.213	313.543 ±2.792	281.708±1. 281
C₂ (mg/L)	32.69±0.44	30.79±0.98	26.50±1.22
Q_{M fit} (mg/g)	1274.84±2.33	1218.69±1.34	1125.00±5.46

Table 5: Numerical values of the parameters (n, N_M, C₁, C₂ and Q_{M, fit}) associated with the BELMS scenario.

The following section explores the physical implications of the steric factors involved in the adsorption process. By examining the spatial arrangement and dimensions of both adsorbate and adsorbent entities, coupled with an assessment of potential steric hindrances, we can gain a deeper understanding of the underlying mechanisms governing adsorption. The impact of molecular size, shape, orientation and the parallel/perpendicular nature of linking will be meticulously inspected to identify preferred adsorption configurations and elucidate the factors driving molecular assembly. These insights are instrumental in the rational design of highly efficient adsorbents, optimization of adsorption processes and accurate prediction of adsorbate behavior in various applications, including catalysis, separations, and environmental remediation. Ultimately, a comprehensive understanding of steric effects provides a foundational framework for unraveling the intricate molecular-scale interactions that underpin adsorption phenomena.

5.1. Stereographic visualization

5.1.1. Steric numbers (n₁, n₂) and (N_m1, N_m2)

Stereographic visualization plays a crucial role in evaluating uranium adsorption by offering detailed insights into the spatial distribution and interaction of MB ions on the surface’s linking points. This technique enables researchers to observe the three-dimensional arrangement of MB ions revealing how they are distributed across different areas of the adsorbent and identifying regions of high or low adsorption activity. Figure 3 depicts the thermal agitation impact on (n₁, n₂).

In general, the steric number offers valuable insights into the geometrical arrangement (i.e., parallel or non-parallel) of anchored molecules at the atomic level. The fact n < 1 signifies a multi-anchorage process (or a parallel anchorage configuration) where the adsorbed molecule is attached parallel to the adsorbent surface with multiple anchor points. Conversely, n > 1 corresponds to the number of docked adsorbate molecules per adsorption site where the adsorption is multimolecular with each active site capable of simultaneously adsorbing various molecules. Note that the adsorbate molecules are expected to be anchored in a non-parallel position. From the plot, we clearly observe that the estimated values exceed unity with respect to temperature. With an increase in molecular agitation, (n₁, n₂) undergo a significant reduction. This suggests that the stereography describing the surface attachment phenomenon is strongly controlled by the thermal factor. For such a scenario, we witness based on (n₁, n₂) values that more than one MB species interacts with a single binding site indicating complex docking behavior. It points to a situation where multiple MB entities are adsorbing onto a single active site of the NaSS-DMA hydrogel. This often occurs in multilayer linking where once the primary layer of MB entities is fully adsorbed, additional MB entities start to interact with the already adsorbed molecules instead of directly with the hydrogel surface. This could also indicate that MB entities might adopt a tilted or oriented arrangement relative to the area, rather than lying flat. The inclinaison would allow multiple dye entities to interact with a single docking cavity, facilitating denser packing and multi-layer formation, particularly at higher concentrations. This behavior is typically controlled by both steric effects and molecular interactions. The MB entities due to their structure, may interact through π-π stacking, van der Waals forces, or electrostatic interactions in these multi-layered formations. As temperature amplifies and molecular agitation grows, the system's thermal energy likely disrupts these multi-molecular interactions causing n₁ and n₂ to diminish.

The densities of receptor sites represent the number of available adsorption sites per unit mass of the adsorbent which can be occupied by dye species during the linking process. Figure 4 illustrates the thermal agitation impact on N_m1 and N_m2.

We observe a strong sensitivity of these stereographic metrics across the temperature range. As molecular agitation intensifies, N_m1 increments while N_m2 progressively decreases. Numerical estimation shows that for the selected temperature range, N_m1 is always lower than N_m2. Based on the bimodal-energy linking monolayer scenario for MB retention onto NaSS-DMA hydrogel, the stereographic metrics represent the receptor site abundance for high- and low-energy binding sites, respectively. A strong sensitivity of these metrics to temperature is observed where incrementing molecular fluctuation leads to a rise in high-energy sites while low-energy sites, gradually decreases. This behavior can be attributed to the differing adsorption dynamics at the two types of binding sites. At higher temperatures, thermal energy enables MB entities to more easily overcome energy barriers and access the high-energy sites leading to an increase in their occupancy. Conversely, the weaker interaction forces at low-energy sites become less effective under intensified molecular agitation causing a reduction in adsorption. Numerical estimations consistently show that N_m1 is always lower than N_m2 reflecting the limited availability of high-energy sites in comparison to the more abundant low-energy sites. This shift in site occupancy as temperature increments highlights the critical role of thermal factors in modulating the adsorption process within the bimodal-energy framework where strong binding sites dominate at elevated temperatures while weak sites govern the retention at lower temperatures.

5.1.2. Concentration evaluation

The saturation adsorption capacity is crucial for determining the maximum potential of NaSS-DMA hydrogel to remove MB. It defines the upper limit of docking representing the point at which all available binding sites on the hydrogel are fully occupied by adsorbate species and no further adhesion can occur under the given conditions. It is mathematically expressed as follows [42]:

$$\:{\varvec{Q}}_{\varvec{s}\varvec{a}\varvec{t}}=2\varvec{n}{\varvec{N}}_{\varvec{m}}$$

6

Figure 5 depicts the molecular kinetic energy impact on the concentrations at half saturation.

From the figure, it can be seen that the adsorbed quantities at saturation reduces with rising temperature. This behavior is ascribed to the drastic modification in steric metrics as discussed in previous plots. These factors reflect the enhanced capacity of the adsorption sites to accommodate multiple MB entities as temperature declines likely due to decreased molecular motion and improved accessibility and stronger interaction with the docking cavities. As a result, the stereographic effect becomes predominant playing a significant role in governing the adsorption process by influencing how MB entities orient and pack onto the NaSS-DMA hydrogel area.

In Fig. 6, we have displayed the impact of kinetic energy on adsorption capacity.

In the retention of MB onto NaSS-DMA hydrogel, we clearly observe the reduction in adsorption capacity with increasing temperature. At higher temperatures, molecular motion intensifies. This increased motion can overcome the binding forces holding MB entities on the hydrogel’s cavities leading to desorption or reduced retention capacity. In other words, the reduced capacity indicates that MB entities have less tendency to stay bound to pores as temperature rises which is a characteristic of exothermic processes where heat drives the system towards desorption. If the adsorption is primarily driven by weak forces (like van der Waals interactions or hydrogen bonds), these forces weaken at elevated temperatures, causing MB to detach more easily.

5.2. Energetic evaluation

To fully understand and optimize the adsorption of MB, it is crucial to thoroughly evaluate the energetic factors involved. This includes assessing how different empirical conditions affect the microscopic mechanisms governing MB docking. By meticulously analyzing these energetic evaluations, researchers can fine-tune the experimental factors to achieve more effective and efficient adsorption processes.

In Fig. 6 we have plotted the response of adhesion energies to thermal agitation.

The estimated adhesion energies, as presented in the plot, do not exceed 38 kJ/mol. This finding aligns with previous literature and strongly suggests that the adsorption process is primarily driven by weak van der Waals forces. These non-specific intermolecular interactions, characterized by their relatively low energy barriers, are the dominant forces governing the attachment of the adsorbate molecules to the adsorbent surface. Such physical interactions described encompass coordination exchange (40 kJ/mol), hydrogen bonds (< 30 kJ/mol), dipole binding interactions (2 − 29 kJ/mol), van der Waals interactions (4 − 10 kJ/mol), and hydrophobic binding (5 kJ/mol) [39, 41, 43]. More importantly, the increase in adsorption energies with temperature in the case of MB adhesion onto NaSS-DMA hydrogel following a dual-energy monolayer retention can be explained by the enhanced mobility and activation of the free species under intensified heat levels. As temperature rises, the kinetic energy of MB increments allowing them to overcome potential energy barriers more easily and interact with higher-energy sites on the NaSS-DMA surface. This leads to a greater proportion of linking at the higher-energy sites, consistent with the dual-energy model where adsorption occurs at both low- and high-energy sites, the latter becoming more significant as temperature increases.

Figure 7 describes the adsorption behavior of MB entities onto NaSS-DMA surface.

5.3. Thermodynamic evaluation

To optimize all factors influencing adsorption, it is essential to accompany docking analysis with a thorough thermodynamic evaluation. Thermodynamics provides crucial insights into the energy changes and stability of adsorption processes revealing how temperature, pressure and other variables affect adsorption efficiency and capacity.

Internal energy

Using the grand canonical partition function, the equation for the calculation of the internal energy, based on the BELMS assumptions is given by :

$$\:{E}_{int}={k}_{B}T{N}_{m1}\frac{(\frac{c}{{C}_{1}}{)}^{{n}_{1}}Ln\left(\frac{c}{{Z}_{tr}}\right)-{n}_{1}(\frac{c}{{C}_{1}}{)}^{{n}_{1}}Ln\left(\right(\frac{c}{{C}_{1}}{)}^{{n}_{1}})}{1+(\frac{c}{{C}_{1}}{)}^{{n}_{1}}}+{k}_{B}T{N}_{m2}\frac{(\frac{c}{{C}_{2}}{)}^{{n}_{2}}Ln\left(\frac{c}{{Z}_{tr}}\right)-{n}_{2}(\frac{c}{{C}_{2}}{)}^{{n}_{2}}Ln\left(\right(\frac{c}{{C}_{2}}{)}^{{n}_{2}})}{1+(\frac{c}{{C}_{2}}{)}^{{n}_{2}}}$$

7

Figure 8 shows the variation of internal energy against concentration with respect to operating temperatures.

The analysis clearly shows a strong correlation between temperature, concentration and internal energy. The calculated Eint associated with the retention of MB onto NaSS-DMA hydrogel exhibit negative values. Furthermore, these values demonstrate a notable decrease when the temperature is increased from 295 K to 305 K. This validates the spontaneous alongside exothermic characteristics of the MB elimination processes. As molecular collisions intensify, the internal energy of the system increases in absolute value. This increase is also sensitive to concentration [40, 42].

Gibbs free enthalpy

Within the framework of the BELMS model, we derived the equation for the Gibbs free enthalpy of the investigated system. It analytical formula is as follows :

$$\:G={k}_{B}T{n}_{1}{N}_{m1}\frac{Ln\left(\frac{c}{{Z}_{tr}}\right)}{1+(\frac{c}{{C}_{1}}{)}^{{n}_{1}}}+{k}_{B}T{n}_{2}{N}_{m2}\frac{Ln\left(\frac{c}{{Z}_{tr}}\right)}{1+(\frac{c}{{C}_{2}}{)}^{{n}_{2}}}$$

8

Figure 9 shows the variation of Gibbs free enthalpy against concentration with respect to operating temperatures.

From the plot, the estimated G values exhibited negative signs. This suggests a decrease in the feasibility of the adsorption process at higher temperatures while confirming the spontaneous and exothermic nature of MB removal using NaSS-DMA hydrogel. Gibbs free enthalpy and internal energy are complementary thermodynamic functions that provide valuable insights for both theoretical and experimental research in optimizing dye removal [41]. Indeed, G offers precise information on the spontaneity and feasibility of the linking phase while E_int sheds light on the energy changes within the system such as heat exchange and molecular interactions.

Configurational entropy

The configurational entropy associated with the MB retention processes by NaSS-DMA hydrogel holds a crucial function in illustrating the dynamic interplay between order and disorder on the nanostructured surface of the material. As MB entities interact with the docking cavities, the arrangement of free species on its outermost area is considerably influenced by both the concentration and the temperature at which the reaction occurs. We have reached the mathematical expression of the entropy and it is written as follows :

$$\:\frac{S}{{k}_{B}}={N}_{m1}\left(Ln\left(1+\left((\frac{c}{{C}_{1}}{)}^{{n}_{1}}\right)\right)-\frac{\left((\frac{c}{{C}_{1}}{)}^{{n}_{1}}\right).Ln\left(\left((\frac{c}{{C}_{1}}{)}^{{n}_{1}}\right)\right)}{1+\left((\frac{c}{{C}_{1}}{)}^{{n}_{1}}\right)}\right)+{N}_{m2}\left(Ln\left(1+\left((\frac{c}{{C}_{2}}{)}^{{n}_{2}}\right)\right)-\frac{\left((\frac{c}{{C}_{2}}{)}^{{n}_{2}}\right).Ln\left(\left((\frac{c}{{C}_{2}}{)}^{{n}_{2}}\right)\right)}{1+\left((\frac{c}{{C}_{2}}{)}^{{n}_{2}}\right)}\right)$$

9

Figure 10 reflects the variation of entropy against concentration with respect to operating temperatures.

Based on the displayed curve, we clearly see that as MB concentration increments across the temperature range, the entropy levels increase dramatically, reach a maximum and then decrease gradually. This behavior is more pronounced at higher temperature (T = 315 K). The observed trend in S can be physically explained by the interplay between molecular interactions and thermal effects during the docking process. Initially, as the concentration of MB increases, the system's configurational entropy rises significantly due to the greater number of possible molecular arrangements on the hypergel surface. This reflects increased disorder as more dye entities are attached. However, once a critical concentration is reached (C = 70 mg/L), the capturing points become saturated, limiting the number of new accessible configurations. This implies an entropy reduction as the system moves toward a more ordered state. at higher thermal levels (case of 315 K), the thermal energy enhances molecular mobility and collision frequencies amplifying the randomness in the initial stages of linking which explains the more pronounced rise in entropy. However, as the system approaches saturation, thermal agitation also accelerates the transition to a more ordered state causing entropy to decrease gradually.

A single layer model with two energy levels, derived from statistical physics treatment and data fitting, successfully predicts the microscopic topography and geometry of docked MB dye species onto NaSS-DMA hydrogel. The steric and energetic parameters extracted from this optimal framework have facilitated insightful physical interpretations and discussions regarding the docking reaction. These metrics provide valuable information about the molecular arrangement, binding interactions and thermodynamics involved at surface adhesion across three different temperatures. Based on the numerical estimations the thermodynamic inspection indicated a feasible and spontaneous process. Physical forces, mainly driven by van der Walls interaction, are expected to participate in the adsorption. To address the environmental and health concerns associated with MB dye pollution, further research is imperatively recommended to identify novel adsorbents that are more cost-effective and environmentally friendly. These alternative materials can potentially be investigated and utilized for the efficient removal of dyes and other harmful substances from the environment.

Acknowledgment

The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project (Grant No.PNURSP2024R10), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Islam, T., Repon, M. R., Islam, T., Sarwar, Z., & Rahman, M. M. (2023). Impact of textile dyes on health and ecosystem: A review of structure, causes, and potential solutions. Environmental Science and Pollution Research, 30(4), 9207-9242.
Alsukaibi, A. K. (2022). Various approaches for the detoxification of toxic dyes in wastewater. Processes, 10(10), 1968.
Sharma, J., Sharma, S., & Soni, V. (2021). Classification and impact of synthetic textile dyes on Aquatic Flora: A review. Regional Studies in Marine Science, 45, 101802.
Al-Tohamy, R., Ali, S. S., Li, F., Okasha, K. M., Mahmoud, Y. A. G., Elsamahy, T., ... & Sun, J. (2022). A critical review on the treatment of dye-containing wastewater: Ecotoxicological and health concerns of textile dyes and possible remediation approaches for environmental safety. Ecotoxicology and Environmental Safety, 231, 113160.
Karsten, A. S. J. (2015). Criminal liability: negligence and environmental health: opinion. Occupational Health Southern Africa, 21(5), 19-23.
Gou, Z., Hopla, G. A., Yao, M., Cui, B., Su, Y., Rinklebe, J., ... & Sun, Y. (2022). Removal of dye pollution by an oxidase derived from mutagenesis of the Deuteromycete Myrothecium with high potential in industrial applications. Environmental Pollution, 310, 119726.
Haridevamuthu, B., Murugan, R., Seenivasan, B., Meenatchi, R., Pachaiappan, R., Almutairi, B. O., ... & Arockiaraj, J. (2024). Synthetic azo-dye, Tartrazine induces neurodevelopmental toxicity via mitochondria-mediated apoptosis in zebrafish embryos. Journal of Hazardous Materials, 461, 132524.
Uddin, F. (2021). Environmental hazard in textile dyeing wastewater from local textile industry. Cellulose, 28(17), 10715-10739.
Pérez-Ibarbia, L., Majdanski, T., Schubert, S., Windhab, N., & Schubert, U. S. (2016). Safety and regulatory review of dyes commonly used as excipients in pharmaceutical and nutraceutical applications. European Journal of Pharmaceutical Sciences, 93, 264-273.
Khan, I., Saeed, K., Zekker, I., Zhang, B., Hendi, A. H., Ahmad, A., ... & Khan, I. (2022). Review on methylene blue: Its properties, uses, toxicity and photodegradation. Water, 14(2), 242.
Oz, M., Lorke, D. E., Hasan, M., & Petroianu, G. A. (2011). Cellular and molecular actions of methylene blue in the nervous system. Medicinal research reviews, 31(1), 93-117.
Varghese, P., Abdel‐Rahman, A. T., Akberali, S., Mostafa, A., Gattuso, J. M., & Carpenter, R. (2008). Methylene blue dye—a safe and effective alternative for sentinel lymph node localization. The breast journal, 14(1), 61-67.
Scigliano, G., & Scigliano, G. A. (2021). Methylene blue in covid-19. Medical Hypotheses, 146, 110455.
Garcia, M. T. J., Gonçalves, T. P., Martins, É. S. F., Martins, T. S., de Abreu Fantini, M. C., Minarini, P. R. R., ... & Lopes, L. B. (2018). Improvement of cutaneous delivery of methylene blue by liquid crystals. International Journal of Pharmaceutics, 548(1), 454-465.
Ihaddaden, S., Aberkane, D., Boukerroui, A., & Robert, D. (2022). Removal of methylene blue (basic dye) by coagulation-flocculation with biomaterials (bentonite and Opuntia ficus indica). Journal of water process engineering, 49, 102952
Hoang, N. T., Manh, T. D., Nguyen, V. T., Nga, N. T. T., Mwazighe, F. M., Nhi, B. D., ... & Nguyen, D. D. (2022). Kinetic study on methylene blue removal from aqueous solution using UV/chlorine process and its combination with other advanced oxidation processes. Chemosphere, 308, 136457.
Li, Q., Li, Y., Ma, X., Du, Q., Sui, K., Wang, D., ... & Xia, Y. (2017). Filtration and adsorption properties of porous calcium alginate membrane for methylene blue removal from water. Chemical Engineering Journal, 316, 623-630.
Hashem, A. H., Saied, E., & Hasanin, M. S. (2020). Green and ecofriendly bio-removal of methylene blue dye from aqueous solution using biologically activated banana peel waste. Sustainable Chemistry and Pharmacy, 18, 100333.
Xia, Y., Zhang, S., Tang, X., Yan, B., & Zheng, H. (2024). Selective adsorption of methylene blue dye by a flocculation sludge-derived adsorbent prepared by carboxymethyl chitosan-based flocculants. International Journal of Biological Macromolecules, 134997.
Sotelo, S., Oyarce, E., Roa, K., Boulett, A., Pizarro, G., & Sánchez, J. (2024). Sodium lignosulfonate as an extracting agent of methylene blue dye using a polymer-enhanced ultrafiltration technique. International Journal of Biological Macromolecules, 275, 133567.
Uddin, J., Idrees, M., Ahmed, H., Batool, S., Rahman, T. U., Mehmood, S., ... & Musharraf, S. G. (2024). Biodegradation and decolorization of methylene blue, reactive Black-5, and toluidine blue-O from an aqueous solution using the polyphenol oxidase enzyme. Frontiers in Sustainable Food Systems, 7, 1320855.
Teğin, İ., Demirel, M. F., Alacabey, İ., & Yabalak, E. (2024). Investigation of the effectiveness of waste nut shell–based hydrochars in water treatment: a model study for the adsorption of methylene blue. Biomass Conversion and Biorefinery, 14(9), 10399-10412.
Umesh, A. S., Puttaiahgowda, Y. M., & Thottathil, S. (2024). Enhanced adsorption: reviewing the potential of reinforcing polymers and hydrogels with nanomaterials for methylene blue dye removal. Surfaces and Interfaces, 104670.
Eldeeb, T. M., Aigbe, U. O., Ukhurebor, K. E., Onyancha, R. B., El-Nemr, M. A., Hassaan, M. A., ... & El Nemr, A. (2024). Adsorption of methylene blue (MB) dye on ozone, purified and sonicated sawdust biochars. Biomass Conversion and Biorefinery, 14(8), 9361-9383.
Khan, I., Ali, N., Jing, Z., Khan, A., Ali, F., Hhan, F., ... & Nawaz, A. (2024). Biopolymer‑carbonaceous composites, progress, and adsorptive mitigation of water pollutants. International Journal of Biological Macromolecules, 133379.
Haider, M. I. S., Liu, G., Yousaf, B., Arif, M., Aziz, K., Ashraf, A., ... & Pikon, K. (2024). Synergistic interactions and reaction mechanisms of biochar surface functionalities in antibiotics removal from industrial wastewater. Environmental Pollution, 124365.
Holliday, M. C., Parsons, D. R., & Zein, S. H. (2024). Agricultural pea waste as a low-cost pollutant biosorbent for methylene blue removal: adsorption kinetics, isotherm and thermodynamic studies. Biomass Conversion and Biorefinery, 14(5), 6671-6685.
El Jery, A., Alawamleh, H. S. K., Sami, M. H., Abbas, H. A., Sammen, S. S., Ahsan, A., ... & Al-Ansari, N. (2024). Isotherms, kinetics and thermodynamic mechanism of methylene blue dye adsorption on synthesized activated carbon. Scientific Reports, 14(1), 970.
Kumari, S., Singh, S., Lo, S. L., Sharma, P., Agarwal, S., & Garg, M. C. (2024). Machine learning and modelling approach for removing methylene blue from aqueous solutions: Optimization, kinetics and thermodynamics studies. Journal of the Taiwan Institute of Chemical Engineers, 105361.
Kamatchi, T., Kumaresan, P., & Suresh, G. (2024). Characterizing the molecules of methylene blue doped glycine magnesium chloride (MDGMC) semi-organic crystal in virtue of quantum computational and analytical approach for photonics. Journal of Materials Science: Materials in Electronics, 35(3), 213.
Malashin, I., Tynchenko, V., Gantimurov, A., Nelyub, V., & Borodulin, A. (2024). Optimizing Neural Networks for Chemical Reaction Prediction: Insights from Methylene Blue Reduction Reactions. International Journal of Molecular Sciences, 25(7), 3860.
Yu, H., Zhang, Y., Wang, L., Tuo, Y., Yan, S., Ma, J., ... & Han, L. (2024). Experimental and DFT insights into the adsorption mechanism of methylene blue by alkali-modified corn straw biochar. RSC advances, 14(3), 1854-1865.
Ganthavee, V., Fernando, M. M., & Trzcinski, A. P. (2024). Monte Carlo Simulation, Artificial Intelligence and Machine Learning-based Modelling and Optimization of Three-dimensional Electrochemical Treatment of Xenobiotic Dye Wastewater. Environmental Processes, 11(3), 1-31.
Salunkhe, B., & Schuman, T. P. (2021). Super-adsorbent hydrogels for removal of methylene blue from aqueous solution: dye adsorption isotherms, kinetics, and thermodynamic properties. Macromol, 1(4), 256-275.
Zhu, M., Wang, Y., Ferracci, G., Zheng, J., Cho, N. J., & Lee, B. H. (2019). Gelatin methacryloyl and its hydrogels with an exceptional degree of controllability and batch-to-batch consistency. Scientific reports, 9(1), 6863.
Salunkhe, B., Schuman, T., Al Brahim, A., & Bai, B. (2021). Ultra-high temperature resistant preformed particle gels for enhanced oil recovery. Chemical Engineering Journal, 426, 130712.
Alyousef, H., Yahia, M. B., & Aouaini, F. (2020). Statistical physics modeling of water vapor adsorption isotherm into kernels of dates: Experiments, microscopic interpretation and thermodynamic functions evaluation. Arabian Journal of Chemistry, 13(3), 4691-4702
Diel, J. C., da Boit Martinello, K., da Silveira, C. L., Pereira, H. A., Franco, D. S., Silva, L. F., & Dotto, G. L. (2022). New insights into glyphosate adsorption on modified carbon nanotubes via green synthesis: Statistical physical modeling and steric and energetic interpretations. Chemical Engineering Journal, 431, 134095.
Knani, S., Mabrouk, N., Alanazi, S. T., & Kechaou, N. (2022). Study of moisture adsorption isotherms characteristics of banana and thermodynamic properties using statistical physics formalism. Drying Technology, 40(16), 3425-3433.
Aouaini, F., Bouzgarou, S., Bouzid, M., Nasr, S., Choukaier, D., & Ben Lamine, A. (2022). CO2 adsorption by molecular sieve 10A°, experimental and theoretical examination via statistical physics: modeling macroscopic and microscopic investigation. Separation Science and Technology, 57(16), 2532-2542
Amrhar, O., El Gana, L., & Mobarak, M. (2021). Calculation of adsorption isotherms by statistical physics models: a review. Environmental Chemistry Letters, 19(6), 4519-4547.
Oueslati, K., Naifar, A., Al-mugren, K. S., Aouaini, F., & Lamine, A. B. (2024). Theoretical assessment of the adsorption mechanism of Reactive Red 141 on metal hydroxide: water remediation via statistical physics modelling. Surfaces and Interfaces, 104631.
Khemis, I. B., Aouaini, F., Al-mugren, K. S., Knani, S., Graba, B., & Lamine, A. B. (2024). Theoretical study of a putative adsorption mechanism of arginine and glutamate on goldfish 5.24 and zebrafish Z06: Statistical physics modeling, thermodynamic study, and docking simulation. Journal of Molecular Structure, 139862.
Khemis, I. B., Aouaini, F., Bukhari, L., Alruwaili, A., Znaidia, S., & Lamine, A. B. (2023). Investigation of the adsorption mechanism of two nitro musk odorants on OR1A1: Advanced modeling and thermodynamic study. Journal of Molecular Liquids, 390, 123017.

No competing interests reported.

Model	Expression of Q_e	Expression of partition function z_gr
MLSSE	\(\:{Q}_{e}=\frac{n{N}_{m}}{1+{\left(\frac{{c}_{1/2}}{c}\right)}^{n}}\)	\(\:{z}_{gr}=1+{e}^{\beta\:\left(\mu\:+\epsilon\:\right)}\)
BELMS	\(\:{Q}_{e}=\frac{{n}_{1}{N}_{{m}_{1}}}{1+{\left(\frac{{c}_{1}}{c}\right)}^{{n}_{1}}}+\frac{{n}_{2}{N}_{{m}_{2}}}{1+{\left(\frac{{c}_{2}}{c}\right)}^{{n}_{1}}}\)	\(\:{z}_{1gr}=1+{e}^{\beta\:\left(\mu\:+{\epsilon\:}_{1}\right)}\) \(\:{z}_{2gr}=1+{e}^{\beta\:\left(\mu\:+{\epsilon\:}_{2}\right)}\)
TELMS	\(\:{Q}_{e}=n.{N}_{M}\frac{{\left(\frac{c}{{c}_{1/2}}\right)}^{n}+2{\left(\frac{c}{{c}_{1/2}}\right)}^{2n}+3{\left(\frac{c}{{c}_{1/2}}\right)}^{3n}}{1+{\left(\frac{c}{{c}_{1/2}}\right)}^{n}+{\left(\frac{c}{{c}_{1/2}}\right)}^{2n}+{\left(\frac{c}{{c}_{1/2}}\right)}^{3n}}\)	\(\:{z}_{gr}=1+{e}^{\beta\:\left(\mu\:+\epsilon\:\right)}+{e}^{2\beta\:\left(\mu\:+\epsilon\:\right)}+{e}^{3\beta\:\left(\mu\:+\epsilon\:\right)}\)
DELSL	\(\:{Q}_{e}=n{N}_{m}\frac{{\left(\frac{c}{{c}_{1}}\right)}^{n}+2{\left(\frac{c}{{c}_{2}}\right)}^{2n}}{1+{\left(\frac{c}{{c}_{1}}\right)}^{n}+{\left(\frac{c}{{c}_{2}}\right)}^{2n}}\)	\(\:{z}_{gr}=1+{e}^{\beta\:\left(\mu\:+{\epsilon\:}_{1}\right)}+{e}^{\beta\:\left(2\mu\:+{\epsilon\:}_{1}+{\epsilon\:}_{2}\right)}\)

Statistical Physics Quantification of Steric, Energetic and Thermodynamic Adsorption Attributes of Methylene Blue onto Super-Adsorbent Hydrogel (NaSS-DMA monomers) for water detoxification

Status:

Journal Publication

Version 1

Abstract

Figures

1. Introduction

2. Experimetal methodology

3. Conceptual framework of statistical physics theory

4. Numerical development of adsorption isotherms

4.1. Consistency between empirical points and theory

4.2. Convergence criterion for accuracy evaluation

5. Results and Interpretations

5.1. Stereographic visualization

5.1.1. Steric numbers (n₁, n₂) and (N_m1, N_m2)

5.1.2. Concentration evaluation

5.2. Energetic evaluation

5.3. Thermodynamic evaluation

6. Conclusion and recommendations

Declarations

References

Additional Declarations

Status:

Journal Publication

Version 1

Convergence criteria	Analytical formula
Reduced Chi-Sqr	\(\:{\chi\:}_{red}^{2}=\frac{\left(\sum\:_{i=1}^{n}\frac{{\left({Q}_{i,exp}-{Q}_{i,\:model}\right)}^{2}}{{\sigma\:}_{i}^{2}}\right)}{(n-p)}\) (6)
R²	\(\:{R}^{2}=1-\left[1-\left(\frac{\sum\:i{\left({Q}_{i,\:exp}-\overline{{Q}_{i,\:exp}}\right)}^{2}-\sum\:i{\left({Q}_{i,\:exp}-{Q}_{i,\:model}\right)}^{2}}{\sum\:i{\left({Q}_{i,\:exp}-\overline{{Q}_{i,\:exp}}\right)}^{2}}\right)\right]\left[\frac{n-1}{n-p}\right]\) (7)

Model	MLSSE	BELMS	TELMS	DELSL
T = 295 K
\(\:{\varvec{\chi\:}}_{\varvec{r}\varvec{e}\varvec{d}}^{2}\)	5.71091	0.29552	4.60099	8.01409
\(\:{\varvec{R}}_{\varvec{a}\varvec{d}\varvec{j}}^{2}\)	0.98401	0.99987	0.98749	0.98401
		T = 305 K
\(\:{\varvec{\chi\:}}_{\varvec{r}\varvec{e}\varvec{d}}^{2}\)	9.65244	0.67803	2.25872	7.19764
\(\:{\varvec{R}}_{\varvec{a}\varvec{d}\varvec{j}}^{2}\)	0.98623	0.99965	0.97865	0.98656
		T = 315 K
\(\:{\varvec{\chi\:}}_{\varvec{r}\varvec{e}\varvec{d}}^{2}\)	8.45109	0.84679	5.94460	9.55823
\(\:{\varvec{R}}_{\varvec{a}\varvec{d}\varvec{j}}^{2}\)	0.98264	0.99777	0.98883	0.97757

Statistical Physics Quantification of Steric, Energetic and Thermodynamic Adsorption Attributes of Methylene Blue onto Super-Adsorbent Hydrogel (NaSS-DMA monomers) for water detoxification

Status:

Journal Publication

Version 1

Abstract

Figures

1. Introduction

2. Experimetal methodology

3. Conceptual framework of statistical physics theory

4. Numerical development of adsorption isotherms

4.1. Consistency between empirical points and theory

4.2. Convergence criterion for accuracy evaluation

5. Results and Interpretations

5.1. Stereographic visualization

5.1.1. Steric numbers (n1, n2) and (Nm1, Nm2)

5.1.2. Concentration evaluation

5.2. Energetic evaluation

5.3. Thermodynamic evaluation

6. Conclusion and recommendations

Declarations

References

Additional Declarations

Status:

Journal Publication

Version 1

5.1.1. Steric numbers (n₁, n₂) and (N_m1, N_m2)