# Mercury Bonding to Xerogel: The Interface Fractal Dynamics of the Interaction between Two Complex Systems

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{1}= 1.604 ± 0.2798, the fractal dimension of the square mask is D

_{2}= 1.596 ± 0.0460, and the lacunarity is 0.0402.

## 1. Introduction

^{0}) and methyl mercury compounds (MeHg). Dental amalgam compounds include upwards of 50% mercury in its elementary form [12]. As soon as the mercury has been absorbed, it remains in the body for a long time because it has a reduced elimination rate. A considerable part of the absorbed amount is collected in the neurological tissues, kidneys, and liver. All mercury varieties have a high degree of toxicity that act in terms of nephrotoxicity, gastro or intestinal toxicity, and neurotoxicity on vital human organs.

## 2. Theoretical Part

#### 2.1. Fractal Parameters

#### 2.1.1. Fractal Dimension

#### 2.1.2. Lacunarity

#### 2.2. Fractal Mathematical Model

_{t}is the transmitted fractal field variable, t is the temporal variable, Q

_{i}is the incident fractal field variable, and A is a parameter independent of the fractality degree in the resolution scales space, by which it is possible to alternate the distinct selfstructuring regimes of individually existent complex systems. This particular result was achieved from the general differential equation in the scale resolutions space:

_{i}, Q

_{t}, and A play the role of dimensionless variables. Regarding Equation (2), it is specified that, at all scale resolutions, the first derivative with respect to the time of the transmitted fractal field variable, (dQ

_{t})/dt, is determined both by the difference between the incident fractal field variable and the transmitted fractal field variable, (Q

_{i}− Q

_{t}), and by another important term: the saturation component, noted as (AQ

_{t})/(1 + (Q

_{t})

^{2}).

_{t}) is noted as a fractal potential function, which is a function that features as a significant category in dynamic fractal systems, more recently generically referred to as “gradient fractal systems” [17,21].

_{t}) describes is comportment covered by a pair of potential wells. The comportment in discussion admits that, from the perspective of progress to equilibrium and, obviously, the steadiness of the equilibrium circumstances, the fractal system governed by Equation (7) is of a similar manner to that of the fractal oscillator related to the fractal potential formula. The certainty values of V(Q

_{t}) can be considered actual states/positions of equilibrium, but at the same time, the maximum limits are within the domain of the values, which can be assimilated with unsteady equilibrium positions.

_{i}, Q

_{t}, and A, when also considering the possible values of the constant A in the range of interest:

_{t}), for four values of the constant A.

_{t},Q

_{i}) as a function of two variables: the transmitted fractal field variable (Q

_{t}), and the incident fractal field variable (Q

_{i}), respectively.

_{i}and Q

_{t}, each for a different value of the constant A are presented in Figure 3.

_{t},Q

_{i},A), with two distinct orientations concerning the axes of the variables Q

_{t}and Q

_{i}. Constant A has positive values between 1 and 14, which can be seen on the colored bar on the right with distinct colors from blue to red-brown.

_{i}= F(Q

_{t}) at all scale resolutions:

_{i}, the function of two variables, Q

_{t}, the transmitted fractal field, and the constant A, respectively, for the two variation domains of Q

_{t}can be seen, with one explicitly noted over the other (see graph a).

_{i}, the function of two variables, Q

_{t}, the transmitted fractal field, respectively, and the constant A for two distinct variation domains of Q

_{t}are shown. In Figure 5a, the transmitted fractal field is ${Q}_{t}\in \left[0,10\right]$, and in (b), it is ${Q}_{t}\in \left[-10,10\right]$.

_{i},Q

_{t},A). In this axis system, the surface Q

_{i}= Q

_{i}(Q

_{t},A) was plotted. This is a double wrinkle catastrophe-type multifractal field area in terms of 3D dimensions (Figure 5) (see the mathematical standard event described in [25,27]). It can be remarked that the reversal curves shown in Figure 6 are correctly assimilated, with some transitions effectuated into the multiscale ideal space [29,30].

_{i}, into a situation on fractal bi-stability comportment is shown.

_{t}= f(A, Q

_{i}) variation.

_{t}= f(A) for values of A greater than 8, can be found in Figure 7. The value of the potential Qt increases rapidly with the increase in the constant A.

## 3. Results and Discussion

#### Fractal Analysis of SEM Image

- (a)
- Magnification (power of amplification) is (251×)—251 times;
- (b)
- Magnification (amplification power) is (1003×)—1003 times.

^{−1}. Another significant modification of the spectrum was the intensification of the stretching vibration band of the C-S-C from the phenothiazine heterocycle at 1324 cm

^{−1}and the appearance of two vibration bands at 668 cm

^{−1}and 684 cm

^{−1}, which is characteristic of the co-ordinative bonds of sulfur with mercury. In conclusion, the mercury was predominantly retained within the xerogels by co-ordination bonds with sulfur of phenothiazine, imine bonds, and the amine groups of chitosan.

_{1}= 1.604 ± 0.2798, the fractal dimension of the square mask is D

_{2}= 1.596 ± 0.0460, and the lacunarity is 0.0402.

_{1}= 1.71 for the green regression line and d

_{2}= 1.74 for the blue regression line.

_{1}= 1.604 ± 0.2798, the fractal dimension of a square mask is D

_{2}= 1.596 ± 0.0460, and the lacunarity is 0.0402. The fractal dimension is much smaller, but the lacunarity is greater due to the fact that the xerogels used in the current article have more voids/interstices, such that they are able to fix mercury in larger quantities.

## 4. Conclusions

_{1}= 1.604 ± 0.2798, the fractal dimension of a square mask is D

_{2}= 1.596 ± 0.0460, and the lacunarity is 0.0402. The fractal dimension is small, a little over 1.5, but the lacunarity is greater due to the fact that the xerogels used in the current article have more voids/interstices, such that they are able to fix mercury in larger quantities.

## 5. Materials and Methods

#### 5.1. The Hydrogels/Xerogels Materials, Synthesis and Characterization

#### 5.2. Mercury Recovery Ability

#### 5.3. Equipment and Methods

^{−1}resolution. Origin8 Pro 8.0 software was utilized to process the recorded spectra. The NMR investigations were executed on the spectrometer (Bruker Avance Neo (400 MHz)) provided with a space/probe-type instrument based on four 5 mm diameter cores and unbiased z-axis-gradient detection. Both spectrums, photoluminescence and UV-Vis absorption, were realized by using a spectrophotometer (PerkinElmer LS 55) and an Agilent Cary 60 UV-Vis spectrophotometer, respectively, on the solid specimens. The SEM pictures were produced by using a scanning electron microscope (SEM EDAX—Quanta 200) at a smaller energy of 20 Kev for the electrons [7]. The EDX can be utilized for qualitative and quantitative investigation, permitting us to identify all types of existing elements, like the concentration percentage within the specimen. Moreover, as with conventional SEM procedures, the EDX techniques are nondestructive and necessitate reduced specimen physical preparation, which does not deteriorate the evaluated sample in any way. In this paper, the SEM-EDX spectrum of the 6X-Hg image (Figure 9b) successfully confirmed the presence of mercury in the investigated xerogel specimen.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**A 3D representation of fractal potential for 14 values of constant A. (

**a**) central position, (

**b**) rotation by 30 degrees.

**Figure 5.**A 3D representation of the incident fractal field Q

_{i}, and the function of two variables, Q

_{t}and A, for two variation domains of Q

_{t}. (

**a**) transmitted fractal field between 0 and 10, (

**b**) transmitted fractal field between −10 and 10.

**Figure 6.**The dependence of the transmitted fractal field as a function of the incident fractal field.

**Figure 10.**FTIR spectra of two different compounds; the first is the simple 6X xerogel, and the second is the mercury-recovery-treated xerogel, coded 6X-Hg.

**Figure 12.**(

**a**) The original image filtered with a median filter; (

**b**) the image of interest in grayscale and without luminance; (

**c**) original image filtered with Wiener filter.

**Figure 13.**(

**a**) An image of the binary version for lacunarity calculation; (

**b**) defined mask for lacunarity calculation.

**Figure 15.**Representation of the voxels corresponding to the verified pixels in the evaluated image.

**Figure 16.**Fractal dimensions provided by multiple linear regression. Green regression line is the use of rectangular mask. Blue regression line is the use of square mask.

FD1 | Standard Deviation 1 | FD2 | Standard Deviation 2 | Lacunarity |
---|---|---|---|---|

1.604 | ±0.2798 | 1.596 | ±0.0460 | 0.0402 |

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**MDPI and ACS Style**

Paun, M.-A.; Paun, V.-A.; Paun, V.-P.
Mercury Bonding to Xerogel: The Interface Fractal Dynamics of the Interaction between Two Complex Systems. *Gels* **2023**, *9*, 670.
https://doi.org/10.3390/gels9080670

**AMA Style**

Paun M-A, Paun V-A, Paun V-P.
Mercury Bonding to Xerogel: The Interface Fractal Dynamics of the Interaction between Two Complex Systems. *Gels*. 2023; 9(8):670.
https://doi.org/10.3390/gels9080670

**Chicago/Turabian Style**

Paun, Maria-Alexandra, Vladimir-Alexandru Paun, and Viorel-Puiu Paun.
2023. "Mercury Bonding to Xerogel: The Interface Fractal Dynamics of the Interaction between Two Complex Systems" *Gels* 9, no. 8: 670.
https://doi.org/10.3390/gels9080670