# The Study on Mathematical Simulation and Analysis of the Molecular Discrete System of the Sulfurated Eucommia Ulmoides Gum

^{1}

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(This article belongs to the Section Engineering Mathematics)

## Abstract

**:**

## 1. Introduction

_{n}-, but they have different molecular structures. The molecular structure expression of NR is cis-1,4-polyisoprene, as shown in Figure 1a, and the molecular structure expression of EUG is trans-1,4-polyisoprene, as shown in Figure 1b. The properties of the two materials are absolutely different due to their different molecular chain configurations. Therefore, natural rubber is an outstanding elastomer at room temperature, while EUG is a hard plastic at room temperature [2]. So, the promotion and application of EUG is limited to the early stage of its development. However, in recent years, the boundaries of EUG application fields have been continuously broadened by multiple explorations by academics.

## 2. Theoretical Basis

#### 2.1. Basic Principles

#### 2.1.1. Equations of Motion

#### 2.1.2. Boundary Conditions

#### 2.1.3. Basic Solution of Motion Equation

- (1)
- Verlet’s algorithm: Using the positions of the first two moments of the particle, Taylor’s formula calculates the position of the following moment. Although this method decreases the complexity of the calculations, the results are inaccurate. In Equation (3), the specific expressions are given [16]:$${r}_{i}(t+\delta t)=2{r}_{i}(t)-{r}_{i}(t-\delta t)+\delta {t}^{2}\frac{{F}_{i}(t)}{m}$$
_{i}(t), t denotes the position and time of particle i, respectively. - (2)
- The Velocity Verlet algorithm, which is based on the Verlet algorithm with appropriate improvements to give the position, velocity, and acceleration rate of the particles, while also having the advantages of a greater computational accuracy and a moderate computational volume is now more widely used, and its expression is given by Equations (4) and (5) [17,18].$${r}_{i}(t+\delta t)={r}_{i}(t)+v(t)\delta t+\frac{F(t)}{2m}\delta {t}^{2}$$$${v}_{i}(t+\delta t)=\frac{d{r}_{i}}{dt}=v(t)+\frac{1}{2m}\left[F(t)+F(t+\delta t)\right]\delta t$$

#### 2.2. Force Field

#### 2.3. Ensemble

- (1)
- In the canonical ensemble (NVT), the total momentum and the number of particles are kept constant. This is analogous to placing an isolated isoenergetic system in a virtual constant temperature, maintaining both the system and the heat source at a constant temperature, and calibrating the system’s kinetic energy using the particle rate.
- (2)
- Isothermal–isobaric ensemble (NPT): The number of particles N, the pressure P, and the temperature T are kept constant in this system. However, fluctuations in its system energy E and system volume V are possible.

#### 2.4. Simulation Programs

## 3. Construction of MS Model of EUG

#### 3.1. Composition and Structure of EUG

_{2}–CH = C–CH

_{2}–, which is a discrete system composed of a large number of isoprene monomers. Additionally, then the molecular structure in Figure 4 was imported into Material Studio 2005 version software, and the molecular structure obtained after structure cleaning and geometry optimization is shown in Figure 5. From Figure 5, the carbon–carbon bond length of the EUG monomer is about 1.540, the hydrocarbon bond length is about 1.140, the carbon–carbon bond angle is about 119°, and the hydrocarbon bond angle is about 109°.

#### 3.2. EUG Model Construction and Reliability Verification

- Since EUG polymers can reflect the real EUG, it is necessary to determine the minimum degree of polymerization that can reflect the real EUG, utilizing the Materials Visualizer module in MS (Materials Studio 2004) software to construct molecular chain polymerization degrees of 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50, respectively, for EUG molecular models [25,26], with the goal of calculating the solubility parameter (δ) for the polymers of different lengths by varying the length of the polymer. Then, using Forcite, geometry and energy optimization were performed sequentially to find the configuration with the lowest molecular potential energy and to eliminate any local unreasonable structures, such as molecular overlap and offset, that may exist during the modelling process to ensure that subsequent kinetic calculations proceed smoothly [27]. Figure 7 depicts the constructed EUG models with eleven distinct polymerization degrees.
- On this basis, the lowest energy conformation was selected for annealing in the temperature range of 200–600K with a temperature gradient of 50 K for five cycles at 50 ps.
- The molecular dynamics of the annealed EUG model were optimized by (1) running molecular dynamics simulations at 50 ps in the NVT ensemble to further relax the EUG chains and (2) running molecular dynamics simulations at 50 ps in the NPT ensemble to achieve kinetic equilibrium.

^{3}. When the degree of polymerization was in the range of 5~10, the difference between the density and the real value was relatively large, but it was still not more than 9%, which was within the acceptable range; when the degree of polymerization was in the range of 15~50, the difference between the density and the real value was relatively small (only 3~5%), and when N = 30, the density ρ

_{MD}= 0.88 g/cm

^{3}is the closest one to the empirical value ρ

_{Ex}. = 0.93–0.94 g/cm

^{3}[28], with a difference of only 5.4%, which is also highly in line with the actual situation.

#### 3.3. Determination of the Minimum Aggregation Degree of the EUG Model

_{coh}) is the energy absorbed by 1 mol of molecules coming together, i.e., the energy required for a substance to overcome the intermolecular forces. Cohesion energy density (CED) is the cohesion energy per unit volume. The cohesion energy density is very useful for predicting the solubility of polymer compounds, and properties such as tensile strength, compressibility, thermal expansion coefficient, and the wettability of the polymers are all related to their cohesion energy density. The solubility parameter δ was obtained by using the square root of the cohesion energy density (CED), which is shown in Equation (6).

_{V}—molar heat of evaporation; RT—expansion work performed during vaporization; V—molar volume.

_{test}= 16.50, and also, with the literature value δ

_{Lit}. = 16.20~17.0(J/cm

^{3})

^{1/2}[25,26]; the specific parameters are shown in Table 1, so it can be considered that N = 30 is the minimum degree of aggregation of EUG. This conclusion is consistent with Qian Liu et al.’s findings [25]. Additionally, the subsequent simulation would also directly apply this conclusion for the subsequent model computational analysis.

## 4. Construction of SEUG MD Model

#### 4.1. Construction of SEUG Models

- The Materials Visualizer module was applied to SEUG single chain models [29] with varying degrees of cross-linking, utilizing sulphur bridges (C-S-S-C) as cross-linking bonds, as illustrated in Figure 10, where the degree of cross-linking is calculated by using Equation (6). Since Packmol [29] has the advantage of a fast computational speed for large system molecules in the packing process, we decided to randomly combine the resulting pdb. files in Packmol into a 100-monomer SEUG model and export it as a pdb. file. The generated SEUG molecular configuration with a 20% cross-linking degree was conformed, as shown in Figure 11.
- Using the geometry optimization module and energy minimization in the Forcite module, the configuration of SEUG molecules with the lowest molecular potential energy was found.
- For molecular dynamics optimization, an annealing treatment with a 50 K temperature gradient for 10 cycles and 50 ps in the temperature range of 100 K–600 K was used until a global energy minimum SEUG molecular conformation was found.
- Using the generated models, molecular dynamics simulations were performed: (1) molecular dynamics simulations at 100 ps in the NVT system further relaxed the EUG molecular chains; (2) molecular dynamics simulations at 100 ps in the NPT system achieved kinetic equilibrium and obtained a stable SEUG molecular structure.

_{CL}denotes the total number of cross-linked bonds; N

_{mono}denotes the number of monomers.

#### 4.2. Reliability Verification of SEUG Model

_{g}can be used to demonstrate that the SEUG model is accurate at various cross-linking levels. At the glass transition temperature, an amorphous polymer transitions from a glassy state to a highly elastic state. It is a crucial indicator of a polymeric material’s stability. Fox and Flory’s specific volume–temperature curve method [30] is one of the most common and reliable methods for determining the glass transition temperature T

_{g}of a material in simulations of molecular dynamics. Below, T

_{g}, the empty volume of the polymer, does not change significantly as the temperature rises; however, when the polymer becomes glassy, the volume changes rapidly. Consequently, NPT simulations were utilized to determine specific volumes at various target temperatures. The segmented linear fitting of the data on both sides of the inflection point produced two straight lines (below and above T

_{g}). The horizontal coordinates of the inflection point were utilized to calculate the glass transition temperature of the material. Figure 12 depicts the temperatures at which the glass transition occurs for various models of cross-linking.

_{g}to climb as the DC increases. Figure 13 shows the precise fluctuation relationship between T

_{g}and DC.

_{g}increases as the degree of cross-linking increases. When the DC is 0–40%, the T

_{g}of SEUG slowly increases with an increasing DC, indicating that SEUG is in a light cross-linking state, and the formed cross-linking points are insufficient to form a complete cross-linking network structure, but they can only form a local cross-linking network structure. When DC = 40–60%, the rate of T

_{g}of SEUG accelerates with an increasing DC, indicating that SEUG is in a medium cross-linking state at this time, and EUG has formed a complete cross-linking network structure at this time. As the cross-linking degree increases, the T

_{g}of the curve rises further, and the rising speed is obviously accelerated. T

_{g}increases significantly when the DC reaches 80%, demonstrating that SEUG is currently in a strong cross-linking condition. When the cross-linking degree reaches the critical turning point, the crystals of EUG are fully destroyed, all of the intrinsic crystals in SEUG are destroyed, SEUG is completely changed into an elastomer, and T

_{g}decreases to −28.6 °C. According to the findings of Professor Rui-Fang Yan’s study [19], the related mechanical behavior is comparable to that of natural rubber (NR). As a result, the validity of the SEUG MD model can be established.

#### 4.3. Radial Distribution Function of SEUG Model

- Intramolecular C-C radial distribution function of SEUG.

- 2.
- Intermolecular C-C radial distribution function of SEUG.

#### 4.4. Mechanical Property Parameters of SEUG Model

_{11}and C

_{12}. To simplify the computation further, λ and 2μ are assumed to be C

_{11}and C

_{11}-C

_{12}, respectively, and the formulas of the Lamé constants and elastic coefficient matrix with stress–strain relationships are provided in (9) [32,33].

_{12}-C

_{44}) can be used to evaluate the ductility of a material; the higher the value is, the greater the ductility is, conversely, the lower the value is, the greater the brittleness is. K/G, the ratio of bulk modulus to shear modulus, indicates the system’s toughness [34]. The variation pattern of K, G, E, v, C

_{12}-C

_{44}, and K/G of SEUG with cross-linking degrees are plotted in Table 3, as shown in Figure 17. This was achieved to illustrate the pattern of variation in the mechanical parameters of SEUG with different cross-linking degrees.

_{12}–C

_{44}, and K/G exhibit similar trends with varying degrees of cross-linking; they all increase and then decrease as the DC increases. This suggests that at DC = 80%, SEUG has attained the critical degree of cross-linking. This is because, initially, when DC = 0–40%, SEUG was weakly cross-linked, and the vulcanized juniper gum system had not formed a complete cross-linked network structure; with the increasing degree of cross-linking, the cross-linked network expands, and its mechanical properties gradually improve, but the crystalline area in the vulcanized gum system is still dominant, and the overall properties are biased toward plastic; at a DC of 40~80%, SEUG is moderately cross-linked. Although the crystalline area and three-dimensional cross-linked network still coexist in the system, the three-dimensional cross-linked network predominates. As a result, the deformation resistance, plasticity, and toughness of SEUG are significantly enhanced, and SEUG undergoes reversible changes from being amorphous to crystalline or crystalline to amorphous, i.e., rubber–plastic duality. This quality has also been extensively utilized in the development of shape memory materials [35]. When the cross-linking degree of SEUG exceeds the critical cross-linking degree, however, most or all of the crystallization in the system is replaced by the amorphous region, and the excessively cross-linked network structure affects the elastic recovery ability of SEUG, which exhibits the properties of a hard elastomer. In order to achieve more desirable mechanical properties in vulcanizates, it is necessary to control the degree of cross-linking between 40% and 80%, which is consistent with the findings of Yan Ruifang et al. [36].

## 5. Test

#### 5.1. Materials and Recipes

#### 5.2. Synthesis Process of SEUG

#### 5.3. Test of Tensile Strength

#### 5.4. Cross-Link Density Test

#### 5.5. Test Results and Discussion

#### 5.5.1. Tensile Strength

#### 5.5.2. Cross-link Density Test

## 6. Conclusions

- MD models of EUG with polymerization degrees from 5 to 50 were constructed, and the reliability of the EUG models was verified by calculating the density and solubility parameters of the EUG models with different polymerization degrees. The solubility parameter of 16.50, which is more consistent with the experimental test results and empirical values, was obtained, and furthermore, the minimum polymerization of 30 was determined, which can reflect the real situation of EUG.
- SEUGs with cross-linking degrees of 20%, 40%, 60%, 80%, and 100% were constructed to test the effects of different cross-linking degrees on the SEUG performance from a microscopic perspective, and T
_{g}increases with increasing cross-linking degree. When DC is 40%, SEUG is in mild cross-linking, which greatly improves its viscosity. When DC = 40–60%, SEUG is in moderate cross-linking; the T_{g}of the curve further increases, and the rate of the increase is obviously faster. When DC = 100%, the intrinsic crystals in EUG are all destroyed, SEUG is in the elastic cross-linking stage, and SEUG completely becomes an elastomer, which is consistent with the research conclusion of Professor Yan Ruifang [2]. Therefore, it can be proven that the MD model of SEUG is reliable. - The intra- and intermolecular radial distribution function curves of SEUG models with varying degrees of cross-linking were analyzed, and it was determined that sulfidation had a relatively large effect on the intra-molecular radial distribution of SEUG, and that as sulfidation increased, the C-C bond was replaced by the C-S-S-C bond, and the molecular distribution became more dense. The effect on the intermolecular radial distribution was not so apparent because it was mixed with other substances, and this is consistent with the objective evidence.
- By calculating and analyzing the mechanical property parameters of SEUG with varying degrees of cross-linking, it is demonstrated that from the perspective of the mathematical simulation method, the suitable cross-linking degree of SEUG should be controlled to be between 40% and 80% when SEUG has rubber–plastic duality and the most ideal mechanical properties, which can serve as a reference for the ratio of EUG and cross-linking agent in the vulcanization process of EUG.
- The variation pattern of vulcanization tensile strength and cross-linking density of SEUG by different degrees of cross-linking with the dosage of S is consistent with the molecular dynamics simulation, which shows that increasing the dosage of S within a certain degree is beneficial to the cross-linking of eucalyptus gum beyond a certain limit, and then it has the opposite effect. It also shows that the simulation of the molecular dynamics of SEUG is effective.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Macromolecular structures of NR and EUG. (

**a**) The chemical structure of NR; (

**b**) the chemical structure of EUG.

**Figure 2.**Schematic diagram of periodic boundary conditions. (

**a**) In the three-dimensional example, molecules can freely traverse the faces of each multidimensional data set; (

**b**) Intermolecular forces adopt the nearest mirror method.

**Figure 7.**MS models of EUG with different polymerization degrees. (

**a**) EUG model with aggregation degree of 5; (

**b**) EUG model with aggregation degree of 10; (

**c**) EUG model with aggregation degree of 15; (

**d**) EUG model with aggregation degree of 20; (

**e**) EUG model with aggregation degree of 25; (

**f**) EUG model with aggregation degree of 30; (

**g**) EUG model with aggregation degree of 35; (

**h**) EUG model with aggregation degree of 40; (

**i**) EUG model with aggregation degree of 45; (

**j**) EUG model with aggregation degree of 50.

**Figure 9.**MD simulated and experienced values of solubility parameters for different degrees of polymerization of EUG.

**Figure 10.**Vulcanized dulcimer chains with different cross-linking degrees. (

**a**) SEUG molecular chain model with 0% cross-linking degree; (

**b**) SEUG molecular chain model with 20% cross-linking degree; (

**c**) SEUG molecular chain model with 40% cross-linking degree; (

**d**) SEUG molecular chain model with 60% cross-linking degree; (

**e**) SEUG molecular chain model with 80% cross-linking degree; (

**f**) SEUG molecular chain model with 100% cross-linking degree.

**Figure 11.**SEUG discrete system consisting of 100 SEUG monomers cross-linked with C-S-S-C bonds (DC = 20%).

**Figure 15.**Intramolecular radial distribution functions of SEUG with different degrees of cross-linking.

**Figure 16.**Intermolecular radial distribution functions of SEUG with different degrees of cross-linking.

**Figure 17.**Relationship between cross-linking degree and mechanical parameters: (

**a**) relationship between cross-linking degree and bulk modulus; (

**b**) relationship between cross-linking degree and shear modulus; (

**c**) relationship between cross-linking degree and elastic modulus; (

**d**) relationship between cross-linking degree and Poisson ratio; (

**e**) relationship between cross-linking degree and

**C**; (

_{12}-C_{44}**f**) relationship between cross-linking degree and K/G.

Polymer | Degrees of Polymerization | NetMass | δ_{MD}(J/cm ^{3})^{1/2} | δ_{Lit}.(J/cm ^{3})^{1/2} | δ_{test}(J/cm ^{3})^{1/2} |
---|---|---|---|---|---|

EUG | 30 | 2045.59 | 16.525 | 16.20~17.0 | 16.50 |

DC Content (%) | C_{12}(GPa) | C_{44}(GPa) | C_{12}-C_{44}(GPa) | K (GPa) | G (GPa) | E (GPa) | ν | K/G |
---|---|---|---|---|---|---|---|---|

0 | 2.705 | 1.724 | 0.981 | 3.092 | 1.724 | 4.501 | 0.305 | 1.793 |

20 | 3.150 | 1.930 | 1.220 | 3.495 | 1.930 | 5.057 | 0.310 | 1.811 |

40 | 3.543 | 1.950 | 1.593 | 3.885 | 1.950 | 5.158 | 0.323 | 1.992 |

60 | 3.758 | 2.030 | 1.728 | 4.086 | 2.030 | 5.378 | 0.325 | 2.013 |

80 | 4.540 | 2.041 | 2.499 | 4.867 | 2.041 | 5.491 | 0.345 | 2.384 |

100 | 3.524 | 2.010 | 1.514 | 3.855 | 2.010 | 5.300 | 0.318 | 1.918 |

EUG | S | Zinc Oxide | Stearic Acid (ZA) | Accelerator CZ | Nano-Silica | Naphthenic Oil | Gummaron Resin |
---|---|---|---|---|---|---|---|

100 | Variants | 4.5 | 1.5 | 1 | 20 | 4.5 | 4.5 |

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**MDPI and ACS Style**

Yan, S.; Guo, N.; Jin, X.; Chu, Z.; Yan, S.
The Study on Mathematical Simulation and Analysis of the Molecular Discrete System of the Sulfurated Eucommia Ulmoides Gum. *Mathematics* **2023**, *11*, 964.
https://doi.org/10.3390/math11040964

**AMA Style**

Yan S, Guo N, Jin X, Chu Z, Yan S.
The Study on Mathematical Simulation and Analysis of the Molecular Discrete System of the Sulfurated Eucommia Ulmoides Gum. *Mathematics*. 2023; 11(4):964.
https://doi.org/10.3390/math11040964

**Chicago/Turabian Style**

Yan, Simeng, Naisheng Guo, Xin Jin, Zhaoyang Chu, and Sitong Yan.
2023. "The Study on Mathematical Simulation and Analysis of the Molecular Discrete System of the Sulfurated Eucommia Ulmoides Gum" *Mathematics* 11, no. 4: 964.
https://doi.org/10.3390/math11040964