Exploring bimodal HDPE synthesis using single- and dual-site metallocene catalysts: a comprehensive review of the Monte Carlo method and AI-based approaches

Abstract

The alteration of the molecular weight distribution (MWD) in high-density polyethylenes (HDPEs) can effectively address processing challenges. However, widening the MWD may have an adverse effect on the mechanical properties. To overcome this, the generation of a bimodal MWD involving low and high molecular weight polymers simultaneously is preferred. Catalytic polymerization and macromolecular engineering design offer viable approaches to achieve this objective. This study explores various production methods, including gas phase, slurry, and solution processes, for synthesizing HDPE with bimodal MWD. It investigates critical variables such as pressure, temperature, initial reactant concentrations (monomer, co-monomer, hydrogen), as well as the utilization of single- and dual-site metallocene catalysts and co-catalysts, which significantly impact the microstructure of bimodal HDPE. In addition, a comprehensive examination of simulation approaches for HDPE synthesis with bimodal MWD is presented, employing deterministic and stochastic methodologies such as moment equations, Monte Carlo simulations, and artificial intelligence (AI) techniques. Detailed insight is provided regarding the simulation algorithms specifically developed for ethylene copolymerization with various co-monomers. A comparative analysis of the advantages and disadvantages of each method is conducted, along with a discussion on the potential application of these methods in future research endeavors.

Introduction

Polyolefins encompass a wide range of polymer materials derived exclusively from carbon and hydrogen atoms. These polymers are synthesized through the copolymerization of alkenes with the general chemical formula C_nH_2n. The polyolefin family includes numerous materials, such as polyethylene, polypropylene, copolymers of ethylene and propylene, high-density diene olefins, as well as ethylene-propylene elastomers and butadiene [1, 2]. Polyethylenes (PEs) possess distinct classifications based on their density and substructural characteristics. Within this context, three primary categories of PEs exist: low-density polyethylene (LDPE), linear low-density polyethylene (LLDPE), and high-density polyethylene (HDPE). A comprehensive summary of the properties exhibited by all three PE types is provided in Table 1. Given the paramount importance of the catalyst in ethylene polymerization and copolymerization, it becomes feasible to attain the desired microstructure of the synthesized PE through diligent control of both the process parameters and the specific catalyst employed [3, 4–5].

Table 1. Comparative analysis of principal PE species: evaluation of density, microstructure, and catalysts employed in synthesis [6, 7–8]

Type	Density ($g{/\text{cm}}^3$)	Microstructure properties	Catalysts
LDPE	0.910–0.940	High content of short and long chain branching	Radical polymerization (without catalyst)
LLDPE	0.915–0.925	High content of short chain branching	Ziegler-Natta, Metallocene, and Phillips
HDPE	0.941–0.971	Low content of short chain branching	Ziegler-Natta, Metallocene

Owing to their limited melt flow index and challenging processability, PEs with narrow molecular weight distributions are not well-suited for industrial applications involving the fabrication of diverse materials, such as plastic films, bottles, cables, and pipes, which are typically produced via the extrusion process [9]. In order to surmount the processing obstacles and challenges pertaining to PE extrusion, one viable methodology involves the synthesis of polyethylenes characterized by a wide-ranging distribution of molecular weights. The acquisition of a broad molecular weight distribution (MWD) becomes imperative to ensure the facilitation of polymer processing. Utilizing metallocene single and dual-site catalysts presents an advantageous approach in the production of polyolefins featuring a broad MWD, targeted chemical composition distribution (CCD), as well as the formation of elongated and short-chain branches along the backbone of the polymer chains [10, 11]. PEs with bimodal MWD possess advantageous attributes associated with different segments within their structure. For example, the low molecular weight fraction of the PE contributes to high elastic modulus and favorable processability, while the high molecular weight fraction influences the ultimate mechanical properties, resistance against slow crack growth (SCG), and environmental stress crack resistance (ESCR). Figure 1 visually represents the distinct regions corresponding to the characteristic features of two PE materials exhibiting wide and bimodal MWD profiles [12].

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Fig. 1

Correlating properties with regions in polyethylene variants exhibiting broad and bimodal MWDs [12]

This research presents a comprehensive evaluation of the computational modeling of ethylene and α-olefins copolymerization in the presence of hydrogen, with a specific focus on the utilization of single and dual-site metallocene catalysts. Employing a diverse array of methodologies, such as Moment equations, Monte Carlo simulations, and AI-based algorithms, the research endeavors to achieve the production of bimodal HDPE. Consequently, this review paper provides an intricate analysis and comparison of pivotal factors that influence reaction kinetics, microstructural properties, CCD, and MWD in the resulting copolymers.

Literature findings

Classification of methodologies for the production of bimodal HDPEs in industrial and academic environments

The categorization of HDPE synthesis, which exhibits a bimodal MWD, is predicated upon differentiating between the gas phase, slurry, and solution processes. This classification is based on the specific conditions within the reactor, such as temperature and pressure, as well as the underlying mechanisms and flow of reactants during the polymerization process. Table 2 delineates the process conditions pertaining to the synthesis methodologies employed for the production of bimodal HDPE. The industrial production of bimodal HDPE can be achieved through gas phase processes that employ Ziegler-Natta catalysts and utilize either single or dual reactors. Prominent examples of such processes encompass Univation technology and the INEOS method by Innovene^TMG. These gas phase techniques dynamically expand the range of possibilities for bimodal HDPE production in an industrial scale, thereby facilitating the customization of specific properties and applications of the resulting polymer [13, 14–15]. In the slurry classification, various industrial techniques are employed for the manufacturing of bimodal HDPE. LyondellBasell implements the Hostalen Advanced Cascade Process (ACP), Mitsui Chemical adopts the CX process, INEOS utilizes the Innovene^TMS loop reactor process, and Chevron-Phillips employs the MarTECH process. Also, the Borstar process utilizes gas and slurry phase reactors to efficiently produce bimodal HDPE, employing advanced polymerization technology [14, 16, 17, 18, 19, 20, 21–22]. Dow Chemical developed the Dowlex solution process, utilizing 2 CSTRs and ISOPAR E solvent for the production of bimodal HDPE. The process operates at 25–30 bar and 150–200 °C. Sclairtech process employs either 2 CSTRs or 1 CSTR with 1 PFR, employing cyclohexane solvent. It operates at 100–130 bar and 300 °C. Similarly, the Sclairtech AST process utilizes 2 CSTRs with a light hydrocarbon (HC) solvent and operates at 140 bar and temperatures above 200 °C, enabling the bimodal HDPE production [13, 14–15, 23].

Table 2. Industrial processes and their corresponding conditions for bimodal HDPE synthesis

Process	Description	References
Gas phase	• One or two fluidized bed reactors • Temperature: ${80}^{\circ }\mathrm{C}<T<{100}^{\circ }\mathrm{C}$ • Pressure: $10atm<P<30atm$ • No need to dissolve monomer, co-monomer, and hydrogen in the dilute phase of hydrocarbon • Minimal formation of deposits and thickening on the reactor wall • Absence of dilute hydrocarbon can lead to sticking of polymer to the reactor wall • Higher presence of hot spots, resulting in heterogeneous product formation • Lower conversion rate of monomers compared to slurry polymerization • Higher likelihood of unexpected product changes	[13, 24, 25, 26, 27, 28, 29, 30–31]
Slurry	• One or two CSTRs or loop reactors • Hexane as a diluent hydrocarbon • Temperature: $80{}^{\circ }\mathrm{C}<T<100{}^{\circ }\mathrm{C}$ • Pressure; $10atm<P<30atm$ • Presence of dilute phase prevents hot spots • Improved heat transfer • Higher conversion rate of monomers • Formation of deposits and thickening on the reactor wall • Potential challenges in slurring monomers and produced polymer in the hydrocarbon phase	[1, 24, 25–26, 32, 33, 34, 35, 36, 37–38]
Solution	• Autoclave or CSTR reactors • Toluene or cyclohexane as reaction media • Temperature: $40{}^{\circ }\mathrm{C}<T<300{}^{\circ }\mathrm{C}$ • Pressure: $100atm<P<150atm$ • Higher polymerization rate due to the absence of catalyst support • Higher pressure and temperature requirements • Necessity to dissolve monomers and polymers in the reaction solution • Separation process and high cost of solvents	[23, 24, 25–26, 39, 40, 41, 42–43]

Upon conducting a comprehensive analysis of the industrial methodologies employed for the manufacture of bimodal HDPE, it becomes imperative to present a robust scholarly report of its synthesis based on the prior research conducted. Single and dual-site metallocene catalysts have been the subject of extensive research due to their distinctive site functionalities. Conventionally, the first site in these catalysts triggers polymerization reactions, leading to the formation of low molecular weight chains as a consequence of its high hydrogen participation. On the other hand, the second site plays a crucial role in enabling the production of high molecular weight long-chain polymers through effective incorporation of co-monomers. This particular characteristic offers a means to synthesize HDPEs with a broad and bimodal MWD, as well as desired CCDs. Furthermore, the introduction of varying branch lengths into the copolymer backbones allows for the modification of their physical properties and enhances their overall versatility [5, 44].

Heiland et al. observed that the Hafnium catalyst exhibited superior performance compared to the zirconium catalyst in ethylene polymerizations and copolymerizations with ethylene and 1-butene. The Hafnium-catalyzed polymers exhibited significantly higher molar masses, up to ten times greater than those synthesized with zirconium. Additionally, Hafnium demonstrated enhanced co-monomer incorporation abilities with lower copolymerization parameters at 30 °C. Interestingly, the simultaneous use of both Hafnium and zirconium catalysts resulted in the formation of independent polymers by each catalyst, leading to a bimodal molecular weight distribution [45]. Recent research successfully synthesized LLDPE using a Ti(OBu)₄/AlEt₃ and SiO₂-supported Et(Ind)₂ZrCl₂ dual-functional catalytic system. It offered efficient copolymerization of ethylene and 1-butene, improving process control. AlEt₃ as the sole co-catalyst enhanced polymerization activity and LLDPE properties. The quantity of Ti(OBu)₄ influenced 1-butene incorporation, affecting molecular weights, melting points, and crystallinity. However, the molecular weight distribution remained narrow like metallocene catalysts [46].

The thermal behavior of PE reactor alloys polymerized by a hybrid catalyst, consisting of a Ziegler-Natta (ZN) catalyst and a late transition metal (LTM), was investigated. Effects of catalyst composition (LTM/ZN), polymerization temperature, and loading temperature on catalyst activity, vinyl content, viscosity average molecular weight (${\overline{M}}_v$), and thermal properties were analyzed. Two melting peaks indicated the presence of two lamellar structures (110–140 °C, 60–90 °C), shifting lower with higher LTM/ZN ratio. Increasing LTM proportion improved activity and vinyl content. Rheological properties confirmed polymer blend homogeneity, while dynamic moduli analysis indicated low molecular weight chains [47]. In another study, researchers focused on synthesizing blends of linear and branched PE using hybrid catalysts. These catalysts consisted of Nickel diimine late transition metal and Cp₂TiCl₂ metallocene, supported on TIBA-modified MgCl₂nEtOH. The supported LTM catalysts showed high polymerization activities at lower temperatures, whereas the metallocene catalyst exhibited the opposite trend. Thermal analysis revealed that the LTM catalyst produced fewer crystallizable branched chains, while the metallocene catalyst produced highly crystallizable linear chains. Rheological behavior studies indicated a mixture of linear and branched chains with varying relaxation activation energies [48]. In coordinative chain transfer polymerization, a new dual catalytic system was developed to control the formation of long-chain branches. Growing polymers were transferred from the main catalyst to the vinyl-producing catalyst via a chain transfer agent, releasing macromers. These macromers were incorporated into the polymer structure, yielding a branch-on-branch microstructure. The reaction was efficient, resulting in the formation of long branches and changes in rheological properties. Microstructural analysis showed a decrease in short-chain branches with the addition of the vinyl-producing catalyst due to steric hindrance. This mechanism has potential for large-scale production of branch-on-branch polyolefins [49]. The challenge of forming long-chain branches (LCBs) in polyolefins led to the use of a tandem catalytic system in coordinative chain transfer polymerization. This system created a branch-on-branch microstructure by repeatedly releasing and incorporating growing chains. Through the analysis of thermal data, it was discovered that the reduction of polymerization temperature from 80 to 40 °C resulted in a decrease in lamellae thickness and a transformation of crystal growth from 3D spherulites to 2D disk-like crystals. These findings emphasize the sensitivity of thermal analysis in quantifying LCBs [50].

Zhou et al. examined two metallocene catalysts (n-BuCp)₂ZrCl₂ on MAO-treated polymer particles with varying levels of 2-hydroxyethyl methacrylate for gas-phase ethylene copolymerization. Temperature and 1-hexene concentration impacted polymerization activities, leading to a maximum yield of 23,200 kg PE/(mol Zr h⁻¹) for ethylene and 1-hexene copolymerization. The resulting copolymer particles displayed distinct concentric spherical shells, implying influences beyond mass transfer on their internal structure. Chemical changes in catalytic sites were identified as the main driver for activation and deactivation, rather than physical processes [51]. In another study, researchers developed a synthetic approach for producing high-yield Ni(II) complexes using 2,5- and 2,6-substituted unbridged and 1,4-dithiane bridged ligands. These complexes were utilized as catalysts for gas phase ethylene polymerization and in homogeneous solution. Ligand structure, hydrogen, and temperature were studied for their impact on polymerization performance. Supported catalysts showed moderate to high activities, producing a range of PEs from HDPE to LLDPE without 1-olefin co-monomers. The 2,5-complexes produced low molecular weight PE with double bonds, while the 2,6-complexes generated high molecular weight PE [52].

Chen et al. demonstrated the significance of hydrogen concentration (63–75% volume) during polymerization for the successful synthesis of bimodal MWD PE using a single slurry reactor. This approach resulted in a broader MWD compared to the two-step process or multiple reactors [34]. Moreover, hydrogen concentration played a role in bimodal PE production using a dual hybrid Metallocene/chromium catalytic system, regulating molecular weight and enhancing chain transfer. Hydrogen was observed to have a significant impact on the metallocene sites, resulting in a reduction in polymer chain length, while the chromium centers exhibited less sensitivity to its presence. With an increase in the partial pressure of hydrogen in the polymerization reactor, both the hybrid catalyst and the physical mixture exhibited a more pronounced bimodal MWD [5]. Liu et al. achieved the production of bimodal and broad MWDs in PE by modifying the quantity of trimethylaluminum co-catalyst employed in the slurry process. The introduction of the co-catalyst led to a transformation from a unimodal to a bimodal distribution, primarily caused by enhanced incorporation of 1-butene and hydrogen [3, 4]. Rahaman et al. investigated the effect of slurry copolymerization conditions on copolymer properties using titanium-magnesium-supported Ziegler-Natta catalysts. Elevated hydrogen pressure led to a decrease in molecular weight, storage modulus, and melting temperature, while causing an increase in molecular weight distribution, crystallinity, tensile modulus, yield stress, and strain at break. Ethylene pressure and polymerization temperature had negligible influences. The study highlighted hydrogen's role as an effective chain transfer agent in controlling copolymer properties during the polymerization process [53]. Disentangled broad/bimodal PE with inverted SCB distribution was synthesized using a supported hybrid catalyst composed of a Poly [styrene-co-(acrylic acid)] (PSA) coating on SiO₂ particles, along with immobilized Fe(acac)₃/2,6-bis [1-(2-isopropylanilinoethyl)]-pyridine on the PSA layer. The catalyst enabled the production of low molecular weight linear polyethylene and high molecular weight polyethylene with SCBs, utilizing C4-C22 α-olefin as a co-monomer. Solely utilizing ethylene as the monomer resulted in a PE with a broad MWD and an inverted distribution of SCBs. The study also investigated the crystallization behavior, finding that fast crystallization led to a noticeable disentangled state. Toluene solution in polymerization yielded the highest degree of disentanglement for the bimodal/broad polyethylene [54]. Figure 2 presents a graphical depiction of the proposed mechanism for the synthesis of PE exhibiting a bimodal MWD, utilizing a hybrid catalyst.

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Fig. 2

Graphical representation of the proposed mechanism for the synthesis of PE with bimodal MWD utilizing a hybrid catalyst [54]

In the solution process, the exclusive confinement of reactions within the solvent medium simplifies the inherent complexities of the slurry process. However, it is important to note that the requirement to dissolve all components and polymer chains in the solvent during the solution process necessitates the utilization of elevated pressure and temperature levels. As a consequence, there is an observed increase in pressure within the reactor when compared to the slurry process [23, 39, 40, 41, 42–43]. Dual-site hybrid catalysts consisting of Ti (On-Bu)_4, (C₅H₅)₄Zr, and methylaluminoxane were utilized to promote the ethylene dimerization and copolymerization with 1-butene in toluene as a solvent. The resulting copolymers exhibited a narrow distribution of molecular weight (2.5–3.5 ${\overline{M}}_w$/${\overline{M}}_n$), making them suitable for molding and extrusion processes. By employing H₂ and organometallics such as Et₂Zn, AlMe₃, and Al(i-Bu)₃, adjustments were made to the molecular weight within the range of 10–500 × 10³ g/mol. These catalysts facilitated the production of polymers with diverse melting points (120–137 °C) and varying levels of crystallinity through the process of ethylene polymerization [55].

Estimating properties of bimodal HDPE

The density of polyethylene contributes significantly to the classification of its diverse grades and provides insights into the phenomena of SCG and ESCR. When the molecular weight of PE increases, it leads to heightened entanglements and structural irregularities. As a result, the folding of polymer chains into crystal cells becomes challenging, resulting in a decrease in the density of PE. Furthermore, an augmented co-monomer content and the presence of multiple short chain branches along the polymer backbone disrupt the chains' regularity and diminish the overall crystallinity of HDPE [56, 57, 58, 59–60]. The evaluation and prediction of PE resin density, synthesized by using metallocene catalysts, plays a crucial role in thoroughly understanding its overall properties. These properties encompass mechanical strength, processing characteristics, and SCG. DesLauriers et al. proposed Eq. (1) as a mathematical expression to establish a connection between the weight average molecular weight (${\overline{M}}_w$) and the density ($\rho$) of linear PE homopolymers [61].

1

$$\rho =1.0748-0.0241\times \text{log}{\overline{M}}_w$$

The inclusion of shorter branches within the copolymer's backbone disrupts the regularity of the molecular structure, resulting in a decrease in both crystallinity and density of HDPE. Hence, comprehensive calculations were carried out to evaluate the variations in density ($\mathrm{\Delta }\rho$) relative to the average short chain branching per 1000 carbon atoms ($\overline{\mathit{SCB}}$) in HDPEs. The mathematical representation of the variations in density and $\overline{\mathit{SCB}}$ relationship, originally proposed by DesLauriers et al., can be seen in Eq. (2). In addition, Tian et al. introduced a novel mathematical expression, referred to as Eq. (3), with the objective of utilizing data on ${\overline{M}}_w$ and $\overline{\mathit{SCB}}$ to achieve precise density estimation of HDPE [61, 62–63].

2

$$\mathrm{\Delta }\rho =0.0115{\overline{\mathit{SCB}}}^{0.4722}$$

3

$$\rho =1.0748-0.0241\times \text{log}{\overline{M}}_w-0.0115{\overline{\mathit{SCB}}}^{0.4722}$$

Previous research has focused on examining the relationship between density and crystallinity of synthesized copolymers in order to estimate the melting point temperature ($T_m$) of bimodal HDPE. The introduction of tie molecules as a concept has shed light on the correlation between HDPE density and $T_m$ in scholarly literature. In their study, Tian et al. aimed to further investigate the connection between HDPE density and its corresponding $T_m$ by utilizing parameter $\rho$ to fit the $T_m$ data. Hence, to ascertain the Tm of bimodal HDPE, Eq. (4) was employed [63].

4

$$T_m\left({}^{\circ },\mathrm{C}\right)=151.79-7.9\times {10}^8\text{exp}(-\rho /0.05488)$$

The presence of crystallinity in polymers arises from intricate mechanisms involving chain folding and the arrangement of polymer chains within crystal cells. These processes play a pivotal role in determining the degree of crystallization, which is often measured as a percentage. Polymers, specifically PEs, are commonly categorized based on their density and crystallinity characteristics. It is worth highlighting that HDPE generally exhibits a crystallization percentage in the range of 60 to 80%. Conversely, LDPE tends to display a crystallization range spanning approximately 40 to 55%. The successful formation of a crystal in PEs necessitates a specific set of structural transformations. Key among these is the folding process of the polymer chain, which must be twice as extensive as its internal lamellar thickness (L_c). Furthermore, following the folding, an additional protrusion is required to match the thickness of the amorphous region (L_a). As a result, the length of the critical tie molecule responsible for facilitating crystal formation can be quantified as 2L_c + L_a, taking into consideration both the folding and leaving mechanisms [64, 65–66]. Moyassari et al. used molecular dynamics simulations to investigate the impact of incorporating short-chain branches in the high molar mass fraction of bimodal PE. The branched systems exhibited higher entanglement density, tie chain concentration, elastic modulus, yield stress, and toughness compared to linear systems. Despite having lower crystal thickness and content, the branched systems showed improved strain hardening capabilities and a reduced presence of voids. Overall, the study emphasized the positive influence of short-chain branches on the topological characteristics and mechanical behavior of bimodal polyethylene [67].

Numerous researchers have previously discovered a robust correlation between the experimental melt index (MI) data obtained in laboratory settings and the theoretical relationships governing the MI of HDPEs synthesized using metallocene catalytic systems. McAuley et al. introduced a notable equation, denoted as Eq. (5), within this particular academic domain [27, 56, 68, 69–70]. This equation establishes a robust relationship between the MI and ${\overline{M}}_w$ of HDPEs, encompassing a wide range of MWDs spanning from narrow to wide.

5

$$MI={(\frac{{\overline{M}}_w}{111525})}^{-3.472}$$

Bimodal criteria

Researchers have recently taken into consideration the development and validation of a criterion for quantifying bimodal CCD in ethylene/1-olefin copolymers, with the aim of improving mechanical and rheological properties. The criterion was established based on Stockmayer's distribution equations and was validated through simulations and experiments using crystallization analysis fractionation (CRYSTAF) and crystallization elution fractionation (CEF) techniques applied to ethylene/1-octene copolymer blends. The investigation also examined the influence of copolymer mass fraction and number average molecular weight (${\overline{M}}_n$), determined by metallocene catalysts, on CCD bimodality. It provides insights into the factors contributing to bimodal CCD, such as unequal mass fractions, ${\overline{M}}_n$, and significant variations in co-monomer content. Additionally, the criterion offers guidance for synthesizing copolymer blends with desired CCD characteristics [71]. Nele et al. developed a theoretical framework to characterize the MWD of polymers produced through linear polymerizations under steady-state or quasi-steady-state conditions. The study identified the conditions required for bimodal MWD in generalized NS-Schulz-Flory distributions and demonstrated the dependence of bimodal nature on the chosen representation, allowing for the calculation of a bimodal index. Numerical algorithms were devised to compute regions exhibiting bimodal behavior by using weighted sums of Nth-order convolutions of the Schulz-Flory distribution to describe the MWD. The results demonstrated that the index was responsive to changes in the polymerization process, operating conditions, and catalyst selection. This theoretical framework enables the classification of different polymer materials, polymer catalysts, and polymerization processes, facilitating the determination of appropriate operating conditions for the production of bimodal or multimodal polymer resins [72]. The production of polyolefins with broader MWDs, varied chemical composition, and increased long-chain branching was investigated through the combination of metallocene single-site catalysts. Compared to traditional Ziegler-Natta catalysts, the resulting resins exhibited enhanced properties, particularly when bimodal MWDs were achieved, which demonstrated favorable mechanical and rheological characteristics. Soares et al. established a necessary condition for producing polyolefins with bimodal MWDs using two single-site catalysts. Additionally, a bimodality index (BI) was introduced as a quantitative measure of MWD bimodality [73]. According to the criterion presented, PEs with bimodal MWDs can be obtained when the number-average chain lengths ($r_{w1}$ and $r_{w2}$) produced by the two single-site catalysts satisfy the Eq. (6).

6

$${\rho }_w=\frac{{(r_{w1}-r_{w2})}^2}{{2r}_{w1}{r}_{w2}}$$

The researchers accurately evaluated the extent of the MWD by utilizing Eq. (6), which incorporates ${\rho }_w$ as a parameter representing the chain length distribution index. The study revealed that a threshold value of ${\rho }_w=1$ signified the transition to a bimodal MWD, with ${\rho }_w<1$ indicating an unimodal distribution and ${\rho }_w>1$ indicating bimodality. Moreover, the proposed BI offered a metric for assessing MWD bimodality, where negative values represented unimodal polyolefins, zero indicated MWDs at the initial stage of bimodality, and a maximum value of 10 denoted a well-separated bimodal MWD with identical peak heights [73].

Analytical exploration of polymerization site and accurate determination of initial concentrations of ethylene, co-monomer, and hydrogen

The gas-phase polymerization procedure encompasses both the gaseous phase consisting of monomer, co-monomer, hydrogen, and nitrogen, as well as the polymer phase comprising amorphous and crystalline regions. Consequently, successful polymerization entails the adsorption of monomer, co-monomer, and hydrogen present in the gaseous phase onto the amorphous polymer phase. The ability of the reactants to dissolve in the crystalline phase of the polymer is limited, thus necessitating the occurrence of the polymerization process within the amorphous phase. Thus, the reactants in the gaseous form initially come into contact with the surface of the developing polymer particles. Subsequently, they dissolve in the amorphous phase and position themselves near the active site of the catalyst to initiate the polymerization reaction. The slurry process encompasses three distinct phases, namely, the gas phase comprising monomer, co-monomer, and hydrogen; the liquid phase consisting of dilute hydrocarbon, monomer, co-monomer, and hydrogen; and finally, the polymer phase encompassing both amorphous and crystalline regions [24, 25–26]. In the present procedure, the requisite reactants undergo a transfer from a pressurized gaseous phase to a diluent hydrocarbon phase. Subsequent to this transition, the reactants dissolve within the amorphous domain of the polymer matrix and establish near the catalyst's active center. Consequently, the polymerization process predominantly takes place within the amorphous phase of the polymer [1, 31, 74, 75, 76, 77–78]. Figure 3 depicts a schematic illustrating the dissolution of monomer, co-monomer, and hydrogen within the polymeric shell, followed by their penetration onto the catalyst surface, initiating the polymerization process.

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Fig. 3

Dissolution of monomer, co-monomer, and hydrogen within polymeric shell and their absorption onto catalyst surface

The olefin solution polymerization process is regarded as possessing a relatively simpler nature when compared to both the gas phase and slurry processes. This can be primarily attributed to the distinctive existence of two different phases namely, gas and liquid. In this process, under conditions of elevated temperature and pressure, ethylene, co-monomer, and hydrogen are deliberately introduced into the solution from the gas phase. This introduction of components result in their proficient dissolution within the polymerization medium, thus designating the solvent medium as the site where the polymerization reaction takes place. In all the processes, it is imperative to utilize gas–liquid and gas-polymer phase equilibrium diagrams of the various species under the intended operating conditions. These diagrams enable the assessment of concentrations and mass transfers occurring from the gas phase to the site of polymerization. A customized equilibrium phase diagram accurately depicts the current state of a system, taking into account only the specific system conditions such as temperature and pressure. It is not affected by the initial concentrations present in the system. In the processes of the gas phase, slurry, and solution, ethylene, co-monomer, and hydrogen exhibit limited solubility in both the amorphous phase of polymer (gas phase and slurry processes) as well as the solvent (solution process), which serve as the sites for polymerization. Hence, it is recommended to employ Henry's law to determine the initial concentration of these components [31, 74, 76, 77, 79, 80, 81, 82, 83–84]. As a consequence, the correlation between the partial pressure ($P_A$) of a particular component (A) in the gas phase and its molar fraction ($x_A$) in the liquid phase can be aptly described by Eq. (7), which incorporates the Henry constant ($H_A$).

7

$$P_A.H_A=x_A$$

A considerable number of researchers have published findings on Henry constants concerning gas–liquid and gas-polymer interactions. The quantification of solvation for hydrogen, ethylene, 1-hexene, and a gas mixture containing ethylene and 1-hexene in PE was conducted using an intelligent gravimetric analyzer (IGA). Furthermore, predictive models incorporating state equations and equilibrium phase diagrams were employed to characterize the adsorption of various species within the structure of semi-crystalline PE. It is noteworthy that the investigation yielded a reduction in the molecular adsorption of ethylene and 1-hexene within the amorphous phase of PE as the temperature increased. In contrast, the adsorption behavior of hydrogen demonstrated an inverse relationship [80]. Figure 4 illustrates the quantity of ethylene, 1-hexene, and hydrogen adsorbed per gram of polyethylene. In a separate investigation, the solubility of ethylene, 1-butene, and 1-hexane in four PE samples with varying crystallinity percentages and branch structures was determined using the gravimetric method. The study revealed that the volume of the polymer exhibited a linear relationship with the increasing proportion of adsorbed co-monomers. The solubility of ethylene, within a temperature range of 30–90 ${}^{\circ }\mathrm{C}$ and a pressure range of zero to 3.5 MPa, was adequately described by Henry's law. The Henry's law constants obtained fell within the range of 0.005 to 0.014 ($gr{C}_2{H}_4/(gramorphousPE\times MPa)$). Moreover, it was observed that the solubility of co-monomers in all four PE samples was significantly higher than that of ethylene. Additionally, the Henry's constants exhibited a decrease with rising temperature and increasing crystallinity [82].

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Fig. 4

Adsorption quantities of ethylene (a), 1-hexene (b), and hydrogen (c) in the amorphous phase of PE (amPE) measured in grams of adsorbent per gram of polymer at varied temperature and pressure [80]

Neto et al. proposed the utilization of Henry constants as a precise means to determine the concentration of ethylene, 1-butene, and hydrogen in the polymer phase, which serves as a copolymerization site in slurry processes, as well as in n-hexane when acting as a diluent. Furthermore, the researchers attempted to estimate the densities of n-hexane and 1-butene using the Rackett equation, but found its applicability limited beyond the critical point of ethylene and hydrogen [85]. To overcome this limitation, they employed a solution consisting of diluted amounts of hydrogen and ethylene in conjunction with n-hexane. Remarkably, the obtained results remained consistent regardless of the concentration of gaseous components. Table 3 presents the Henry's constants for gas–liquid and gas–solid equilibria, as well as the density of components, as dependent variables of temperature [56]. Chen et al. conducted a scientific investigation employing previously published empirical findings concerning the mole fraction of ethylene and hydrogen. The primary aim was to determine the Henry constants associated with the dissolution of both hydrogen and ethylene in propane fluids under conditions of supercritical and non-supercritical states. The model developed by the researchers exhibited a notable level of agreement with the experimental data obtained from phase equilibrium diagrams pertaining to the molar fraction of ethylene and hydrogen [17].

Table 3. Henry’s constants for gas–liquid, gas–solid equilibria, and density of components as a function of temperature (${}^{\circ }\mathrm{C}$)

Component	$H_A^{\mathit{lv}}$($\mathit{atm}$)	$H_A^{\mathit{vs}}$(${\mathit{atm}}^{-1}$)	Density ($gr.{\mathit{lit}}^{-1}$)
Ethylene	$48.56+0.6340T$	$0.00116+0.000004T$	$680.0635-0.7343\text{T}$
1-Butene	$1.388+0.035T+0.0009T^2$	$0.0271+0.000183T$	$637.2882-1.6351\text{T}$
Hydrogen	$1563.64+6.355T$	$0.967841exp[-12.323-567.44/(T+273.15)]$	$678.6159-1.0421\text{T}$
n-Hexane	$0.967841exp[9.236-2697.54/(T+224.36)]$	$0.967841exp[-10.591+2529.6/(\text{T}+273.15)]$	$686.1365-1.0466\text{T}$
PE	-	-	$976-0.6\text{T}$

As demonstrated in Table 3, the relationship between the partial pressure ($P_A$) and molar fraction ($x_A$) of component A in the gas and liquid phases has been previously delineated through Eq. (7), which incorporates the utilization of the Henry constant ($H_A^{\mathit{lv}}$). It is postulated that the Henry constant ($H_A^{\mathit{vs}}$) for the polymer-component combination is equivalent to the binary Henry's constant due to the typically low concentration of components in the polymer phase. Moreover, Eq. (8) elucidates the interdependence between the partial pressure of component A in the gas phase and its corresponding weight fraction ($w_A$) in the polymer phase, in accordance with the foundational principles outlined by Henry's law.

8

$$P_A.H_A^{\mathit{vs}}=w_A$$

In the context of the solution process, the accurate determination of monomer and hydrogen concentrations in the solvent medium, specifically cyclohexane, requires referencing gas–liquid phase diagrams obtained from previous research investigations. Zhuze et al. conducted a study aimed at examining the solubility characteristics of ethylene in the cyclohexane phase across a range of temperatures under constant pressures. The results demonstrated a significant decrease in ethylene solubility in cyclohexane with increasing temperature, while maintaining a constant pressure [86]. Snijder et al. employed a methodology involving the measurement of pressures, temperature, cyclohexane quantity in the equilibrium cell, and gas volumes of two vessels to ascertain the solubility of hydrogen in cyclohexane. The solubility was quantified using the Henry constant (Eqs. (9) and (10)), which served as a measure of the extent to which hydrogen dissolved in cyclohexane under the specific conditions [87].

9

$$K_{H_2}=\frac{(C_{H_{2,l}}/C_{H_{2,g}})}{\mathit{RT}}$$

10

$$K_{H_2}=1.89\times {10}^{-4}exp(\frac{-454}{T})$$

In the given equations, $K_{H_2}$ represents the Henry constant, denoting the equilibrium constant for the partitioning of hydrogen between the liquid and gas phases. $C_{H_{2,l}}$ refers to the concentration of hydrogen in the liquid phase, while $C_{H_{2,g}}$ signifies the concentration of hydrogen in the gas phase. The variable $T$ denotes the temperature of the system, expressed in Kelvin.

Kinetic model

The deliberate employment of either single or dual-site metallocene catalysts in the copolymerization process between ethylene and diverse co-monomers aims to produce HDPE with a bimodal MWD. The first single-site catalyst or first site of a dual-site metallocene catalyst demonstrates decreased co-monomer incorporation capability, but shows enhanced hydrogen participation, leading to the formation of linear copolymer chains with decreased molecular weights. In contrast, the second single-site catalyst or second site of the dual-site metallocene catalyst exhibits higher co-monomer incorporation, accompanied by reduced transfer reactions and decreased hydrogen participation. Consequently, this catalyst or site facilitates the generation of copolymers with shorter branches along the copolymer backbone and higher molecular weights. As a consequence, the utilization of these specific catalyst types results in a bimodal MWD [88]. Table 4 presents the comprehensive kinetic model that elucidates the copolymerization mechanism between ethylene ($M_1$) and co-monomer ($M_2$), employing either a dual-site metallocene catalyst or two distinct single-site metallocene catalysts. The model takes into consideration various crucial reactions, encompassing catalyst activation, chain initiation, chain propagation, spontaneous deactivation, spontaneous chain transfer, chain transfer by hydrogen, and chain transfer by monomer and co-monomer. In the presented kinetic model, the variable "$i$" can be linked with either the first single-site or second single-site metallocene catalyst. Furthermore, it can be corresponded to the first or second site of a dual-site metallocene catalyst.

Table 4. Different types of reactions involved in the copolymerization of ethylene and co-monomer using a single or dual-site Metallocene catalysts [56, 57–58, 68, 89, 90, 91–92]

[See PDF for image]

The chemical reaction occurring in the first row is closely linked to a specific site on the catalyst called $C^i$. To initiate this reaction, alkyl aluminum (A) must be employed as a co-catalyst for activation. As a result of this reaction, an active catalyst referred to as $P_0^i$, is generated. This active catalyst possesses the capability to initiate the copolymerization process. During the chain initiation step, depicted in the second row, the active site $P_0^i$ is subject to attack by either $M_1$ or $M_2$. This event subsequently leads to the formation of active chains, specifically $P_{0,1}^i$ and $P_{0,2}^i$. These chains are characterized by unit lengths that encompass the termination of either $M_1$ or $M_2$. As indicated in the third row, it is observed that the two species arising from the chain initiation reaction exhibit the ability to engage in propagation reactions, employing either monomer $M_1$ or $M_2$, thereby inducing a progressive elongation of the chain. After spontaneous deactivation occurs, both propagating chains ($P_{n,1}^i$ or $P_{n,2}^i$) manifest as deactivated or inactive chains ($D_n^i$). This is accompanied by an irreversibly deactivated catalyst ($C_d^i$) that cannot be reactivated or reused, as depicted in row (4). The control of the copolymer's molecular weight and the conversion of an active propagation chain ($P_n^i$) into an inactive dead chain can be accomplished by employing spontaneous transfer reactions and hydrogen chain transfer. Additionally, the catalyst with an active site (fifth and sixth rows) can be preserved throughout the process. In contrast to other transfer reactions, transfer by monomer generates an active chain composed of a single-length active catalyst ($P_1^i$) and a deactivated dead chain ($D_n^i$).

Methods for simulating HDPE synthesis with bimodal MWD

Deterministic methods

The simulation of polymerization processes can be classified into two distinct categories, namely deterministic and stochastic approaches. Deterministic simulation uses equations to achieve mass equilibration and determine average system properties like molecular weight and branching index. It relies on mathematical models that consider reaction kinetics, monomer concentrations, and catalyst activity. By solving these equations, average characteristics of the system can be revealed. However, deterministic simulation may overlook the randomness and variability present in real-world systems. The method of Moment equations, as a deterministic approach, is widely employed in the simulation of polymerization systems. It achieves equation simplification through the definition and rewriting of the derivative of mass balance, primarily relying on moment equations. The utilization of moment equations leads to a reduction in the complexity of the system, thereby enhancing computational efficiency. This approach facilitates valuable insights into the behavior and average properties of polymerization processes, with particular relevance to large-scale or complex systems [91, 93, 94]. Figure 5 provides a graphic representation showcasing the application of the moment equations within the context of λ (representing the mass balance of living and growing chains ($\dot{P_n}$)) and µ (representing the mass balance of dead chains ($P_n$)) to generate a bimodal distribution. This rudimentary visual demonstration exemplifies the general technique commonly employed for simulating the synthesis of various polymers.

[See PDF for image]

Fig. 5

Graphical illustration of the application of moment equations for simulating the polymerization process to attain bimodal distribution

Multiple scholarly articles have been published regarding the synthesis and simulation of PEs exhibiting bimodal MWD. A compilation and summary of select scholarly endeavors and investigations in this domain are presented in Table 5. Chatzidoukas and colleagues embarked on an insightful endeavor wherein they engaged in mathematical modeling to investigate the industrial copolymerization of ethylene and 1-butene. Their objective was to synthesize PE with a bimodal WWD within a gas-phase fluidized bed reactor. To achieve this, they employed two distinct single-site Metallocene catalysts, each with unique properties and capabilities. Additionally, a co-catalyst, alkyl aluminum, was employed in conjunction with the catalysts for enhanced catalytic activity. The first catalyst facilitated the production of long-chain PE and the second catalyst excelled in hydrogen participation and was capable of generating shorter polymer chains compared to the first catalyst. Through their investigations, the researchers observed that specific combinations and proportional variations of the two Metallocene catalysts yielded distinct types of PE with bimodal MWDs. Furthermore, CCDs varied based on the proportions of the catalysts employed in the copolymerization process (Fig. 6) [57]. In a recent study, researchers employed mathematical simulation techniques to investigate the synthesis of PE with bimodal MWDs using the Borstar technology. The simulation was conducted in both a supercritical solvent loop slurry reactor and a gas-phase reactor. The four-site Ziegler-Natta kinetic model accurately predicted the MWD in both reactors and explored the impact of hydrogen and propane feed rates on the bimodal MWD. The study comprehensively assessed various process variables, such as co-monomer to ethylene concentration, hydrogen to ethylene concentration, residence time in each reactor, and microstructure properties of the final polymer, yielding highly favorable outcomes. This model demonstrated its efficacy in representing the Borstar process and displayed potential for further advancement in dynamic simulations during startup and grade transitions [17]. Another investigation focused on developing a predictive model for the short-chain branching distribution (SCBD) of bimodal PE in the Borstar process. A comprehensive process model was established to calculate SCBD and MWD simultaneously, with good alignment between predicted and plant data. An updated structure-performance model (SPM) was developed, accounting for the effect of SCBD, which revealed that considering only the average SCB content underestimates SCG performance. The integrated model allowed for assessing the impact of different operating conditions on SCBD, MWD, and SCG performance of bimodal PE produced in the Borstar process [63].

Table 5. Overview of recent studies utilizing deterministic methods for simulating PE synthesis with bimodal MWD

Process	Description	Authors
Gas phase	A pseudo-homogeneous model integrating moment equations and hydrodynamic models accurately predicts LLDPE production in a fluidized bed reactor, including melt flow index and polymer properties [27].	(Alizadeh et al. 2004)
Gas phase	Investigating the feasibility of a mathematical model for gas-phase catalytic copolymerization to produce tailor-made polyolefins with bimodal MWD properties using two single-site catalysts and optimized operating conditions in a single reactor [57].	(Chatzidoukas et al. 2007)
Gas phase	Efficient regulation of PE's MWD in gas-phase fluidized bed reactors was achieved by utilizing a predictive control algorithm, considering the influence of hydrogen dynamics [95].	(Haj ali et al. 2010)
Gas phase	MWD of PE in gas-phase reactors was controlled by adjusting the monomer and hydrogen feeds, with trade-offs involving the purge, production rate, and the time needed to achieve the desired MWD [96].	(Haj ali et al. 2011)
Two gas-phase	An investigation on dynamic simulation and control of industrial gas-phase ethylene polymerization reactors for LLDPE production, highlighting melt flow index, dispersion index, and MWD prediction, validated with plant data for process controllability assessment [97].	(Kazerooni et al. 2019)
Slurry	Slurry ethylene polymerization modeling was studied using metallocene/MAO catalysts, observing correlation between experimental results and calculations of catalyst activity and molecular weight, accurate predictions of molecular weight behavior under different conditions, and discussions about reaction time, pressure, catalyst concentration, and co-catalyst ratio [98].	(Young et al. 2002)
Slurry	The slurry polymerization of ethylene using Cp2ZrCl2/MAO catalyst showed that higher temperatures caused a decrease in molecular weight due to lower activation energy in propagation, but elevated temperatures boosted the reaction rate peak due to initiation with highest activation energy [99].	(Ahmadi et al. 2007)
Loop slurry	The new mathematical model accurately simulated industrial slurry-phase olefin polymerization loop reactors, considering operating conditions, reactor behavior, polyolefin properties, and employing dynamic balances to calculate concentrations, temperature profiles, and heat removal based on an ideal CSTR configuration with a product removal unit, while utilizing a multi-site Ziegler-Natta kinetic scheme to determine polymer properties and simulate various operating policies [100].	(Touloupides et al. 2010)
Slurry	The advancements in multiscale modeling and simulations of slurry-phase ethylene polymerization processes, focusing on Zeigler-Natta catalysts, product qualities, and microstructures, were discussed and guidelines for implementing efficient models in industrial reactors for optimization studies were proposed [18].	(Thakur et al. 2020)
Borstar	A rigorous model for simulating the Borstar bimodal PE process was presented, incorporating thermodynamic modeling, multi-site Ziegler-Natta kinetics, and reactor modeling with accurate prediction of process variables and polymer properties observed and the effects of hydrogen and solvent inflow rates on the MWD of the PE captured [17].	(Chen et al. 2014)
Borstar	A model was developed to predict the SCBD of bimodal PE and its correlation with SCG performance, achieving accurate predictions of SCBD and MWD and highlighting the importance of considering SCBD for estimating SCG resistance, with comprehensive assessment of the effects of operating conditions on SCBD and SCG performance [63].	(Tian et al. 2015)

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Fig. 6

Bimodal MWD of PE attained through moment equation method utilizing two single-site metallocene catalyst [57]

Stochastic approaches

Stochastic methods represent a distinct computational approach that distinguishes itself from deterministic methodologies by circumventing the need to solve differential equations in a given system. The accurate calculation of probabilities in stochastic methods relies on a comprehensive understanding of the reaction mechanism, encompassing kinetic rate constants and the computation of reaction rates. This necessitates exact information regarding the kinetics and rate of the reaction within the polymerization system to ensure precise determination of the probabilities associated with each individual reaction occurring at specific times. The kinetic Monte Carlo algorithm is widely regarded as one of the foremost stochastic methodologies, notable for its efficacy in analyzing the intricate microstructural attributes and spatial topologies exhibited by polymer chains. Moreover, it serves as a valuable tool for investigating polymerization processes featuring complex molecular architectures, enabling the examination of various microstructural aspects, including but not limited to the molecular weight distribution, copolymer composition distribution, and sequence lengths of monomers, as well as the presence of long and short chain branching, among others [101, 102, 103, 104, 105–106].

The Monte Carlo method has gained significant popularity in the field of polymerization reaction engineering for the kinetic simulation of polymerization processes. Within this domain, numerous algorithms have been developed for Monte Carlo simulations in polymerization systems, with a majority being rooted in the Gillespie algorithm [107]. This algorithm is a stochastic simulation technique developed specifically for simulation chemical reactions, including polymerization reactions. In contrast to deterministic methods that rely on differential equations, the Gillespie algorithm takes into account the probabilistic nature of reactions. It follows a sequential procedure comprising system initialization, calculation of reaction propensities, generation of random numbers, determination of reaction times and types, and system update. By incorporating individual reaction rates, the algorithm offers a more precise depiction of dynamics and fluctuations in polymerization systems. This capability enables realistic simulations of system behavior and facilitates the exploration of diverse polymerization phenomena [103, 107, 108].

The Monte Carlo simulation methodology involves the utilization of an initial transformation process to systematically convert concentration values of species, including catalysts, co-catalyst, ethylene, co-monomer, hydrogen, live chain, and dead chain, into numerical values. The determination of the initial number of these species is specifically accomplished by applying Eq. (11), which incorporates the simulation volume ($V$) as a parameter.

11

$$X_S=[\text{S}]N_{\mathit{Av}}V$$where $X_S$ denotes the quantity of reactive species corresponding to the concentration of $[\text{S}]$, and $N_{\mathit{Av}}$ symbolizes the Avogadro number. Additionally, it is crucial to transform all remaining species concentrations into species numbers, as delineated in the aforementioned equation. The size of the simulation volume in Monte Carlo simulation directly affects the accuracy of the polymerization process. A larger volume captures more polymerization events and provides a comprehensive view of the system but requires more computational time. In contrast, a smaller volume speeds up reactions but sacrifices result accuracy. Choosing an appropriate simulation volume is crucial for balancing accuracy, computational efficiency, and meaningful outcomes in Monte Carlo simulation [88, 108, 109, 110, 111, 112–113]. Furthermore, the experimental rate constants ($k^{\mathit{exp}}$) must undergo conversion into Monte Carlo constants ($k^{\mathit{MC}}$), which are indispensable for the computation of reaction rates within the simulation framework, as outlined in Eq. (12) to (14).

12

$$\begin{array}{cc}{k}^{\mathit{MC}}=k^{\mathit{exp}}& \mathrm{Unimolecular}\mathrm{reaction}\end{array}$$

13

$$\begin{array}{cc}{k}^{\mathit{MC}}=\frac{k^{\mathit{exp}}}{N_{\mathit{Av}}\times V}& \mathrm{Bimolecular}\mathrm{reaction}\mathrm{between}\mathrm{different}\mathrm{species}\end{array}$$

14

$$\begin{array}{cc}{k}^{\mathit{MC}}=\frac{2k^{\mathit{exp}}}{N_{\mathit{Av}}\times V}& \mathrm{Bimolecular}\mathrm{reaction}\mathrm{between}\mathrm{similar}\mathrm{species}\end{array}$$

Consequently, the rate of the ith reaction ($R_i$) at time t is determined by the multiplication of the Monte Carlo constant by the number of species ($X_{S1}$ and $X_{S2}$) involved in the reactions, as expressed in Eq. (15).

15

$$R_i=k^{\mathit{MC}}.X_{S1}.X_{S2}$$

Following the computation of the rate pertaining to the ith reaction, the probability of the aforementioned reaction occurring within the copolymerization medium at time t is determined by dividing the rate of the specific reaction ($R_i$) by the total ($n$) rate of all reactions (${\text{R}}_{\text{total}}$) occurring within a specified time frame. To generate a sequence of non-repetitive semi-random numbers ($r_1$), the Mersenne Twister algorithm, proposed by Mersenne and Matsumoto [114], is employed. Additionally, Eq. (16) is utilized to select the specific reaction type, utilizing the random numbers generated during the given time period. Furthermore, an additional semi-random number ($r_2$) is necessary to determine the time interval ($\tau$) between two consecutive reactions, as expressed in Eq. (17).

16

$$\sum _{i=1}^{n-1}{P}_i<r_1\le \sum _{i=1}^n{P}_i$$

17

$$\tau =\frac{1}{R_{\mathit{total}}}ln\left(\frac{1}{r_2}\right)$$

The coding of the copolymerization process involving ethylene and various co-monomers to synthesize bimodal HDPE can be implemented using several programming languages, such as C++, Fortran, Pascal, or MATLAB. The selection of a programming language depends on various factors, such as the complexity of the system, the number of involved reactions, the data storage requirements, the programmer's proficiency, the specifications of the computer system used (including RAM capacity, CPU performance, hard memory size, and operating system specifications), and other pertinent considerations. In cases where the copolymerization system consists of a significant number of reactions and produces diverse outputs, it is recommended to adopt the C++ programming language due to its object-oriented nature. C++ offers advanced features for managing complex systems, enabling efficient organization and manipulation of objects representing the reactions and their associated variables. This object-oriented programing (OOP) approach facilitates the development of modular and reusable code structures, enhancing the overall clarity and maintainability of the software. Figure 7 provides a graphical representation of the Monte Carlo method employed to simulate the copolymerization of ethylene and co-monomer, resulting in the synthesis of bimodal HDPE via dual or single-site catalysts.

[See PDF for image]

Fig. 7

Visualization of the Monte Carlo technique for simulating the copolymerization of ethylene and co-monomer, resulting in the synthesis of bimodal HDPE using dual or single-site catalysts

Numerous scholarly papers have undertaken comprehensive investigations into the application of Monte Carlo simulation within the scope of polymerization processes. The dynamic and optimized Monte Carlo algorithm was utilized to investigate the kinetics of bulk and radical polymerization of methyl methacrylate (MMA) under isothermal and adiabatic conditions. The simulation outcomes revealed that elevating the average temperature, considering various heating policies, resulted in higher values of conversion at which the Gel effect occurs. Furthermore, implementing a step increase in temperature during the conversion process of approximately 30% enabled the achievement of a narrower molecular weight distribution [109]. The impact of bi-functional initiators was examined through Monte Carlo simulation to analyze their influence on the free-radical polymerization of styrene. The findings revealed that the application of bifunctional initiators led to enhanced monomer conversion and increased molecular weight of the resulting polymers. Moreover, the use of bi-functional initiators resulted in a narrower MWD in comparison to monofunctional initiators [115].

The impact of varying feed composition percentages on the microstructure of copolymers synthesized in copolymerization systems with reactivity ratios exceeding unity was investigated successfully through the utilization of the Monte Carlo method. Furthermore, an analysis was performed on the influence of reactivity ratios and initial feed composition on the microstructure of copolymers formed through free-radical integration using Monte Carlo simulations. Deviations from the azeotrope point in terms of azeotropy and composition drifts were effectively predicted by the model [116, 117]. The atom transfer radical polymerization (ATRP) of styrene, considering both chain-length-dependent and diffusion-controlled termination, was simulated using the high-efficiency Monte Carlo method. Additionally, a thorough examination of the termination modes in ATRP polymerization was conducted, resulting in strong concordance with experimental data [118, 119].

A Monte Carlo-based program was developed and employed for the purpose of tailoring ethylene/1-hexene copolymers through semi-batch single-site metallocene catalyzed copolymerization. The program was utilized to monitor and evaluate various architectural features of the copolymers, including copolymer composition, ethylene sequence length, longest ethylene sequence length, and number-average degree of polymerization. The program created controlled feeding strategies in order to achieve copolymers with customized co-monomer distributions, and the advantages of controlled feeding in comparison to uncontrolled feeding were examined. The simulation results demonstrated the critical importance of computerized feeding in order to attain desired microstructures. Besides, an evaluation of the crystallization and composition distribution of crystals within the copolymer chains was conducted using the crystallization analysis fractionation technique (CRYSTAF). Furthermore, potential future applications of the feeding policy to other polymerization reactions were explored, and the benefits of optimizing feed composition were highlighted [120].

The effect of hydrogen and co-catalyst concentration on ethylene polymerization using Ziegler-Natta catalysts was investigated by means of the Monte Carlo simulation algorithm. It was determined that the catalyst-induced overall MWD is at least twice as broad as the MWD observed for individual catalyst sites (Fig. 8). It is evident that the rise in hydrogen concentration does not lead to deactivation of the catalyst's active sites, but rather acts as a potent agent in reducing the molecular weights produced by each catalyst site. Additionally, a comparable outcome was observed with an increase in co-catalyst concentration [121]. A novel Monte Carlo method was used to simulate LDPE polymerization in a tubular reactor, focusing on topological features and kinetic mechanisms. The study provided detailed insights into the chain structure, accurately identifying various chain types and characterizing branches on branch. These findings enhanced understanding of polymer chain architecture and improved the prediction of microstructure-related properties. According to the findings, around sixty percent of the observed instances involved branches on the branch, measuring between 1 to 5 carbons in length. Moreover, the analysis revealed that the longest branch on the branch contained approximately 50 carbon atoms [122]. The copolymerization of ethylene with α-olefin co-monomers in a slurry process was explored using a dynamic Monte Carlo simulation. Two series autoclave reactors with a dilute n-hexane phase and Ziegler-Natta catalysts were employed. Thermodynamic calculations determined species solubility and concentration in the gas, liquid, and polymer phases. Analysis of MWD and SCB produced accurate results. MWD results for each reactor and the overall bimodal distribution are illustrated in Fig. 9. Comparison with experimental data demonstrated the model's effectiveness across different operational conditions. The model shows promise for future applications in process optimization and product control at various scales [78].

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Fig. 8

Overall MWD and individual MWDs of Ziegler-Natta catalyst sites [121]

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Fig. 9

MWD in the first reactor (red dash line), second reactor (blue dash line), and overall bimodal distribution (solid line) of the process [78]

The kinetics of ethylene and 1,9-decadiene solution copolymerization were investigated in a semi-batch reactor with a constrained geometry catalyst, as studied by Brandão et al. The resulting MWDs were described using a Monte Carlo model that incorporated macromonomer reincorporation through pendant double bonds from diene incorporation. Model parameters were estimated through a hybrid approach involving particle swarm optimization, parameter identifiability procedures, and the Gauss-Newton method. The predicted long-chain branching frequencies were validated through Monte Carlo simulation. Average molecular weights and ethylene feed flow rates were accurately estimated using both methods, thereby emphasizing the occurrence of long-chain branching at higher molecular weights and the improvement of viscoelastic properties in the synthesized PEs (Fig. 10) [123]. In a different study, the researchers used adaptive orthogonal collocation (CAOC) and Monte Carlo methods to calculate MWD for ethylene/1,9-decadiene copolymers. The MWDs obtained were compared to experimental data and showed a decrease in macromonomer incorporation rate as the radius of gyration increased. Both CAOC and Monte Carlo methods produced similar MWDs, but CAOC was more computationally efficient. The rate of macromonomer reincorporation was found to be influenced by factors such as unsaturation content, chain length, and radius of gyration. The experimental results showed that sigmoidal and exponential functions provided good fits for the measured MWDs [124].

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Fig. 10

Re-participation of active macromonomers through pendant double bonds in a growing chain [123]

Simon et al. conducted research on the use of two single-site catalysts to produce polyolefins. One catalyst generated linear chains, while the other contributed to the formation of both linear and long-chain branched (LCB) chains. Using the Monte Carlo method, the study investigated how to control the molecular structure of polyolefins, with a specific focus on the presence of LCBs. The combined use of these catalysts aimed to achieve precise control over the molecular architecture of branched polyolefins, where the linear catalyst formed linear chains and the LCB catalyst introduced branching to create the desired branched polyolefin structures. The findings included predictions on chain length distribution, the number and location of free pendants, the length of internal chain segments, and the connection points of branches to individual chains. Additionally, the model demonstrated the flexible manipulation of the LCB/linear catalyst ratio as a means to control the overall level of long-chain branching and the composition of distinct chain family members. This approach, utilizing combined metallocenes, effectively showcased the design of polyolefin microstructures [125]. The synthesis of PE with a bimodal MWD by dual-site metallocene catalysts was simulated using a developed kinetic model of Monte Carlo. Through kinetic investigation of polymerization systems with metallocene catalysts, it was observed that long branches were formed at the end of the polymer chain. This occurred when dead chains having unsaturated bonds at their ends participated in the growth reaction of another chain, resulting in the production of long branches in the growing polymer chain. Consequently, the MWD of the polymer became bimodal. The Monte Carlo model provided valuable information that complemented the results of the kinetic model, allowing for the mapping of chain microstructure and modeling of the polymerization reactor. The proposed methodology was evaluated through simulation of eight different dual-site catalyst systems [126]. Figure 11 illustrates the results of a Monte Carlo simulation that was conducted to investigate the bimodal MWD and branching frequency in PE synthesis using dual-site metallocene catalysts, as described in the aforementioned research.

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Fig. 11

Monte Carlo simulation of bimodal MWD and branching frequency in PE synthesis with dual-site metallocene catalysts [126]

In recent research, a Monte Carlo method accurately simulated the copolymerization of ethylene and 1-butene using a dual-site metallocene catalyst in the presence of hydrogen. Higher consumption rates of ethylene and 1-butene were observed at the second catalyst site, while the first site exhibited faster hydrogen transfer rates. The copolymers synthesized at the second site had a higher percentage of 1-butene. The overall weight average and number average molecular weights increased with increasing 1-butene concentration, while they decreased with increasing hydrogen concentration. The overall dispersity of the copolymers was slightly higher compared to each catalyst site. Higher 1-butene concentrations resulted in wider bimodal MWDs, whereas higher hydrogen concentrations led to taller peaks at shorter chain lengths. Finally, as the temperature increased, the molecular weight decreased, the bimodal MWD narrowed, the thickness and weight fraction of crystals increased, and the density of HDPE also increased [88]. Figure 1) depicts the comprehensive bimodal MWDs generated by the dual-site metallocene catalyst, which are influenced by the concentration ratios of ethylene to 1-butene (Fig. 12a) and ethylene to hydrogen (Fig. 12b).

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Fig. 12

Overall bimodal MWDs of dual-site metallocene catalyst with various concentration ratios of ethylene to 1-butene (a) and ethylene to hydrogen (b) [88]

Artificial intelligence-based (AI) approaches

Artificial intelligence (AI) methodologies have been extensively utilized in the domain of polymerization processes. Machine learning algorithms have proven to be valuable tools for effectively predicting the outcomes of polymerization reactions, optimizing reaction conditions, and designing catalysts tailored to the specific requirements of the polymerization process. The incorporation of AI techniques significantly contributes to the advancement of data analytics and the identification of intricate patterns in polymerization processes, thereby enhancing the monitoring and control mechanisms governing these processes. Furthermore, AI-driven models enable the synthesis of novel polymers with precise properties by accurately predicting the relationships between the structure and properties of polymers. Collectively, the employment of AI methodologies offers promising prospects for enhancing efficiency, productivity, and innovation within the realm of polymerization processes and the development of new polymers.

One notable approach rooted in artificial intelligence involves the utilization of artificial neural networks (ANNs) to simulate and optimize polymerization processes. ANNs serve as a computational model derived from the black box paradigm, adept at mimicking the intricate mechanisms observed in biological neural systems, such as the human brain. ANNs exhibit remarkable capabilities in addressing a wide range of nonlinear problems. Through the analysis of diverse examples and data, ANNs possess the capacity to discern and establish perfect correlations between input and output variables. ANNs possess the ability to detect patterns through necessary training sets and their corresponding responses. Comprised of interconnected neurons, ANNs function as a cohesive system to address specific problems. Moreover, ANNs learn from prior examples and refine the learning process by adapting biases and weights to attain an optimal learning pattern [127].

The process of simulating and optimizing copolymerization processes utilizing ANN encompasses three stages: training, validation, and testing. In the training stage, the ANN is trained using a diverse set of input variables and corresponding output values to minimize the disparity between predicted and actual outputs. The validation step assumes great significance as it delineates the network's capacity for generalization and establishes a criterion for concluding the training process, should the generalization performance cease to exhibit further enhancement. In order to establish a trustworthy and precise ANN model for copolymerization process prediction, it is imperative to undergo a testing phase wherein the ANN's performance is assessed using a new dataset consisting of real-world copolymerization processes. These test steps play a pivotal role in guaranteeing the development of a dependable and highly accurate ANN model for copolymerization process forecasting. Typically, the training process involves utilizing 70% of the datasets, while 15% are allocated for validation and an additional 15% for testing purposes [128, 129, 130, 131–132].

The utilization of ANN is viable for the simulation of the copolymerization process involving ethylene and diverse co-monomers, aiming to obtain HDPE exemplified by a bimodal MWD. This approach entails the acquisition of input and output data from experimental records or alternative simulation techniques like Monte Carlo or Moment equations. The ANN can serve as both a forward model and an inverse model, employing the developed training algorithm. Figure 13 illustrates the forward model, which consists of five input layers representing the initial conditions of the copolymerization process: ethylene concentration, co-monomer concentration, hydrogen concentration, co-catalyst concentration, and reaction temperature. The model includes hidden and output layers that generate weight fractions of chains ($w$) and SCBs at specific chain lengths (log DP_w). For this purpose, it is necessary to employ a set of selected MWDs and SCBs at specific chain lengths as the target matrix for training the ANN in programming languages such as MATLAB, Python, or similar alternatives [127].

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Fig. 13

Forward model algorithm with input layer representing initial reaction conditions, hidden layers, and output layer depicting weight fraction of copolymers ($\mathit{w}$) and SCBs

The training of ANN entails two key stages: feed-forward and backpropagation of errors. In the feed-forward stage, the model forms an optimized relationship between the input layer (which holds the dataset) and the output layer (which houses the target data). This relationship is determined through the manipulation of weights and biases. During the backpropagation stage, the model rectifies errors by propagating them backward through the network. In the feed-forward model, the optimization of weights and biases is performed to minimize the discrepancy between the calculated weight fractions and the desired network targets. Then, the input data sets (e.g., ethylene concentration) are multiplied by the weights and added to the bias term. This combination is processed through the sigmoid activation function to generate output values in the hidden layer neurons [130]. In Eq. (18), the values of neurons in the hidden layers and output layers can be calculated based on the input layer data, as well as the values of weights and biases. Furthermore, Eq. (19) demonstrates the resulting net output from each neuron, which is obtained by applying the sigmoid function.

18

$$O_{n,j}^{l+1}=B_j^{l+1}+\sum _{i=0}^i{W_{j,i}^{l+1}X}_{n,j}^l$$

19

$$X_{n,j}^{l+1}=\frac{1}{1+\text{exp}({-O}_{n,j}^{l+1})}$$

In Eqs. (18) and (19), the notations $W_{j,i}^{l+1}$,$B_j^{l+1}$, $X_{n,j}^l$, $O_{n,j}^{l+1}$, and $X_{n,j}^{l+1}$ respectively represent the weight for layer $l+1$ in row $i$ and column $j$, the bias for layer $l+1$ in column $j$, the net value of the final output of the neuron from layer $l$ in column $j$ and data set $n$, the output value of the neuron for layer $l+1$ in column $j$ and data set $n$, and the net output value of the final neuron for layer $l+1$ in column $j$ and data set $n$. Upon the completion of net value computation for the output neurons, encompassing predicted weight fractions and SCBs at specific chain lengths, the ensuing phase involves the initiation of error backpropagation. During this step, the algorithm's error is calculated utilizing the sum of squared error function, facilitated by Eq. (20).

20

$$E=\frac{1}{N}\sum _{n=1}^N\sum _{j=1}^{j_{\mathit{max}}}{{(t}_{n,j}^m-X_{n,j}^m)}^2$$

In Eq. (20), $N$, $X_{n,j}^m$, and $t_{n,j}^m$ represent the total number of data sets used for training, the net value of the final output calculated by the ANN, and the real value of the final output neuron in the last layer (target) respectively.

Upon evaluation of the errors, it is necessary to proceed with the weight updates based on the generated errors. The aim of such updates is to minimize the discrepancy between the values computed by the ANN and the real outputs. So, it is necessary to compute the partial derivative of the overall errors with respect to the derivative of each weight and aim for their minimum values. Initiating the weight updating process involves subtracting the computed values of the partial derivative of the total error with respect to the derivative of the corresponding weight from the old weight, resulting in a new weight value [127, 130, 132] (Eq. (21)).

21

$$W_{j,i,new}^{l+1}=W_{j,i}^{l+1}-\mu \times \frac{\partial E}{\partial W_{j,i}^{l+1}}$$where $W_{j,i,new}^{l+1}$ represents the updated weight in one iteration of the neural network and $\mu$ represents the adaptive factor, typically assumed as 0.001; a higher adaptive factor leads to a faster learning process. The Levenberg-Marquardt error backpropagation method is a highly effective approach for optimization and updating of weights and biases in ANNs. It combines the Gauss-Newton method with partial derivative minimization to achieve efficient convergence of neural network output values during training. This approach focuses on enhancing the model's learning capabilities and enhancing its prediction accuracy [127, 130, 133]. Equations (22) and (23) elucidate the iterative process of employing the Levenberg-Marquardt backpropagation algorithm to compute errors and adjust weights throughout the learning process.

22

$$\mathrm{\Delta }W=-{[J^{T}J+\mu I]}^{-1}{J}^{T}e$$

23

$$W_{j,i,new}^{l+1}=W_{j,i}^{l+1}+\mathrm{\Delta }W$$

Equations (22) and (23) introduce the variables $\mathrm{\Delta }W$, $J$, $J^T$, $I$, and $e$, representing the matrix of weight changes, the Jacobian matrix of error derivatives with respect to weight changes, the transpose of the Jacobian matrix, the unit matrix, and the error matrix, respectively. Figure 14 illustrates a concise and precise depiction of the ANN model utilizing the Levenberg-Marquardt algorithm for simulating the copolymerization process involving ethylene and co-monomer.

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Fig. 14

Concise and simplified representation of the ANN model utilizing the Levenberg-Marquardt algorithm to simulate the copolymerization process of ethylene and co-monomer

Charoenpanich et al. developed two ANN models, namely the forward and inverse models, to describe the ethylene/1-butene copolymerization process using a catalyst with two site types. The study utilized a mathematical model incorporating bimodal MWD and CCD, and employed 15 discrete data points from MWD and CCD at chosen chain lengths as the output layer (target) and input data containing concentrations and copolymerization conditions to train the feed-forward ANN. The forward model exhibited accurate estimation of copolymer properties based on the given polymerization conditions. On the other hand, the inverse model identified suitable conditions for achieving desired copolymer microstructures. It was observed that multiple solutions were generally obtained; however, by fixing a key parameter, unique solutions could be obtained. This approach demonstrated an efficient method for material design through the manipulation of polymerization conditions and had the potential for extension to more complex catalyst systems [134]. Figure 15 displays the comparison of bimodal MWD and CCD between the kinetic model and forward model of the ANN. In another study conducted by this group, an ANN feed-forward model was utilized to determine the average properties of copolymers synthesized through the copolymerization of ethylene and 1-butene. The focus of this study was to calculate the average copolymer composition, as well as ${\overline{M}}_n$, ${\overline{M}}_w$, and yield, using the forward model. In contrast to previous research that considered discretized data such as weight fractions of bimodal MWD and CCD at specific chain lengths, this study concentrated solely on the average copolymer properties. Furthermore, an inverse model was employed, where inputs such as ${\overline{M}}_n$, ${\overline{M}}_w$, yield, and average copolymer composition were used to estimate the initial conditions (ethylene, 1-butene, hydrogen, co-catalyst concentration, catalyst precursor, and temperature) as the output layer for the copolymerization process. The forward model was robust against noise, while the inverse model had multiple solutions and was sensitive to noise. Despite differences in estimated conditions, similar microstructures were achieved. These models offer insights for selecting suitable conditions. However, the inverse model requires further optimization for improved robustness [131].

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Fig. 15

Comparing bimodal MWD and CCD between the kinetic model and forward model of the ANN [134]

In other studies, a novel approach was presented for estimating kinetic rate constants in olefin polymerization using metallocene catalysts. The polymerization rate was modeled using the method of moments, and the prediction of reaction rate constants at various time points was enabled through the training of ANN. Actual polymerization data was used to test the model, resulting in satisfactory results. The comparison of different metallocene catalysts and the need for minimal experimental samples for training were facilitated by this method. Independence from operational conditions was exhibited, along with the capability to estimate rate constant Arrhenius parameters. Moreover, the incorporation of molecular weight information demonstrated potential in predicting chain transfer constants [129]. Robust models were developed using ANNs to predict the average molecular weight and activity of ethylene polymerization with multisite catalysts. The reliability and quality of the models were confirmed using the Leverage method, with only a few data points falling outside the models' applicability domain. A comparison with other artificial intelligence approaches showed that the proposed models were outperformed by them in terms of performance and robustness. The potential of neural networks as reliable models with reasonable accuracy for estimating the performance of multisite catalysts in ethylene polymerization was highlighted [135].

ANNs successfully predicted chain length, polydispersity, and temperature in zirconocene-catalyzed polymerization. The ANN results aligned with theory, uncovered trends, and identified unfavorable regions. The reverse-training approach enabled reaction rate prediction based on chain length and polydispersity, advancing kinetic deduction. ANNs effectively analyzed intricate olefin polymerizations, demonstrating machine learning's potential in approximating outcomes and comprehending reactions. Catalyst analysis revealed a suboptimal region associated with low rates and increased polydispersity. Lower termination rates yielded higher molecular weight, while the ratio of propagation to termination rates influenced the ${\overline{M}}_n$. The study faced constraints of incomplete data and industrial secrecy, but prediction and reverse training benefit experimental design and process optimization [136]. The determination of copolymerization parameters for generating ethylene/1-octene copolymers with targeted chain microstructures was accomplished through the application of an ANN within defined conditions. The ANN, utilizing a feed-forward network incorporating a backpropagation learning methodology and a 5–6-6–3 architecture, demonstrated proficient prediction capabilities, closely aligning with the experimental data obtained from both the training and testing datasets. These outcomes affirmed the proficient and accurate estimation of copolymerization parameters facilitated by the proposed approach [137]. In another work, ANN was employed to model and analyze the mechanical properties of PE, PP, and their blends. The ANN accurately predicted the stress–strain curve of PE/PP blends with different ratios and showcased the improved tensile behavior of PE when combined with strong-load-bearing PP. The ANN's capacity to perform nonlinear mapping without adjustments suggests its potential as a cost-effective and time-saving tool in polymer characterization [128].

In the study conducted by Amnuaykijvanit et al., the copolymerization kinetic model was utilized to calculate the chain microstructures of ethylene/1-butene copolymers produced under specific polymerization conditions. However, estimating polymerization conditions based on desired microstructures was not feasible using the kinetic model. To address this, the researchers developed two denoising autoencoder models, Model A and Model B, based on ANN. These models aimed to estimate suitable polymerization conditions for desired microstructures, including MWD, CCD, and average molecular weight. Model A considered MWD and CCD, while Model B included average microstructures and polymer yield. Both autoencoder models successfully estimated polymerization conditions with acceptable mean square error (MSE). Model B showed slightly better performance, demonstrating lower MSE values [138]. In other work, a comparison was conducted on four global optimization techniques (genetic algorithm (GA), particle swarm optimization (PSO), improved ant colony optimization (ACO), and modified artificial bee colony optimization (ACB)) for the purpose of identifying alternative polymerization conditions to produce specific ethylene/1-butene copolymers utilizing a dual-site catalyst. It was determined that the modified artificial bee colony optimization method exhibited the smallest number of incorrect solutions when optimizing MWDs, CCDs, and yield. The findings of this study suggested that the modified artificial bee colony method was deemed the most effective approach in generating multiple solutions with minimal inadequacies, even in the presence of random noise. Additionally, it was observed that the proposed approach could be extended to various catalysts and polymerization conditions, and the objective function could be fine-tuned to achieve a balance between property and cost considerations within industrial systems [139].

The microstructure of ultra-high molecular weight polyethylene (UHMWPE) was accurately predicted using post-metallocene multi-site catalysts through the successful implementation of the Imperialist Competitive Algorithm (ICA). The polymerization conditions were optimized, leading to the determination of the kinetic constants necessary for predicting the bimodal MWD of UHMWPE. It was revealed that the MWD of UHMWPE was predicted by one site of the catalyst, while the remaining three sites were responsible for synthesizing PE chains with high molecular weights. The results demonstrated that the combined kinetic-ICA model represented a flexible, precise, and robust strategy for modeling ethylene polymerization with post-metallocene multi-site catalysts [140].

Advantages and disadvantages of Moment, Monte Carlo, and AI-based approaches

This study presents a novel approach that incorporates kinetic modeling, moment equations, Monte Carlo simulations, and AI-based algorithms for the precise prediction of the microstructure and properties of polyolefins, particularly emphasizing bimodal HDPE. Hence, it is crucial to acknowledge and address both the strengths and limitations associated with each of the aforementioned methods employed in predicting the microstructure and ultimate properties of bimodal HDPE.

The method of moments is a widely utilized approach in simulating copolymerization processes, offering advantages and disadvantages that shape its applicability and limitations. This reaction-based kinetic technique involves taking into account the kinetics constants and mass balance principles related to the derivatives of species and reactants. By applying this method, the original set of mass balance equations, which encompass different types of species like small molecules, dormant chains, alive chains, and dead chains, are simplified into smaller and more manageable sets of differential equations. Moreover, the method of the moments allows for the estimation of average characteristics of polymers, such as average molecular weight, average MWD, and average chemical composition. This enables a thorough evaluation of their overall behavior. This method is frequently favored for the simulation of polymerization processes due to its propensity for expediting the attainment of average outcomes. Its straightforward coding in different programming languages adds to its wide acceptance in both academic and industrial polymerization processes [94].

Nevertheless, there are certain limitations that hinder the method of moments from offering a comprehensive understanding of copolymerization processes. Significantly, the method of moments is notably unable to predict instantaneous events such as instantaneous reaction rates, instantaneous concentration profiles of different species, MWD and CCD at different times and conversions, as well as the distribution of side-chain branches throughout the entire polymerization process. Moreover, the efficacy of this method diminishes significantly in reactions where the kinetic constants exhibit dependency on the chain length. This is due to the fact that as the chain length undergoes modifications throughout the polymerization process, the corresponding kinetic constants also undergo alterations. It is worth mentioning that there can be difficulties in obtaining precise sets of differential equations for complex distributions of species. Although this method is effective for adjusting model parameters and providing an overall review of processes, it becomes challenging when used to investigate polymerization processes that result in complex distributions [94, 141].

The Monte Carlo method presents numerous advantages over the method of moments in the context of simulating copolymerization processes. In contrast to the method of moments, the Monte Carlo method necessitates solely the initial concentration of species and the kinetic rate constants related to the reactions implemented in the polymerization process. The procedure adopted by this method for simulating the polymerization process relies on the probabilistic properties of reactions. It follows a systematic sequential approach that encompasses system initialization, computation of reaction propensities, generation of random numbers, identification of reaction interval times, and subsequent system update [142, 143–144].

The Monte Carlo method offers numerous specifications that make it attractive for simulating copolymerization processes. It relies on simple assumptions, eliminating the need for solving complicated sets of differential equations. Additionally, it provides a highly accurate and detailed representation of both the micro and macro structure of polymers. The utilization of the Monte Carlo method allows for the instantaneous calculation of reaction rates and concentrations of all species involved in a copolymerization process. This capability facilitates a comprehensive examination of the chain length distribution at different levels of reaction conversion, encompassing even low conversion regions. Furthermore, it provides instantaneous information about multiple parameters such as MWD, CCD, and the distribution of short and long chain branches at any given time. These features make the Monte Carlo method particularly suitable for studying complex systems, including those characterized by chain length-dependent rate constants as well as processes that deviate from the assumption of steady-state, such as radical polymerization [142, 143, 145, 146–147].

Nonetheless, the implementation of the Monte Carlo method is inherently more complex when compared to the method of moments. The proficiency and expertise of the programmer responsible for writing the codes assumes a paramount significance in guaranteeing optimal efficiency of the simulation. As the complexity of the polymerization processes increases, the program execution time might also slightly lengthen. Additionally, powerful computers with high storage capacity may be required to handle simulations with extensive data. Moreover, careful selection of the simulation volume is crucial as it directly impacts the execution time of the simulation [146, 148]. In summary, the Monte Carlo method offers significant advantages, such as its flexibility, accuracy, and ability to simulate complex polymerization processes. While it requires more advanced coding skills and may impose computational demands, when properly utilized, it serves as a valuable tool in investigating complex polymerization processes.

AI-based approaches, specifically those employing ANNs, have garnered substantial recognition in simulating polymerization processes. ANNs have emerged as a promising methodology for effectively simulating and optimizing these intricate processes. By providing a phenomenological comprehension of the modeled process, ANNs enable efficient utilization of the available data without necessitating protracted time or financial investments [128, 129]. ANNs possess the capability to be constructed using either directly acquired experimental data or data derived from alternative simulation techniques such as moment equations or Monte Carlo simulations. This unique characteristic enables ANNs to autonomously organize themselves and effectively identify intricate correlations between polymerization conditions and pertinent structural attributes, including but not limited to ${\overline{M}}_w$, ${\overline{M}}_n$, MWD, CCD, MI, density, and dispersity. Hence, ANNs are highly suitable for the process of optimizing outcomes acquired through alternative simulation methods or the optimization of initial values and conditions in polymerization reactions as well as pre-designing the microstructure of copolymers with desired properties. Typically, ANNs are coded in Python or MATLAB programming languages, offering comparatively simpler implementation compared to the Monte Carlo method. Moreover, MATLAB program provides convenient access to pre-built toolboxes specific to ANNs, simplifying the determination of the number of input layers, hidden layers, and output layers [128, 149, 150–151].

While employing ANNs for modeling and simulating systems may seem straightforward, the dependability of the network in generating trustworthy outputs is contingent upon the caliber and extent of the training data. Moreover, in complex polymeric systems, a comprehensive and precise dataset must be supplied to the ANN, one that exhibits a close correlation with the predictive parameters. Typically, AI-based methodologies are unable to directly derive output data solely from input data consisting of kinetic constants and initial conditions. In contrast to the Moment and Monte Carlo methods that rely on initial conditions and reaction kinetics to anticipate process outputs, the utilization of ANN necessitates the availability of a series of accurate input and output data. The network learns from this data and subsequently acquires the capability to predict novel outputs based on new input data [127, 131, 134, 129, 152].

Future perspectives in simulating bimodal HDPE synthesis

Recent academic publications have observed a growing interest in the utilization of Monte Carlo and ANN methods for simulating the synthesis of bimodal HDPE, as compared to the commonly employed Moment method. Figure 16 presents the percentage of scholarly papers published between 2000 and 2024 that have employed simulation methods such as Moment, Monte Carlo, and ANNs. The graph reveals an increase in the percentage of articles employing Monte Carlo and ANN methods, while the utilization of the Moment method has shown a slight decline. Taking into account the data obtained from Fig. 16, it can be inferred that future perspectives on simulating HDPE synthesis offer numerous opportunities and proposals for investigating the synthesis of this polymer with modified MWDs. This exploration aims to effectively address processing challenges while retaining the desired mechanical properties. Additionally, catalytic polymerization and macromolecular engineering design can be further explored as viable approaches to achieve bimodal MWD. This involves combining the ratio of two single-site metallocene catalysts capable of simultaneously producing low and high molecular weight polymers while controlling bimodality.

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Fig. 16

Percentage distribution of published papers on HDPE synthesis simulation using Moment, Monte Carlo, and AI techniques from 2000 to 2024

Critical variables such as pressure, temperature, and initial reactant concentrations (monomer, co-monomer, hydrogen) should be examined using the aforementioned simulation methods to understand their impact on the microstructure of bimodal HDPE. Special attention should be given to the development of simulation algorithms specifically tailored for ethylene copolymerization with various co-monomers. Furthermore, it is possible to investigate the simulation of HDPE synthesis across all three processes: gas phase, slurry, and solution, utilizing the aforementioned simulation and optimization methodologies. Subsequently, the resulting properties of the copolymers produced by each method can be comprehensively compared and contrasted. Finally, future research efforts can aim to enhance the accuracy of bimodal HDPE simulation and achieve desired structures with controlled bimodality through the simultaneous utilization of artificial intelligence and Monte Carlo methods.

Conclusion

In conclusion, this review paper has emphasized the potential of utilizing single or dual-site metallocene catalysts to achieve innovative and cost-effective prospects in the synthesis of HDPE with a bimodal MWD. The copolymer microstructure is greatly influenced by important variables such as pressure, temperature, reactant concentrations, and catalysts in the polymerization processes. Various production methods have been explored, including gas phase, slurry, and solution processes, for the synthesis of bimodal HDPE. The objective of this work was to provide insights into the development of metallocene catalysts that possess specific characteristics. These include a first site or first catalyst with high hydrogen participation for producing low molecular weight chains, and a second site or second catalyst with high co-monomer incorporation for synthesizing copolymers with high molecular weights. This catalyst design aims to facilitate a single-step process, eliminating the need for the current two or multi-step processes, in both lab-scale and industrial-scale applications across various production methods.

Simulation methodologies have been employed to verify and ensure the ongoing efficacy of industrial polymerization procedures. This facilitates well-informed decision-making during the initialization stage, offering significant promise for advancing optimal industrial solutions in the copolymerization process for producing HDPE with a bimodal MWD. Valuable insights for HDPE synthesis with bimodal MWD have been provided through simulation approaches, including moment equations, Monte Carlo simulations, and artificial intelligence (AI) techniques. While moment equations have been traditionally used to quickly obtain average results in simulating copolymerization processes, they have limitations in predicting instantaneous events and complex distributions. On the other hand, the Monte Carlo method offers a detailed representation of micro and macro structures, enables instantaneous calculations of reaction rates and concentrations, and allows for comprehensive examinations of chain length distributions and CCD. Although implementation of the Monte Carlo method can be complex, its advantages greatly enhance our understanding of complex systems. AI-based approaches, particularly those employing ANNs, have gained recognition for their ability to simulate and optimize polymerization processes in copolymer synthesis and pre-designing. However, the reliability of ANN outputs relies heavily on the quality and extent of the training data.

By combining the strengths of the Monte Carlo method and AI techniques, significant progress has been made in understanding and optimizing the synthesis of bimodal HDPE. Upon completing the training of the ANN using Monte Carlo simulation or Moment equation outcomes, the ANN is capable of being utilized to acquire desired optimal output data comprising weight fractions of bimodal MWD and SCB data at specific chain lengths. Subsequently feeding these optimal targets as input to the ANN, it generates the corresponding optimal initial conditions for copolymerization. These optimal conditions encompass the ideal monomer concentration, pressure, and temperature necessary to attain the previously mentioned desirable weight fractions of bimodal MWD or SCB data at specific chain lengths. This research opens up new avenues for innovation and economic feasibility in the polymer industry, paving the way for enhanced production processes and the development of high-quality polymer materials.

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Conflict of interest

We hereby state that there is no conflict of interest.

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