Effect of Monomer Mixture Composition on TiCl₄-Al(i-C₄H₉)₃ Catalytic System Activity in Butadiene–Isoprene Copolymerization: A Theoretical Study

Abstract

Divinylisoprene rubber, a copolymer of butadiene and isoprene, is used as raw material for rubber technical products, combining isoprene rubber’s elasticity and butadiene rubber’s wear resistance. These properties depend quantitatively on the copolymer composition, which depends on the kinetics of its synthesis. This work aims to theoretically describe how the monomer mixture composition in the butadiene–isoprene copolymerization affects the activity of the TiCl₄-Al(i-C₄H₉)₃ catalytic system (expressed by active sites concentration) via kinetic modeling. This enables development of a reliable kinetic model for divinylisoprene rubber synthesis, predicting reaction rate, molecular weight, and composition, applicable to reactor design and process intensification. Active sites concentrations were calculated from experimental copolymerization rates and known chain propagation constants for various monomer compositions. Kinetic equations for active sites formation were based on mass-action law and Langmuir monomolecular adsorption theory. An analytical equation relating active sites concentration to monomer composition was derived, analyzed, and optimized with experimental data. The results show that monomer composition’s influence on active sites concentration is well described by a two-step kinetic model (physical adsorption followed by Ti–C bond formation), accounting for competitive adsorption: isoprene adsorbs more readily, while butadiene forms more stable active sites.

1. Introduction

Butadiene and isoprene rubbers (BDR and IR) are widely used as raw materials for the production of rubber technical products [1]. Compared to each other, these rubbers have both advantages and disadvantages. The general advantages of BDR are its outstanding resilience, excellent flexibility at low temperature (better than that of IR), superior resistance to abrasion, cut growth, and flex cracking [1,2]. That is why it is used in truck tire tread composition [2]. However, its limitations are inferior processability, poor resistance to oil and gasoline, and very little resistance to heat and ozone [2]. Therefore, it is rarely used alone, but blended with various proportions of styrene butadiene rubber, natural rubber and IR [1]. IR exhibit good inherent tack, high compounded gum tensile properties, and good hot tensile properties [1].

An alternative to rubbers based on blends of BDR and IR are rubbers based on the copolymer of butadiene and isoprene (divinylisoprene rubber—DIVR) [3]. DIVR combines the properties of the corresponding homopolymers: the elasticity of IR and the wear resistance of BDR. Quantitatively, these properties also depend on the composition of DIVR. Thus, the task of optimizing the composition of DIVR for each specific application area arises. This task may seem trivial, since it is obvious that the composition of DIVR is determined by the composition of the monomer mixture. However, as shown in studies [4,5], the monomer mixture composition not only affects the composition of DIVR but also nonlinearly influences the synthesis rate of DIVR and its molecular weight characteristics. The task of predicting the synthesis patterns of DIVR is complicated by the fact that this synthesis proceeds using a heterogeneous multisite Ziegler–Natta catalytic system, and the formation history of this catalytic system affects both the number of different types of its active sites and their activity [4,5]. The rate of a heterogeneous catalytic process, such as DIVR synthesis, usually depends nonlinearly on the reaction conditions. Prediction of such processes based solely on experimental data is unreliable, as it generally relies on linear extrapolation of the observed results. A more reliable approach is based on mathematical modeling, specifically kinetic modeling of the process, since catalysis is, by definition, a kinetic phenomenon. Considering the complex nature of heterogeneous catalysis, the kinetic model should account for adsorption and desorption of reagents. Furthermore, such a kinetic model must explicitly consider the interaction of reagents with the catalytically active surface of solid particles—the active sites of catalytic systems. Therefore, within this kinetic model, the activity of the heterogeneous catalytic system is explicitly characterized by the concentration of active sites and the rate constants of reactions involving them.

Taking all of the above into account, the task of theoretically describing the influence of the monomer mixture composition in the butadiene–isoprene copolymerization (DIVR synthesis) on the activity of the catalytic system (quantitatively expressed through the concentration of its active sites) within the framework of kinetic modeling becomes highly relevant. The objective of this work is to solve this problem for the classical and industrially applied Ziegler–Natta catalytic system—TiCl₄-Al(i-C₄H₉)₃. This theoretical description will subsequently enable the development of a reliable kinetic model of DIVR synthesis, describing the synthesis rate, molecular weight characteristics, and composition of DIVR, which will allow the practical application of this model to address applied problems related to the design of DIVR synthesis reactors and the intensification of this process. This underlines the practical significance of achieving the stated goal of the work. Moreover, creating this theoretical description will advance the evaluation of active sites’ concentrations in heterogeneous catalytic systems based on the rates of reactions involving that involve them. This approach differs from the currently accepted strictly experimental methods for measuring active sites concentrations, such as (1) X-ray absorption spectroscopy [6,7,8]; (2) infrared spectroscopy [6,7]; (3) scanning transmission electron microscopy [6,7]; (4) probe molecule spectroscopy [7,9,10]; (5) Mössbauer spectroscopy [10,11]; (6) adsorption and desorption methods [10,11,12]; (7) environmental scanning transmission electron microscopy with a high-angle annular dark-field [13]; and (8) titration with catalytic poisons [14].

It is impossible to achieve the objectives of this work without understanding the nature of active sites in Ziegler–Natta heterogeneous catalytic systems, the mechanism of their formation, and their performance in polymerization and copolymerization processes. Despite the industrial importance of Ziegler–Natta catalytic systems, the nature of their active sites remains a subject of debate, as their structure still cannot be directly observed [15,16,17]. The current understanding is primarily based on density functional theory calculations [15,16]. Ziegler–Natta catalytic systems have a multisite nature, meaning that particles within the same catalytic system may contain active sites with different structures, distinguished by their stability, chain propagation rate, and capacity for copolymerization [17]. For example, on the surface of supported catalysts such as TiCl₄ + MgCl₂ + cocatalyst (where the catalytically active TiCl₄ component is supported on the inert MgСl₂ matrix), both mononuclear active sites—octahedrally coordinated Ti(IV) sites on the MgСl₂ crystallites—and polynuclear clusters containing Ti(III) can exist [16]. The multisite nature of active sites directly affects polymerization kinetics and the molecular weight distribution of the polymer [17]. Sites of a single type produce polymer fractions with a narrow molecular weight distribution (described by the Flory distribution), whereas a combination of different site types together yield a polymer with a broad molecular weight distribution [17]. The activity of active sites depends on reaction conditions; the relative contribution of different types of sites to polymer formation can change with temperature [17].

The structure of the catalyst particle surface plays a major role in the formation of active sites [18]. This role can also be illustrated with supported TiCl₄/MgCl₂ catalysts. Surface defects and areas of the MgCl₂ support with low coordination numbers stabilize titanium active complexes, increasing their reactivity [18]. Such regions on the catalytic particle’s surface are most likely to anchor propagating polymer chains. In its pure form, MgCl₂ is inactive as a catalyst support, and its particles have a low surface area [19]. The surface of MgCl₂ can be activated both physically and chemically by reacting MgCl₂ with alcohols (e.g., ethanol), but the alcohol must subsequently be removed from the reaction system, as it acts as a catalytic poison [19]. Ethanol is chemically removed from the MgCl₂ surface using weak Lewis acids: TiCl₂, triethylaluminum, triisobutylaluminum, and ethylaluminum dichloride [19]. It has been found that the concentration of ethanol in the catalytic system significantly affects the performance characteristics and properties of polyethylene [19], polypropylene [20], and the propylene/1-hexene copolymer [20].

In addition to the chemical nature of the support, steric hindrances also influence the activity and stereoselectivity of the active sites [21]. For example, realizing the propagation of an isotactic polymer chain requires sterically restricted migratory monomer insertion into the propagating polymer chain at the active site (the Cossee mechanism [22]) [21]. To achieve this, the MgCl₂ support must adsorb at least two adsorbate molecules on its surface on opposite sides of the active site [21].

The formation of active sites in Ziegler–Natta catalysts begins with a sequence of complexation and activation (alkylation/reduction) reactions, initiated by contact between transition metal compounds, such as titanium chlorides (TiCl₃ or TiCl₄), and aluminum alkyls. Here, TiCl₄ (or TiCl₃) acts as an electron donor, while the aluminum alkyl acts as an acceptor. The concentration of the cocatalyst directly determines the number of sites formed [16,23]. The most common cocatalyst is triethylaluminum [16]. The type of cocatalyst affects the deactivation rate of the reaction system [23]. For example, it was found that when triisobutylaluminum was used as a cocatalyst in a TiCl₄/MgCl₂-based catalytic system, the rate of ethylene polymerization noticeably decreased over time, whereas the use of triethylaluminum as a cocatalyst maintained a constant polymerization rate for a long period [23]. This difference was explained by the smaller molecular volume of triethylaluminum compared to triisobutylaluminum, facilitating its diffusion to the active sites of the catalytic system [23].

The combination of a cocatalyst with electron donors (Lewis bases) significantly changes both the rate of formation of active sites and their catalytic activity, as well as the properties of the resulting polymers [16]. However, the activity of the catalytic system is not constant. Even after the formation of active sites, their quantity and types continue to evolve. Some examples of such an evolution are listed below.

• Under the influence of hydrodynamic factors (turbulent velocity pulsations in the reaction system, which cause significant shear stresses within it [24]) and chemical factors (propagation of polymer chains in the pores of catalyst particles, which wedges these particles apart [25]), the particles of the catalytic system undergo dispersion. Dispersion of the catalytic system particles during polymerization exposes previously hidden active surfaces [26,27].
• Changes in temperature and reaction time shift the equilibrium between active sites of different types [26].
• Introduction of comonomers accelerates activation by creating additional, more reactive sites [28].
• Complex mechanism of active sites formation. For example, a two-step mechanism for active sites formation has been described, whereby initially a limited number of active sites are formed, and then, as the catalytic system particles disperse and their surface reorganizes, active sites of new types appear [28,29].

The functioning of active sites consists of the propagation, transfer, and termination of polymer chains [17,30]. The functioning of active sites enables their existence to be fixed; therefore, within the framework of this article, it is important to consider not only the formation of active sites but also their functioning. As a result of polymer chain propagation near active sites, polymer globules are formed [30]; these become entangled due to diffusion of their amorphous segments, forming filamentous structures [30]. At a random moment during propagation, a chain may detach from the active site and from the surface of the catalytic system particles [30]. Detached chains move away from the surface due to convective transport and diffusion, which makes it difficult to establish a relationship between the structure of the active site and the structure of the polymer chain synthesized with its participation [30]. During polymerization, each active site synthesizes a large number of polymer chains. For example, in polypropylene synthesis, each active site is capable of synthesizing approximately twenty thousand polypropylene chains per hour, each containing about 7.5 × 10³ monomeric units, which corresponds to an average molecular weight of ≈3 × 10⁵ g/mol [30].

The evolution of active sites, expressed by the disappearance of one type of active sites and the appearance of another type during polymerization, leads to the synthesis of polymer chains with different molecular weights and chain tacticity at various polymerization stages [31], since active sites of different types differ in stereoselectivity [31], and different copolymer chains of varying composition are synthesized at different stages of copolymerization [28]. For example, active sites of the TiCl₄/MgCl₂ catalytic system formed at late stages of butadiene and isoprene copolymerization due to exposure of new surfaces of the catalytic system particles have a higher rate constant for propylene chain propagation and synthesize chains of a statistical copolymer with reduced crystallinity, whereas active sites located on the initial surface of the catalytic system particles continue to synthesize copolymer chains with a higher degree of blockines [28].

Thus, the kinetics of copolymerization of monomers in the presence of Ziegler–Natta catalytic systems is determined by the number, types, and structure of active sites. The number of active sites influences the copolymerization rate, while their diversity causes heterogeneity in the molecular weight distribution of the copolymer, as well as heterogeneity in the distributions of composition and tacticity of the segments.

Kinetic modeling of the polymerization and copolymerization processes of monomers on Ziegler–Natta catalysts is based on accounting for the multisite nature of the catalytic system [17,31]. One of the key aspects of such models is the description of the interaction of reagents with active sites, including adsorption of monomer molecules on the surface of the solid particles of the catalytic system, desorption of these molecules, and coordination of monomer molecules to the active sites with subsequent migratory insertion into the polymer chain.

The construction and verification of kinetic models rely on experimental data on copolymerization rates and molecular weights, which allows determination of individual reaction rate constant values, as well as concentrations of active sites. The use of kinetic models and numerical methods for solving their equations, such as the Monte Carlo method, provides the possibility to reproduce the dynamics of the process, taking into account changes in the distribution of active sites types and their evolution over time [29]. At the same time, the development of statistical models is underway, within which the activity of catalytic systems is predicted using machine learning methods [32].

Thus, to achieve the goal of the work, the following tasks were decided (in their solution, the physicochemical nature of the formation and functioning of active sites of multisite heterogeneous Ziegler–Natta catalytic systems, described above, were taken into account).

To achieve the objective of this work, the following tasks were undertaken.

• The concentrations of active sites (μ) of the TiCl₄-Al(i-C₄H₉)₃ catalytic system during the butadiene–isoprene copolymerization were calculated at various monomer mixture compositions (q) based on experimental data on the copolymerization rate taken from [5]. Here, q = [M₁]/([M₁] + [M₂]), [M₁] is the concentration of butadiene in the monomer mixture, [M₂] is the concentration of isoprene in the monomer mixture, q = 1 corresponds to the homopolymerization of butadiene, and q = 0 corresponds to the homopolymerization of isoprene (here and below, […]—denotes concentration) These calculated concentrations are hereafter referred to as experimental and denoted as μ_exp.
• Based on the Langmuir monomolecular adsorption theory [33] and the mass-action law, a kinetic model for the formation of active sites in the catalytic system was developed. As a result of the analytical solution of the system of kinetic equations, an equation was obtained which established a direct functional relationship between the concentration of active sites μ and the composition of the monomer mixture q. The concentration values calculated using this equation are hereafter referred to as calculated concentrations and denoted as μ_calc.
• An analysis of the obtained equation was performed to determine the ranges of kinetic parameters of the active sites’ formation process for which the dependence of μ_calc on q matches, in shape, the dependence of μ_exp on q.
• Based on the results of solving the third task, specific quantitative values of the kinetic parameters of the active sites’ formation process were determined within the allowable parameter ranges, for which the dependence of μ_calc on q quantitatively coincides with the dependence of μ_exp on q.

2. Materials and Methods

In addressing the first task, we relied on the experimental results reported in [5]. In that study [5], by solving the inverse problem of the molecular weight distribution of the copolymer, it was established that active sites of up to four types operate during the butadiene–isoprene copolymerization. Active sites of different types are understood as catalytically active fragments of the catalytic system particle surface that possess distinct structures. Due to this structural diversity, the propagation of polymer chains at active sites of different types proceeds with varying ratios of chain propagation rate to chain termination rate. As a result, active sites of different types produce copolymer chains with varying molecular weights, leading to a broadening of the molecular weight distribution of the copolymer [34]. The number of active sites types depends on the value of q: at q = 1, all four types of active sites are active, whereas at q < 1, only the second and third types of active sites remain active. This suggests that even small amounts of isoprene deactivate the first and fourth types of active sites.

In [5], two methods of copolymerization were considered. In Method 1, the TiCl₄-Al(i-C₄H₉)₃ catalytic system was prepared separately and aged at 0 °C for 30 min, after which it was introduced into a 500 cm³ flask containing a solution of the monomer mixture in toluene. A magnetic stirrer was used to ensure continuous agitation of the reaction system during polymerization. Toluene served as the solvent both in the preparation of the catalytic system and in the butadiene–isoprene copolymerization. Copolymerization was carried out at the same initial total monomer concentration, while the molar ratio of the monomers varied in different experiments. The copolymerization was performed under the following initial conditions: total monomer concentration [M] = 1.5 mol/L; catalytic system preparation conditions—[TiCl₄] = 5 mmol/L, [Al(i-C₄H₉)₃]/[TiCl₄] = 1.4; polymerization temperature 25 °C.

Method 2 differed from Method 1 in that the mixing of the catalytic system with the monomer mixture solution in toluene was carried out not in the flask, but immediately before the flask in a tubular turbulent diffuser–constrictor apparatus (Figure 1). The flow velocity of the reaction mixture through the apparatus was 0.9 m/s. The resulting reaction mixture was then introduced into the flask for polymerization. In both copolymerization methods, methanol was added to the flask 60 min after the start of the process to terminate the reaction.

In addressing the first task, theoretical results from studies [35,36] were also utilized. In [35], a kinetic model of butadiene homopolymerization involving the TiCl₄-Al(i-C₄H₉)₃ catalytic system was developed. In [36], a kinetic model of isoprene homopolymerization with the same catalytic system was developed. The classical model of chain reactivity in homo- and copolymerization is the chain-end model [37], according to which the rate constants of reactions involving active polymer chains are determined by the type of the terminal unit of these chains. Therefore, based on the terminal unit model, the rate constants of chain propagation reactions in butadiene and isoprene homopolymerization established in [35,36] were employed in this work to calculate the rates of self-propagation reactions in copolymerization (i.e., the addition of butadiene to active chains with terminal butadiene units and the addition of isoprene to active chains with terminal isoprene units). The rate constants of cross-propagation reactions were calculated based on the rate constants of self-propagation reactions and the corresponding copolymerization constants determined in [5].

3. Results

3.1. Calculation of the Concentration of Active Sites of the TiCl₄-Al(i-C₄H₉)₃ Catalytic System

The task of calculating the concentrations of active sites of the TiCl₄-Al(i-C₄H₉)₃ catalytic system was addressed as follows. First, the experimental initial rate of copolymerization W was calculated for each experiment as the product of the total monomer concentration and the initial values of the tangent of the slope angle of the copolymer yield versus time dependencies reported in [5].

The copolymerization rate W was related to the concentration of active sites of the TiCl₄-Al(i-C₄H₉)₃ catalytic system through the kinetic model equations of the butadiene–isoprene copolymerization. These equations were formulated based on the mass-action law and classical kinetic schemes of coordination copolymerization [5], which include the following reactions.

1. Chain propagation:

$${\mathit{R}}_{\mathit{xy}}+{\mathit{M}}_{\mathit{z}}\stackrel{{\mathit{k}}_{\mathit{pxyz}}}{\to }{\mathit{R}}_{\mathit{xz}}$$

2. Chain transfer to monomer:

$${\mathit{R}}_{\mathit{xy}}+{\mathit{M}}_{\mathit{z}}\stackrel{{\mathit{k}}_{\mathit{Mxyz}}}{\to }{\mathit{P}+\mathit{R}}_{\mathit{xz}}.$$

3. Chain transfer to cocatalyst:

$${\mathit{R}}_{\mathit{xy}}+\mathit{C}\stackrel{{\mathit{k}}_{\mathit{Axy}}}{\to }{\mathit{P}+\mathit{R}}_{\mathit{x}}.$$

4. Deactivation of active sites:

$${\mathit{R}}_{\mathit{xy}}\stackrel{{\mathit{k}}_{\mathit{txy}}}{\to }\mathit{P}.$$

5. Interconversion of active sites of different types:

$${\mathit{R}}_{\mathit{xy}}\stackrel{{\mathit{k}}_{\mathit{cxsy}}}{\to }{\mathit{R}}_{\mathit{sy}}.$$

In the symbolic notation of the reactions, the degrees of polymerization of the chains are intentionally omitted to avoid complicating the representation.

Here, R represents the active copolymer chains; M represents the monomer molecules; P represents the inactive copolymer chains; and C is the cocatalyst molecule (Al(i-C₄H₉)₃); k is the rate constant of the corresponding reaction; x and s = 1, 2, 3, 4 represent the type of active site at the active chain end; y and z = 1, 2 are indices reflecting the type of the chain terminal unit or the type of monomer molecule involved in the reaction: 1 corresponds to a butadiene terminal unit or butadiene molecule and 2 corresponds to an isoprene terminal unit or isoprene molecule.

The copolymerization rate W is equal to the rate of change of the total monomer concentration; that is, according to the presented kinetic scheme, the copolymerization rate W equals the sum of the rates of chain propagation and chain transfer to monomers. However, when calculating the copolymerization rate W, the chain transfer to monomers can be neglected, as it is usually several times lower than the chain propagation rate [5].

Based on the above, the equation relating the concentration of active sites to the copolymerization rate of butadiene and isoprene at q < 1 was obtained by solving the following system of Equations (1)–(4).

1. Equation relating the concentration of active sites to the copolymerization rate (according to the mass-action law):

$$\sum _{\mathit{z}=1}^2\sum _{\mathit{y}=1}^2\sum _{\mathit{x}=2}^3{\mathit{k}}_{\mathit{pxyz}}{\mu }_{\mathit{xy}}\left[{\mathit{M}}_{\mathit{z}}\right]=\mathit{W},$$

(1)

where ${\mu }_{\mathit{xy}}$ is the concentration of active chains propagating on active sites of type x with terminal unit of type y (the total concentration of active sites equals the total concentration of active chains, since each active site corresponds to one active chain); [M_z] is the initial concentration of monomer of type z.

The experimental value of the initial copolymerization rate W was determined by the following equation:

$$W=\left[M\right]{\left.{\displaystyle \frac{dU}{dt}}\right|}_{t=0},$$

where [M] = 1.5 mol/L is the total initial monomer concentration; $U$ is the experimentally measured time-dependent copolymer yield (the data for all q values were taken from [5]); and t is the copolymerization time (t = 0 indicates that the derivative was calculated at the initial segment of the $U$).

2. Next, we have an equation which establishes a direct proportionality between the experimental relative activity S_x of active sites of different types x and their chain propagation rate:

$${\displaystyle \frac{{\sum }_{z=1}^2\sum _{y=1}^2{k}_{p3yz}{\mu }_{3y}\left[M_z\right] }{{\sum }_{z=1}^2{\sum }_{y=1}^2{k}_{p2yz}{\mu }_{2y}\left[M_z\right] }}={\displaystyle \frac{S_3}{S_2}}.$$

(2)

The relative activities S_x are defined as the areas under the peaks in the activity distribution of the catalytic system according to kinetic heterogeneity (the ratio of the chain termination rate to the chain propagation rate), which was determined in study [5] by solving the inverse problem of the polymer molecular weight distribution.

3. The equations below express the equality of cross-propagation rates on active sites of each type (which essentially represent the quasi-steady-state conditions for the concentrations of active chains terminated with butadiene and isoprene units propagating on active sites of each type):

$$k_{p221}{\mu }_{22}\left[M_1\right]=k_{p212}{\mu }_{21}\left[M_2\right],$$

(3)

$$k_{p321}{\mu }_{32}\left[M_1\right]=k_{p312}{\mu }_{31}\left[M_2\right].$$

(4)

The rate constants $k_{p211}$ and $k_{p311}$ were taken from the model presented in [35]. The rate constants $k_{p222}$ and $k_{p322}$ were taken from the model presented in [36]. The rate constants of cross-propagation reactions were calculated using the copolymerization constants $r_1=k_{px11}/k_{px12},$ $r_2=k_{px22}/k_{px21}$. In study [5], the calculation of copolymerization constants by the Fineman–Ross method [38] showed that r₁ = r₂ = 1.

In the system of Equations (1)–(4), written for each value of q, there are four unknowns: ${\mu }_{21}$, ${\mu }_{22}$, ${\mu }_{31}$, ${\mu }_{32}$. Since the system also contains four equations, it is closed and can be solved with respect to these unknowns.

For this purpose, the concentrations ${\mu }_{x1}$ were first expressed from Equations (3) and (4) as follows:

$${\mu }_{x1}={\displaystyle \frac{k_{px21}\left[M_1\right]}{k_{px12}\left[M_2\right]}}{\mu }_{x2}.$$

(5)

Equation (5) was then substituted into Equation (1), and the resulting equation was transformed accordingly:

$$\begin{gathered} \sum _{z=1}^2\sum _{x=2}^3\left(k_{px1z}{\mu }_{x1}+k_{px2z}{\mu }_{x2}\right)\left[M_z\right]=W, \\ \sum _{z=1}^2\sum _{x=2}^3\left(k_{px1z}{\displaystyle \frac{k_{px21}\left[M_1\right]}{k_{px12}\left[M_2\right]}}{\mu }_{x2}+k_{px2z}{\mu }_{x2}\right)\left[M_z\right]=W, \end{gathered}$$

$$\sum _{x=2}^3{\mu }_{x2}\sum _{z=1}^2\left(k_{px1z}{\displaystyle \frac{k_{px21}\left[M_1\right]}{k_{px12}\left[M_2\right]}}+k_{px2z}\right)\left[M_z\right]=W.$$

(6)

Equation (5) was substituted into Equation (2), and the resulting equation was transformed accordingly:

$$\begin{gathered} {\displaystyle \frac{{\sum }_{z=1}^2\left[M_z\right]{\sum }_{y=1}^2{k}_{p3yz}{\mu }_{3y} }{{\sum }_{z=1}^2\left[M_z\right]{\sum }_{y=1}^2{k}_{p2yz}{\mu }_{2y} }}={\displaystyle \frac{S_3}{S_2}}, \\ {\displaystyle \frac{{\sum }_{z=1}^2\left[M_z\right]\left(k_{p31z}\frac{k_{p321}\left[M_1\right]}{k_{p312}\left[M_2\right]}{\mu }_{32}+k_{p32z}{\mu }_{32}\right)}{{\sum }_{z=1}^2\left[M_z\right]\left(k_{p21z}\frac{k_{p221}\left[M_1\right]}{k_{p212}\left[M_2\right]}{\mu }_{22}+k_{p22z}{\mu }_{22}\right) }}={\displaystyle \frac{S_3}{S_2}}, \\ {\displaystyle \frac{{\mu }_{32}{\sum }_{z=1}^2\left[M_z\right] \left(k_{p31z}\frac{k_{p321}\left[M_1\right]}{k_{p312}\left[M_2\right]}+k_{p32z}\right)}{{\mu }_{22}{\sum }_{z=1}^2\left[M_z\right]\left(k_{p21z}\frac{k_{p221}\left[M_1\right]}{k_{p212}\left[M_2\right]}+k_{p22z}\right) }}={\displaystyle \frac{S_3}{S_2}}, \end{gathered}$$

$${\mu }_{22}\sum _{z=1}^2\left[M_z\right]\left(k_{p21z}{\displaystyle \frac{k_{p221}\left[M_1\right]}{k_{p212}\left[M_2\right]}}+k_{p22z}\right)={\displaystyle \frac{S_2}{S_3}}{\mu }_{32}\sum _{z=1}^2\left[M_z\right] \left(k_{p31z}{\displaystyle \frac{k_{p321}\left[M_1\right]}{k_{p312}\left[M_2\right]}}+k_{p32z}\right).$$

(7)

Equation (7) was substituted into Equation (6), and the resulting equation was subsequently transformed:

$$\left(1+{\displaystyle \frac{S_2}{S_3}}\right){\mu }_{32}\sum _{z=1}^2\left[M_z\right] \left(k_{p31z}{\displaystyle \frac{k_{p321}\left[M_1\right]}{k_{p312}\left[M_2\right]}}+k_{p32z}\right)=W,$$

$${\mu }_{32}={\displaystyle \frac{S_{3}W}{{\sum }_{z=1}^2\left[M_z\right] \left(k_{p31z}\frac{k_{p321}\left[M_1\right]}{k_{p312}\left[M_2\right]}+k_{p32z}\right)}}.$$

(8)

The following transformation was applied here

$$1+{\displaystyle \frac{S_2}{S_3}}={\displaystyle \frac{S_3+S_2}{S_3}}={\displaystyle \frac{1}{S_3}},$$

This is because $S_3+S_2=1$, according to the normalization condition.

The monomer concentrations were expressed through the composition of the monomer mixture for further transformations:

$$\left[M_1\right]=q\left[M\right],\left[M_2\right]=\left(1-q\right)\left[M\right].$$

(9)

Furthermore, it was taken into account that since r₁ = r₂ = 1, the following equalities hold:

$$k_{pz11}=k_{pz12}, k_{pz22}=k_{pz21}.$$

(10)

Equation (8) was transformed considering Equations (9) and (10):

$$\begin{gathered} {\mu }_{32}={\displaystyle \frac{S_{3}W}{{\sum }_{z=1}^2\left[M_z\right] \left(k_{p311}\frac{k_{p322}q}{k_{p311}\left(1-q\right)}+k_{p322}\right)}}, \\ {\mu }_{32}={\displaystyle \frac{S_{3}W}{k_{p322}\left(\frac{q}{\left(1-q\right)}+1\right){\sum }_{z=1}^2\left[M_z\right] }}, \end{gathered}$$

$${\mu }_{32}={\displaystyle \frac{\left(1-q\right)S_{3}W}{k_{p322}\left[M\right] }}.$$

(11)

Equation (5) was transformed considering Equations (9) and (10):

$${\mu }_{x1}={\displaystyle \frac{k_{px22}q}{k_{px11}\left(1-q\right)}}{\mu }_{x2}.$$

(12)

Equation (11) was further transformed using Equation (12):

$${\mu }_{31}={\displaystyle \frac{q{S}_{3}W}{k_{p311}\left[M\right] }}.$$

(13)

The equations for calculating μ_2y were obtained from Equations (11) and (13), taking into account the symmetry of the problem with respect to the index x:

$${\mu }_{21}={\displaystyle \frac{q{S}_{2}W}{k_{p211}\left[M\right] }},$$

(14)

$${\mu }_{22}={\displaystyle \frac{\left(1-q\right)S_{2}W}{k_{p222}\left[M\right] }}.$$

(15)

The total concentration of active sites ${\mu }_{exp}$ for each composition of the reaction mixture q was calculated by summing Equations (11) and (13)–(15):

$${\mu }_{exp}=\sum _{x=2}^3\left({\displaystyle \frac{q}{k_{px11} }}+{\displaystyle \frac{\left(1-q\right)}{k_{px22} }}\right)S_x{\displaystyle \frac{W}{\left[M\right]}}.$$

(16)

Here, ${\mu }_{exp}$, S_x, and W are quantities dependent on q; k_px11, k_px22, [M] are quantities independent of q. The values of ${\mu }_{exp}$ calculated from Equation (16) based on experimental data from [5] for all q < 1 are presented in Figure 2.

To derive the equation relating the concentration of active sites to the polymerization rate of butadiene at q = 1, the concentration of active sites of each type μ_x1 was expressed through the relative activity of the active sites and the polymerization rate, and then these concentrations were summed:

$${\mu }_{x1}={\displaystyle \frac{S_{x}W}{k_{px11}\left[M\right]}},$$

$${\mu }_{exp}=\sum _{x=1}^4{\displaystyle \frac{S_x}{k_{px11}}}{\displaystyle \frac{W}{\left[M\right]}}.$$

(17)

The values of ${\mu }_{exp}$ calculated using Equation (17) based on the experimental data from [5] for q = 1 are also presented in Figure 2.

3.2. Development of a Kinetic Model for the Formation of Active Sites in the TiCl₄-Al(i-C₄H₉)₃ Catalytic System

Based on the Langmuir monomolecular adsorption theory [33], a kinetic model for the formation of active sites in the TiCl₄-Al(i-C₄H₉)₃ catalytic system was developed. The formulation of the kinetic model equations was guided by the following postulates:

• Adsorption occurs at adsorption sites on the surface of the adsorbent. In this case, the adsorbent consists of particles of the TiCl₄-Al(i-C₄H₉)₃ catalytic system. The adsorbate comprises molecules of butadiene or isoprene. Adsorption is competitive, meaning that butadiene and isoprene molecules adsorb at the same adsorption sites. An active site is formed after the adsorption of a monomer molecule onto the surface of the catalytic system particles and the formation of a Ti-C bond, where C is the carbon atom of the butadiene or isoprene molecule.
• Each adsorption site can adsorb only one adsorbate molecule.
• The adsorption process is reversible and at equilibrium, with its rate determined by the rate of reaching equilibrium. The formation of the Ti-C bond is also assumed to be reversible.
• Interaction between adsorbate molecules in the adsorbed state is absent.

Since the number of stages and the rate-limiting step in the formation of active sites are unknown a priori, two kinetic schemes for developing the kinetic model were considered.

Initially, it was assumed that the adsorption of monomer molecules and the formation of the Ti-C bond occur in a single stage; thus, the kinetic model was formulated based on the following kinetic scheme.

$$M_1+A \begin{array}{c}{k}_1\\ \rightleftarrows \\ k_{-1}\end{array} { R}_1,$$

(18)

$$M_2+A \begin{array}{c}{k}_2\\ \rightleftarrows \\ k_{-2}\end{array} R_2,$$

(19)

where A is the adsorption site, k are the rate constants of the respective stages, and R_y are the active sites formed involving butadiene molecules (y = 1) and isoprene molecules (y = 2).

The following kinetic model of single-stage active site formation, formulated according to the mass-action law, corresponds to this scheme:

$$\begin{gathered} {\displaystyle \frac{d\left[M_1\right]}{dt}}=-k_1\left[M_1\right]\left[A\right]+k_{-1}{\mu }_1, \\ {\displaystyle \frac{d\left[M_2\right]}{dt}}=-k_2\left[M_2\right]\left[A\right]+k_{-2}{\mu }_2, \\ {\displaystyle \frac{d{\mu }_1}{dt}}=k_1\left[M_1\right]\left[A\right]-k_{-1}{\mu }_1, \\ {\displaystyle \frac{d{\mu }_2}{dt}}=k_2\left[M_2\right]\left[A\right]-k_{-2}{\mu }_2, \\ {\displaystyle \frac{d\left[A\right]}{dt}}=-k_1\left[M_1\right]\left[A\right]+k_{-1}{\mu }_1-k_2\left[M_2\right]\left[A\right]+k_{-2}{\mu }_2 \end{gathered}$$

where ${\mu }_y$ are the concentrations of active sites formed that involve butadiene molecules (y = 1) and isoprene molecules (y = 2).

Considering that this system of equations does not exhibit significant nonlinearity, the reaction system is expected to approach a single stable equilibrium state. At this state, the time derivatives of the concentrations of all species are zero, and the system reduces to two equations:

$$k_1\left[M_1\right]\left[A\right]-k_{-1}{\mu }_1=0,$$

(20)

$$k_2\left[M_2\right]\left[A\right]-k_{-2}{\mu }_2=0.$$

(21)

The system of Equations (20) and (21) is complemented by the conservation law (since one active site forms from one adsorption site, the total concentration of adsorption sites and active sites remains constant):

$${\mu }_1+{\mu }_2+\left[A\right]=[A{]}_0,$$

(22)

where $[A{]}_0$ is the initial concentration of adsorption sites.

As a result, a closed system of three equations with three unknowns is obtained. The concentrations of active sites were expressed from Equations (20) and (21) and substituted into Equation (22), yielding an equation that can be used to calculate the concentration of adsorption sites remaining free after the formation of active sites:

$${\mu }_1={\displaystyle \frac{k_1\left[M_1\right]\left[A\right]}{k_{-1}}},$$

(23)

$${\mu }_2={\displaystyle \frac{k_2\left[M_2\right]\left[A\right]}{k_{-2}}},$$

(24)

$$\begin{gathered} {\displaystyle \frac{k_1\left[M_1\right]\left[A\right]}{k_{-1}}}+{\displaystyle \frac{k_2\left[M_2\right]\left[A\right]}{k_{-2}}}+\left[A\right]=[A{]}_0, \\ \left[A\right]={\displaystyle \frac{[A{]}_0}{K_1\left[M_1\right]+K_2\left[M_2\right]+1}}, \end{gathered}$$

where $K_1=k_1/k_{-1},$ $K_2=k_2/k_{-2}.$

Substituting the obtained result into Equations (23) and (24) and summing the resulting equations, an equation for calculating the concentration of active sites was obtained:

$$\begin{gathered} {\mu }_{calc}=K_1\left[M_1\right]\left[A\right]+K_2\left[M_2\right]\left[A\right], \\ {\mu }_{calc}={\displaystyle \frac{K_1\left[M_1\right]+K_2\left[M_2\right]}{K_1\left[M_1\right]+K_2\left[M_2\right]+1}}[A{]}_0, \end{gathered}$$

$${\mu }_{calc}={\displaystyle \frac{K_{1}q+K_2(1-q)}{K_{1}q+K_2(1-q)+\frac{1}{\left[M\right]}}}[A{]}_0.$$

(25)

Thus, Equation (25) shows the dependence of ${\mu }_{calc}$ on q for a single-stage formation of active sites.

The option of active site formation in two stages was also considered.

Adsorption of monomer molecules at adsorption sites (Stage 1, physical):

$$M_1+A \begin{array}{c}{k}_{f1}\\ \rightleftarrows \\ k_{-f1}\end{array} M_1^{*},$$

(26)

$$M_2+A \begin{array}{c}{k}_{f2}\\ \rightleftarrows \\ k_{-f2}\end{array}{ M}_2^{*},$$

(27)

Formation of the Ti-C bond (Stage 2, chemical):

$$M_1^{*} \begin{array}{c}{k}_1\\ \rightleftarrows \\ k_{-1}\end{array}{ R}_1,$$

(28)

$$M_2^{*} \begin{array}{c}{k}_2\\ \rightleftarrows \\ k_{-2}\end{array} R_2,$$

(29)

where $M_1^{*}$ and $M_2^{*}$ are the adsorbed molecules of butadiene and isoprene, respectively.

The following kinetic model of two-stage active sites formation, formulated according to the mass-action law, corresponds to this scheme:

$$\begin{gathered} {\displaystyle \frac{d\left[A\right]}{dt}}=-k_{f1}\left[A\right]\left[M_1\right]-k_{f2}\left[A\right]\left[M_2\right]+k_{-f1}\left[M_1^{*}\right]+k_{-f2}\left[M_2^{*}\right], \\ {\displaystyle \frac{d\left[M_1^{*}\right]}{dt}}=k_{f1}\left[A\right]\left[M_1\right]-k_{-f1}\left[M_1^{*}\right]-k_1\left[M_1^{*}\right]+k_{-1}{\mu }_1, \\ {\displaystyle \frac{d\left[M_2^{*}\right]}{dt}}=k_{f2}\left[A\right]\left[M_2\right]-k_{-f2}\left[M_2^{*}\right]-k_2\left[M_2^{*}\right]+k_{-2}{\mu }_2, \\ {\displaystyle \frac{d{\mu }_1}{dt}}=k_1\left[M_1^{*}\right]-k_{-1}{\mu }_1, \\ {\displaystyle \frac{d{\mu }_2}{dt}}=k_2\left[M_2^{*}\right]-k_{-2}{\mu }_2. \end{gathered}$$

When the reaction system reaches equilibrium, the original system of five equations reduces to a system of four equations:

$$k_{f1}\left[A\right]\left[M_1\right]-k_{-f1}\left[M_1^{*}\right]=0,$$

(30)

$$k_{f2}\left[A\right]\left[M_2\right]-k_{-f2}\left[M_2^{*}\right]=0,$$

(31)

$$k_1\left[M_1^{*}\right]-k_{-1}{\mu }_1=0,$$

(32)

$$k_2\left[M_2^{*}\right]-k_{-2}{\mu }_2=0.$$

(33)

The system of Equations (30)–(33) is complemented by the conservation law:

$${\mu }_1+{\mu }_2+\left[M_1^{*}\right]+\left[M_2^{*}\right]+\left[A\right]=[A{]}_0.$$

(34)

As a result, a closed system of five equations with five unknowns is obtained. The concentrations of adsorbed butadiene and isoprene molecules were expressed from Equations (30) and (31) and substituted into Equations (32) and (33), from which the concentrations of active sites were derived. All expressed concentrations were then substituted into Equation (34), yielding an equation for calculating the concentration of adsorption sites remaining free after the formation of active sites:

$$\begin{gathered} \left[M_1^{*}\right]={\displaystyle \frac{k_{f1}}{k_{-f1}}}\left[A\right]\left[M_1\right], \\ \left[M_2^{*}\right]={\displaystyle \frac{k_{f2}}{k_{-f2}}}\left[A\right]\left[M_2\right], \end{gathered}$$

$${\mu }_1={\displaystyle \frac{k_1}{k_{-1}}}\left[M_1^{*}\right]={\displaystyle \frac{k_1}{k_{-1}}}{\displaystyle \frac{k_{f1}}{k_{-f1}}}\left[A\right]\left[M_1\right],$$

(35)

$${\mu }_2={\displaystyle \frac{k_2}{k_{-2}}}\left[M_2^{*}\right]={\displaystyle \frac{k_2}{k_{-2}}}{\displaystyle \frac{k_{f2}}{k_{-f2}}}\left[A\right]\left[M_2\right],$$

(36)

$$\begin{gathered} {\displaystyle \frac{k_1}{k_{-1}}}{\displaystyle \frac{k_{f1}}{k_{-f1}}}\left[A\right]\left[M_1\right]+{\displaystyle \frac{k_2}{k_{-2}}}{\displaystyle \frac{k_{f2}}{k_{-f2}}}\left[A\right]\left[M_2\right]+{\displaystyle \frac{k_{f1}}{k_{-f1}}}\left[A\right]\left[M_1\right]+{\displaystyle \frac{k_{f2}}{k_{-f2}}}\left[A\right]\left[M_2\right]+\left[A\right]=[A{]}_0. \\ \left[A\right]={\displaystyle \frac{[A{]}_0}{K_1{K}_{f1}\left[M_1\right]+K_2{K}_{f2}\left[M_2\right]+K_{f1}\left[M_1\right]+K_{f2}\left[M_2\right]+1}}, \end{gathered}$$

where $K_{f1}=k_{f1}/k_{-f1},$ $K_{f2}=k_{f2}/k_{-f2},$ $K_1=k_1/k_{-1},$ $K_2=k_2/k_{-2}.$

Substituting this result back into Equations (35) and (36) and summing the resulting expressions yields the equation for calculating the concentration of active sites:

$$\begin{gathered} {\mu }_{calc}=K_1{K}_{f1}\left[M_1\right]\left[A\right]+K_2{K}_{f2}\left[M_2\right]\left[A\right], \\ {\mu }_{calc}={\displaystyle \frac{K_1{K}_{f1}\left[M_1\right]+K_2{K}_{f2}\left[M_2\right]}{K_1{K}_{f1}\left[M_1\right]+K_2{K}_{f2}\left[M_2\right]+K_{f1}\left[M_1\right]+K_{f2}\left[M_2\right]+1}}[A{]}_0, \end{gathered}$$

$${\mu }_{calc}={\displaystyle \frac{K_1{K}_{f1}q+K_2{K}_{f2}\left(1-q\right)}{\left(1+K_1\right)K_{f1}q+\left(1+K_2\right)K_{f2}\left(1-q\right)+\frac{1}{\left[M\right]}}}[A{]}_0.$$

(37)

where [M] = [M₁] + [M₂] = 1.5 mol/L.

Equation (37) thus shows the dependence of ${\mu }_{calc}$ on q for the two-stage formation of active sites.

3.3. Determination of the Kinetic Parameter Range for the Active Sites Formation Process Based on the Shape of the Dependence of Active Sites’ Concentration ${\mu }_{calc}$ on Monomer Mixture Composition q

The experimental active sites concentration ${\mu }_{exp}$ versus monomer mixture composition q exhibits a minimum point at q_min. This dependence is also convex downward. For the shapes of the experimental ${\mu }_{exp}$ and calculated ${\mu }_{calc}$ dependences of active site concentration on monomer mixture composition q to coincide, the following conditions must be satisfied.

1. Minimum point condition:

$${\displaystyle \frac{d{\mu }_{calc}}{dq}}<0 \begin{array}{ccc}\mathrm{before}& q_{min}& \mathrm{and} \end{array}{\displaystyle \frac{d{\mu }_{calc}}{dq}}>0 \begin{array}{cc}\mathrm{after}& q_{min}\end{array}$$

(38)

2. Condition for downward convexity of the dependence:

$${\displaystyle \frac{d^2{\mu }_{calc}}{d{q}^2}}>0.$$

(39)

In the case of single-stage active site formation:

$$\begin{gathered} {\displaystyle \frac{d{\mu }_{calc}}{dq}}={\displaystyle \frac{\left(K_1-K_2\right)\left(K_{1}q+K_2\left(1-q\right)+\frac{1}{\left[M\right]}\right)-\left(K_1-K_2\right)\left(K_{1}q+K_2\left(1-q\right)\right)}{{\left(K_{1}q+K_2\left(1-q\right)+\frac{1}{\left[M\right]}\right)}^2}}[A{]}_0= \\ ={\displaystyle \frac{(K_1-K_2)[A{]}_0}{\left[M\right]{\left(K_{1}q+K_2(1-q)+\frac{1}{\left[M\right]}\right)}^2}}. \end{gathered}$$

(40)

$${\displaystyle \frac{d^2{\mu }_{calc}}{d{q}^2}}={\displaystyle \frac{d}{dq}}{\displaystyle \frac{d{\mu }_{calc}}{dq}}=-{\displaystyle \frac{2(K_1-K_2{)}^2[A{]}_0}{\left[M\right]{\left(K_{1}q+K_2(1-q)+\frac{1}{\left[M\right]}\right)}^3}}.$$

(41)

According to Equation (40), the dependence of ${\mu }_{calc}$ on q cannot have a minimum point because the denominator of this dependence is always positive, and the numerator is independent of q (i.e., it is either strictly positive or strictly negative for all q). Since the region of the ${\mu }_{exp}$ dependence on q where ${\mu }_{exp}$ increases is significantly larger than the region where ${\mu }_{exp}$ decreases (see Figure 2), it was subsequently assumed that the correct form of the ${\mu }_{calc}$ dependence on q satisfies the condition.

$${\displaystyle \frac{d{\mu }_{exp} }{dq}}>0.$$

(42)

Condition (42) was transformed taking into account Equation (40):

$${\displaystyle \frac{(K_1-K_2)[A{]}_0}{\left[M\right]{\left(K_{1}q+K_2\left(1-q\right)+\frac{1}{\left[M\right]}\right)}^2}}>0,$$

$$K_1>K_2.$$

(43)

Condition (39), taking into account Equation (41), will take the following form:

$$-{\displaystyle \frac{2(K_1-K_2{)}^2[A{]}_0}{\left[M\right]{\left(K_{1}q+K_2\left(1-q\right)+\frac{1}{\left[M\right]}\right)}^3}}>0.$$

The numerator of the left side of this condition is always positive; therefore, this condition is equivalent to the following:

$$K_{1}q+K_2(1-q)+{\displaystyle \frac{1}{\left[M\right]}}<0.$$

(44)

It is evident that condition (44) cannot be satisfied, since each term on its left side is non-negative.

Thus, the model of active site formation in the catalytic system TiCl₄-Al(i-C₄H₉)₃, developed on the basis of a single-stage kinetic scheme for active site formation, cannot describe the experimental dependence of ${\mu }_{exp}$ on q even qualitatively (i.e., in terms of the shape of this dependence) for any values of K₁ and K₂.

In the case of two-stage active site formation

$$\begin{gathered} {\displaystyle \frac{d{\mu }_{calc}}{dq}}=[A{]}_0{\displaystyle \frac{(K_1{K}_{f1}-K_2{K}_{f2})\left(\frac{1}{\left[M\right]}+(1+K_1)K_{f1}q+(1+K_2)K_{f2}(1-q)\right)}{{\left(\frac{1}{\left[M\right]}+(1+K_1)K_{f1}q+(1+K_2)K_{f2}(1-q)\right)}^2}} \\ --[A{]}_0{\displaystyle \frac{\left(\right(1+K_1)K_{f1}-(1+K_2\left)K_{f2}\right)(K_1{K}_{f1}q+K_2{K}_{f2}(1-q\left)\right)}{{\left(\frac{1}{\left[M\right]}+(1+K_1)K_{f1}q+(1+K_2)K_{f2}(1-q)\right)}^2}}= \end{gathered}$$

$$=[A{]}_0{\displaystyle \frac{K_1{K}_{f1}(1+K_{f2}[M\left]\right)-K_2{K}_{f2}(1+K_{f1}[M\left]\right)}{\left[M\right]{\left(\frac{1}{\left[M\right]}+(1+K_1)K_{f1}q+(1+K_2)K_{f2}(1-q)\right)}^2}},$$

(45)

$${\displaystyle \frac{d^2{\mu }_{calc}}{d{q}^2}}={\displaystyle \frac{d}{dq}}{\displaystyle \frac{d{\mu }_{calc}}{dq}}=-2\left[A{]}_0\right((1+K_1)K_{f1}-(1+K_2)K_{f2})\times$$

$$\times {\displaystyle \frac{\left(K_1{K}_{f1}\right(1+K_{f2}\left[M\right])-K_2{K}_{f2}(1+K_{f1}\left[M\right]\left)\right)}{\left[M\right]{\left(\frac{1}{\left[M\right]}+(1+K_1)K_{f1}q+(1+K_2)K_{f2}(1-q)\right)}^3}}.$$

(46)

The denominator in Equation (45) is always positive, and the numerator does not depend on q; therefore, the dependence of ${\mu }_{calc}$ on q in the case of two-stage active site formation will also not have a minimum point. Condition (42) will be satisfied only if the numerator in Equation (45) is positive; that is, when the following condition is met:

$$K_1{K}_{f1}(1+K_{f2}[M\left]\right)>K_2{K}_{f2}(1+K_{f1}[M\left]\right),$$

$${\displaystyle \frac{K_1}{K_2}}>{\displaystyle \frac{K_{f2}(1+K_{f1}[M\left]\right)}{K_{f1}(1+K_{f2}[M\left]\right)}}.$$

(47)

The denominator in Equation (46) is positive since 1/[M] > 0, and all other terms in the denominator are at least non-negative for any value of 0 < q < 1. Therefore, condition (39), taking into account Equation (46), is equivalent to the following condition:

$$2\left[A{]}_0\right((1+K_1)K_{f1}-(1+K_2)K_{f2})\times (K_1{K}_{f1}(1+K_{f2}[M\left]\right)-K_2{K}_{f2}(1+K_{f1}[M\left]\right))<0.$$

(48)

Here, the “>” sign changes to “<” because the minus sign in condition (48) was eliminated by multiplying both sides of the inequality by −1. Condition (48) is equivalent to the following set of two systems of inequalities:

$$\left[\begin{array}{c}\left\{\begin{array}{}\mathrm{(49)}& K_1{K}_{f1}(1+K_{f2}[M\left]\right)-K_2{K}_{f2}(1+K_{f1}[M\left]\right)>0,\mathrm{(50)}& (1+K_1)K_{f1}-(1+K_2)K_{f2}<0,\end{array}\right.\\ \left\{\begin{array}{}\mathrm{(51)}& K_1{K}_{f1}(1+K_{f2}[M\left]\right)-K_2{K}_{f2}(1+K_{f1}[M\left]\right)<0,\mathrm{(52)}& (1+K_1)K_{f1}-(1+K_2)K_{f2}>0.\end{array}\right.\end{array}\right.$$

The square bracket denotes a collection (i.e., the union of solution sets of inequalities), while the curly brace denotes a system (i.e., the intersection of solution sets of inequalities).

The solution sets of inequalities (47) and (51) do not intersect. Therefore, examining the solution set of the system of inequalities (51) and (52) is meaningless (in this region, the dependence of ${\mu }_{calc}$ on q decreases rather than increases). In the remaining system of inequalities (49) and (50), inequality (49) is equivalent to inequality (47), while inequality (50) simplifies to the following inequality:

$${\displaystyle \frac{K_{f1}}{K_{f2}}}<{\displaystyle \frac{1+K_2}{1+K_1}}.$$

(53)

Thus, the dependence of ${\mu }_{calc}$ on q, calculated using Equation (37) and derived under the assumption that active sites formation is a two-stage process, is identical in form to the dependence of ${\mu }_{exp}$ on q when the following two conditions are simultaneously satisfied.

1. The dependence of ${\mu }_{calc}$ on q is increasing (inequality (47)):

$${\displaystyle \frac{K_1}{K_2}}>{\displaystyle \frac{K_{f2}(1+K_{f1}[M\left]\right)}{K_{f1}(1+K_{f2}[M\left]\right)}}.$$

2. The dependence of ${\mu }_{calc}$ on q is convex downward (inequality (53)):

$${\displaystyle \frac{1+K_2}{1+K_1}}>{\displaystyle \frac{K_{f1}}{K_{f2}}}.$$

Finding the general solution to the system of inequalities (47) and (53) is quite challenging. It is overly cumbersome, and this complexity obscures the physical meaning of the solution. Therefore, instead of the general solution to the system of inequalities (47) and (53), all possible asymptotic solutions were found. By an asymptotic solution of the system of inequalities (47) and (53), we mean a solution of this system obtained not over the entire set of parameter values K_f1 > 0, K_f2 > 0, K₁ > 0, K₂ > 0, but within certain subsets of these parameter values. For each parameter, the range of values was divided into two subsets, the physical meaning of which is easy to interpret:

1.

$K_{f1}\left[M\right]>>1$—the adsorption/desorption equilibrium of butadiene is shifted toward adsorption, i.e., butadiene is adsorbed by adsorption sites with high efficiency.

2.

$K_{f1}\left[M\right]<<1$—the adsorption/desorption equilibrium of butadiene is shifted toward desorption, i.e., butadiene is adsorbed by adsorption sites with low efficiency.

3.

$K_{f2}\left[M\right]>>1$—the adsorption/desorption equilibrium of isoprene is shifted toward adsorption, i.e., isoprene is adsorbed by adsorption sites with high efficiency.

4.

$K_{f2}\left[M\right]<<1$—the adsorption/desorption equilibrium of isoprene is shifted toward desorption, i.e., isoprene is adsorbed by adsorption sites with low efficiency.

5.

$K_1>>1$—the equilibrium of formation/breaking of the bond between the Ti atom of the catalytic system and the C atom of butadiene is shifted toward bond formation, i.e., butadiene forms stable active sites.

6.

$K_1<<1$—the equilibrium of formation/breaking of the bond between the Ti atom of the catalytic system and the C atom of butadiene is shifted toward bond breaking, i.e., butadiene forms unstable active sites.

7.

$K_2>>1$—the equilibrium of formation/breaking of the bond between the Ti atom of the catalytic system and the C atom of isoprene is shifted toward bond formation, i.e., isoprene forms stable active sites.

8.

$K_2<<1$—the equilibrium of formation/breaking of the bond between the Ti atom of the catalytic system and the C atom of isoprene is shifted toward bond breaking, i.e., isoprene forms unstable active sites.

Such boundaries of parameter subsets allow us, for each subset, to determine which terms in the sums $1+K_{f1}\left[M\right],$ $1+K_{f2}\left[M\right],$ $1+K_1,$ $1+K_2$ can be neglected, thereby simplifying inequalities (47) and (53) (this transformation is hereafter referred to as asymptotic transformation).

As a result, the entire set of parameter values K_f1 > 0, K_f2 > 0, K₁ > 0, K₂ > 0 was divided into 16 subsets, some of which admit asymptotic solutions. The asymptotically transformed inequalities (47) and (53) for various parameter subsets are presented in

Table 1. Asymptotically transformed inequalities (47) and (53) for various subsets of parameter values K_f1 > 0, K_f2 > 0, K₁ > 0, K₂ > 0.

	Boundary of the Subset (Stage 1)	Kf1[M] >> 1, Kf2[M] >> 1	Kf1[M] >> 1, Kf2[M] << 1	Kf1[M] << 1, Kf2[M] >> 1	Kf1[M] << 1, Kf2[M] << 1
Boundary of the Subset (Stage 2)
K1>> 1, K2 >> 1		subset 1 K1/K2 > 1, K2/K1 > Kf1/Kf2	subset 2 K1/K2 > Kf2[M], K2/K1 > Kf1/Kf2	subset 3 K1/K2 > 1/(Kf1[M]), K2/K1 > Kf1/Kf2	subset 4 K1/K2 > Kf2/Kf1, K2/K1 > Kf1/Kf2
K1>> 1, K2 << 1		subset 5 K1/K2 > 1, 1/K1 > Kf1/Kf2	subset 6 K1/K2 > Kf2[M], 1/K1 > Kf1/Kf2	subset 7 K1/K2 > 1/(Kf1[M]), 1/K1 > Kf1/Kf2	subset 8 K1/K2 > Kf2/Kf1, 1/K1 > Kf1/Kf2
K1 << 1, K2 >> 1		subset 9 K1/K2 > 1, K2 > Kf1/Kf2	subset 10 K1/K2 > Kf2[M], K2 > Kf1/Kf2	subset 11 K1/K2 > 1/(Kf1[M]), K2 > Kf1/Kf2	subset 12 K1/K2 > Kf2/Kf1, K2 > Kf1/Kf2
K1<< 1, K2 << 1		subset 13 K1/K2 > 1, 1 > Kf1/Kf2	subset 14 K1/K2 > Kf2[M], 1 > Kf1/Kf2	subset 15 K1/K2 > 1/(Kf1[M]), 1 > Kf1/Kf2	subset 16 K1/K2 > Kf2/Kf1, 1 > Kf1/Kf2

.

Analysis of the asymptotically transformed inequalities (47) and (53) for all 16 subsets showed that asymptotic solutions to inequalities (47) and (53) exist only in 7 subsets.

In subset 2 ($K_{f1}\left[M\right]>>1,$ $K_{f2}\left[M\right]<<1,$ $K_1>>1,$ $K_2>>1$), from the inequality $K_1/K_2>K_{f2}\left[M\right]$, it follows that $K_1>>K_2;$ from the inequality $K_2/K_1>K_{f1}/K_{f2}$, it follows that $K_1<<K_2.$ These two inequalities are mutually exclusive; therefore, there are no asymptotic solutions in subset 2.

In subset 3 ($K_{f1}\left[M\right]<<1,$ $K_{f2}\left[M\right]>>1,$ $K_1>>1,$ $K_2>>1$): from the inequality $K_1/K_2>1/\left(K_{f1}\right[M\left]\right)$ it follows that $K_1>>K_2;$ from the inequality $K_2/K_1>K_{f1}/K_{f2}$ it follows that $K_1<<K_2.$ These two inequalities are mutually exclusive; therefore, there are no asymptotic solutions in subset 3.

In subset 4 ($K_{f1}\left[M\right]<<1,$ $K_{f2}\left[M\right]<<1,$ $K_1>>1,$ $K_2>>1$), from the inequality $K_2/K_1>K_{f1}/K_{f2}$, it follows that $K_1/K_2<K_{f2}/K_{f1},$ which together with the other inequality in this subset $K_1/K_2>K_{f2}/K_{f1}$ are mutually exclusive. Therefore, there are no asymptotic solutions in subset 4.

In subset 6 ($K_{f1}\left[M\right]>>1,$ $K_{f2}\left[M\right]<<1,$ $K_1>>1,$ $K_2<<1$), the inequality $1/K_1>K_{f1}/K_{f2}$ is not satisfied for any parameter values in this subset. Therefore, there are no asymptotic solutions in subset 6.

In subset 9 ($K_{f1}\left[M\right]>>1,$ $K_{f2}\left[M\right]>>1,$ $K_1<<1,$ $K_2>>1)$, the inequality $K_1/K_2>1$ is not satisfied for any of the parameter values in this subset. Therefore, there are no asymptotic solutions in subset 9.

In subset 10 ($K_{f1}\left[M\right]>>1,$ $K_{f2}\left[M\right]<<1,$ $K_1<<1,$ $K_2>>1$), from the inequality $K_2>K_{f1}/K_{f2}$, it follows that $K_{f2}\left[M\right]>K_{f1}\left[M\right]/K_2$. This inequality, together with the other inequality $K_1/K_2>K_{f2}\left[M\right]$, forms the system $K_1/K_2>K_{f2}\left[M\right]>K_{f1}\left[M\right]/K_2,$ which has no solution, since $K_{f1}\left[M\right]>>1,$ and $K_1<<1.$ Therefore, there are no asymptotic solutions in subset 10.

In subset 11 ($K_{f1}\left[M\right]<<1,$ $K_{f2}\left[M\right]>>1,$ $K_1<<1,$ $K_2>>1$), the inequality $K_1/K_2>1/\left(K_{f1}\right[M\left]\right)$ is not satisfied for any parameter values in this subset. Therefore, there are no asymptotic solutions in subset 11.

In subset 12 ($K_{f1}\left[M\right]<<1,$ $K_{f2}\left[M\right]<<1,$ $K_1<<1,$ $K_2>>1$), from the inequality $K_2>K_{f1}/K_{f2}$, it follows that $1/K_2<K_{f2}/K_{f1}$. This inequality, together with the other inequality $K_1/K_2>K_{f2}/K_{f1}$, forms the system $1/K_2<K_{f2}/K_{f1}<K_1/K_2,$ which has no solution since $K_1<<1.$ Therefore, there are no asymptotic solutions in subset 12.

In subset 14 ($K_{f1}\left[M\right]>>1,$ $K_{f2}\left[M\right]<<1,$ $K_1<<1,$ $K_2<<1$), the inequality $1>K_{f1}/K_{f2}$ is not satisfied for any parameter values in this subset. Therefore, there are no asymptotic solutions in subset 14.

For all other subsets, particular asymptotic solutions of inequalities (47) and (53) were proposed (

Table 2. Particular asymptotic solutions of inequalities (47) and (53) for various subsets of parameter values K_f1 > 0, K_f2 > 0, K₁ > 0, K₂ > 0 at [M] = 1.5 mol/L.

Boundaries of Parameter Subsets	Kf1[M] >> 1, Kf2[M] >> 1	Kf1[M] >> 1, Kf2[M] << 1	Kf1[M] << 1, Kf2[M] >> 1	Kf1[M] << 1, Kf2[M] << 1
K1>> 1, K2 >> 1	${\mathit{K}}_{\mathit{f}1}=10^1 \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},$ $K_{f2}=10^3 \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},$ $K_1=10^2,$ $K_2=10^1$	No solutions	No solutions	No solutions
K1>> 1, K2 << 1	$K_{f1}=10^1 \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},,$ $K_{f2}=10^3 \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},$ $K_1=10^1,$ $K_2=10^{-1}$	No solutions	$K_{f1}=10^{-2} \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},$ $K_{f2}=10^2 \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},$ $K_1=10^3,$ $K_2=10^{-2}$	$K_{f1}=10^{-3} \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},$ $K_{f2}=10^{-1} \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},$ $K_1=10^1,$ $K_2=10^{-2}$
K1 << 1, K2 >> 1	No solutions	No solutions	No solutions	No solutions
K1<< 1, K2 << 1	$K_{f1}=10^1 \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},$ $K_{f2}=10^2 \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},$ $K_1=10^{-1},$ $K_2=10^{-2}$	No solutions	$K_{f1}=10^{-1} \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},$ $K_{f2}=10^2 \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},$ $K_1=10^{-1},$ $K_2=10^{-2}$	$K_{f1}=10^{-2} \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},$ $K_{f2}=10^{-1} \mathrm{L}/\mathrm{m}\mathrm{o}\mathrm{l},,$ $K_1=10^{-1},$ $K_2=10^{-3}$

), including examples of specific parameter values K_f1, K_f2, K₁, K₂ satisfying these inequalities. Figure 3 shows the dependencies of ${\mu }_{calc}$ on q corresponding to these particular asymptotic solutions.

3.4. Determination of Specific Quantitative Values of Kinetic Parameters of the Active Sites Formation Process, at Which the Dependence of μ_calc on q Quantitatively Coincides with the Dependence of μ_exp on q

The particular asymptotic solutions of inequalities (47) and (53) from Table 2, which provide qualitative agreement between the forms of the dependencies μ_exp on q and μ_calc on q, were used as initial approximations for solving the inverse problem of finding particular asymptotic solutions of inequalities (47) and (53) that ensure quantitative coincidence of the dependencies μ_exp on q and μ_calc on q. The inverse problem was solved using an optimization algorithm implemented in the FindMinimum operator of the Wolfram software system (Mathematica 12.0). During the solution of the inverse problem, the following function was minimized:

$$F={\left({\mu }_{exp}-{\mu }_{calc}\right)}^2{\displaystyle \frac{{\mathrm{L}}^2}{{\mathrm{m}\mathrm{o}\mathrm{l}}^2}}++10^r\left({\left(\mathit{lg}{K}_1\right)}^2+{\left(\mathit{lg}{K}_2\right)}^2+{\left(\mathit{lg}{\displaystyle \frac{K_{f1}}{\frac{\mathrm{L}}{\mathrm{m}\mathrm{o}\mathrm{l}}}}\right)}^2+{\left(\mathit{lg}{\displaystyle \frac{K_{f2}}{\frac{\mathrm{L}}{\mathrm{m}\mathrm{o}\mathrm{l}}}}\right)}^2\right),$$

(54)

where the second term is the regularization term (r is the regularization coefficient). Prior to this, dependence (37) was expressed in the following form:

$${\mu }_{calc}={\displaystyle \frac{10^{X_1}10^{X_{f1}}q+10^{X_2}10^{X_{f2}}\left(1-q\right)}{\left(1+10^{X_1}\right)10^{X_{f1}}q+\left(1+10^{X_2}\right)10^{X_{f2}}\left(1-q\right)+\frac{1}{\left[M\right]}}}10^{X_0},$$

where X₀ = lg[A]₀, X_f1 = lg(K_f1/(L/mol)), X_f2 = lg(K_f2/(L/mol)), X₁ = lgK₁, X₂ = lgK₂.

Thus, during the minimization of function (54), we obtained the parameters X₀, X_f1, X_f2, X₁, X₂. This representation of dependence (37) allowed us, firstly, to search for the minimum of function (54) within the domain of physically meaningful values [A]₀ > 0, K_f1 > 0, K_f2 > 0, K₁ > 0, K₂ > 0, and secondly, to vary not the exact values of the parameters [A]₀, K_f1, K_f2, K₁, K₂, but their orders of magnitude, i.e., to search for [A]₀, K_f1, K_f2, K₁, K₂ over very wide ranges. Since the search for [A]₀, K_f1, K_f2, K₁, K₂ was conducted over broad intervals, to avoid obtaining unreasonably large or small values of X₀, X_f1, X_f2, X₁, X₂ (e.g., ±1000), a regularization term was introduced into function (54). The purpose of the regularization term is to exclude inadequate solutions. That is, the farther the values of X₀, X_f1, X_f2, X₁, X₂ are from zero, the less likely these values will be accepted as a solution. Empirically, it was established that for this problem that the regularization coefficient should be taken as r = −9 (this value was varied in steps of 1). At lower values of r, the values of X₀, X_f1, X_f2, X₁, and X₂ obtained from minimizing function (54) were unreasonably high or low. At higher values of r, the contribution of the regularization term to the value of function (54) became too large, and the possibility of minimizing the first term of this function was lost. Thus, the result of minimizing function (54) is regularized, or, more simply, ordered. This result is not purely mathematical but incorporates ordering dictated by a priori chemical knowledge.

The minimization of function (54) was performed in two variants.

Variant 1. Particular asymptotic solutions of inequalities (47) and (53) from Table 2 were used as initial approximations. The initial approximation for X₀ varied from −4 to −2. No constraints were imposed on the values of X_f1, X_f2, X₁, X₂, i.e., the values of X_f1, X_f2, X₁, X₂ could move from one solution subset to another.

Variant 2. This variant differed from Variant 1 in that constraints were imposed on the values of X_f1, X_f2, X₁, X₂ during the minimization of function (54) to prevent the solution from leaving the subset in which the initial approximation lies.

As a result of minimizing function (54) according to Variant 1, the same minimum was found in solution subset 7 for all initial approximations. This is the global minimum (F = 4.42 × 10⁻⁹).

By minimizing function (54) according to Variant 2, the following results were achieved:

• No solutions were found in subsets 1, 8, 15, and 16 that provided quantitatively exact agreement between the dependencies μ_exp on q and μ_calc on q (specifically, none were found because they might exist if other initial approximations within these subsets were used);
• Local minima were found in subsets 5 and 13 with values (F = 2.81 × 10⁻⁸ and F = 1.54 × 10⁻⁸ respectively);
• No local minimum was found in subset 7 (!), despite the fact that the global minimum found in Variant 1 minimization lies in subset 7. This is probably due to the fact that during the optimization algorithm implementation, when searching for the global minimum, the values of X_f1, X_f2, X₁, X₂ initially located in subset 7 leave it and then re-enter it; this supports the idea that solutions in subsets 1, 8, 15, and 16 may exist but were not found with the chosen initial approximations.

Particular asymptotic solutions (values of [A]₀, K_f1, K_f2, K₁, K₂) that ensure quantitative coincidence of the dependencies of μ_exp on q and μ_calc on q are presented in

Table 3. Particular asymptotic solutions of inequalities (47) and (53) that ensure quantitatively exact coincidence of the dependencies μ_exp on q and μ_calc on q for various subsets.

Boundaries of Subsets of Parameter Values	${\mathit{K}}_{\mathit{f}1}\left[\mathit{M}\right]\gg 1,$ ${\mathit{K}}_{\mathit{f}2}\left[\mathit{M}\right]>>1$	${\mathit{K}}_{\mathit{f}1}\left[\mathit{M}\right]\gg 1,$ ${\mathit{K}}_{\mathit{f}2}\left[\mathit{M}\right]<<1$	${\mathit{K}}_{\mathit{f}1}\left[\mathit{M}\right]\ll 1,$ ${\mathit{K}}_{\mathit{f}2}\left[\mathit{M}\right]>>1$	${\mathit{K}}_{\mathit{f}1}\left[\mathit{M}\right]\ll 1,$ ${\mathit{K}}_{\mathit{f}2}\left[\mathit{M}\right]<<1$
$K_1>>1,$ $K_2>>1$	Qualitative solution exists; no exact quantitative solution found	No solutions	No solutions	No solutions
$K_1>>1,$ $K_2<<1$	$[A{]}_0=10^{-3.85} \mathrm{mol}/\mathrm{L}$ (method 1 *) $[A{]}_0=10^{-3.55} \mathrm{mol}/\mathrm{L}$ (method 2) $K_{f1}=10^{1.18} \mathrm{L}/\mathrm{mol}$ $K_{f2}=10^{3.16} \mathrm{L}/\mathrm{mol},$ $K_1=10^{1.18}$, $K_2=10^{-1.17}$ local minimum F = 2.81 × 10−8	No solutions	$[A{]}_0=10^{-3.57} \mathrm{mol}/\mathrm{L}$ (method 1) $[A{]}_0=10^{-3.43} \mathrm{mol}/\mathrm{L}$ (method 2) $K_{f1}=10^{-0.31} \mathrm{L}/\mathrm{mol},$ $K_{f2}=10^{0.91} \mathrm{L}/\mathrm{mol},$ $K_1=10^{0.25}$, $K_2=10^{-1.07}$ global minimum F = 4.42 × 10−9	Qualitative solution exists; no exact quantitative solution found
$K_1<<1,$ $K_2>>1$	No solutions	No solutions	No solutions	No solutions
$K_1<<1,$ $K_2<<1$	$[A{]}_0=10^{-2.66} \mathrm{mol}/\mathrm{L}$ (method 1) $[A{]}_0=10^{-2.53} \mathrm{mol}/\mathrm{L}$ (method 2) $K_{f1}=10^{1.18} \mathrm{L}/\mathrm{mol},$ $K_{f2}=10^{2.16} \mathrm{L}/\mathrm{mol},$ $K_1=10^{-1.18},$ $K_2=10^{-2.16}$ local minimum F = 1.54 × 10−8	No solutions	Qualitative solution exists, no exact quantitative solution found	Qualitative solution exists, no exact quantitative solution found

* The copolymerization achieved via method 1 and method 2 is described in the Experimental Section.

. The dependencies of μ_calc on q corresponding to these particular asymptotic solutions, which ensure quantitatively exact agreement between the dependencies of μ_exp on q and μ_calc on q, are shown in Figure 4.

Thus, according to the conducted analysis, it can be concluded that during the formation of active sites in the TiCl₄-Al(i-C₄H₉)₃ catalytic system, one of the following qualitative scenarios may exist:

1.

Monomer molecules are easily adsorbed on the surface of the TiCl₄-Al(i-C₄H₉)₃ catalytic system particles; active sites formed by butadiene molecules are stable (i.e., the bond between the Ti atom of the catalytic system and the C atom of butadiene is stable), while active sites s formed by isoprene molecules are unstable (i.e., the bond between the Ti atom of the catalytic system and the C atom of isoprene is unstable). This description corresponds to subset 5 ($K_{f1}\left[M\right]>>1,$ $K_{f2}\left[M\right]>>1,$ $K_1>>1,$ $K_2<<1$).

2.

Isoprene molecules are easily adsorbed, while butadiene molecules are adsorbed with difficulty; active sites formed by butadiene are stable, and active sites formed by isoprene are unstable. This description corresponds to subset 7 ($K_{f1}\left[M\right]<<1,$ $K_{f2}\left[M\right]>>1,$ $K_1>>1,$ $K_2<<1$).

3.

Monomer molecules are easily adsorbed but form unstable active sites; the stability of active sites formed by butadiene should be higher than that of active sites formed by isoprene. This description corresponds to subset 13 ($K_{f1}\left[M\right]>>1,$ $K_{f2}\left[M\right]>>1$ $K_1<<1,$ $K_2<<1$).

All of the solutions indicate that isoprene molecules should adsorb better on the surface of TiCl₄-Al(i-C₄H₉)₃ particles than butadiene molecules, while butadiene molecules should form more stable active sites than isoprene molecules.

It should also be noted that the developed two-stage model of active sites formation was unable, under any conditions, to describe the presence of a minimum point in the experimental dependence of μ_exp on q. The search for the reasons behind this phenomenon is a promising direction for further research. Among the possible causes, the following can be noted:

1. The wedging effect of active copolymer chains, which leads to dispersion of the TiCl₄-Al(i-C₄H₉)₃ catalytic system particles. Experimentally, it has been established that the dependence of the particle size of the TiCl₄-Al(i-C₄H₉)₃ catalytic system on q also exhibits a minimum point [39].

2. The difference between the real adsorption process of butadiene and isoprene and the process described by the Langmuir monomolecular adsorption theory [33]. This difference may have the following causes.

2.1. The biographical nonuniformity of the surface of TiCl₄-Al(i-C₄H₉)₃ particles (different crystal facets have different properties, and crystals contain defects). Within the framework of the model presented in this work, this means that the equilibrium constants K_f1 and K_f2 will be functions of the concentration of adsorbed butadiene and isoprene molecules; in classical theoretical models, called adsorption isotherms, which relate the fraction of occupied adsorption sites to the pressure of the adsorbed gas, this leads to the dependence of the adsorption constant on the pressure of the adsorbed gas. In this case, adsorption is described not by the Langmuir isotherm, where the adsorption constant is constant [33], but, for example, by the Freundlich [40], Zeldovich [41], or Sips isotherms [42,43].

2.2. Induced nonuniformity of the surface of TiCl₄-Al(i-C₄H₉)₃ particles (the heat of adsorption depends on the number of adsorbed monomer molecules on the surface of TiCl₄-Al(i-C₄H₉)₃ particles, indicating interactions between adsorbed molecules [33]). To describe the induced nonuniformity, the model of active sites formation in TiCl₄-Al(i-C₄H₉)₃ must take into account the change in binding energy of adsorbed molecules depending on the number of adsorbed molecules due to adsorbate–adsorbate interactions. Such models include the Ising model (in which the adsorbed molecular layer corresponds to a two-dimensional lattice gas model), which considers lateral interactions between molecules in lattice sites [44]. The Ising model is closed in the Bragg–Williams approximation, which assumes that the interaction energy of nearest adsorbed molecules is estimated as if the molecules were randomly distributed among adsorption sites, i.e., this energy is estimated as the average interaction energy [45]. Adsorbate–adsorbate–adsorbent interactions are also accounted for in the Fowler–Guggenheim isotherm [46,47]. The King model also describes induced nonuniformity of particles’ surfaces [33].

2.3. Adsorption is multidentate. In this case, butadiene and isoprene molecules may require different numbers of adsorption sites. More compact molecules in this respect can adsorb in the gaps between bulkier molecules. Such adsorption is described by the semi-competitive model [48].

2.4. The adsorption rate depends on the concentration of the adsorbed substance, i.e., there exist different adsorption regimes (at low concentrations, the orientation of adsorbed molecules relative to the surface of solid particles occurs in one way, while at high concentrations their orientation is different) [33].

4. Conclusions

The theoretical influence of the monomer mixture composition q in the butadiene–isoprene copolymerization on the concentration of active sites of TiCl₄-Al(i-C₄H₉)₃ catalytic system μ has been described (q = [M₁]/([M₁] + [M₂]), [M₁] is the concentration of butadiene in the monomer mixture and [M₂] is the concentration of isoprene in the monomer mixture).

The theoretical description was developed according to the following algorithm:

1.

Based on experimental values of the copolymerization rate from study [5] and known values of chain propagation rate constants [35,36] and copolymerization constants from study [5], the concentrations of active sites of the TiCl₄-Al(i-C₄H₉)₃ catalytic system were calculated at various values of q. The experimental dependence of μ on q has the following features: 1. The minimum concentration of active sites μ corresponds to some value of q in the range q = 0.2–0.6, and the concentration μ predominantly increases as the butadiene concentration in the monomer mixture increases. 2. The dependence of μ on q is nonlinear, with (${\displaystyle \frac{d^2{\mu }_{calc}}{d{q}^2}}>0$).

2.

The kinetic model equations for the formation of active sites in the TiCl₄-Al(i-C₄H₉)₃ catalytic system were written. These kinetic equations were formulated based on the mass-action law and the Langmuir monomolecular adsorption theory for two variants of the kinetic scheme: one-stage and two-stage (physical stage—adsorption; chemical stage—formation of the Ti-C bond). The adsorption of butadiene and isoprene molecules on the catalytic system TiCl₄-Al(i-C₄H₉)₃ surface was considered competitive. The equation expressing the theoretical dependence of μ on q was obtained by analytically solving the system of kinetic model equations under the assumption of equilibrium in all stages of the formation of active sites.

3.

The obtained equation was analyzed. It was found that the kinetic model based on the one-stage kinetic scheme cannot even qualitatively describe the experimental dependence of μ on q with the described features. The analogous kinetic model based on the two-stage kinetic scheme satisfactorily describes this experimental dependence (except for the existence of the minimum concentration of active sites μ).

4.

The domains of kinetic parameter values for active sites formation were established, at which the theoretical dependence of μ on q reproduces the corresponding experimental dependence both qualitatively and quantitatively. Qualitatively, this occurs under the condition that isoprene adsorbs better than butadiene, but butadiene forms more stable active sites than isoprene. Quantitatively, this is ensured by any of the three sets of equilibrium rate constants for the stages of active sites formation found in this work.

The results of this work can be generalized to other heterogeneous catalytic systems used for binary copolymerization. Essentially, this work has developed a methodology for rapid assessment of the mechanism of active site formation in heterogeneous catalytic systems of binary copolymerization. This methodology is based on experimental data describing the relationship between the copolymerization rate W and the composition q of a multidimensional mixture. The shape of the dependence of W on q is identical to the shape of the dependence of μ on q. According to this methodology, three possible variants can be identified, each of which provides information about the mechanism of active site formation in the catalytic system.

Variant 1. If the dependence of μ on q has no minima or maxima and is convex upward, then the simplest mechanism of active site formation in such a catalytic system is a simple one-step mechanism. Active sites may also form via a more complex mechanism; however, there is no basis for assuming a more complex mechanism of active site formation for such a catalytic system. The activity of such a catalytic system is close to maximal for most values of q.

Variant 2. If the dependence of μ on q has no minima or maxima and is convex downward, then the simplest mechanism of active site formation in such a catalytic system is a two-step mechanism. The activity of such a catalytic system is close to minimal for most values of q.

In variants 1 and 2, the maximum and minimum activity of the catalytic system are observed at q = 0 and q = 1 (the maximum activity can occur at either of these two points, and the minimum activity can also occur at either of these two points).

Variant 3. If the dependence of μ on q has at least one maximum or minimum point at q ≠ 0 and q ≠ 1, then the simplest mechanism of active site formation of such a catalytic system is more complex than a two-step mechanism.

The obtained results will allow us, in the future, to draw conclusions about the complexity of the mechanism of active sites’ formation in heterogeneous catalytic systems of copolymerization based on the shape of the experimental dependence of μ on q (and, correspondingly, based on the shape of the experimental dependence of W on q).