A Novel Method for Dynamic Molecular Weight Distribution Determination in Organometallic Catalyzed Olefin Polymerizations

Abstract

In this study, a mathematical model for the time evolution of molecular weight distribution (MWD) was developed. This temporal model is based on the well-known Ziegler–Natta polymerization mechanism and reaction kinetics by the parametric solving of related differential equations. However, due to the generality of the reactions involved, the model can be extended to the other type of catalysts, such as metallocenes, Phillips, etc. The superiority of this model lies in providing the possibility of a more precise prediction over the active sites and kinetic parameters using a simple mathematical equation, which leads to improved reactor design in large-scale production. The model uses a function to develop a methodology for MWD calculations. In this way, the transient response is limited to the first few minutes of the reaction; however, it is important as it demonstrates the establishment of the final MWD. According to the results, almost for practical conditions with negligible transfer resistances, the time dependency of the MWD has a transient interval, depending on the kinetic constants of polymerization reactions. Increasing the time to infinity results in an increase in MW and a widening in MWD, which confirms the experimental plots well. In short, the main advantage of our proposed model over the previous ones is its ability to predict the MWD even before the completion of the polymerization reaction. The results of the present model match well with those of the well-known Schulz–Flory distribution, which only predicts the final molecular weight distribution, thus confirming that the model is reliable and generalizable.

1. Introduction

Polyolefins are one of the essential ingredients used in the production of artefacts such as pipes, tanks, polymer films, electrical conduits, electrical insulators, etc. [1,2,3,4]. Industrial catalytic production of polyolefins began in the early 1950s using Ziegler–Natta and Phillips catalysts. These catalysts were continuously improved to achieve higher yields, advanced particle morphology, and better control over the polymer microstructure [5,6,7]. In fact, despite the industrial success of metallocene catalysts in the late 1980s, Ziegler–Natta and Phillips catalysts remain the major contributors to the commercialization of polyolefins. Therefore, understanding how these catalysts produce polyolefins is important from both an industrial [8,9] and an academic point of view [10,11,12,13,14]

The kinetics of polymerization by Ziegler–Natta catalysts using simple mathematical models to predict the reaction rate and average molecular weight have been investigated previously [15,16,17,18]. However, an important property of the final polymer is molecular weight distribution (MWD), which affects many physical properties [19,20,21].

Several models were developed to predict the molecular weight distribution [22,23,24] and average molecular weight by using the method of moments as the most popular approach [25,26]. Kiparissides et al. [27] considered a generalized multisite mechanism to develop a mathematical model, simulating the polymer chain molecular characteristics during ethylene/alpha-olefin copolymerization in CSTRs (continuously stirred tank reactors).

MWD and short-chain branching distribution are usually modeled using two approaches: the method of moments and the Monte-Carlo algorithm [28]. Bontu developed a model for the micro- and meso-scale level of olefin polymerization using the method of moments. The model was based on the polymer flow model, alongside the intrinsic kinetic model obtained in the moments approach. In this model, the effect of mass transfer and kinetic parameters on the temperature profile and monomer concentration, and subsequently on the catalyst activity, polymer molecular weight [29,30,31] and MWD were examined [32]. Fan et al. developed a mathematical model for polymerization activity and active center concentrations in the first 5 min of polymerization. They found a two-stage kinetic for the formation of active centers. In addition, the polymer particle morphology and MWD were modeled as a function of time [33]. However, the suggested model was based on the catalyst active centers’ behavior in the first 5 min of polymerization. Furthermore, Fan and coworkers studied the kinetics and mechanism of ethylene polymerization over a TiCl₄/MgCl₂ catalyst. Here, the polymer MWD was determined in the first 10 min of polymerization considering the Ti content’s effect on the number and structure of active sites formed [34].

There are some other models that were developed for MWD estimation at the end of polymerization [26,35,36]. The polymerization kinetics and the effect of reactor residence time on polymer microstructure, i.e., polymer molecular weight and MWD, were studied thoroughly by Soares et al. [37]. A dynamic model was developed to predict the polymer MWD, temperature profile, and polymer production rate in an industrial fluidized bed reactor during ethylene copolymerization [38]. In this model, the reactor parameters and polymer properties were estimated from the two-phase (bubble and emulsion) dynamic model. Chen et al. predicted the dynamic MWD during grade transition in a continuously stirred tank reactor. In this work, to improve the derived system of non-linear equations, a new moving finite element method was proposed [39]. Biegler and coworkers developed a model based on Flory distribution to predict the dynamic MWD of an industrial HDPE slurry process in CSTRs. This model was based on an equation-oriented (EO) framework. In this methodology, the parameter estimation problem was addressed [40]. Abedi et al. examined the influence of the polymerization time on the molecular weight and molecular weight distribution of Ziegler–Natta propylene polymerization [41]. Their results showed that the reaction time has a remarkable effect on the molecular weight and molecular weight distribution.

Existing experimental data show that, as the polymerization reaction proceeds, the MWD widens and the polymer molecular weight (MW) increases. This is not represented in the previous models as they only predict the final molecular weight. Therefore, in this research, we aimed to fill this gap. In this sense, a novel method was developed to determine the polymer molecular weight distribution (MWD) over time by using simple and comprehensive equations, as a key parameter in the modeling and prediction of the characteristics of the final polyolefin product.

2. Results and Discussion

The cumulative molecular weight distribution, based on

Table 1. Normal Range of Kinetic Constants.

Kinetic Constant	Acceptable Range
kp	10–5000 s−1/(mol/L)
ki	0–500 s−1/(mol/L)
ktβ	0–5 s−1
ktH	0–1000 s−1/(mol/L)
ktAl	0–50 s−1/(mol/L)
ktM	0–100 s−1/(mol/L)
kd	0–100 s−1
kdI	0–1000 s−1/(mol/L)
[M]	0–30 mol/L
[H2]	0–1 mol/L
[Al]	0–1 mol/L
[I]	0–1 mol/L
KP	0–150,000 s−1
KI	0–15,000 s−1
KT	0–4000 s−1
KD	0–1000 s−1

according to various references [25,42,43,44,45,46,47], is presented in Figure 1. We find that the cumulative molecular weight distribution curve is shifted to the left of the cumulative distribution curve at steady state (infinity time or Schulz–Flory distribution). This means that the average molecular weight increases with elapsing time until it reaches a steady-state situation and final distribution. Additionally, as seen in Figure 1, the molecular weight distribution is not totally developed after 20 s from the beginning of the reaction. The most developed molecular length is about 1800, while, as time approaches infinity, this value approaches 10,000.

The normalized differential distributions of Figure 1 are presented in Figure 2. As seen, by decreasing polymerization time, the peak height of molecular weight distribution increases, which means that the MWD becomes narrower (since the total area under the curve must be equal to 1). Additionally, similarly to Figure 1, since the molecular weight distribution cannot develop totally at the time 20 s, and the most developed molecular length is about 1800, one could see a spike in differential molecular weight distribution around a molecular length of 1800. By tending time to infinity, this spike completely disappears, because of the development of molecular weights.

As seen in Figure 1 and Figure 2, according to selected kinetic constants, the time-dependent distribution is close to the steady-state distribution (F_inf), even in the early stages. This reality is also obvious from the rapidly converging PDI to the final value (almost 2) in Figure 3. Therefore, if we ignore the first few minutes of reaction, when the molecular weight distribution function is identical and time-independent, only the number of polymer dead chains increases with time.

Another parameter that cannot be described by Schulz–Flory distribution is the development of average molecular weight (or its twin, average molecular length). As seen in Figure 4, the average molecular length (or weight) increases with time to reach its steady value. This final value is predicted by Schulz–Flory distribution.

Cumulative MWD as a function of time is given in Figure 5. Obviously, as time goes on, the MWD approaches the Schulz–Flory distribution.

The molecular weight distribution is illustrated in Figure 6. The peak of distribution approaches the higher molecular weights values and finally reaches the Schulz–Flory distribution. In fact, the main advantage of this model is its ability to predict the molecular weight distribution during the polymerization reaction. Although the transition period is less than a few minutes, according to the values of K_P, K_I, K_T, and K_D, it may rarely exist for an hour or more; however, in commercial catalysts, this period is very short.

Gel permeation chromatography (GPC) instruments often express a weight distribution of molecular weight, using Equation (34) discussed in the Materials and Methods Section, which we may write as:

$$\frac{{\mathrm{dF}}_{\infty }\left(\mathrm{m}\right)}{\mathrm{d} \log \left(\mathrm{m}\right)}=\frac{{\mathrm{dF}}_{\infty }\left(\mathrm{m}\right)}{\mathrm{d} \log \left(\mathrm{m}/\mathrm{dm}\right)}=-\left(\mathrm{m} \ln 10\right)\left[\mathrm{m}+\left({\mathrm{m}}^2+\mathrm{Q}\right)\left(1-\mathrm{Q}\right)\right]{\mathrm{Q}}^{\mathrm{m}-1}$$

(1)

Additionally, from Equation (37) we have:

$$\overline{{\mathrm{r}}_{\mathrm{n}}}=\frac{1}{1-\mathrm{Q} }\to \mathrm{Q}=1-\frac{1}{\overline{{\mathrm{r}}_{\mathrm{n}}} }$$

(2)

Using Equations (1) and (2), one could deconvolute a real final GPC curve of a sample [25,48,49,50,51]. As an example, the GPC graph of a polypropylene produced by conventional MgCl₂-supported TiCl₄ catalyst [51], deconvoluted using our proposed method, is shown in Figure 7. For the aforementioned GPC curve, the deconvolution results are given in

Table 2. Molecular Weight Distribution Deconvolution Result.

	Site 1	Site 2	Site 3	Site 4	Site 5
Weight Percentage of Each Site	21.88	27.18	26.02	14.70	10.22
Mn,avg (g/mol)	17,940	38,690	95,430	252,470	724,230
Mw,avg (g/mol)	35,877	77,377	190,860	504,940	1,448,500

.

Note that the weight percentage of each site presented in Table 2 is the weight percentage of the polymer produced by that catalyst site. Determining the weight percentage of individual sites requires the kinetic constants (K_P, K_I, K_T, and K_D) for each site type, and is beyond the scope of this paper.

3. Materials and Methods

The MWD model developed here is based on the method of moments equations. There are several proposed kinetic mechanisms and equations in the literature [52,53,54], but the most accepted one for single- or multi-site polymerization of α-olefins by Ziegler–Natta catalysts is as follows [55]:

$$\begin{array}{ll}initiation:C_j^{*}+M\stackrel{k_{i,j}}{\to }{P}_{1,j}& \hfill \left(\mathrm{a}\right)\\ propagation:P_{r,j}+M\stackrel{k_{p,j}}{\to }{P}_{r+1,j}& \hfill \left(\mathrm{b}\right)\\ transfer:\\ \beta -\mathrm{transfer}: P_{r,j}\stackrel{k_{t\beta ,j}}{\to }{C}_j^{*}+D_{r,j}& \hfill \left(\mathrm{c}\right)\\ \mathrm{to} \mathrm{hydrogen}: P_{r,j}+H_2\stackrel{k_{tH,j}}{\to }{C}_j^{*}+D_{r,j}& \hfill \left(\mathrm{d}\right)\\ \mathrm{to} \mathrm{monomer}: P_{r,j}+M\stackrel{k_{tM,j}}{\to }{C}_j^{*}+D_{r,j}& \hfill \left(\mathrm{e}\right)\\ \mathrm{to} \mathrm{co}-\mathrm{catalyst}: P_{r,j}+Al\stackrel{k_{tAl,j}}{\to }{C}_j^{*}+D_{r,j}& \hfill \left(\mathrm{f}\right)\\ deactivation:\\ \mathrm{live} \mathrm{polymers}: P_{r,j}\stackrel{k_{d,j}}{\to }{C}_{d,j}+D_{r,j}& \hfill \left(\mathrm{g}\right)\\ \mathrm{active} \mathrm{centers}: C_j^{*}\stackrel{k_{d,j}}{\to }{C}_{d,j}& \hfill \left(\mathrm{h}\right)\\ \mathrm{poisoning}: P_{r,j}+I\stackrel{k_{dI}}{\to }{C}_{d,j}+D_{r,j}& \hfill \left(\mathrm{i}\right)\end{array}$$

(3)

where M represents the concentration of α-olefin monomer; C_j* and C_d,j represent the vacant active site and deactivated site of type j; and P_r,j and D_r,j represent living and dead polymer with chain length of r associated with the active site of type j, respectively. In addition, Al denotes the concentration of aluminum alkyl and H₂ represents the concentration of hydrogen, which is the most important transfer agent in Ziegler–Natta polymerization that significantly controls the molecular weight and, in some situations, the rate of polymerization [42,56]. Moreover, the chain transfer to hydrogen and aluminum alkyl (co-catalyst) is shown through the transfer reactions proportional to the square root of the concentration of transfer agents [50,55,57]. The complicated poisoning process is simplified in this model, such that it cannot considerably deteriorate the accuracy of the model [43]. In addition, the activation process is very fast; thus, it is assumed that before the catalyst enters the reactor, it is activated and therefore the role of Al in transfer reactions appears.

Note that this scheme represents kinetics of each type of the active sites individually without any interrelation. Although this mechanism is well accepted in polymerization reaction modeling, its microstructural simulation ability is limited [55]. It is worth mentioning that the main difference between single-site and multi-site models is the presence of linear summation over the number of active sites in the latter. The number of active centers is usually from two to five and one could easily expand the single-site catalyst equations to a multi-site catalyst by methods of linear algebra and the superposition of physical phenomena.

Moreover, the aforementioned mechanism shows that the chain growth is initiated by the insertion of the first monomer to the active site, followed by chain propagation. In addition, the chain transfer reactions, which include spontaneous beta chain transfer, transfer to hydrogen, transfer to monomer, and transfer to co-catalyst, occur simultaneously, leading to the production of dead polymer chains. In addition, these chains can be produced through growing chain deactivation or reaction with poisons and impurities. Additionally, each active site may undergo a deactivation reaction [51,55,58,59], which goes against the goal of a high catalyst reuse [60,61,62,63].

A few assumptions have been made in the polymerization mechanism to simplify the next derivations, which are as the following:

• Site activation by alkyl aluminum assumed to be instantaneous (i.e., reaction between catalyst and co-catalyst). Therefore, the site activation mechanism was ignored.
• Transfer reactions assumed to form the same site type, C*, which originally formed by activation of catalyst with the co-catalyst.

Using Equation (1) and its sub-equations, the material balance equations can be written as follows. Note that to avoid unnecessary complexity, one site equation is written without index j, and, as stated before, they could be integrated easily for a multi-site model. Therefore, we have:

$$\begin{array}{l}\frac{d{P}_r}{dt}=k_{p}M(P_{r-1}-P_r)-(k_{t\beta }+k_{tH}{H}_2+k_{tAl}Al+k_{tM}M+k_{dI}I+k_d)P_r\\ \mathrm{or} \mathrm{simply}\hspace{1em}\frac{d{P}_r}{dt}=K_P{P}_{r-1}-(K_P+K_T+K_D)P_r\end{array}$$

(4)

Moreover, for the chain length of one, it is a bit different:

$$\frac{d{P}_1}{dt}=K_I{C}^{*}-(K_P+K_T+K_D)P_1$$

(5)

For dead polymer chains we could write:

$$\frac{d{D}_r}{dt}=(k_{t\beta }+k_{tH}{H}_2+k_{tAl}Al+k_{tM}M+k_{dI}I+k_d)P_r=(K_T+K_D)P_r$$

(6)

where K_P, K_T, K_I, and K_D are

$$\begin{array}{ll}{K}_P=k_{p}M& \hfill \left(\mathrm{a}\right)\\ K_I=k_{i}M& \hfill \left(\mathrm{b}\right)\\ K_T=(k_{t\beta }+k_{tH}{H}_2+k_{tAl}Al+k_{tM}M)& \hfill \left(\mathrm{c}\right)\\ K_D=(k_{dI}I+k_d)& \hfill \left(\mathrm{d}\right)\end{array}$$

(7)

Additionally, for C* we have:

$$\frac{d{C}^{*}}{dt}=-(K_I+K_D)C^{*}+K_T{Y}_0$$

(8)

To make further calculations easier, we define a variable, named P₀, by the following formula:

$$P_0=(K_I/ K_P)C^{*} \mathrm{equivalents} \mathrm{to} K_P{P}_0=K_I{C}^{*}$$

(9)

The substitution of Equation (9) in Equation (5) leads to:

$$\frac{d{P}_1}{dt}=K_P{P}_0-(K_P+K_T+K_D)P_1$$

(10)

This is similar to Equation (4) for higher-order living polymer chains equations. Additionally, assuming constant values for K_P and K_I, Equation (8) can be rewritten as:

$$\frac{d{P}_0}{dt}=-(K_I+K_D)P_0+\frac{K_I}{K_P}{K}_T{Y}_0$$

(11)

In addition, Y₀ is the zero moment of living polymer chain distribution and using the general definition of qth moment of a distribution function f(x) by:

$${\mu }_q={\displaystyle {\sum }_{x=1}^{\infty }{x}^q}f(x)$$

This leads to:

$$Y_0={\displaystyle {\sum }_{r=1}^{\infty }{P}_r}$$

Considering this definition, we obtain

$$\frac{d{Y}_0}{dt}=\frac{d{P}_1}{dt}+{\displaystyle {\sum }_{r=2}^{\infty }\frac{d{P}_r}{dt}}$$

Additionally, using Equations (4) and (5), we easily show that:

$$\frac{d{Y}_0}{dt}=K_I{C}^{*}-(K_T+K_D)Y_0=K_P{P}_0-(K_T+K_D)Y_0$$

(12)

In addition [23],

$$\frac{d{Y}_1}{dt}=K_I{C}^{*}+K_P{Y}_0-(K_T+K_D)Y_1=K_P{P}_0+K_P{Y}_0-(K_T+K_D)Y_1$$

(13)

$$\frac{d{Y}_2}{dt}=K_I{C}^{*}+K_P{Y}_0+2K_P{Y}_1-(K_T+K_D)Y_2=K_P{P}_0+K_P{Y}_0+2K_P{Y}_1-(K_T+K_D)Y_2$$

(14)

Similar equations could be written for dead polymer chains distribution by denoting X_q as qth moment of dead polymer chains distribution. Therefore, we have:

$$\frac{d{X}_q}{dt}={\displaystyle {\sum }_{r=1}^{\infty }{r}^q\frac{d{D}_r}{dt}}={\displaystyle {\sum }_{r=1}^{\infty }(K_T+K_D)}{r}^q{P}_r=(K_T+K_D)Y_q$$

(15)

Or, for zero to second moments, we could write:

$$\begin{array}{ll}\frac{d{X}_0}{dt}=(K_T+K_D)Y_0& \hfill \left(\mathrm{a}\right)\\ \frac{d{X}_1}{dt}=(K_T+K_D)Y_1& \hfill \left(\mathrm{b}\right)\\ \frac{d{X}_2}{dt}=(K_T+K_D)Y_2& \hfill \left(\mathrm{c}\right)\end{array}$$

(16)

Equations (8)–(16) can be written as a linear first-order ordinary differential equation by using a matrix form of differential equations. However, it should be noted that the values of K_P, K_I, K_D, and K_T are functions of time. The concentrations of monomer, hydrogen, other transfer agents, and impurities are not constant in the different positions of a growing polymer particle. In addition, the rate parameters are not constant over the growing particle because the temperature differs in practice in different positions of a growing polymer particle under the reaction conditions. However, in some cases of mild conditions of slurry ethylene or propylene polymerization, the temperature gradient inside the growing particle could be neglected and concentrations could remain approximately constant.

Note that the average molecular weights of polymerization and dispersity (PDI) could be found by the following Equations [64]:

$${\overline{M}}_w=mw\frac{X_2+Y_2}{X_1+Y_1}=mw\frac{Z_2}{Z_1}$$

(17)

$${\overline{M}}_n=mw\frac{X_1+Y_1}{X_0+Y_0}=mw\frac{Z_1}{Z_0}$$

(18)

$$PDI=\frac{{\overline{M}}_w}{{\overline{M}}_n}=\frac{(X_2+Y_2)(X_0+Y_0)}{{(X_1+Y_1)}^2}=\frac{Z_2{Z}_0}{Z_1{}^2}$$

(19)

In Equations (17) and (18), mw denotes the molecular weight of monomer. In most polymerization reactions, dead polymer moments are much bigger than their corresponding live moment (X_j >> Y_j) [43,54], so sometimes in Equations (17)–(19), live moments can be ignored.

Since the main purpose of this research is to develop a methodology for the molecular weight distribution calculation, the following equation is presented:

$$f(m)=\frac{\mathrm{weight} \mathrm{of} \mathrm{polymer} \mathrm{with} \mathrm{chain} \mathrm{lenghts} \mathrm{from} m+1 \mathrm{to} \infty }{\mathrm{total} \mathrm{weight} \mathrm{of} \mathrm{polymer}}=1-F(m)$$

(20)

In Equation (20), F(m) denotes the cumulative distribution of molecular weight. By such definition of f(m) we could easily write:

$$f(m)=\frac{{\displaystyle {\sum }_{r=m+1}^{\infty }r(P_r+D_r)}}{{\displaystyle {\sum }_{r=1}^{\infty }r(P_r+D_r)}}=\frac{{\displaystyle {\sum }_{r=m+1}^{\infty }r(P_r+D_r)}}{Y_1+X_1}$$

(21)

Therefore, by differentiating Equation (21) we have:

$$\frac{df(m)}{dt}=\dot{f}(m)=\frac{{\displaystyle {\sum }_{r=m+1}^{\infty }r({\dot{P}}_r+{\dot{D}}_r)}}{Y_1+X_1}-\frac{{\dot{Y}}_1+{\dot{X}}_1}{Y_1+X_1}f(m)$$

(22)

It is obvious from Equation (22) that by finding an expression for ${\displaystyle {\sum }_{r=m+1}^{\infty }r({\dot{P}}_r+{\dot{D}}_r)}$, an equation for the molecular weight distribution is obtained. By simultaneous solution of this equation with the moment differential equations, time distribution of molecular weight is achieved. Using Equations (4), (6), and (10):

$$\begin{array}{l}{\displaystyle {\sum }_{r=m+1}^{\infty }r({\dot{P}}_r+{\dot{D}}_r)}={\displaystyle {\sum }_{r=m+1}^{\infty }r\left\{\left[K_P{P}_{r-1}-(K_P+K_T+K_D)P_r\right]+\left[(K_T+K_D)P_r\right]\right\}}\\ \Rightarrow \boxed{{\displaystyle {\sum }_{r=m+1}^{\infty }r({\dot{P}}_r+{\dot{D}}_r)}=K_P{\displaystyle {\sum }_{r=m+1}^{\infty }r(}{P}_{r-1}-P_r)} , m\ge 1\end{array}$$

(23)

By manipulating Equation (23), along with some simplification, the results show that:

$${\displaystyle {\sum }_{r=m+1}^{\infty }r({\dot{P}}_r+{\dot{D}}_r)}=K_P\left[(m+1)P_m+(P_{m+1}+P_{m+2}+\dots +P_{\infty })\right]=K_P\left[m{P}_m+{\displaystyle {\sum }_{r=m}^{\infty }{P}_r}\right]$$

(24)

Writing the above equation in matrix form together with some algebraic manipulations, Equations (25) and (26) are finally obtained:

$$C^{*}=\frac{C_0^{*}}{K_I+K_T}\left[K_T{e}^{-K_{D}t}+K_I{e}^{-(K_I+K_T+K_D)t}\right]$$

(25)

Additionally,

$$Y_0=\frac{C_0^{*}}{K_I+K_T}{K}_I\left[e^{-K_{D}t}-e^{-(K_I+K_T+K_D)t}\right]$$

(26)

Deriving molecular weight distribution equation needs defining $X_1$, $Y_1$, ${\dot{X}}_1$, ${\dot{Y}}_1$, and the right-hand term in Equation (24) is defined as:

$$A(t)=K_P\left[m{P}_m+{\displaystyle {\sum }_{r=m}^{\infty }{P}_r}\right]$$

This is performed by the Laplace transform, which finally yields:

$$\begin{array}{l}A\left(\mathrm{s}\right)=\frac{K_I{C}_0^{*}}{K_I+K_T}G(s)H(s)\hspace{1em}\mathrm{in} \mathrm{which}\\ \hspace{1em}G(s)=\left[(m+1)+\frac{K_P}{(K_T+K_D)+s}\right]\left[\frac{K_T}{K_D+s}+\frac{K_I}{(K_I+K_T+K_D)+s}\right]\\ \hspace{1em}H(s)={\left[\frac{K_P}{(K_P+K_T+K_D)+s}\right]}^m\end{array}$$

(27)

The equation can be written in time domain as:

$$\begin{array}{l}A(t)={ℒ}^{-1}A(s)=\frac{K_I{C}_0^{*}}{K_I+K_T}{\displaystyle {\int }_0^{t}G(t-\tau )H(\tau )d\tau }\\ \hspace{1em}\hspace{1em}=\frac{K_I{C}_0^{*}}{K_I+K_T}\frac{K_P{}^m}{(m-1)!}\left\{\begin{array}{l}\left[(m+1)K_T+K_P\right]e^{-K_{D}t}{\displaystyle {\int }_0^t{\tau }^{m-1}{e}^{-(K_P+K_T)\tau }d\tau }\\ +\left[(m+1)K_I-K_P\right]e^{-(K_I+K_T+K_D)t}{\displaystyle {\int }_0^t{\tau }^{m-1}{e}^{-(K_P-K_I)\tau }d\tau }\end{array}\right\}\end{array}$$

(28)

Referring to the definition of the lower, incomplete Gamma function (γ function), we may write:

$$\boxed{A(t)=\frac{K_I{C}_0^{*}}{(K_I+K_T)(m-1)!}\left\{\begin{array}{l}\left[(m+1)K_T+K_P\right]{\left(\frac{K_P}{K_P+K_T}\right)}^m{e}^{-K_{D}t}\gamma (m,(K_P+K_T)t)\\ +\left[(m+1)K_I-K_P\right]{\left(\frac{K_P}{K_P-K_I}\right)}^m{e}^{-(K_I+K_T+K_D)t}\gamma (m,(K_P-K_I)t)\end{array}\right\}}$$

(29)

Now, to find f(m,t), we refer to Equation (22), which could be written as:

$$\frac{df(m,t)}{dt}=\dot{f}(m,t)=\frac{A(t)}{Y_1+X_1}-\frac{{\dot{Y}}_1+{\dot{X}}_1}{Y_1+X_1}f(m,t)=\frac{A(t)}{Z_1(t)}-\frac{{\dot{Z}}_1(t)}{Z_1(t)}f(m,t)$$

(30)

Since Equation (30) is a linear first-order ordinary differential equation, it could be easily solved by finding its integral factor, as follows:

$$f(m,t)=\frac{{\displaystyle {\int }_0^{t}A(\tau )d\tau +C(m)}}{Z_1(t)}$$

(31)

At the beginning of the polymerization (time zero), there is no live or dead polymer chain; therefore, the value of $F(m,t)=1-f(m,t)$ must be equal to 1 for every m equal to or bigger than 1. Therefore, C(m) must be zero for every m ≥ 1.

Now the remaining parameter to find the molecular weight distribution is the sum of the live and dead first polymer moments (value of Z₁), which finally results in:

$$\boxed{Z_1(t)=Y_1(t)+X_1(t)=\left(\frac{K_I{C}_0^{*}}{K_I+K_T}\right)\left\{\frac{K_I-K_P}{K_I+K_T+K_D}\left[1-e^{-(K_I+K_T+K_D)t}\right]+\frac{K_P+K_T}{K_D}(1-e^{-K_{D}t})\right\}}$$

(32)

In that way, combining Equations (20), (29), (31), and (32), we may write:

$$\boxed{\boxed{F(m,t)=1-\frac{\left\{\begin{array}{l}\frac{\left[(m+1)K_T+K_P\right]}{K_D(m-1)!}\left[\begin{array}{l}{\left(\frac{K_P}{K_P+K_T+K_D}\right)}^m\gamma (m,(K_P+K_T+K_D)t)\\ -{\left(\frac{K_P}{K_P+K_T}\right)}^m{e}^{-K_{D}t}\gamma (m,(K_P+K_T)t)\end{array}\right]\\ +\frac{\left[(m+1)K_I-K_P\right]}{(K_I+K_T+K_D)(m-1)!}\left[\begin{array}{l}{\left(\frac{K_P}{K_P+K_T+K_D}\right)}^m\gamma (m,(K_P+K_T+K_D)t)\\ -{\left(\frac{K_P}{K_P-K_I}\right)}^m{e}^{-(K_I+K_T+K_D)t}\gamma (m,(K_P-K_I)t)\end{array}\right]\end{array}\right\}}{\left\{\frac{K_P+K_T}{K_D}(1-e^{-K_{D}t})+\frac{K_I-K_P}{K_I+K_T+K_D}\left[1-e^{-(K_I+K_T+K_D)t}\right]\right\}}}}$$

(33)

At the end of the reaction:

$$\begin{array}{l}F(m,t=\infty )=1-\frac{\left\{\frac{\left[(m+1)K_T+K_P\right]}{K_D}{\left(\frac{K_P}{K_P+K_T+K_D}\right)}^m+\frac{\left[(m+1)K_I-K_P\right]}{(K_I+K_T+K_D)}{\left(\frac{K_P}{K_P+K_T+K_D}\right)}^m\right\}}{\left\{\frac{K_P+K_T}{K_D}+\frac{K_I-K_P}{K_I+K_T+K_D}\right\}}\\ \Rightarrow F(m,t=\infty )=1-\left[1+m\left(1-\frac{K_P}{K_P+K_T+K_D}\right)\right]{\left(\frac{K_P}{K_P+K_T+K_D}\right)}^m\\ \Rightarrow \boxed{\boxed{F(m,t=\infty )=1-\left[1+m(1-Q)\right]Q^m}}\end{array}$$

(34)

It is obvious from Equation (34) that the final molecular weight distribution is identical to the one predicted by Schulz–Flory distribution [64]. Q is the growth probability.

$$\boxed{PDI=\frac{{\overline{M}}_w}{{\overline{M}}_n}=1+Q=1+\frac{K_P}{K_P+K_T+K_D}}$$

(35)

Equation (35) shows that polydispersity index is between one and two for single site polymerization; this is a bit less than the value two, which was reported before, based on Flory’s most probable distribution of chain lengths [48,58,59]. Therefore, Equation (35) once again confirms the validity of the proposed methodology.

The average molecular weight and PDI can be derived with respect to the presented model as:

$$\boxed{\boxed{\begin{array}{l}{\overline{M}}_w=mw\frac{(x-1)(2x-1)A-2(x+y)B+(y+1)(2y+1)C}{\left(1-x\right)A+\left(1+y\right)C}\\ {\overline{M}}_n=mw\frac{(1-x)A+(1+y)C}{A+C}\\ PDI=\frac{\left[(x-1)(2x-1)A-2(x+y)B+(y+1)(2y+1)C\right]\left(A+C\right)}{{\left[(1-x)A+(1+y)C\right]}^2}\\ A=K_I\frac{1-e^{-(K_I+K_T+K_D)t}}{K_I+K_T+K_D}, B=K_P\frac{1-e^{-(K_T+K_D)t}}{K_T+K_D}, C=K_T\frac{1-e^{-K_{D}t}}{K_D}\\ x=\frac{K_P}{K_I},y=\frac{K_P}{K_T},\underset{t\to 0}{\lim }PDI=1\end{array}}}$$

(36)

Although from Equation (34) we expect that, when time tends to infinity, the values of average molecular weights and polydispersity are identical to equations derived by Schulz–Flory distribution; however, by simplifying Equation (36), we could write:

$$\begin{array}{l}{{\overline{M}}_w|}_{t=\infty }=mw\frac{1+Q}{1-Q} , {{\overline{M}}_n|}_{t=\infty }=mw\frac{1}{1-Q} , {PDI|}_{t=\infty }=1+Q\\ \mathrm{where} Q=\frac{K_P}{K_P+K_T+K_D}\end{array}$$

(37)

This is identical to the previous equations and justifies the validity of the proposed methodology and derived equations.

4. Conclusions

A new protocol for time-dependent molecular weight distribution estimation was developed for catalytic olefin polymerizations, using traditional heterogeneous Ziegler–Natta, metallocene, and Phillips catalysts. The proposed method, in addition to its practical characteristics, can estimate molecular weight and its distribution at any time of olefin polymerization experiments. According to the method, the molecular weight distribution at infinity obeys the well-known Schulz–Flory distribution, but the polymerization progresses especially at the initial times, resulting in a deviation from Schulz–Flory distribution. Our model predicts both the molecular weight increase and molecular weight distribution widening, which confirms the experimentally observed trend well. This affirms the reliability and generalizability of the proposed model. Our next goal is to extend the methodological results to other metal-catalyzed olefin polymerizations, as well as other, more complex substrates.