Exploring Some Kinetic Aspects of the Free Radical Polymerization of PDMS-MA

Abstract

This study conducts a theoretical exploration of the free radical polymerization of polydimethylsiloxane homopolymers with a methyl methacrylate end group (PDMS-MA). To achieve this, a methodology is developed to model and simulate experimental data previously reported by one of the authors. The model incorporates a typical chain reaction mechanism, encompassing initiation, propagation, chain transfer, and termination stages. The resulting ordinary differential equations from this mechanistic approach are numerically integrated over time. Employing a semi-heuristic procedure, the study derives estimated values for the diffusive steps of termination (k_td) and propagation (k_pd). Methodological accuracy is assessed through a comparison of the mathematical model results and experimental data. This evaluation includes the estimation of conversion and the average molecular weight (both number (M_n) and weight (M_w)) at three distinct monomer concentrations, considering a 4.78% mol initiator-to-monomer ratio. The theoretical results obtained from this study contribute to a better understanding of the evolution of diffusion coefficients. These coefficients play a crucial role in influencing the behavior of the studied reactions, particularly in relation to the absence of the autoacceleration effect in these reactions.

1. Introduction

Polydimethylsiloxane (PDMS) is an exceptionally versatile material known for its remarkable properties. It exhibits hydrophobicity, repelling water, and possesses desirable surface characteristics, including low surface tension, excellent lubrication, and a soft tactile sensation. PDMS also offers advantageous bulk physical properties, such as high permeability, electrical strength, compressibility, and a low boiling point. In addition, it demonstrates favorable chemical properties, including exceptional thermal stability, minimal chemical reactivity, low fire hazard, and limited environmental metal hazards [1,2].

PDMS, particularly at a high molecular weight, serves as an effective sealant, finding utility in lithographic processes. Its applications range from catheters and oxygenator membranes to kitchen molds for various items, showcasing a wide array of intriguing properties that make it attractive for numerous applications [1,2].

Several studies have explored applications of polydimethylsiloxanes (PDMS). Gutowski et al. [3] investigated the use of PDMS as the base material for formulating high-performance structural adhesives, sealants, and multifunctional coatings. In a similar vein, Zolper et al. [4] delved into the rheological properties, elastohydrodynamic film thickness, and friction coefficients of commercially available lubricants, both polyalphaolefin (PAO) and PDMS-based, aiming to establish connections between the molecular structure and lubricant performance.

In another line of research, Zhang et al. [5] synthesized waterborne polyurethane (WPU) prepolymers, incorporating toluene diisocyanate (TDI), trimethylolpropane (TMP), polydimethylsiloxane (PDMS), N-methyldiethanolamine (N-MDEA), and glycerin monostearate (GMS). These prepolymers were chain-extended with aminoethylaminopropyl polydimethylsiloxane (AEAPS) to produce fluorine-free water repellents.

Additionally, Liu et al. [6] prepared flexible polymer composite films through a spin-coating process, enhancing the thermal conductivity and electrical insulation by incorporating short carbon fibers (SCFs) and boron nitride (BN) particles into the PDMS matrix.

As evident from the research mentioned in the preceding paragraphs, investigations into PDMS have been multifaceted and conducted across various fields owing to its diverse applications [7,8]. Although studies on the kinetics and diffusion of PDMS have attracted significant interest, the relationship between these aspects is not fully understood [9,10]. To address this gap in our understanding, efforts have been made to establish connections between the experimental data and theoretical concepts, with the goal of enhancing our understanding of the observed phenomena during the chemical production of PDMS [11].

Furthermore, low-molecular-weight PDMS-MA finds extensive application as a lubricant and antifoam agent in microfluidic applications. Its utilization as a comonomer is also well-documented in the scientific literature [12,13,14]. The production of high-molecular-weight PDMS-MA is relatively straightforward through a free radical method.

Despite the extensive studies on the polymerizations of monomers with small molecules, like methyl methacrylate (MMA) or styrene, there is a noticeable lack of research concerning the homopolymerization of macromonomers. A recent study by Ning Ren et al. in 2021 [15] emphasized the importance of studying the kinetic aspects of macromonomers, specifically investigating the impact of topology on their polymerization using ring-opening metathesis polymerization. In the field of free radical polymerization, exploring the relationship between polymeric radicals and the termination coefficient is an intriguing subject. The length of radicals can trigger diffusive effects, leading to distinctive behaviors in the evolution of termination coefficients [16].

During the polymerization of macromonomers, the length of these molecules is expected to trigger the well-known autoacceleration effect, a characteristic consequence of the diffusive limitations observed in free radical polymerizations. An example related to this topic was reported by Darras et al. in 2007, covering the uses of PDMS as a monomer and as a comonomer in the formation of interpenetrating networks (IPNs). They reported the occurrence of the autoacceleration phenomenon during the production of PDMS/polyAcRf6 IPNs, attributing it to steric hindrance that hindered the mobility of PDMS oligomers, interfering with the formation of the network [17].

Conducting research that includes the kinetic studies of macromonomer polymerization and an examination of its kinetic coefficients is a topic that can deepen our understanding of these reactions. Consequently, it can lead to proposals for enhancing their practical applications. Employing simulation studies to explore this process can foster improvements in both conversion and molecular weight, offering valuable insights.

The present study investigates the kinetic behavior of the diffusive step, specifically examining the termination and propagation coefficients, during the polymerization of PDMS-MA macromonomers in a solution. Our focus is on the kinetic aspects related to PDMS-MA macromonomers with a molecular weight of 5000 g mol⁻¹. These macromonomers were synthesized previously by Morales-Cepeda, who determined both the conversion and molecular weights [18,19].

To gain a comprehensive understanding of the reactions, our initial step involved exploring various modeling options using a semi-heuristic approach. This research focuses on examining the relationship between the coefficients and factors such as molecular weight and monomer conversion, in addition to explaining the theoretical results in the absence of the autoacceleration phenomenon despite the length of the monomers involved.

2. Methods

The methodology employed to establish the model involved using a kinetic scheme with the kinetic coefficients determined in a semi-heuristic way, as described in the following paragraphs.

A typical kinetic scheme of free radical polymerization, which considered the initiation, propagation, chain transfer, and termination stages (Equations (1)–(5)), was employed to model and simulate the reactions reported by Morales-Cepeda [18,19]. It is relevant to mention that Morales-Cepeda also reported additional homopolymerizations using a prepolymerization procedure. However, the methods involving prepolymerization, where this process is carried out, are beyond the scope of our current study. Our exclusive focus is on analyzing conventional polymerization reactions, excluding the use of previously polymerized materials.

$$\mathit{I} \stackrel{ \mathit{f}{\mathit{k}}_{\mathit{d} } }{\to }\mathbf{2}{\mathit{R}}^{*}$$

(1)

$${\mathit{R}}^{*}+\mathit{M} \stackrel{ {\mathit{k}}_{\mathit{i} }}{\to }{\mathit{P}}_{\mathbf{1}}^{*}$$

(2)

$${\mathit{P}}_{\mathit{n}}^{*}+\mathit{M} \stackrel{ {\mathit{k}}_{\mathit{p} }}{\to }{\mathit{P}}_{\mathit{n}+\mathbf{1}}^{*}$$

(3)

$${\mathit{P}}_{\mathit{n}}^{*}+\mathit{M} \stackrel{ {\mathit{k}}_{\mathit{t}\mathit{r}\mathit{m} }}{\to }{\mathit{D}}_{\mathit{n}}+{\mathit{P}}_{\mathbf{1}}^{*}$$

(4)

$${\mathit{P}}_{\mathit{n}}^{*}+{\mathit{P}}_{\mathit{m}}^{*} \stackrel{ {\mathit{k}}_{\mathit{t} }}{\to }{\mathit{D}}_{\mathit{n}}+{\mathit{D}}_{\mathit{m}}$$

(5)

where f is the initiator efficiency, I, R*, M, P₁*, P_n*, P_m* D_n, and D_m represent the initiator, initiator fragment, the monomer, the active or live chain of one unit, active or live chain of length, n, the active or live chain of length, m, the inactive or death chain of length, n, and the inactive or death chain of length, m, respectively. k_d, k_trm, k_p, and k_t are the overall or global kinetic coefficients of decomposition of the initiator, transfer to monomer, propagation and termination, respectively.

The Euler method was employed to solve the differential equations related to the monomer and initiator concentrations, as well as the inactive moments derived from the aforementioned scheme. In order to avoid stability issues, the quasi-steady-state assumption was applied to the active chains. The iterative solution of these equations was implemented in Fortran and the corresponding equations are presented in Equation (6) to Equation (13):

$$\left[\mathit{I}\right]=\left[\mathit{I}\right]+\left(-{\mathit{k}}_{\mathit{d}}\left[\mathit{I}\right]\right)\times \mathit{s}\mathit{t}\mathit{e}\mathit{p}$$

(6)

$$\left[\mathit{M}\right]=\left[\mathit{M}\right]+\left(-{\mathit{k}}_{\mathit{p}}\left[\mathit{M}\right]{\mathit{P}}^{*}-{\mathit{k}}_{\mathit{t}\mathit{r}\mathit{m}}\left[\mathit{M}\right]{\mathit{P}}^{*}\right)\times \mathit{s}\mathit{t}\mathit{e}\mathit{p}$$

(7)

$${\mathit{\lambda }}_{\mathbf{0}}={\left(\frac{\mathbf{2}\mathit{f}{\mathit{k}}_{\mathit{d}}\left[\mathit{I}\right]}{{\mathit{k}}_{\mathit{t}}}\right)}^{\mathbf{0.5}}$$

(8)

$${\mathit{\lambda }}_{\mathbf{1}}=\left(\frac{\mathbf{2}\mathit{f}{\mathit{k}}_{\mathit{d}}\left[\mathit{I}\right]+{\mathit{\lambda }}_{\mathbf{0}}\left[\mathit{M}\right]({\mathit{k}}_{\mathit{p}}+{\mathit{k}}_{\mathit{t}\mathit{r}\mathit{m}})}{{\mathit{k}}_{\mathit{t}\mathit{r}\mathit{m}}\left[\mathit{M}\right]+{\mathit{k}}_{\mathit{t}}{\mathit{\lambda }}_{\mathbf{0}}}\right)$$

(9)

$${\mathit{\lambda }}_{\mathbf{2}}=\left(\frac{\mathbf{2}\mathit{f}{\mathit{k}}_{\mathit{d}}\left[\mathit{I}\right]+{\mathit{\lambda }}_{\mathbf{0}}\left[\mathit{M}\right]{\mathit{k}}_{\mathit{p}}+{\mathit{\lambda }}_{\mathbf{0}}\left[\mathit{M}\right]{\mathit{k}}_{\mathit{t}\mathit{r}\mathit{m}}+\mathbf{2}{\mathit{\lambda }}_{\mathbf{1}}\left[\mathit{M}\right]{\mathit{k}}_{\mathit{p}}}{{\mathit{k}}_{\mathit{t}\mathit{r}\mathit{m}}\left[\mathit{M}\right]+{\mathit{k}}_{\mathit{t}}{\mathit{\lambda }}_{\mathbf{0}}}\right)$$

(10)

$${\mathit{\mu }}_{\mathbf{0}}={\mathit{\mu }}_{\mathbf{0}}+\left({\mathit{\lambda }}_{\mathbf{0}}\left[\mathit{M}\right]{\mathit{k}}_{\mathit{t}\mathit{r}\mathit{m}}+{{\mathit{\lambda }}_{\mathbf{0}}}^{\mathbf{2}}{\mathit{k}}_{\mathit{t}}\right)\times \mathit{s}\mathit{t}\mathit{e}\mathit{p}$$

(11)

$${\mathit{\mu }}_{\mathbf{1}}={\mathit{\mu }}_{\mathbf{1}}+\left({\mathit{\lambda }}_{\mathbf{1}}\left[\mathit{M}\right]{\mathit{k}}_{\mathit{t}\mathit{r}\mathit{m}}+{\mathit{\lambda }}_{\mathbf{0}}{\mathit{\lambda }}_{\mathbf{1}}{\mathit{k}}_{\mathit{t}}\right)\times \mathit{s}\mathit{t}\mathit{e}\mathit{p}$$

(12)

$${\mathit{\mu }}_{\mathbf{2}}={\mathit{\mu }}_{\mathbf{2}}+\left({\mathit{\lambda }}_{\mathbf{2}}\left[\mathit{M}\right]{\mathit{k}}_{\mathit{t}\mathit{r}\mathit{m}}+{\mathit{\lambda }}_{\mathbf{0}}{\mathit{\lambda }}_{\mathbf{2}}{\mathit{k}}_{\mathit{t}}\right)\times \mathit{s}\mathit{t}\mathit{e}\mathit{p}$$

(13)

where [I] is the initiator concentration; [M] is the monomer concentration; ${\mathit{\lambda }}_{\mathbf{0}}$, ${\mathit{\lambda }}_{\mathbf{1}}$, ${\mathit{\lambda }}_{\mathbf{2}}$ are the zero (which represents total polymeric radicals, P*), one, and two moments of the active or live polymer chains, respectively; and ${\mathit{\mu }}_{\mathbf{0}}$, ${\mathit{\mu }}_{\mathbf{1}}$, ${\mathit{\mu }}_{\mathbf{2}}$ are the zero, one, and two moments of the inactive or death polymer chains, respectively. The step size (step) was equal to 1 × 10⁻⁷ h.

Equations (14)–(16) are used to calculate M_n, M_w, and the monomer conversion.

$${\mathit{M}}_{\mathit{n}}=\mathit{P}{\mathit{M}}_{\mathit{P}\mathit{D}\mathit{M}\mathit{S}-\mathit{M}\mathit{A}}\times \left(\frac{{\mathit{\lambda }}_{\mathbf{1}}+{\mathit{\mu }}_{\mathbf{1}}}{{\mathit{\lambda }}_{\mathbf{0}}+{\mathit{\mu }}_{\mathbf{0}}}\right)$$

(14)

$${\mathit{M}}_{\mathit{w}}=\mathit{P}{\mathit{M}}_{\mathit{P}\mathit{D}\mathit{M}\mathit{S}-\mathit{M}\mathit{A}}\times \left(\frac{{\mathit{\lambda }}_{\mathbf{2}}+{\mathit{\mu }}_{\mathbf{2}}}{{\mathit{\lambda }}_{\mathbf{1}}+{\mathit{\mu }}_{\mathbf{1}}}\right)$$

(15)

$$\mathit{C}\mathit{o}\mathit{n}\mathit{v}\mathit{e}\mathit{r}\mathit{s}\mathit{i}\mathit{o}\mathit{n}=\frac{\left[{\mathit{M}}_{\mathbf{0}}\right]-\left[\mathit{M}\right]}{\left[{\mathit{M}}_{\mathbf{0}}\right]}$$

(16)

where PM_PDMS-MA is the macromonomer’s molecular weight and [M₀] is the initial monomer concentration.

As an initial approximation for modeling purposes, the overall kinetic coefficients used for the polymerization stages of PDMS-MA were assumed to be the same as those for the chemical step of MMA polymerization (without considering the diffusion effects). This assumption was made based on the fact that macromonomers possess an MMA-derived molecule at their termination, which is responsible for the reactions involved in the polymerization stages. The specific values of these kinetic coefficients can be found in

Table 1. Kinetic coefficients and parameters at 60 °C for preliminary simulations of this work. k_p, k_trm, and k_t were considered the chemical step kinetic coefficients of MMA polymerization.

Kinetic Coefficients/Parameter	Value	Reference
kd	31.87 × 10−3 h−1	[20]
f	0.5	[21]
kp0	3 × 106 L mol−1 h−1	[22]
kt0	3 × 1010 L mol−1 h−1	[22]
ktrm0	14.4 L mol−1 h−1	[23]

.

After analyzing the results obtained in the preliminary simulations, a sensitivity analysis was conducted to assess the impact of the kinetic coefficients on the target outcomes. The derived conclusion highlights the significance of the kinetic coefficients of termination and propagation. During the sensitivity analyses conducted before selecting the parameters for estimation, it was observed that varying the k_trm coefficient did not result in significant variations in the results. Consequently, various values of k_p and k_t were suggested and tested across different conversion intervals of the simulated experimental reactions. The selected values were those that yielded theoretical results aligning with the dispersion observed in the experimental data.

Once the best values were selected, semi-empirical equations were established that related the values of k_p and k_t to some variables in the data generated and/or introduced by the model, finding a relationship of k_t with M_w and a relationship of k_p with the initial molecular weight [M_w0] and M_w.

Ultimately, the study progressed to calculate k_td and k_pd. This accomplishment was attained through solving Equations (17) and (18).

$$\frac{\mathbf{1}}{{\mathit{k}}_{\mathit{t}}}=\frac{\mathbf{1}}{{\mathit{k}}_{\mathit{t}\mathbf{0}}}+\frac{\mathbf{1}}{{\mathit{k}}_{\mathit{t}\mathit{d}}}$$

(17)

$$\frac{\mathbf{1}}{{\mathit{k}}_{\mathit{p}}}=\frac{\mathbf{1}}{{\mathit{k}}_{\mathit{p}\mathbf{0}}}+\frac{\mathbf{1}}{{\mathit{k}}_{\mathit{p}\mathit{d}}}$$

(18)

The subscript “0” indicates that the coefficient corresponds to the chemical step.

It is important to note that, despite the simplicity of developing this model, the theoretical results offer semi-quantitative information relevant to the objectives of our work. This was achieved without the necessity of applying complex models or the excessive use of computational resources. Due to the model incorporating values of the chemical step equal to those coefficients reported for the polymerization of MMA, adapting this model would be necessary to extend its applicability to other types of macromonomers.

3. Results

3.1. Initial Simulation Results

In the initial simulations, the use of chemical kinetic coefficients as overall kinetic coefficients resulted in a significant deviation from the experimental results by the model. Figure 1a–c show the outcomes from both the modeling and experimental data. The graphical representations provide a comparison of the experimental and modeling data concerning the evolution of conversion.

Upon reviewing the results presented in Figure 1a–c, it became apparent that exploring new values for the coefficients was necessary. In the cases analyzed, the theoretical results significantly overestimated the trend of the dispersion of the experimental data over time.

3.2. k_t and k_p Implementation Equations

As previously mentioned, in the sensitivity analysis where various values for the kinetic coefficients were tested, it was observed that both k_t and k_p significantly influenced the model results. After selecting values for k_t and k_p that best aligned with each of the experimental trends studied, mathematical regressions were conducted, resulting in the semiempirical Equations (19) and (20), where the molecular weights must be considered in g mol⁻¹.

$$k_p=1.281e^{\left(-3\times {10}^{-7}\times M_{w0}\right)}\times {M_w}^{0.9543}$$

(19)

$$k_t=1.33\times {10}^9{e}^{-8\times {10}^{-6}\times {M_w}^{-1}}$$

(20)

The final implementation in the program was carried out with the following considerations:

The initial concentrations of the monomer and initiator for each case were input.

The chemical coefficients were considered as initial values for the overall coefficients, assuming no polymer existed at this stage. Except for k_t and k_p, these coefficients were consistently considered throughout the execution of the model. k_t and k_p were calculated with Equations (19) and (20).

The active and inactive moments were calculated.

The average molecular weights were determined.

The results from this implementation in the model are shown in Figure 2, Figure 3 and Figure 4, where a reasonable congruence with the experimental data becomes apparent.

By employing Equations (19) and (20) to model the studied reactions, a notably improved fit was achieved compared to using only the chemical kinetic coefficients for calculating k_t and k_p. This ensured that the theoretical results reasonably aligned with the dispersion observed in the reported experimental results. Such an alignment justifies the interpretation of the theoretical findings that are described in this work.

The theoretical results reveal that the k_td value remains nearly constant under the studied conditions and is lower than the k_t0 value. This is illustrated in Figure 5a, where the graph depicts values related to the termination coefficients for the reaction, initiated with 0.098 mol L⁻¹ of the monomer. The comparison between the values of k_t0 and k_td indicated that the reaction under analysis was controlled by the diffusive step.

Similarly, the results of the propagation (Figure 5b) show that the diffusive stage also controls the propagation.

Another relevant aspect of the kinetic analysis is the evolution of P* [24]. In free radical polymerizations, primarily conducted in bulk, it is common to observe a rapid increase in P*, leading to the phenomenon known as autoacceleration. This can be explained as follows: as k_t tends to decrease significantly, P* tends to increase abruptly, manifesting itself as a sudden acceleration of the polymerization rate as well as the curves obtained from conversion versus time. However, as depicted in Figure 6, the reactions studied in this work exhibit a different behavior.

In Figure 6, it can be observed that, for the three monomer concentrations used, the evolution of P* tends to decrease from the initial moments of polymerization and continues to decrease throughout the reaction. This pattern coincides with the absence of autoacceleration.

P* is associated with the changes in the distance between radicals, determined through Equation (21) [25]. The results consistently demonstrate that a higher P* corresponds to a smaller calculated distance, as depicted in Figure 7.

$$Distance={\left(\frac{1}{P^{*}{N}_A}\right)}^{1/3}$$

(21)

where N_A is the Avogadro number. To maintain consistency between the units, P* must be considered in mol m⁻³.

The decrease observed in all studied reactions, as shown in Figure 6, suggests that the diffusive limitations are not significant enough to trigger the appearance of the autoacceleration effect.

Ultimately, the diffusion coefficient of polymeric radicals (D_s) was calculated using the obtained k_td value and estimated data. This procedure involved solving the Smoluchowski equation expressed in Equation (22), where “σ” represents the separation distance assumed to correspond to an instantaneous termination, equal to the size of a monomeric unit segment in MMA (6.9 × 10⁻¹⁰ m) [26].

$${\mathit{k}}_{\mathit{t}\mathit{d}}=\mathbf{4}\mathit{\pi }\mathit{\sigma }{\mathit{D}}_{\mathit{s}}$$

(22)

To utilize the k_td units depicted in Figure 5a, employ N_A and the conversion factor from liters to cubic meters as specified in Equation (23):

$${\mathit{D}}_{\mathit{s}}=\frac{{\mathit{k}}_{\mathit{t}\mathit{d}}\left(\frac{\mathbf{0.001}{\mathit{m}}^{\mathbf{3}}}{\mathit{L}}\right) }{\mathbf{4}\mathit{\pi }\mathit{\sigma }{\mathit{N}}_{\mathit{A}}}$$

(23)

D_s represents the diffusion at which a short polymeric radical can move through the solvent, transitioning from one location to another. In other words, it quantifies the translational movement of polymers in the reaction medium. Its value is influenced by various factors, such as the temperature, free volume, and specific properties of the diffusing molecules. Although the estimation of D_s can involve complex equations, in this work, we initiated the process from the k_td estimation. This approach enabled us to calculate D_s by solving Equation (22), thereby obtaining Equation (23).

The resulting value for D_s is 2.67 × 10⁻⁹ m² h⁻¹. It is worth mentioning that this value of D_s is of the same order of magnitude as the self-diffusion coefficient experimentally measured at 60 °C for PDMSs [9] with molecular weights similar to those studied in this work. This observation is also useful for justifying the theoretical work developed in this study.

4. Discussion

Numerous studies on the kinetics of free radical polymerization reactions have converged on a consensus, indicating that the overall kinetic coefficients for the various stages of these reactions can be determined by considering two pivotal factors: the kinetic coefficient associated with the chemical step and the kinetic coefficient corresponding to the diffusive step. These coefficients are denoted in Equations (17) and (18) for the termination and propagation stages, respectively [16,24,25,26]. The chemical step occurs when participating species are in close proximity and capable of reacting, while the diffusive step concerns the journey and challenges that participating species face in order to meet and initiate their respective reactions. These steps are exemplified in Figure 8a,b, where the diffusive and chemical steps for termination and propagation are represented, respectively.

Figure 8 illustrates, by a dashed line, the path that a short polymeric radical—considered to be that which diffuse in the reaction medium—would have to take in order to react. This can involve either a termination reaction with a long radical (a) or a propagation reaction with some monomer molecule (b).

Figure 8a suggests that most reactions occur between a short radical and a long radical. Due to the difference in their sizes, the short radical is the one in motion (diffusion), while the long radical is considered immobile. Therefore, the termination reaction can only occur when the short radical successfully reacts with the reactive terminal unit of the long radical. The consideration that termination occurs between a short radical and a long radical is based on the short–long termination approach [27]. This implies that the majority of termination events take place between short and long radicals. The latter is considered immobile due to its significantly smaller diffusion coefficient compared to the former. In this context, “immobile” indicates that there is no center of mass movement, only segmental movements.

In this context, various issues can arise, potentially hindering the short radical from following the appropriate path to execute the termination reaction with the long radical. These issues can include an increase in viscosity or impediments caused by segments of long radicals. In this work, we globally identified these challenges as diffusive limitations, and their repercussions should lead to a decrease in the most affected coefficients, such as k_t.

While Figure 8b indicates that a polymeric radical can react with a macromonomer, it is important to note that it can undergo a propagation reaction either with a nearby long radical or with a short radical during its diffusion.

It is indeed true that the diffusive coefficients of the various stages can significantly impact the evolution of polymerization reactions. Among these stages, the diffusive coefficient of the termination stage has garnered attention in theoretical and experimental studies related to the mathematical modeling of polymerization reactions. This is primarily due to its profound influence on the overall behavior and evolution of k_t [16]. It has been observed that a decrease in this coefficient tends to produce the autoacceleration effect.

In contrast to the experimental results examined in this study, other investigations primarily focusing on bulk polymerization reported the presence of autoacceleration phenomena [28,29,30]. In the results presented in this work, no autoacceleration is observed. Although diffusion controls the reaction over the entire investigated range, as indicated by the lower value of k_td compared to k_t0 (Figure 5a), P* does not display the same upward trend observed in several bulk free radical polymerization reactions of vinyl monomers.

In a previous study, Victoria-Valenzuela et al. [25] developed a model that included a methodology for describing the diffusive step of the kinetic rate coefficients. This model was based on geometric considerations and the application of the Einstein diffusion equation to simulate the bulk polymerization of MMA under various reaction conditions. Additionally, the authors proposed a theoretical explanation for the onset of the autoacceleration effect. Using the results of their work, they derived Equation (24) to calculate k_td:

$$k_{td}=\left(\frac{1}{t_{td}}\right)\left(N_A{v}_r{\gamma }_{kt}p/26\right)$$

(24)

where t_td represents the characteristic time for a short radical to displace a distance equivalent to the separation distance between a short radical and the reactive terminal unit of a long radical. v_r is the volume associated with the reaction, γ_kt is the probability that the trajectory of a short radical is correct for colliding with the terminal unit of a long radical, and p is the probability of the molecules being correctly oriented to form a C–C bond.

Among the conclusions drawn from their study, it was noteworthy that the diffusive stage of the termination reactions in the MMA polymerizations they investigated was determined by the difficulty short radicals had in successfully colliding with the reactive terminal unit of the long radicals during the reaction. In other words, as the reaction progressed, γ_kt decreased in such a way that a decrease in k_t occurred, closely related to the autoacceleration phenomenon. They also reported that the onset of the autoacceleration phenomenon was related to a nearby region where a transition from chemical control to diffusive control occurred. This was explained by comparing k_td and k_t0. Initially, in their results, the value of k_t0 was lower than k_td. However, as the reaction progressed, k_td decreased until it acquired values lower than k_t0. The area where this change occurred coincided with the onset of the autoacceleration phenomenon in the experimental data they simulated.

However, as depicted in Figure 5a in this work, even though the k_td value is lower than k_t0, there is no observable decrease in k_td values over time, nor a transition from chemical to diffusive control, nor is there an abrupt increase in the concentration of radicals (as shown in Figure 6). Therefore, the autoacceleration phenomenon was not manifested, aligning with the hypothesis developed by Victoria-Valenzuela et al. (2016) [25]. Summarizing the points mentioned, it becomes apparent that, although both the termination and propagation stages are controlled by diffusion, the values of their respective coefficients remain nearly constant. This conceptual implication suggests that the species involved experience diffusive limitations from the beginning of the reaction, possibly due to the length of the monomers used. However, these limitations do not increase significantly enough to cause a decrease in k_t during the course of the studied polymerizations.

Regarding k_p, while it is commonly assumed that k_p is primarily influenced by diffusive limitations only at high conversions in polymerizations of small molecules [31] like MMA, Ren et al. 2021 delved into the topological effect on the kinetics and mechanisms of macromonomer polymerization. They concluded that, as the reaction progresses, a brush segment forms adjacent to the polymerization center, acting as a hindrance to propagation like a “wall”. This results in the molecular-weight dependence of the propagation rate, highlighting the topological effect on macromonomer polymerization and revealing the mechanistic differences between the polymerization of small polymerizable molecules and macromonomers. These findings align with the results obtained in our work, where it is evident that diffusive limitations affect k_p.

With the estimation of k_td and the absence of autoacceleration noted, it became appropriate to apply the Smoluchowski equation to estimate D_s. Otherwise, this would have necessitated the experimental determination of a significant number of parameters for use in an equation, such as that of Vrentas and Duda [32]. This calculus was justified considering that Victoria-Valenzuela et al. (2016) observed that, when using the Smoluchowski equation to calculate k_td, the theoretical outcomes aligned with the results from their model when assuming γ_kt equaled one. This assumption implied that, in concept, all termination reactions between short and long radicals were successful. Consequently, they presented theoretical results where the autoacceleration phenomenon did not occur. Hence, the rationale for employing Equation (22) in this study to determine the value of D_s is justified.

Finally, another aspect explored involved employing Equations 19 and 20 in an independent analysis. It was observed that, when M_w reached a value of 1 × 10⁷ gmol⁻¹, k_p became identical to k_p0. Meanwhile, no significant variation was noted with the value of k_t. This suggests that, for the autoacceleration phenomenon to occur, the initial values of M_w should be around the order of magnitude of 10⁷ g mol⁻¹ before further increases. This progression leads to a shift from chemical to diffusive control during the propagation stage, likely triggering autoacceleration. However, it is crucial to continue both experimental and theoretical work on these matters to validate this hypothesis.

5. Conclusions

This study utilized a comprehensive approach employing a semiheuristic procedure to analyze the free radical polymerizations of PDMS-MA macromonomers in benzene. Through the modeling approach, we investigated the potential behavior of the kinetic coefficients involved. This analysis is expected to enhance the understanding of these reactions and, over time, serve as a foundation for optimizing this type of synthesis by delving into the kinetic aspects involved. Furthermore, the methodology employed was designed to simplify the exploration, avoiding complex equations and procedures, thereby making it more accessible and straightforward.

The adjustments made to the model revealed that the diffusive step governed the reaction. This conclusion arises from the observation that the theoretical values of k_td and k_pd consistently remain lower than the values of k_t0 and k_p0.

As deduced findings in this work, it is noteworthy that, in contrast to the behavior observed in several bulk polymerization reactions, the results presented here indicate that, despite termination being controlled by diffusion, there is no abrupt increase in P* or the appearance of the autoacceleration phenomenon. This was attributed to the theoretical understanding that a transition from the chemical to the diffusive stage did not occur.

The model also facilitated the estimation of both the distance between radicals during polymerization and the diffusion coefficient of short polymeric radicals.

The authors aim for this work to contribute to the understanding of these reactions and serve as a basis for the development of more detailed studies, allowing for the generation of new hypotheses justified by theoretical findings.