International Journal of Polymer Science

Volume 2018, Article ID 7803702, 16 pages

https://doi.org/10.1155/2018/7803702

## Deterministic Approaches for Simulation of Nitroxide-Mediated Radical Polymerization

^{1}Planta Piloto de Ingeniería Química (PLAPIQUI), Universidad Nacional del Sur (UNS) – CONICET, 8000 Bahía Blanca, Argentina^{2}Departamento de Ingeniería Química, Universidad Nacional del Sur (UNS), 8000 Bahía Blanca, Argentina

Correspondence should be addressed to Mariano Asteasuain; ra.ude.iuqipalp@niausaetsam

Received 26 April 2018; Accepted 12 June 2018; Published 9 August 2018

Academic Editor: Hossein Roghani-Mamaqani

Copyright © 2018 Mariano Asteasuain. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Since its development in the last decades, controlled radical polymerization (CRP) has become a very promising option for the synthesis of polymers with controlled structure. The design and production of tailor-made materials can be significantly improved by developing models capable of predicting the polymer properties from the operating conditions. Nitroxide-mediated polymerization (NMP) was the first of the three main variants of CRP to be discovered. Although it has lost preference over the years against other CRP alternatives, NMP is still an attractive synthesis method because of its simple experimental implementation and environmental friendliness. This review focuses on deterministic methods employed in mathematical models of NMP. It presents an overview of the different techniques that have been reported for modelling NMP processes in homogeneous and heterogeneous media, covering from the prediction of average properties to the latest techniques for modelling univariate and multivariate distributions of polymer properties. Finally, an outlook of model-based design studies of NMP processes is given.

#### 1. Introduction

Controlled radical polymerization (CRP), also called reversible-deactivation radical polymerization (RDRP) according to the IUPAC recommendation [1], has emerged in the last decades as a very promising option for the synthesis of polymers with controlled structure. In CRP systems, the high reactivity of the polymer radicals is regulated by the addition of an agent which establishes a balance between active and temporarily inactive (dormant) chains. This dynamic equilibrium is strongly shifted towards the dormant chains. As a consequence, the number of active chains is very small which reduces the interactions between them. For this reason, the overall termination effect is decreased and all chains grow on average at the same rate. CRP systems are distinguished by a linear increase of molecular weight with conversion and by a low amount of dead polymer chains (~1–10%).

There are two general approaches to establishing the necessary balance between active and inactive species: the first one is based on a reversible termination (deactivation), while the second one is based on a reversible (degenerative exchange) transfer. In both cases, the polymer radical propagates a few times during activation periods, before is converted back to the inactive state. The initiation rate, the contribution of the termination reactions, and the dynamics of the activation/deactivation exchange influence the polymer dispersity. As the exchange rate increases, which means that fewer monomer units are added to the growing active chains per activation period, the polymer dispersity decreases. Therefore, CRP is effective in controlling molecular weights. Besides, the dynamic equilibrium between active and dormant radicals spans the lifetime of the living polymer chains from fractions of seconds to several hours. This allows manipulating the chain structure during the polymerization reaction. In this way, it is possible to synthesize complex molecular structures, such as comb or star polymers or copolymers with well-defined structure (block, gradient, or branched copolymers) [2, 3].

The three best-known and most effective variants of CRP are atom transfer radical polymerization (ATRP), nitroxide-mediated polymerization (NMP), and reversible addition-fragmentation chain transfer polymerization (RAFT).

##### 1.1. Atom Transfer Radical Polymerization

The ATRP technique is based on the rapid capture of the propagating radicals in a deactivation reaction. Effective ATRP catalysts () are generally organo-metallic species formed by a transition metal capable of changing their oxidation number, a complexing ligand , and a counterion which can form a covalent or ionic bond with the metal centre. The capture of the active radicals is performed by the counterion upon its release after the reduction of the transition metal. Since the activation in ATRP is a bimolecular process, the inactive species formed by the polymer radical and the counterion is inherently stable and can only be activated sporadically, according to the activity of the transition metal that acts as a catalyst. The characteristic kinetic step of ATRP processes is

The reduction of the transition metal complex allows the release of the atom (), which captures the propagating radical forming the dormant species . The success of control on the polymerization is in the rate of periodic activation of the dormant species [4].

##### 1.2. Nitroxide-Mediated Polymerization

In NMP, the reaction is also controlled through the reversible termination of the active chains. Here, the mediating agent is a nitroxide that traps the propagating radical, forming a stable radical. The dissociation step of this reversible termination releases the active radical from the dormant species, and this radical can then propagate or terminate with another radical before being deactivated again. The reaction is strongly displaced towards the inactive species. Therefore, most of the polymer chains in the reaction medium are temporarily dormant, and the concentration of radicals capable of propagating or terminating is several orders of magnitude lower than that of conventional radical polymerization [5]. The distinctive reaction of these polymerizations technique is

Here, species is the nitroxide mediator which, upon trapping the radical, forms the dormant species with monomer units. The slow thermal dissociation of the dormant species provides a low concentration of radicals, which allows the termination step to be kept to a minimum.

##### 1.3. Reversible Addition-Fragmentation Chain Transfer

RAFT polymerization differs from the two above in that control over the growth of the polymer chains is achieved by a degenerative chain transfer process. Basically, what happens is that a single active radical site is shared among many chains so that these molecules participate in fewer reactions. In these systems, as in conventional radical polymerization, a concentration of radicals in the pseudostationary state is established through the initiation and termination processes. Through bimolecular transfer processes, a minimal number of growing radicals undergoes a chain interchange with dormant species through an intermediate adduct. Control over molecular weight and dispersity is provided by chain transfer agents (CTAs) which exchange the active radical centre between the growing chains. Good control requires that the exchange is rapid compared to propagation [6]. The distinctive kinetic step is

The species acts as a chain transfer agent by transferring the radical centre from the chain with monomer units to the chain with units. In an intermediate step, a radical adduct of 2 branches, , is generated. The success of this polymerization is given by a rapid exchange, that is, a large , to keep the concentration of active radicals low [4].

The three main variants of CRP, ATRP, NMP, and RAFT have their own advantages and disadvantages [7]. Although NMP was initially restricted to a limited number of monomers, the development of new nitroxides and alkoxyamines has allowed using this technique with almost all monomers, with exception of vinyl acetate and vinyl chloride. This can be achieved with few nitroxides such as SG1 (and the BlocBuilder MA alkoxyamine) or TIPNO, which are commercially available [5]. Since the development of the different CRP techniques, NMP lost rapidly the preference against RAFT and ATRP because of several advantages of the latter two, such as the capability of polymerizing more monomers and having more convenient reaction temperatures. However, NMP still remains as an attractive synthesis method because of its thermal activation mechanism, a monocomponent control system, a very simple purification of the product, and no environmental issues.

The development of CRP has made it possible to control key polymer properties such as the molecular weight distribution (MWD), the copolymer composition distribution (CCD), or the length distribution of comonomer sequences (SLD). This feature is outstanding for achieving the production of polymers with prespecified properties [8, 9]. Still, the manipulation of the chain structure is a difficult task since the molecular properties have a strong dependence on the operating conditions, which results in complex interactions between process variables. Thus, the design and production of tailor-made materials can be significantly improved by developing models capable of predicting the polymer properties from the operating conditions [10, 11]. The present review focuses on deterministic methods employed in mathematical models of NMP. It presents an overview of the different techniques that have been reported for modelling NMP processes in homogeneous and heterogeneous media, covering from the prediction of average properties to the latest techniques for modelling univariate and multivariate distributions of polymer properties. Finally, an outlook of model-based design studies of NMP processes is given**.**

#### 2. Deterministic Methods for the Simulation of NMP

Mathematical modelling is undoubtedly a very valuable tool in polymer science. It helps to gain understanding in the kinetic mechanisms of polymer processes and also complements and may partially substitute expensive and time-consuming laboratory experiments. Besides, the model-based design is becoming a new paradigm in polymer processes [12, 13]. Polymer applications are every day more sophisticated and specific. New synthesis methods like CRP allow producing polymers with controlled, complex microstructures. These include the MWD, the CCD, chain functionality, and topology. The polymer chain microstructure determines several properties of the material. Therefore, the combination of experiments and mathematical models allows designing processes for the production of polymers with precise, tailored characteristics [14].

Most of the mathematical techniques used in the simulation of polymer processes can be grouped into two major approaches: deterministic and stochastic methods. Stochastic approaches are mainly represented by the Monte Carlo (MC) method. The majority of the applications of this method use the kinetic Monte Carlo (KMC) technique proposed by Gillespie [15, 16]. A KMC simulation stochastically follows the evolution in time of a full set of molecules of a sample of the reaction system, based on the probabilities of the individual reaction steps of the kinetic mechanism. A KMC simulation tracks explicitly and individually each molecule in the ensemble of chains. Therefore, it can provide extremely detailed information about the polymer microstructure, which is generally not available with deterministic solvers. Besides, the implementation of these methods is simple. On the other hand, due to the statistical nature of the simulation, the reproducibility and reliability of the results depend on the sample size, which needs to be large enough to guarantee accurate outputs. The problem is that large samples require long computational times and large storage capacities. However, the improvement of the computational efficiency of MC implementations is nowadays an area of active research, which includes parallelization, code optimization, and hybrid deterministic-MC approaches [17–20]. Recent reviews summarize relevant works using MC methods in polymer science [21, 22]. In NMP, these methods have been used to predict copolymer SLD [10, 11, 23, 24], chain functionality and the full MWD [25], kinetics of branching [26], and bivariate MWD-CCD in continuous processes with arbitrary residence time distributions [27] or to track the exact position of functional comonomers in the copolymer chains [28]. Besides, the model-based design of copolymer synthesis by NMP using MC models has also been performed [11, 28, 29]. The other variants of CRP have also been studied using MC models [20, 30–41].

On the other hand, deterministic methods are based on the solution of the differential-algebraic system of population balance equations (PBEs) that represent the polymer system. PBEs are balance equations that describe the evolution in specific properties, called internal coordinates, of the population of polymer molecules. For instance, it is usual to characterize copolymer molecules in a copolymerization reaction by the content of each comonomer in the chain. Therefore, a PBE for this system may involve a differential balance equation for the time evolution of the concentration of species , a copolymer molecule with units of monomer 1 and units of monomer 2 (the internal coordinates). PBEs are derived from a kinetic mechanism of the polymer process. Deterministic methods provide unique results, as opposed to the random output of stochastic methods, and generally demand shorter computing times and less storage capacity. Besides, they are usually more suitable for identification and optimization purposes where it is necessary to count with smooth, differentiable structures.

Deterministic methods have a long history in polymer reaction engineering [42]. In particular, there are articles that review deterministic approaches used in mathematical models of the different variants of CRP [12, 43–45]. The present review deepens on deterministic methods employed in mathematical models of NMP. It starts with models describing polymerization rate and average properties only. Then, it discusses techniques for modelling distributions of polymer properties, like the MWD and the SLD. Afterward, it examines complex techniques for predicting multivariate distributions. This is followed by a review of NMP models in heterogeneous media and hybrid deterministic-stochastic methods. Finally, an outlook of model-based design studies of NMP processes is given.

##### 2.1. Average Properties and Polymerization Rate

PBEs constitute a set of equations that is, in principle, infinite in number, because the internal coordinates are distributed properties that theoretically may take values from zero/one to infinity. Since a balance equation must be set for every possible value of the internal coordinates in order to achieve a complete description of the polymer system, an unbounded system of equations is obtained. For instance, in the case of the population, , a mathematical model would need a balance equation for every possible value of and in order to get the full distribution of chain lengths. Although it is sometimes possible to assume a maximum value for a distributed property, the resulting number of equations is still usually large.

Taking advantage that some of their averages can sometimes represent the property distributions, there are techniques that allow limiting the number of model equations. The method of moments, originally proposed by Bamford and Tompa [46], is the most popular method for modelling averages of distributed polymer properties. Considering that polymer species are described only by their chain length, the following equations define the number average molecular weight (), the weight average molecular weight (), and the dispersity () of a homopolymer in a NMP in terms of the leading moments of the MWD:

In these equations, is the monomer molecular weight, and , , and are the moments of order () of the MWD expressed in molar concentration, of propagating radicals, dead polymer, and dormant radicals, respectively. Besides, (7) expresses the monomer conversion () in terms of the first-order moments: where is the monomer molar concentration.

The moments used above are defined as where is the concentration of a polymer species () of chain length .

The method of moments, as applied in the modelling of average molecular weights and conversion, consists in transforming the univariate PBEs of the polymer species (chain length as the only internal coordinate) into balance equations for the moments of their chain length distributions. This procedure replaces the infinite equations of the chain length distribution with just nine equations for the moments of order zero, one, and two of the MWDs of propagating radicals, dead polymer, and dormant radicals. This method is simple and computationally inexpensive [47]. Its drawback, however, is that it loses information on the full shape of the MWD. The process of transforming the PBEs into the moment equations is not complex, but it can be tedious and lengthy for some cases. Recently, Mastan and Zhu [47] presented a tutorial on the use of the method of moments in different polymer systems.

The method of moments has been used extensively to study CRP systems. Examples on the application of this method to ATRP and RAFT can be found elsewhere [12, 48–54]. In particular, several works employed the method of moments for the simulation of NMP processes. One of the pioneering works was presented by Veregin et al. [55]. Assuming a kinetic mechanism consisting only of activation/deactivation equilibrium and propagation reactions, as well as constant nitroxide and living radical concentrations, they obtained an analytical solution for the average molecular weights and dispersity. In spite of the simplicity of the kinetic mechanism, they obtained a good match with experimental data. Later, more comprehensive representations were proposed. Butté et al. [56] presented a moment-based model suitable for handling NMP and ATRP processes. The model predicted polymerization rate and average properties and was validated against experimental data. Zhu [50] developed a kinetic model for NMP processes using the method of moments. This model predicted conversion, average molecular weights, and dispersity. He used the model for investigating the effect of operating variables, such as the concentration of initiator, nitroxide, and monomer, on the process performance. Also, he investigated the influence of the rate constants of initiation, propagation, termination, transfer, and equilibrium between propagating and dormant radicals, highlighting the significant effect of the transfer to monomer reaction in the broadening of the MWD. Bonilla et al. [57] presented a moment-based model for the NMP of styrene. They considered a detailed kinetic mechanism that included chemical initiation, reversible nitroxyl ether decomposition, monomer dimerization, thermal initiation, propagation, the equilibrium between propagating and dormant radicals, alkoxyamine decomposition, rate enhancement, transfer to monomer and dimer, and conventional termination. Nondimensional model equations improved the numerical behavior in the parameter estimation. The model predictions showed good agreement with experimental data. Kruse et al. [58] modeled the NMP of styrene predicting average molecular weights and conversion. Using an Evans-Polanyi description of the activation energy (), the model was fit to experimental data of the reaction at 87°C to obtain parameters of the decoupling reaction of dormant radicals and of the propagation/depropagation reactions. They compared the fit with different side reactions, such as transfer to monomer, monomer thermal initiation, transfer to polymer, and the reaction between active radicals and nitroxide to form a hydroxy amine. They found that the monomer thermal initiation was critical for obtaining a good fit. On the other hand, the effect of transfer to polymer and the reaction between active radicals and nitroxide were found to be negligible. Belincanta-Ximenes et al. [59] used the model developed by Bonilla et al. [57] to simulate polymerization rate, average molecular weights, and concentration of polymeric species in the NMP of styrene over a range of operating conditions. They also performed a parameter sensitivity analysis showing the effects of temperature, activation/deactivation equilibrium constant, and initial concentrations of nitroxide and initiator. The simulated profiles of conversion, number average molecular weight, and dispersity agreed well with experimental data. Roa-Luna et al. [60] updated the model by Bonilla et al. [57] including the diffusion-controlled (DC) effects on bimolecular radical termination, propagation, and activation/deactivation reactions. They employed free volume theory to model the DC effects. The authors found that DC termination enhanced the living behavior of the system, whereas DC propagation, DC activation, and DC deactivation worsened it. By comparing model predictions with experimental data, the overall conclusion was that the slight improvement in model performance by the inclusion of the DC effects did not justify the increased model complexity. A later work by this group [61] postulated that side reactions between the initiator and the nitroxide might reduce the effective level of alkoxyamine produced in the early stages of the reaction. Based on the previous model by Bonilla et al. [57], these side reactions were modelled by introducing a controller efficiency factor that determined the fraction of nitroxide able to produce dormant species. This model refinement improved the model predictions, particularly those of the average molecular weights. Asteasuain et al. [62] used the method of moments to predict average molecular properties in the NMP of styrene in tubular reactors. They obtained a good comparison with experimental data.

Copolymerization processes increase the complexity of the mathematical model. In addition to chain length, copolymer composition and the length of the monomer sequences are important properties to be considered. The extension of the method of moments to copolymerization reactions uses double-index moments to compute the average molecular properties. The balance equations for these moments are derived from PBEs that include two internal coordinates, the number of units of each comonomer in the chain. For these bivariate PBEs, the double-index moments of order may be defined as where is the molar concentration of a copolymer species with units of monomer 1 and units of monomer 2.

Copolymer systems require being careful in the definition of the average weights in terms of the molar concentration of the polymeric chains because there is not a straightforward relationship between chain length and mass as in homopolymers [63]. Based on the appropriate definitions, the following equations express the average molecular weights, in an NMP, in terms of the leading double-index moments:

The dispersity is calculated using (6) as previously. The conversion can also be expressed in terms of the double-index moments as

Equations (10), (11), and (12) involve 18 moments, those of order (2,0), (1,1), (0,2), (1,0), (0,1), and (0,0) for each of the three polymeric species. This is twice the number of moments than in homopolymerization models.

The CCD is also an important property of copolymers. Number and weight average values of this distribution can also be expressed in terms of the double-index moments [64]: where and are the number and weight average copolymer composition of monomer 1 in the copolymer, respectively.

Zhang and Ray [65] used this approach in the development of a comprehensive model for CRP in tank reactors. The model was generic and valid for reversible capping CRP systems (i.e., NMP and ATRP). It was validated using experimental data of NMP of styrene and atom transfer radical copolymerization of styrene and n-butylacrylate. Polymerization in batch, semibatch, and a series of continuous tank reactors was analyzed, getting insight into the strengths and weaknesses of the control of the polymer architecture of each operation mode. These authors later extended this work to analyze polymerizations in tubular reactors [66]. Fortunatti et al. [67] used the double-index moment method to predict average molecular weights and copolymer composition in the nitroxide-mediated copolymerization of styrene and *α*-methyl styrene.

The SLD, that is, the distribution of the length of sequences of each kind of monomer along the polymer chains, is also a significant copolymer property. MC methods are very well suited for giving a detailed insight into this distribution. These methods can predict the explicit sequence of monomers along each copolymer chain in an ensemble of chains [10]. Deterministic models, on the other hand, are in general less exhaustive. They can predict the SLD of the ensemble of chains as a whole, but not of the individual chains. Deterministic models of the SLD are usually set up by formulating a parallel kinetic mechanism that considers the sequences of monomers as the reacting species [68]. For instance, Table 1 shows an extract of a conventional NMP kinetic mechanism, and Table 2 shows the corresponding parallel kinetic mechanism for modelling the SLD [67].