Kinetic models of radical polymerization contain low molecular weight species and macromolecules with different chain lengths. Furthermore, the macromolecules occur in complex architectures.

The simplest implementation is to store the macromolecules in a chain length distribution histogram. If the polymerization model contains linear homopolymers only, this alternative is very memory efficient. We only require a field, in which the number of chains is stored using the index as mapping reference to the chain length. One disadvantage is the time-consuming choice of a discrete macromolecule for executing a reaction.

A second alternative is storing the macromolecules in a vector of pointers to individual macromolecular objects with their own properties. This needs much more memory space, but offers a lot of advantages due to the individual characteristics of the discrete macromolecules. In these objects, complex molecular architectures (e.g., star polymers) can be stored, which can occur when modeling the controlled radical polymerization. Another example is storing the individual composition of each polymer chain in a copolymerization model. The implemented macromolecular species are summarized in Table 1. Another advantage of this mode comes from the much faster choice of the macromolecules as educts while executing the reactions.

Beside elemental reactions, a set of templates for reactions with macromolecular reactants were provided. The reaction templates are summarized in Table 2. In the reaction templates, macromolecular species are both stored in the vector and the histogram mode. Occasionally, both modes are used, e.g., when generating dead polymer in a termination reaction.

The controlled radical polymerization involves a reversible deactivation of propagating polymer chains. This can be modeled using the reaction templates Transfer_PL-PL (ATRP) or Transfer_PL-P and Transfer_P-PL (NMP). For modeling the RAFT polymerization, both an addition and a fragmentation reaction is needed. Table 2 contains a suitable fragmentation template (ChainFragmentation) with a polymer species as educt typed SpeciesMacro_P-o-P, which requires an additional selection step for the specific chain to be transferred. Copolymerization models need templates for recording polymer chain compositions (Table 3). Penultimate effects can optionally be simulated by using the provided reaction templates. The appendices contain the concrete application of the reaction templates for simulating controlled radical polymerization (NMP: Appxs 1, 2; ATPP: Appx 3; RAFT: Appx 4, NMP-Copolymerization: Appx 5).

The C++ STL offers highly efficient vector classes for storing macromolecular species. The vector only contains pointers to the potentially very complex objects. In the aforementioned examples, a macromolecular object has to be transferred from the vector of the educt species to the vector of a product species. Thereby, a macromolecular object is chosen using a random number. Since the arrangement of the object pointers is not relevant for our model, we only perform insert and delete operations at the tail of the vectors. This minimizes the administration overhead of the vector class and gives a measurable performance increase.

The simulation of the controlled radical polymerization with mcPolymer is exemplified first of all by a thermal-initiated nitroxide-mediated styrene homopolymerization in bulk. In addition to the reactions of the styrene homopolymerization (Figure 2, left), the reactions of alkoxyamine activation and the equilibrium of reversible activation of dormant chains (Figure 2, right) are required. The rate constants of styrene homopolymerization (Table 4) are well established in the literature. For modeling the thermal initiation of styrene and the gel effect we used the empirical Hamielec model [26] (Equations 13 and 14). In our simulation we applied the nitroxide 2,2,5-trimethyl-4-(isopropyl)-3-azahexane-3-oxyle (BIPNO, Figure 3). The corresponding rate constants k_d and k_c (Table 5) were determined using time dependent measurements of the free nitroxide and alkoxyamine concentration and applying the persistent radical effect theory [28].

Figure 4 illustrates the simple implementation of the kinetic model as a TCL script. At first, we include the mcPolymerInferface and the Hamielec Model. Afterwards, we have to define the low molecular and macromolecular species and set their initial concentrations. The rate constants are declared as either temperature dependent or fixed. The reaction templates formulate the kinetic model using the formerly declared definitions.

InitSimulation defines the number of molecules n_sx being used in the simulation. In this application, the time interval parameter dt not only defines the frequency of the output of the intermediate results, but also the time stamps for recalculating k_t, applying Equations 13 and 14 as function of the monomer conversion (X).

The Monte Carlo simulation only yields correct results if the concentrations of the involved species are represented by an adequate number of molecules (n_sx). A general statement for the required system size cannot be made since it depends on the concrete reaction conditions. In our example, the monomer concentration scales at 10¹ while the radical concentration P_n^· is about 10⁻⁸.

Therefore, we can derive a minimum number of 10⁸ molecules for the actual simulation. Proper results considering the monomer conversion and the molar mass distribution can already be obtained with 5 × 10⁸ molecules. Evoked by the persistent radical effect [31], we are in a domain of non-stationary radical concentration. This effect does not show up until in a simulation with at least 5 × 10⁹ molecules (Figure 5). At the point of initialization (Equation 10) as well as during the simulation, the actual number of molecules is directly proportional to the total number of particles (n_sx). Here (n_sx = 5 × 10⁸), the propagating polymer chains P_n^· were represented by approximately 20–70 discrete objects in the MC simulation.

A validation of the MC simulation can be carried out by a comparison with the simulation results of the commercial program PREDICI ([PhEt-BIPNO]₀ = 0.02 mol/L; Reaction Time = 1 h). PREDICI is a deterministic differential equation solver, so the formerly discussed simulation parameter does not matter. Comparing computation performance between PREDICI and a Monte Carlo simulator are not very useful, since the computation time of the latter varies dependent on the system size, while PREDICI’s remains constant. The execution time of the model discussed here is approx. 15 seconds in PREDICI (PREDICI Version 6.20.2, Intel Core i5-650, OS: MS Windows 7).

The correctness of the applied kinetic model was proven earlier on the basis of experimental data [19]. Considering the perfect correlation between the simulation by mcPolymer and the results obtained by the PREDICI simulation (Figure 6), we conclude a correct implementation of the kinetic model of the nitroxide-controlled polymerization with mcPolymer. Due to the comparatively high concentrations of the polymer species (PN: 1.45 × 10⁻² mol/L, D: 3.8 × 10⁻³ mol/L) presented in Figure 6, no noise appears in the chain length distribution results.

The computation time depends on the number of molecules (n_sx), the mode of storage of the macromolecular species (histogram/vector), and the concentration of the reactants. In addition to the mere computation time, the speed performance of the simulation can be rated by the number of Monte Carlo steps executed per second. We introduced this performance parameter with the first application of mcPolymer [17]. Later on, it was used by Barmer Kowollik et al. [18] for benchmarking parallelized Monte Carlo simulations and by Broadbelt et al. [32] for performance analysis of nitroxide-controlled copolymerization.

The performance analysis has been processed on the following two systems:

The results of the test runs are summarized in Table 6 and Table 7. As one can see, the number of MC steps is directly proportional to the system size n_sx. Due to persistent radical effect, the radical concentration is not constant and depends on the initial concentration of the alkoxyamine. Therefore, at the same reaction time and higher alkoxyamine concentration, a higher monomer conversation is achieved and thus more Monte Carlo steps are required.

Since the simulation program is executed on one CPU core only, it is not surprising that the Windows system runs faster due to the higher CPU clocking. Using the vector storage mode for the macromolecular species, the program performance increases due to the simpler selection of the macromolecules, but leads to a massive increase of the memory requirements (each polymer object has to be stored individually in the RAM). The latter depend on the initial concentration of the alkoxyamine and the simulation parameter n_sx. Taking the example of [BIPNO]₀ = 0.02 mol/L and n_sx = 10¹⁰, the memory requirements are up to 2.1 GB RAM in vector storage mode compared to 0.014 GB RAM in histogram mode.

The number of Monte Carlo steps executed per seconds (MCpsec) remains constant during the simulation in histogram mode, while it decreases in vector mode. We assume the reason might lie in the increasing number of cache misses. An indication is the stronger correlation between MCpsec und n_sx on the Windows system.

For further improvements of our program, we analyzed the statistical time consumption of the particular parts of a MC simulation step. Using the CPU profiler of the Google Performance Tools [33], we could measure the computation time of every single line of code. The following relevant parts are summarized in Figure 7:

For the computation time only, the propagation reaction and the reversible deactivation of the propagating chain by the nitroxide are important, while all other reactions are negligible. Remarkable is the large difference between the reactions Transfer_PL-P (5) and Transfer_P-PL (6), which are performed almost equally as often. The reason is the vast concentration difference between the propagating chain (macromolecular educt in Transfer_PL-P) and the dormant chains (macromolecular educt in Transfer_P-PL). The number of propagating chains is comparatively small and can be kept in cache, causing a very fast individual educt molecule selection.

The presented kinetic model of the ATRP polymerization is well-known in the literature. Matyjaszewski et al. published a modeling approach of chain-end functionality in atom transfer radical polymerization using PREDICI [34,35]. Based on these models, Soares et al. introduced a Monte Carlo simulation of ATRP [14]. It was validated with experimental data and by the simulation method of moments. Its implementation with MATLAB only allowed comparatively small molecule numbers (n_sx up to 2.7 × 10⁷). This model was extended to the simulation of ATRP with bifunctional initiators [15] and to the modeling of the semibatch copolymerization [36]. Najafi et al. utilized the ATRP model for simulating the chain-length dependent termination [37].

Figure 8 illustrates the applied model of ATRP polymerization. The styrene homopolymerization and the corresponding reaction rates (Table 4) were used identically as in the NMP model presented earlier.

The specific reactions of ATRP on the right side of the scheme were implemented by two elemental reactions using template ‘Elemental_2E2P’; One reaction applying template ‘Initiation’ and two transfer reactions of the macromolecular species using the template ‘Transfer_PL-PL’. All templates were introduced in Table 2. The required reaction rate constants (Table 8) were experimentally determined by Fukuda et al. [38] (k_a) and Matyjaszewski et al. [39] (K = k_a/k_d) for the catalyst Cu^IX/2L (X: Br, L: 4,4'-di-n-heptyl-2,2'-bipyridine).

For the validation of our Monte Carlo implementation, we compared the results with those obtained by a PREDICI simulation. Figure 9 shows a good correlation between the results concerning monomer concentration and molar masses, which are already reached at moderate system sizes regarding n_sx.

Comparing to the nitroxide-controlled polymerization, the ATRP simulation requires a lot more computation time in relation to the system size n_sx (Table 9). This effect can be explained by the reaction rates of the reversible activation equilibrium (k_a/k_d), which are about two orders of magnitude higher than those of nitroxide-controlled polymerization. Another aspect is the higher concentration of the controlled species and therefore that of the dormant chains. Both factors cause an increase in the MC steps at the same reaction time. The performance identifiers MCpsec of both models are contiguous, which is expected since in both cases the relevant reaction templates (NMP: Transfer_PL-P, Transfer_P-PL; ATRP: Transfer_PL-PL) transfer one macromolecular species into another reactant. Additionally, the result points out the comparability of the performance analysis using the parameter MCpsec.

The kinetic model of the Cumyl dithiobenzoate (CDB) mediated methyl acrylate (MA) RAFT polymerization that we applied is illustrated in Figure 10. We determined the corresponding reaction rates (Table 10) by fitting the model to the experimental data [17]. As a side reaction, the model contains a cross termination between the propagating chain and the intermediate radical of the RAFT equilibrium. The ongoing discussion “cross termination vs. stable intermediate model” [40,41,42,43,44] is not the subject of this paper. Perrier et al. demonstrated that both approaches were able to describe the experimental data of the RAFT polymerization adequately [41].

The validation of the kinetic model was performed [17] using experimental data measuring the monomer conversion as function of time, the full molecular weight distributions, and the concentrations of the intermediate RAFT radical with experimental electron spin resonance spectroscopic data.

The RAFT polymerization is an example for the advantages of Monte Carlo methods when simulating polymerization models containing complex macromolecular species. In the main equilibrium of the RAFT polymerization, a species with two polymer chains appears as an intermediate radical. During a MC simulation, we can conserve both polymer chains in a molecular object and, therefore, are able to model a fragmentation of this molecule quite simply.

The mcPolymer interface offers the macromolecular species “SpeciesMacro_P-O-P” (Table 1) for modeling the intermediate radical in the main equilibrium. During the cross termination reaction, a three-arm star polymer is created, which is represented by a macromolecular object of type “SpeciesMacro_P-O<PP” (Table 1).

For modeling the pre-equilibrium, we can use the reaction templates “Transfer_PL-P” and “Transfer_P-PL” as we have already used them for the NMP, while the main equilibrium of the RAFT polymerization can be implemented utilizing the templates “ChainAddition” and “ChainFragmentation” (Table 1).

In contrast to mcPolymer, RAFT polymerization simulations with PREDICI have to simulate every polymer chain separately as a unique pseudo species [9]. This approach also leads to correct results [10], but the analyses are more complex.

As a conclusion, simulating RAFT polymerizations with mcPolymer offers a deep view inside the complex macromolecular structures using a straight model representation, while the produced results are simple for handling and analyses.

Barner-Kowollik et al. presented a parallel high performance Monte Carlo simulation approach for complex polymerization using the example of RAFT polymerization [18]. In this paper, the simulation was compared to PREDICI regarding the influence of the following simulation parameters: the system size (n_sx range between 1 × 10⁹ and 1 × 10¹⁰ particles), the number of parallel processes (4 to 16), and the reaction time between synchronization (2 to 500 s). It was not surprising that the only important factor was the number of particles available for each process. As long the number is large enough to remain in the continuum, the simulation provides correct results independent of the synchronization time.

We implemented a parallel simulation of the formerly discussed RAFT polymerization model with mcPolymer. The system size was set up in the range of 1 × 10⁹–2.5 × 10¹⁰ for each process (10 processes total). After a fixed synchronization time, all molecules, including the complex macromolecular objects are collected, mixed up, and statistically redistributed over the processes. Each process requires approx. 2.5 GB of RAM in vector storage mode (n_sx = 1 × 10¹⁰, 10 processes). Therefore, 25 GB have to be shifted at every single synchronization step. The results in Table 11 make obvious that synchronizations after every 5 min of reaction time (12 cycles) determine the speed of the whole simulation. The simulation was executed with overall 10¹¹ molecules in 11723 s. The results agree with those obtained in a single process set up with 10¹¹ particles, but the latter required 86897 s. In our test runs we could not detect an influence of the synchronization time on the simulation results, which leads to the conclusion that we can generally omit the synchronization steps and merge the individual results (chain length distribution) at the end. The results are qualitatively the same as in the single process—as long as the precondition of every simulation process remaining in the continuum is fulfilled. The speed-up factor reaches 14.3 with 10 processes (Table 11).

Using parallelization, a large number of molecules can be obtained, which can be used to smooth all concentration-time curves. Additionally, the number of polymer species used for statistical analysis is increased. Unfortunately, the underlying single MC simulation is not accelerated, since it still needs a certain system size.

An effective acceleration could be achieved, if we were able to parallelize each Monte Carlo step. As an example, the calculation of the reaction probabilities could be processed in parallel. Since the sequential part of a Monte Carlo step is over 50 percent of the time consumption and the mean execution time of a MC step (e.g., 5.5 × 10⁻⁷ s from 1.81 × 10⁶ MC steps/s; n_sx = 5 × 10⁹) is in dimension of the memory synchronization between the processes, the attempt of parallelization at the level of a single MC step and the common x86 CPU architecture is futile.

The addition reaction of a propagating chain (P_n^·) to a dormant species (macro RAFT), predominantly determines the computation time of the RAFT polymerization with mcPolymer (Figure 11). For modeling the intermediate radical in the main equilibrium, the complex macromolecular species “SpeciesMacro_P-O-P” (Table 1) is built up by two polymer chains. While performing ‘ChainAddition’, three steps are necessary that consume varying computation time:

When performing this reaction, a discrepancy concerning the time of memory access shows up. The concentration of propagating chains P_n^· is a few orders of magnitude smaller than that of the dormant species. Therefore, the small vector can mostly be kept in the CPU cache while the dormant chain vector occupies the larger part of the RAM. Thus, the performance advantage of the Core i5 system is significantly smaller than expected and decreases consistently with increasing system size (Table 12). In contrast to the nitroxide-controlled polymerization, the histogram storage mode is no alternative due to the complex data structure.

A nitroxide-controlled homopolymerization of methyl methacryclate (MMA) is not possible due to the degradation of nitroxide by H-abstraction [49]. The SG1-controlled radical copolymerization of MMA and styrene (S) is therefore of particular interest, since Charleux et al. [50] demonstrated the possibility of performing a nitroxide-controlled polymerization of methyl methacrylate with a small amount of styrene. Besides experimental work, the influence of the copolymer styrene was analyzed by model calculations [51]. Charleux et al. implemented the copolymerization model in PREDICI considering the penultimate effect [51]. Furthermore, Broadbelt et al. created Monte Carlo models of SG1-controlled MMA/S copolymerization, examining the copolymer composition and sequence effects [20,21,22].

We implemented the model of SG1 controlled MMA/S copolymerization published by Charleux et al. [51]. It contains the following reactions:

Formulating the entire 52 reactions and 37 reaction rate constants along with their references would go beyond the scope of this paper and can be found in the appendix of the publication by Charleux et al. [51] and in our appendix material as mcPolymer model (Appendix-5-Example-SG1-MMA-S-copolymerization.tcl). The initial concentrations of the reactants are: [MMA]₀ = 8.48 mol/L, [Styrene]₀ = 0.82 mol/L, [SG1-alcoxyamine]₀ = 0.0273 mol/L and [SG1]₀ = 0.00274 mol/L.

The mcPolymer simulation was validated using a PREDICI model. We examined the time-dependent concentrations of the MMA and S comonomers (Figure 12) and, therefore, the copolymer composition.

Figure 12 shows the agreement of our implementation (mcPolymer) with the PREDICI model considering the concentration-time profile of the essential reactants of the polymerization:

The chain length distributions of the products P-MMA-S-SG1, D and D-Htr determined after two hour reaction time also results in a perfect correlation between our simulator and the PREDICI model (Figure 13).

The model of the nitroxide-controlled copolymerization contains a total of 52 reactions. Therefore, the calculations of reaction probabilities (Equations 2, 6–8) and the selection of the reaction to be executed (Equation 1) are decisive for the simulation speed. Overall, approximately 86% of computation time is consumed for those two steps (Figure 14).

In comparison to our BIPNO-controlled styrene homopolymerization model, just under half of the MCpsec rate is achieved with a similar concentration of dormant chains (Table 7 and Table 13). Since no memory-intensive activities are speed determined, the MCpsec rate remains nearly constant during increasing system size (Table 13). The performance factor between the Core i5 and Opteron system is approximately 1.8 and is independent of the system size.

The discrete recording of the composition of each polymer chain offers a thorough analysis of the molecular and chemical structure of the polymers formed by the copolymerization. Contour plots (Figure 15) are an example of coupled breakdown of chain length and copolymer composition of polymers, known from the MALDI-TOF analytic of copolymers [52,53]. The addition of the number of styrene units (x-axis) with the number of MMA units (y-axis) results in the chain length of the corresponding data point. Due to the concentration relation in the reaction solution and the reactivity ratios ([MMA]₀ = 8.48 mol/L, [styrene]₀ = 0.82 mol/L, r_MMA/S = 0.420 [52], r_S/MMA = 0.517 [52]), MMA is predominantly built in in the polymer chain. Since the simulation was performed with a large number of molecules (n_sx = 5 × 10¹⁰), the frequency distributions are well separated. The chain length and composition distribution of a total number of 1.5 × 10⁷ polymer chains is presented in this plot.