Polym. Bull. (2016) 73:1795–1810 DOI 10.1007/s00289-015-1577-z ORIGINAL PAPER
Optimization of reaction conditions in functionalized polystyrene synthesis via ATRP by simulations and factorial design Ronie´rik P. Vieira1,2 • Liliane M. F. Lona2
Received: 2 August 2015 / Revised: 25 October 2015 / Accepted: 20 November 2015 / Published online: 24 November 2015 Ó Springer-Verlag Berlin Heidelberg 2015
Abstract Atom-transfer radical polymerization (ATRP) is a powerful reversibledeactivation radical polymerization technique that provides polymers with several macromolecular architectures. Global optimization of this type of system is often difficult due to the mathematical models complexity. Based on that, the purpose of this paper is to provide a simplified optimization method to determine the best reaction conditions in bulk styrene ATRP initiated by 2,2,2-tribromoethanol. A kinetic model, with experimental validation, was used to generate the response variables in a full 23 factorial design. Thus, an easy statistic modeling was employed to optimize reaction conditions. It was also performed a traditional global dynamic optimization to prove that our approach could be accomplished without considerable errors. Finally, we demonstrated that the optimum conditions can be obtained in an easy and uncomplicated way, allowing extend it to any polymeric system. Keywords 2,2,2-Tribromoethanol ATRP Factorial design Optimization Simulation
Introduction The potential of self-assembled block copolymers for nanotechnologies is now well recognized [1], and the synthesis of new materials aiming this application has been strongly studied. Atom-transfer radical polymerization (ATRP) is a powerful technique that provides the synthesis of several different macromolecular & Ronie´rik P. Vieira
[email protected] 1
Federal Institute of Education, Science and Technology of South of Minas Gerais – IFSULDEMINAS, Pouso Alegre, Minas Gerais 37550-000, Brazil
2
School of Chemical Engineering, University of Campinas – UNICAMP, Campinas, Sa˜o Paulo 13083-852, Brazil
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architectures: polymers with narrow molecular weight distribution [2], block copolymers [3, 4], random or gradient [5] and functionalized polymers [6–10]. The functionalized polymers synthesis is one of the most important characteristics of the ATRP process, since several other materials could be synthesized from these functionalized polymers usually called ‘‘macroinitiators’’. 2,2,2-Tribromoethanol is a new ATRP initiator with great potential for the synthesis of materials with functional groups, since the produced polymers have hydroxyl and bromine end groups, which can be used as macroinitiators for copolymerization with other monomers, including biocompatible ones, such as lactic acid [11]. In a previous research, our group successfully carried out ATRP experiments of styrene initiated by 2,2,2-tribromoethanol under different conditions, and we also developed a kinetic model to represent this process at several operating temperatures by the determination of kinetic parameters [12], but the process optimization was not performed. Optimizing a process is usually done using as basis the equations generated from deterministic and stochastic modeling [4, 13, 14]. However, the system of equations generated is generally complex, especially for dynamic systems and problems of distributed parameter, which complicates the optimization. In this context, the use of statistical tools such as Design ExpertÒ software allows not only obtaining the response surface curves, but also obtaining simple equations that correlate all system variables. These algebraic equations could be easily programed in Excel Solver, providing the optimum conditions. In this research, the algebraic equations generated by the Design ExpertÒ software was programmed in Excel and a simple optimization problem was formulated aiming to replace the traditional dynamic optimizations, which are extremely difficult to apply in polymerization engineering. The approach of this work is totally unprecedented in ATRP polymerization and the simplicity of the method could be extended to any polymeric system. To validate the purpose of this article, a dynamic optimization for the considered case study was also developed. The main objective of this article is to provide to the readers a new way to optimize polymer systems through computer simulations. Unlike what has been traditionally published (random simulations to find the optimum conditions), it will be shown that the optimal conditions in polymer systems can be obtained simply and quickly by simulations and factorial design.
Methodology Kinetic model To perform all simulations, the generic kinetic model developed in a previous research of our group was used [15]. To obtain such modeling, we carried out a material balance in order to account for the variation of the following species: ‘‘living’’ polymers, ‘‘dead’’ polymers, ‘‘dormant’’ polymers and monomer considering the polymerization mechanism expressed in Scheme 1. Method of moments was also used in order to obtain important polymer properties, such as number-
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Scheme 1 General mechanism of atom-transfer radical polymerization considered in this research, adapted from Shipp and Matyjaszewski [25]
average molecular weights (Mn) and polydispersity index (PDI) (or dispersity). Detailed development of ATRP kinetic models is available in literature with several adaptations for specific cases [15–22]. For the solution of ODE’s system, we used a computer program in Fortran code with the aid of LSODE subroutine developed by Hindmarsh [23]. This subroutine uses the Adams–Moulton method to solve initial value problems, because this method is very effective to solve problems with high numerical stiffness, which is quite common in polymer engineering [15, 24]. Bulk styrene atom-transfer radical polymerization initiated by 2,2,2-tribromoethanol (which acts as a nonfunctional initiator), copper (I) bromide as catalyst and N,N,N0 ,N00 ,N00 -pentamethyldiethyllenetriamine (PMDETA) as ligand was considered as system due to the high potential of using it for the production of macroinitiators and future nanotechnologies application. Moreover, there are no researches in literature dealing with the optimization of this process. All kinetic parameters used in the simulations follow the Arrhenius’ expressions, shown in Table 1. In Table 1 are the common process parameters, with emphasis on the activation rate constant (ka) and deactivation rate constant (kda). Note that kda has a relatively high value compared to the common ATRP parameters, suggesting that the initiator used provides a good control of the polymerization. In addition, equilibrium is assumed to be maintained for both the small molecule (initiator) and the polymer at all times [26]. Furthermore, Table 1 expresses that diffusional limitations on the activation/deactivation process can be neglected, since a relatively active ATRP catalyst is used [27].
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1798 Table 1 Set of expressions used to obtain the kinetic parameters in Styrene ATRP initiated by 2,2,2tribromoethanol as a function of temperature
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Parameter
Expression
References
kp
4.226 9 107 exp(-3910/T)
[28]
ki
1.63 9 106 exp (-12,020/T)
[28]
ktherm
2.19 9 105 exp(-13,800/T)
[28]
kdim
188.97 exp(-1947/T)
[29]
ka
4.014 9 104 exp(-3636/T)
[12]
kda
8.66 9 1016 exp (-8702.5/T)
[12]
kt0
3.820 9 109 exp(-958/T)
[28]
kt
kt0 9 exp(-2 9 (A1X ? A2X2 ? A3X3))
[30]
ktc
0.99 kt
[28]
ktd
0.01 kt
[28]
kt,p
109
[31]
ktr,M
2.310 9 106 exp(-6377/T)
[28]
ktr,D
150
[28]
A1
2.57 - (5.05 9 10-3 T)
[30]
A2
9.56 - (1.76 9 10-2 T)
[30]
-3
A3
-3.03 ? (7.85 9 10
T)
[30]
Table 2 Levels of factors in the simulation design in styrene ATRP initiated by 2,2,2-tribromoethanol at 6 h of polymerization Factor
Lower level (-1)
Higher level (?1)
A: Temperature (°C)
90
110
B: [RX]0 (mol L-1)
0.05
0.1
C: [MtnX]0 (mol L-1)
0.05
0.1
Factorial design Design ExpertÒ 7.0 software (http://www.statease.com/software.html) was used for data analysis. This is a piece of software designed to help with the design and interpretation of multi-factor experiments. The software offers a wide range of designs, including factorials, fractional factorials and composite designs. A full two-level factorial design was proposed to analyze the effects of three different operational variables: Temperature (T), initial concentration of initiator [RX]0 and initial concentration of catalyst [MtnX]0. The effect of these operating variables and their possible interactions on the monomer conversion, numberaverage molecular weight (Mn) and polydispersity index (PDI) were carried out using a 23 factorial design. The levels of each factor indicated in Table 2 were selected based on preliminary studies of this polymerization system [11]. The factors analyzed in this article were: temperature and initial concentrations of catalyst and initiator in the process. The factorial design was considered on two levels, a lower (lower value) and a higher (higher value) for each factor analyzed. These levels were based on some previous works available in the literature with such concentrations
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of reactants, and the temperature levels took into account the excellent control of the ATRP process that is normally obtained in the measured range. The levels expressed in Table 2 were utilized as initial conditions (input data) on the computer program previously developed in FORTRAN code to simulate the ATRP system. It is important to highlight that it was ignored any solubility limits for the ATRP catalyst [28]. The output data were monomer conversion, Mn and PDI. All simulated output data were considered as ‘‘real’’ experimental data due to the careful kinetic modeling validation reported in a previous work of our group [12]. The analysis of variance (ANOVA) provided a study of variation on the results and the test of statistical significance, p value, was determined according to the total error criteria considering a confidence level of 95 %. The influence of a factor was considered significant if the value of critical level (p) was lower than 0.05 discarding the meaningless parameters for p values over 0.05. As a result of the response surface methodology (RSM), an empirical model (system of algebraic equations) encompassing all the operating variables and their binary interactions was calculated by Design Expert software. These algebraic equations are extremely simple to scheduled and resolved through the Excel Solver.
Results and discussion Effect of operating variables (T, [RX]0 and [MtnX]0) and their possible interactions on the monomer conversion, Mn and PDI were studied by simulations and full factorial design. Table 3 shows the full matrix of the design. Columns 2–4 give the variable levels coded in the dimensionless coordinate while columns 5–7 give the simulated values obtained for the selected responses. Analysis of variance (ANOVA) Analysis of variance was performed for estimation of significance and validation of the models suggested by Design Expert software for all response data. Table 4 shows the ANOVA for the monomer conversion, Mn and PDI considering only the significant variables. Table 3 Full 23 factorial design matrix for simulations results using the kinetic model developed by Vieira et al. [15] in styrene ATRP initiated by 2,2,2-tribromoethanol at 6 h of polymerization Run
A temperature
B [RX]0
C [MtnX]0
Conversion (%)
Mn (g mol-1)
PDI 1.27
1
?
?
?
95
10,037
2
–
?
?
72
8624
1.20
3
?
?
–
91
9724
1.13
4
?
–
?
90
19,601
1.10
5
–
–
–
56
13,093
1.15
6
?
–
–
85
17,161
1.20
7
–
?
–
64
7400
1.12
8
–
–
?
64
15,664
1.08
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Table 4 ANOVA for the monomer conversion, number-average molecular weight (Mn) and dispersity (PDI) calculation by the empirical models fitted in styrene ATRP initiated by 2,2,2-tribromoethanol at 6 h of polymerization Source
Sum of squares
df
F value
p value
R2
Pred R2
Adj R2
0.992
0.970
0.987
0.971
0.883
0.949
0.959
0.709
0.905
Monomer conversion Model
1553.40
3
174
0.0001
A: Temperature
1373.67
1
462
0.0001
B: [RX]0
95.43
1
32
0.0048
C: [MtnX]0
84.31
1
28
0.0060
Residual
11.90
4
Number-average molecular weight (Mn) 1.331 9 108
3
44
0.0016
A: Temperature
7
1.723 9 10
1
17
0.0142
B: [RX]0
1.105 9 108
1
111
0.0005
C: [MtnX]0
5.360 9 106
1
5.36
0.0815
Residual
3.997 9 106
4
Model
0.027
4
17.6
0.0202
A: Temperature
2.812 9 10-3
1
7.4
0.0723
B: [RX]0
4.512 9 10-3
1
11.9
0.0409
C: [Mt X]0
3.125 9 10-4
1
0.82
0.4309
BC interaction
0.019
1
50
0.0058
Residual
1.137 9 10-3
3
Model
Dispersity (PDI)
n
Table 4 shows the higher F values calculated imply significant reproducibility for the empirical models in monomer conversion (F value = 174), Mn (F value = 44) and PDI (F value = 17.6). In addition, statistical calculations of the Design Expert software show there were only 0.01, 0.16 and 2.02 % of chance that the F values reported for monomer conversion, Mn and PDI, respectively, could occur due to noise. These results suggested a very good reproducibility of the computational simulations by the empirical modeling. p values lower than 0.05 indicate the model terms are significant for all responses analyzed. It can be seen from Table 4 that temperature (A), initial catalyst (B) and initiator (C) concentrations were significant just for monomer conversion (all p values were lower than 0.05). On the other hand, Mn was too much influenced by the initiator concentration (F value = 111 and p value = 0.0005), confirming what has been described in several ATRP researches [11, 32–34]. Temperature and catalyst concentrations have a low significance on Mn, but it was considered in the final modeling to ensure minor errors. Finally, the PDI was very influenced by the interaction of the initial concentrations of catalyst and initiator (BC interaction), with F value = 50 and p value = 0.0058. Thus, a careful analysis of these factors must be done to obtain the lowest PDI value. p values greater than 0.1 indicate the model terms are not significant, so all other terms are not expressed in Table 4.
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Moreover, the ‘‘Pred R2’’ values are in reasonable agreement with the ‘‘Adj R2’’ values for all models, principally for the monomer conversion (0.970 and 0.987, respectively). Mn and PDI presented some deviations of these values, but it was also small. Thus, the empirical models (Eqs. 2–4) obtained from the factorial design (general Eq. 1) can be used to navigate on the surface. Y^ ¼ b0 þ b1 x1 þ b2 x2 þ b3 x3 þ b12 x1 x2 þ b13 x1 x3 þ b23 x2 x3 þ b123 x1 x2 x3 þ e ð1Þ Conversion ¼ 74:03 þ 1:31 T þ 138:15 ½RX0 þ 129:85 ½Mtn X]0
ð2Þ
Mn ¼ 6680:25 þ 146:78 T 1:4867 105 ½RX0 þ 32740 ½Mtn X0
ð3Þ
PDI ¼ 1:32 þ 1:88 103 T 4:9 ½RX0 5:6 ½Mtn X0 þ 78 ½Mtn X0 ½RX0
ð4Þ
Equations 2–4 are the empirical models generated by the Design Expert software and expresses the monomer conversion (Eq. 2), Mn (Eq. 3) and PDI (Eq. 4) as function of the variables: temperature (T) in Celsius degree; initial concentration of initiator [RX]0 in mol L-1, and initial concentration of catalyst [MtnX]0 in mol L-1. Figure 1 compares the predictions by statistical models (Eqs. 2–4) and the simulated results by the kinetic model (observed values).
Fig. 1 Predicted values by empirical models versus observed values by kinetic modeling for a monomer conversion, b Mn and c PDI in styrene bulk ATRP using 2,2,2-tribromoethanol at 6 h of polymerization
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Figure 1 provides a validation of the method by comparison between the empirical model values and the data generated by the kinetic modeling. In this figure, one can see an almost linear evolution of the predicted values versus observed values for monomer conversion, Mn and PDI. At the same time, the curve is almost coincident with the origin of the graph for all response variables. From Fig. 1a, one can see a determination coefficient equals 0.9924, which is an excellent value for predictive models, confirming its validation to represent monomer conversion without perceptible errors. Some deviations from linearity were observed in the curves expressed in Fig. 1b, c, but determination coefficients were 0.9708 for Mn and 0.9591 for PDI. These values were considered reasonable, since they were higher than 0.9. Thus, the empirical models can be also used to give directions to obtain good predictions of Mn and PDI. Analysis of response surface Response surface methodology is a very useful statistical technique to obtain the optimal conditions of every process. In this present work, we used this methodology in order to achieve the optimum polymer properties, by varying reaction conditions, in a particular case study: Its desirable to synthesize hydroxyl terminated polystyrene for applications as ‘‘macroinitiator’’ in future copolymerization, so the polymer chains must to be with low dispersity. For our purpose, Mn must be close to 10,000 g mol-1, and PDI value must be less than 1.15. In addition, thinking in large-scale production, the reactor residence time is a parameter extremely important, so the polymerization rate must be as faster as possible. Furthermore, it is desirable to maximize the polymerization rate without loss of the polymer architecture control. Figure 2 shows the effect of each one of the reaction variables (temperature, initial concentration of catalyst and initiator) on the monomer conversion. From the analysis of Fig. 2a, b, one can see temperature has the main effect on the monomer conversion. Clearly, the polymerization rate follows the Arrhenius Law and the monomer conversion increases for the same reaction time at higher temperatures, confirming what has been expected [29, 35, 36]. In addition, Fig. 2c illustrates the combined effect of the initial concentration of catalyst and initiator on the monomer conversion considering constant temperature, and one can see high monomer conversions for high initial concentration of reactants. It is important to highlight the influence of initial concentration of reactants is less pronounced than temperature effect, suggesting that the optimization of reaction time should be made by selecting an ideal temperature. However, this factor probably affects the polymer architecture, so a careful analysis of its effect on Mn and PDI is shown in Figs. 3 and 4 in order to determine the best reaction temperature for the case study. Analyzing Fig. 3, one can see temperature and catalyst concentration had a low effect on Mn evolution. So the main effect was related to initial concentration of initiator. The low variation of Mn as temperature goes on, in Fig. 3a, b, occurred due to the increase on the polymerization rate increase, i.e., the consumption consume of monomer accelerated and, as consequence, there was an increase of Mn. But considering the same monomer conversion (for all temperatures on this range) the same Mn value was observed.
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Fig. 2 Response surface plot showing the effect of reaction conditions on the monomer conversion: a temperature and initiator concentration effect (catalyst concentration constant), b temperature and catalyst concentration effect (initiator concentration constant), and c initiator and catalyst effect (temperature constant) in styrene ATRP initiated by 2,2,2-tribromoethanol at 6 h of polymerization
The main effect on the evolution of Mn was the initial concentration of initiator, as illustrated in Fig. 3a, c. These responses surfaces show a strong dependency of Mn in [RX]0; for example, the variation of Mn achieves almost 10,000 g mol-1 when concentration of initiator varies from 0.05 to 0.1 mol L-1. This behavior can be explained due to the high concentration of primary radicals formed in the process beginning that make more polymer chains been generated. Therefore, the increase of the initial concentration of initiator produces high polymer concentration with low Mn and vice versa. Finally, the analysis of the same reaction conditions on PDI is shown in Fig. 4. Figure 4a shows that both temperature and initial concentrations of initiator produced a considerable effect on the PDI. Considering the same temperature in Fig. 4a, one can see the increase of initiator concentration makes a small increase on the PDI. The same result was obtained if concentration of initiator was fixed and varying operating temperature. As discussed previously, both parameters influenced directly on the polymerization rate because the increase of them produce more ‘‘living’’ polymer chains available for termination and chain transfer. The increase in dispersity is due to the fact that with more chains there is less copper per chain to mediate polymerization. The same effect occurs with less Cu(II) added to the
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Fig. 3 Response surface plot showing the effect of reaction conditions on the number-average molecular weight (Mn): a temperature and initiator concentration effect, b temperature and catalyst concentration effect and c initiator and catalyst effect in styrene ATRP initiated by 2,2,2-tribromoethanol at 6 h of polymerization
system. In the case of adding lower oxidation state metal, there is more termination but also more deactivator, leading to the complex plots shown with variations in initiator and catalyst concentrations. However, analyzing Fig. 4b, one can see the variation of catalyst concentration in that range almost does not affect the PDI, suggesting this range of concentration could be used without loss of the polymerization control. Figure 4c shows the combination of initiator and catalyst effects have a nonlinear dependence. From this figure, one can see low concentration of initiator and high concentration of catalyst provide the smallest value of PDI. Global optimization To find the optimum for the case study stated in this paper, we compared the traditional technique of dynamic optimization of the phenomenological models (method 1) with the simple nonlinear optimization of the statistical models (method 2) obtained in this work. As mentioned earlier, is desirable to synthesize polystyrene with Mn close to 10,000 g mol-1, PDI less than 1.15 and fast polymerization rate.
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Fig. 4 Response surface plot showing the effect of reaction conditions on the polydispersity index (PDI): a temperature and initiator concentration effect, b temperature and catalyst concentration effect and c initiator and catalyst effect in styrene ATRP initiated by 2,2,2-tribromoethanol at 6 h of polymerization
Method 1 As described above, many practical situations are modeled abstractly by ordinary differential equations, and ATRP systems have a complex kinetic model that describes monomer conversion, Mn and PDI [15]. The techniques developed for solving these problems are mainly based on solving necessary conditions for local minima. Except for problems possessing certain convexity properties, global solutions are not verifiable. A popular method for deterministic global optimization is the combination of branch-and-bound and relaxation techniques to eliminate systematically proportions of the decision space that theoretically cannot contain the minimum [37]. The problem consists into maximize the monomer conversion at 6 h of polymerization by varying its reaction conditions (temperature, [RX]0, and [MtnX]0) in the phenomenological mathematical model [15]. To do that, an initial value problem must be solved to provide values of reaction conditions that make monomer conversion, Mn and PDI be restricted as stated:
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Maximize X ¼ 1
½M ½M0
subject to : d½M ¼ f ½M; ½RX0 ; ½Mtn X0 ; li ; ki ; di ; kj dt dli ¼ f ½M; ½RX0 ; ½Mtn X0 ; li ; ki ; di ; kj dt dki ¼ f ½M; ½RX0 ; ½Mtn X0 ; li ; ki ; di ; kj dt ddi ¼ f ½M; ½RX0 ; ½Mtn X0 ; li ; ki ; di ; kj dt ½Mð0Þ ¼ ½M0 li ð0Þ ¼ 0 ki ð0Þ ¼ 0 di ð0Þ ¼ 0 i ¼ 0; 1; 2 j ¼ 1; . . .; n 0:05 ½RX0 0:1 0:05 ½Mtn X0 0:1 90 T 110 l þ k1 þ d1 Mn ¼ MWM 1 10;000 l0 þ k0 þ d0 Mw 1:15 PDI ¼ Mn l þ k2 þ d2 Mw ¼ MWM 2 l1 þ k1 þ d1 where di ,ki and li are the moment models obtained by the method of moments and ki are the kinetic rate constants as functions of temperatures. To solve this problem, the software developed by Singer and Barton, named GDOC, was used [37]. The program itself is divisible into four modules: branchand-bound, numerical optimization, numerical integration with parametric sensitivity, and residual evaluation. In order to compute both the objective function and the gradient of the objective function, the associated ODEs for this problem must be numerically solved. Because the problem addressed in the case study is stiff, LSODE was chosen for numerical integration [23]. The software solved this problem within an absolute branch-and bound tolerance of 10-4. GDOC terminated in few CPU seconds with an objective function of 0.9182 (maximum monomer conversion of 91.82 %) at the points ([RX]0 = 0.0881 mol L-1, [MtnX]0 = 0.0532 mol L-1, T = 110 °C). The response values were: -1 Mn = 9950 g mol and PDI = 1.08.
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Method 2 The global optimization in this situation is related only with Eqs. 2–4, from the statistical modeling obtained by the factorial design of this work. This case is very easy to solve, since the equations are algebraic and not stiff differential equations like used in method 1. Again, it is necessary to maximize the monomer conversion in 6 h of polymerization, by varying reaction conditions to obtain the desired polymer properties: Maximize X ¼ 74:03 þ 1:31 T þ 138:15 ½RX0 þ 129:85 ½Mtn X subject to : 6680:25 þ 146:78 T 1:4867 105 ½RX0 þ 32740 ½Mtn X0 10;000 1:32 þ 1:88 103 T 4:9 ½RX0 5:6 ½Mtn X0 þ 78 ½Mtn X0 ½RX0 1:15 0:05 ½RX0 0:1 0:05 ½Mtn X0 0:1 90 T 110 This is an easy nonlinear optimization problem that was solved by Excel Solver, selecting the GRG method of solution to maximize the objective function. The problem maximum is 90.57 % of monomer conversion with reaction conditions: [RX]0 = 0.1 mol L-1, [MtnX]0 = 0.0515 mol L-1, T = 110 °C. The response variables were Mn = 9641 g mol-1 and PDI = 1.15. Table 5 shows a comparison of the two optimization methods used in this research. In Table 5, it appears that the errors associated with temperature and catalyst concentration values determined by the two optimization methods are negligible. Next to this, the error related to the initiator concentration was approximately 13 %, which is a little high compared to others. But when it comes to predictions by models, these errors can be considered small. Thus, it can be said that both methods are useful for determining the optimal reaction conditions for the styrene ATRP polymerization using 2,2,2-tribromoethanol initiator. The main difference between the two methods is associated with the simplicity of the second compared to the first. Method 2 was programmed using Excel Solver in minutes, providing similar results to traditional dynamic optimization, which required a long time programming. Table 5 Summary of optimum reaction conditions obtained by two optimization methods in styrene ATRP initiated by 2,2,2-tribromoethanol at 6 h of polymerization Reaction conditions Temperature
Polymer properties [RX]0
n
[Mt X]0
Conversion
Mn
PDI
Method 1
110
0.0881
0.0532
0.9182
9950
1.08
Method 2
110
0.1
0.0515
0.9057
9641
1.15
Error (%)
0
3.20
1.36
3.11
6.48
13.51
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Conclusions This paper provides a new and useful method for uncomplicated ATRP optimizations. It was used a deterministic model previously developed for the ATRP system to generate response data on a 23 factorial design. Response surfaces plus a system of algebraic equations easy to program in Excel were obtained, allowing the process optimization through a reduced number of simulations. A traditional global dynamic optimization was also applied in the deterministic kinetic modeling to compare it results with the approach of this work. It only needs monomer conversion, numberaverage molecular weights and dispersity data as answers, making this easy method to be implemented in any polymeric system since those data are known. In general, the approach of this paper provides directions for optimal synthesis of polymers. Our finds indicated that: the results of both optimization techniques employed were similar, showing negligible mean errors. The optimization proposal proved quite easy to implement in accessible software such as Excel Solver. One can synthesize polystyrene functionalized with high monomer conversions and predefined properties by a simple adjustment of the experimental data for the targeted optimization technique employed. The optimization approach expressed in this paper cannot only be used in ATRP systems but also in any other polymeric system that has a predefined kinetic model.
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