A combined computational and experimental study on the

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Kinetics, Catalysis, and Reaction Engineering

A combined computational and experimental study on the polymerization of #-caprolactone Raphael P Rosa, Filipe V. Ferreira, Ana Paola K Saravia, Silvana A Rocco, Mauricio Luis Sforça, Rubia F Gouveia, and Liliane Maria Ferrareso Lona Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b03288 • Publication Date (Web): 14 Sep 2018 Downloaded from http://pubs.acs.org on September 14, 2018

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Industrial & Engineering Chemistry Research

A combined computational and experimental study on the polymerization of ε-caprolactone

Raphael P. Rosa1,†, Filipe V. Ferreira1,2,†,*, Ana Paola K. Saravia3, Silvana A. Rocco4, Mauricio L. Sforça4, Rubia F. Gouveia2, Liliane M.F. Lona1

1. School of Chemical Engineering, University of Campinas (UNICAMP), Campinas-SP, Brazil. 2. Brazilian Nanotechnology National Laboratory (LNNano), Brazilian Center for Research in Energy and Materials (CNPEM), Campinas-SP, Brazil. 3. Department of Materials Engineering, Federal University of Paraná (UFPR) – Curitiba – PR- Brazil. 4. Brazilian Biosciences National Laboratory (LNBio), Brazilian Center for Research in Energy and Materials (CNPEM), Campinas-SP, Brazil. * Corresponding Authors: Filipe Vargas Ferreira | School of Chemical Engineering, University of Campinas, UNICAMP, Campinas-SP 13083-970, Brazil. Email: [email protected] †Author Contributions: R.P. Rosa and F.V. Ferreira contributed equally to this work. ABSTRACT This paper reports the study on the synthesis of poly(ε-caprolactone) – PCL by ringopening polymerization (ROP) of ε-caprolactone (CL) monomer with focus on mathematic developing of the growth mechanisms of polymer chain. Kinetics and mathematical modeling of ROP of CL was carried out to replicate the different experimental conditions. The computational results of conversion and molecular weight of the polymer were found to be comparable with the experimental results of nuclear magnetic resonance (NMR) spectroscopy and showed that the polymerization is highly dependent of the moisture (ROH). Moreover, parametric studies have shown how the 1 ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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concentrations of octanoic acid and catalyst affect the conversion and molecular weight of the polymer. The study here presented provides further understanding of synthesis of PCL, reporting mathematical models of PCL synthesis which can be used for predicting the characteristic of this biocompatible and biodegradable polymer. Keywords: ε-caprolactone polymerization, ring-opening polymerization, synthesis, poly(ε-caprolactone), PCL.

1

Introduction

The 21st century can also be referred to as the “Polymer Age” since polymeric materials have permeated our everyday lives, being used in a wide variety of applications ranging from household items to airplane parts

1–3

. A major unforeseen consequence of the

“Polymer Age” is the accumulation of this material and their environmental impact due

to its recalcitrant nature 4. The pollution caused by the polymers in the environment was first considered an aesthetic problem due to plastic waste found in cities, shores, etc. 5. Then, severe environmental problems impacting wildlife were identified, such as intoxication, choking and entanglement 6,7. Undeniably, polymers may persist in the environment for centuries and the discarded plastics are therefore hazardous to the entire ecosystem. The use of biodegradable polymers is an obvious route to minimize the severe environmental problems related to the disposal of polymer into the environment 8,9. Unfortunately, the price of biodegradable polymers is relatively high and the processing difficulties still hinder their wider use

10

. Several attempts have been made to produce cheaper

biodegradable polymers and with properties that match or exceed the properties of the most widely used non-biodegradable polymers 11–13. Nevertheless, these challenges still wait to be overcome 14. A deep understanding of polymerization reaction engineering is 2 ACS Paragon Plus Environment

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a way to develop sustainable polymer with improved properties 15. Thus, the efforts are now concentrated on the study of the synthesis of biodegradable polymers. Poly(ε-caprolactone) (PCL), a well-known biocompatible and biodegradable polymer, has low melting point (59–64 °C) resistance

17

and good drug permeability

16

, good water, oil, solvent and chlorine

18,19

. These properties allow considering PCL

for use in several applications, especially in the pharmaceutical and biomedical fields 20,21

. PCL is synthesized mainly by ring-opening polymerization (ROP) of ε-

caprolactone (CL), as observed in Figure 1. In the case of tin-based catalysts, the polymerization reaction performs best in inert atmosphere over a broad range of temperatures (from 80 to 130 °C) and is remarkably sensitive to monomer to catalyst ratio 22. Inert atmosphere is important to ensure accurate control of the concentration of moisture (source of hydroxyl groups, ROH), which act as co-initiator during the reaction 23. Several other process variables have been reported as playing a significant role in the characteristic of PCL, such as catalyst concentration, time of reaction and hydroxyl groups used as co-initiator 24–26. Hajiali 25 reported that OH groups can be used to control the molecular weight (Mw) of the polymer during the polymerization of PCL. Storey and Taylor 26 reported that when ethylene glycol (EG) is used as OH source, the molecular weight (MW) is determined by the [monomer]/[EG] ratio. On the other hand, without EG the polymerization rate is higher but decreases as stannous octoate (catalyst) increases.

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Figure 1 - Schematic representation of the ring-opening polymerization of εcaprolactone. The computational study of ε-caprolactone using neural network Carlo modeling

28

27

and Monte

has been reported by other authors. However, to the best of our

knowledge, a computational study on polymerization of ε-caprolactone by ROP using the methods of moments has not yet been reported. The methods of moments consider the reaction mass balance equations to determine characteristics of the polymer such as the average molar mass and conversion. This makes the methods of moments more accurate than other methods previously mentioned, which only uses a statistical approach. Some papers have studied the kinetics and modeling of the l-lactide (l-LA) by ROP using the methods of moments 22,23,29–31. Yu et al. 22,29,30 developed models for polymerization of l-LA where the reversible initiation/propagation, transfer/inter polymer chain, transesterification and chain scission side reactions at different temperature were considered. Furthermore, Pladis et al. 23 developed a similar model for the ROP of l-LA using trace amounts of H2O that act as an initiator. However, a method that can be applied for ε-caprolactone and lactides at lower temperatures (below 140 °C) is still missing in literature. Thus, the efforts have been made to mathematically explain the growth mechanisms of polymer chain and the observed characteristic of PCL under different process conditions 32. The use of mathematical models to explain such a robust


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process is still a challenge. In view of the potential applications of PCL in several fields as biocompatible and biodegradable material, an in depth study on the polymerization of CL is an important topic. In this paper we present a computational study on the polymerization of ε-caprolactone using the methods of moments. The computational study was based on the works of Yu et al.

22,29,30

and Pladis et al.

23

which studied the ROP of l-LA. The

polymerization reaction conditions were selected to closely match the experimental conditions. For each reaction time (ranging from 1 to 8 hours), the conversion and molecular weight of the polymer are calculated. The proposed complex process of polymerization was explained based on computational results and validated using experimental results. The study here presented supports further understanding of the synthesis of PCL by ring opening polymerization, which can be used to improve industrial processes of this biodegradable polymer.

2 2.1

Computational Details ROP Kinetics and Mathematical Modeling of the ε-caprolactone

Penzeck et al.33–35 studied both reaction and kinetics mechanism of ring-opening polymerization of various cyclic ester monomers, including the ε-caprolactone and L,Llactide. Based on these studies, we can assume that the polymerization of ε-caprolactone in the presence of Sn(Oct)2 follows the same polymerization mechanism (coordinationinsertion) of the L,L-lactide

22,29,35,36

. Thus, the mechanism of polymerization of

ε-caprolactone can be described according to Figure 2 based on the mechanism of polymerization of L,L-lactide reported by Yu et al. 29.



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Figure 2 - Scheme for ROP of PCL with Sn(Oct)2 based on the mechanism of polymerization of L,L-lactide. Adapted from ref 29. Copyright 2009 American Chemical Society. Four types of reversible reactions are identified in Figure 2: activations, propagations, reversible deactivations and reversible transesterification. Random scission chain reaction was not considered since it is significant at temperatures higher than 140ºC (our study was performed at 130 ºC). Reaction (a) is the activation of the catalyst (Sn(Oct)2) by a hydroxyl source (ROH), forming the –SnOR (I) and the octanoic acid (A). The reaction (b) occurs between the catalysts and the dormant chain Dn

34,37

. The reactions (c) and (d) represent the reversible propagation steps, which

affect the length of the active chains. The reversible chain transfers are represented in reactions (e) and (f). Reaction (e) is the interaction between the hydroxyl source and the active chain, resulting in a dormant chain and a new alkoxide group

29

, whereas the

reaction (f) is a completely dormant chain. The last two reactions (g and h) are side reactions, which affect the molecular weight of polyesters and are called reversible transesterification reactions. The first interaction (reaction g) is between two active chains and it is well described in the literature

38

. The second interaction (reaction h) 6

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was proposed by Yu et al.

29

, who considered the high concentration of ROH and the

large number of dormant chains in the system to better evaluate the molecular weight of the PLA. In both cases (reactions g and h) an active chain end interacts with an internal ester bond in another chain and restarts the reaction. The rate coefficients used in both kinetics and simulation are: catalyst activation (ka1), deactivation rates (ka2), polymer propagation (kp), deactivation rates (kd), reversible transfer rate (ks) and reversible transesterification rate (kte). The reactions (a) and (b) of ε-caprolactone have a similar reaction mechanism as the L-lactide. Therefore, the same values of ka1 and ka2 coefficients were considered in this work. The values of the coefficients rates, kp and kd, used in the propagation and depropagation reactions are independent of chain length and they can be applied to both reactions (c) and (d). The reaction rate of reactions (e) and (f) has the same value (ks). Also, due to the complete equivalence in both directions, the value of ks is the same for both forward and backward reactions, meaning that the Keq,s (equilibrium constant) is equal to 1. Finally, the value of the last rate (kte), is also the same for both forward and backward reactions and the value used in the simulation is available in the literature for PLA 29,39,40. Using the simplified kinetic scheme in Figure 2, and considering a well-stirred, homogeneous batch reactor, the population balances are described by equations 1-8.

(1)

(2)

= − + − +

= + − + −

= − + − +

(3)



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= − + + − + −

= − − +

(4)

(5)

= − + − + − −

+ − + + ! − 1# $ % $

− ! − 1# $ %

(6)

$

= & − + ' − + − −

+ − + + ! − 1# $ %

− λ ) − 1# + λ $ % &

+ $ $

*&$'

$'

$

(7)

* % − ! − 1# $ %

+ ! − 1# $ %

$

$

= − + + − + −

+ + − ) − 1#

(8)

+'

Because Rn and Dn represent the chain length of all species containing tin alkoxide and OH-sources, when n=0 (when the reaction starts) all species of initiator (Sn(Oct)2)0 and cocatalyst (ROH)0 become R0 and D0, respectively. Therefore, a new polymerization mechanism follows the scheme in Figure 3.


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Figure 3 - Polymerization mechanism for PCL with (Sn(Oct)2)0. Adapted from ref 22. Copyright 2011 American Chemical Society. The new population balances, based on equations (a) to (e), are described in equations 9-14. These equations were further used to simulate the ε-caprolactone polymerization.

= − +

= + −

= − +

= − + − + − − +

$

$

+ ! − 1# $ % − ! − 1# $ %

(9)

(10)

(11)

(12)



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= & − + ' − + − −

+ + ! − 1# $ % − λ ) − 1# $

&

+ λ $ % + $ $'

$

*&$'

* %

(13)

− ! − 1# $ % + ! − 1# $ % $

$

= − + + − + + − ) − 1#

+'

(14)

All population balances, the previous (equations 6-8) and new (equations 12-14), were solved using the method of moments. These equations are reported in Supporting Information. The constants values used in the simulation are described in Table 1. The kinetics values of l-lactide polymerization can be used in ε-caprolactone polymerization because the mechanism of synthesis of both polymers is similar when Sn(Oct)2 is the catalyst

35,36,41

. However, since the polymerization of ε-caprolactone is faster than l-

lactide polymerization

36

, the computational results should be validated using

experimental results. Table 1: Rate coefficients used for the ring opening polymerization of ε-caprolactone.

Parameters*

Value

ka1

1 × 10. /. 123 & ℎ&

ka2 kd kp

9.6 × 107 /. 123 & ℎ& 87.5 ℎ&

825.5 /. 123 & ℎ&

Reference 29,36

36

36

36


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1 × 10. /. 123 & ℎ&

ks

6 /. 123 & ℎ&

kte *

29

30

Parameters obtained for polymerization at 130 ºC

2.2

Model and Simulation Technique The code was developed using the Matlab software (version r2017a) and the

algorithms used for the simulation were the ode23s. Explicit methods like regular Runge-Kutta are not able to solve the model equations due to the stiffness of the system. Therefore, the code was developed using the ode23s, which is one of the five ODE internal suits in Matlab software (version r2017a). According to the Matlab internal guide, this numerical solver is based on an implicit modified second and third orders Rosenbrock formula and it is used to solve stiff problems with crude tolerances or with solutions that change rapidly 42,43.

3 3.1

Experimental Section Materials

Stannous octoate (catalyst; Sn(Oct)2), ε-caprolactone (monomer, CL), deuterated chloroform (CDCl3; > 99.8%), and ethanol (Neon, 95%) were purchased from Sigma Aldrich®.

3.2

CL polymerization

The conditions of PCL synthesis were chosen based on the experimental procedures reported in the literature 37,41. The reactions were performed under controlled conditions and all materials employed were careful dried before the preparation of experimental procedures to ensure controlled concentration of moisture and sufficient reproducibility of experiments. 5 ml of CL was introduced into a vial equipped with a magnetic stirrer



Page 12 of 27

and then 10 µL of catalyst (Sn(Oct)2 in toluene 10wt%) was added. The monomer and catalyst mixture was immersed in an oil bath at 130 °C and the polymerization reaction was carried out at five different reaction times (ranging from 1 to 8 hours) under nitrogen atmosphere (flow rate of 5 cm3/min). Polymerization was stopped by adding ethanol, when the polymer was precipitated and the unpolymerized monomer was removed together with the ethanol.

3.3

Characterization

3.3.1 Nuclear magnetic resonance (NMR) The NMR spectra were recorded on an Agilent DD2 spectrometer from Brazilian Biosciences National Laboratory (Brazilian Center for Research in Energy and Materials - CNPEM), operating at Larmor frequency of 499,726 MHz. The proton chemical shifts were reported in parts per million (ppm) and referenced at 0.0 ppm using TMS (tetramethylsilane). The coupling constants were measured in hertz (Hz). To record NMR spectra of each sample, ~20 mg of each sample was dissolved in 0.6 ml of deuterated solvent (CDCl3). 1H NMR spectra were determined in the usual manner for verifying the structure and proving the identity of the compounds. The parameters for 1

H NMR data acquisition were: spectral window of 13 ppm, acquisition time of 4.0s,

32 scans and relaxation delay of 1.5 s. 1H NMR spectra of CL and PCL are shown in Figure 4.


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Figure 4 - 1H-NMR spectra of the monomer CL and polymer PCL. The ring-strain to lactones gives rise to differing chemical shifts between monomer and polymer signals in 1H NMR spectra. Complete separation of some monomer signals and the respective signals from the polymer makes kinetic analysis easier and reduces errors associated with overlapping peaks. Additionally, 1H NMR spectra are quantitative

40

, allowing the determination of the degree of conversion for

lactone polymerizations, as well as assisting in the elucidation of polymerization mechanisms by end-, and junction group analysis. In the 1H NMR spectrum, typical signals obtained for PCL are assigned as follows: δ 4.06 ppm (H-A, multiplet), δ 2.31 ppm (H-C, triplet, nJH-H = 7.50 Hz), δ 1.39 to 1.65 ppm (H-B, multiplet), which characterize the polymer chain. The δ 3.65 ppm (H-D, triplet, nJH-H = 6.84 Hz) was assigned to the -CH2OH ending group. These signals are in agreement with the literature

44,45

. The total areas of the signals were normalized, and the monomer

conversion and the molecular weight were calculated by integrated area ratio as follows. The monomer conversion was determined according to the integrated area ratio of the signal at 4.23 ppm (-CH2O-) with the corresponding signal of the polymer at 4.06 ppm 13 ACS Paragon Plus Environment


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40

. The molecular weight of the PCL was determined analyzing the end of the PCL

chain. According to In't Veld et al.

46

the methylene proton signal at 3.65 ppm in the

spectrum of the PCL shows that the polymer was terminated by hydroxyl-end groups. Thus, the average molecular weight was estimated from the integral ratios of the signals at 4.05 and 3.65 ppm 40.

4 4.1

Results Operating conditions

The ratio between the monomer (ε-caprolactone; 9.02 mol.L-1), and catalyst (Sn(Oct)2; 0.018 mol.L-1), considered here was 500:1 ([M]/[C]). The catalyst reacts with a source of ROH to form the initiator (-SnOR), represented as I, and an acid group, represented as A. The ratio between the acid and catalyst concentration ([A]/[C]) was 0.3, this value was based on the work of Yu et al. 47 and adapted to better fit our experimental results. The reaction time ranged from 1 to 8 hours and the temperature was 130 °C. The concentration of the co-initiator (the source of hydroxyl groups; ROH) was 0.026 mol/L. The values of the rate coefficients are described in Table 1.

4.2

Validation of computational modeling by experimental results

The progress of the PCL synthesis was monitored by 1H-NMR spectroscopy, which revealed that, as expected, the monomer conversion and average molecular weight are affected by reaction time (circles in Figure 5). A monomer conversion greater than 90% was achieved after 4 hours of reaction, and the highest values was reached at 8 hours (99.2%). Low molecular weight PCL was produced in 1-2 hours of reaction (19-35% monomer conversion), and the average molecular weight enhanced as monomer conversion increased. The simulation data are also shown in Figure 5a-b (as lines), which showed that the model was able to predict the experimental results. 14 ACS Paragon Plus Environment

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Figure 5 – (a) Conversion vs time and (b) molecular weight vs conversion of the simulation Moisture concentration: ROH0%=0.026 mol.L-1, ROH-50%=0.013 mol.L-1 and ROH+50%=0.052 mol.L-1. CL polymerization with Sn(Oct)2 at 130 °C. Experimental data: ○. Simulation data: lines. The relationship between concentration of ROH and reaction time for the monomer conversion, average molecular weight of the polymer, concentrations of initiator and octanoic acid are shown in Figure 6. The effect of water concentration on the polymerization was studied using a concentration of ROH0%=0.032 mol.L-1 and other 2 values (+ and - 50%), i.e. we did simulations increasing and decreasing the ROH value in 50%, ROH+50%=0.064 mol.L-1 and ROH 50%=0.018 mol.L-1, respectively. The results showed that the increase in ROH concentration increased the monomer conversion (especially in the first four hours of reaction) and the concentrations of initiator and octanoic acid, while it decreased the average molecular weight of the polymer. The moisture acts on both conversion and molecular weight of PCL. The moisture (ROH source) accelerates the reaction when used as a co-catalyst (as previously discussed). However, the excessive moisture brought into the reactor via the monomer and the catalyst can also open the ring of the CL, resulting in a polymer with



Page 16 of 27

lower molecular weight. Another impact of moisture in the system is related to the octanoic acid and catalyst relationship. The catalyst Sn(Oct)2 has two octoate groups able to form active sites (tin alkoxide bonds)

47

. The initial concentration of acid

impurities observed in the ε-caprolactone was A/C = 0.28. This value is lower than value reported by Yu, et al.

22,29

, which observed A/C = 0.35 for purified l,l-lactide

using the same monomer/catalyst ratio. The excess of moisture decreases the number of actives sites due to increase of the acid formation in the system, affecting the polymerization rate.

Figure 6 - Effect of moisture on (a) monomer conversion, initiator concentrations (b) and (c) octanoic acid and (d) average molecular weight of PCL at different reaction times. Moisture concentration: ROH0%=0.026 mol.L-1, ROH16 ACS Paragon Plus Environment

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50%=0.013

mol.L-1 and ROH+50%=0.052 mol.L-1. CL polymerization with Sn(Oct)2 at

130 °C. The study of living and dormant chains during the CL polymerization reaction is shown in Figure 7. It can be seen that at the end of the reaction there are numerous first order dormant chains, a predominant characteristic of a reversible-deactivation polymerization. According to Penczek et al.

48

, this behavior occurs usually in cyclic

esters due to the predominant characteristics of reversible-deactivation and reversible chain transfer polymerization.

Figure 7: Concentration of different species vs time during the polymerization of CL with Sn(Oct)2 at 130 °C. λ0 is the total number of living chains (propagation units), λ1 the total number of monomeric units, µ0 is the total polymer molecules and µ1 is the total monomeric units in all polymer molecules.



4.3

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Parametric study

A parametric study was carried out to better understand the effect of the catalyst concentration [C] and octanoic acid concentration [A] on the characteristics of PCL. Three different conditions were studied: [A] varying and [C] constant (Figure 8a-b), [C] varying and [A] constant (Figure 8c-d), and both [C] and [A] varying (Figure 8e-f). The results showed that the increase of the [C] accelerated the monomer conversion, whereas the increase of the [A] decreased the monomer conversion. Moreover, the final molecular weight of the polymer was not sensitive to or affected by the variations of [C] and [A]. These results occur because the increase of catalyst increases the formation of –SnOR (initiator; I). When the octanoic acid is increased, the excess of [A] can react with the initiator, deactivating or transforming it into a dormant chain, and consequently decreasing the conversion of the polymer. Similar results were reported by Zhang et al. 49

studying the L,L-lactide polymerization by ROP. Since the same catalysts (Sn(Oct)2)

were used in both L,L-lactide and ε-caprolactone syntheses, the results reported by Zhang et al. can be compared with our findings. Figure 8e-f showed that there is an optimum amount of [C] and [A] to prepare PCL with faster conversion and constant average molecular weight, which was [C] = 0.009 and [A] = 0.0023.


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Figure 8 - Conversion vs time and molecular weight vs conversion for the εcaprolactone polymerization at 130 °C, under different conditions: (a-b) [A] varying and [C] = 1.81x10-3, (c-d) [C] varying and [A]; ratio [A]/[C] = 0.3. (e-f) both [C] and [A] varying. Experimental data in circles and the simulation results in lines.



5

Page 20 of 27

Conclusion The synthesis of PCL by ROP was studied using a combined experimental and

computational approach. This approach allows understanding the growth kinetics mechanisms of a polymer chain of PCL which is suitable for predicting the characteristic of this polymer. By achieving a good agreement between simulated and measured results, we have shown that our model is able to obtain an optimal condition reducing the reaction time, while maintaining the same molar mass. Faster reaction is related to reducing the costs of production, thereby allowing a more competitive PCL. The study here reported is an additional contribution to the research area of biodegradable polymers. It is expected that the mathematical models of PCL synthesis here presented are potentially useful tool for industrial scale production of PCL under controlled conditions and therefore with predictable and reproducible properties.

Notation A = acid [A] = concentration of acid [A0] = initial concentration of acid I = SnOR C = catalyst, Sn(Oct)2 [C] = concentration of catalyst [C0] = initial concentration of catalyst ROH = dormant chains, OH-bearing species,


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[OH] = concentration of cocatalyst, 1-dodecanol [OH0] = initial concentration of cocatalyst, 1-dodecanol Dn = dormant chains with n repeating units [ROHimp] = concentration of OH-bearing impurities ka1, ka2 = reversible catalyst activation rate coefficient kd = depropagation rate coefficient Keq,a = reversible catalyst activation equilibrium constant Keq,c = equilibrium constant of PLA ring-chain equilibrium kp = propagation rate coefficient ks = reversible chain transfer rate coefficient kte = intermolecular transesterification rate coefficient mmon = molar mass of monomer M = monomer [M] = instantaneous monomer concentration [M0] = initial monomer concentration [Meq] = equilibrium monomer concentration Mw = average molecular weight MWD = molecular weight distribution Nc = overall concentration of polymer chains 21 ACS Paragon Plus Environment


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LA = Lactic Acid CL = ε-caprolactone PLA = poly(lactic acid) PCL = poly(ε-caprolactone) PBT = Polybutylene terephthalate R* = active chains [R*] = concentration of active chains R0 = activated catalyst, tin alkoxide Rn = active chains with n repeating units ROP = ring-opening polymerization SEC = size exclusion chromatography t = time T = temperature λ0, λ1, λ2, λ3 = zero, first, second, and third moments of active chains, respectively μ0, μ1, μ2, μ3 = zero, first, second, and third moments of dormant chains, respectively

Acknowledgements The authors acknowledge São Paulo Research Foundation - FAPESP (Grants 2016/09588-9; 2016/19847-1) for financial support. The authors also gratefully acknowledge the support received by LNNano/CNPEM and LNBio/CNPEM.


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Conflict of Interests The authors declare that there is no conflict of interests.

Supporting Information: All moment equations of the active chains and dormant chains corresponding to the ε-caprolactone model can be found in Supporting Information.

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Graphical abstract


A combined computational and experimental study on the

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