Nonlinear Method for Determining Reactivity Ratios of Ring-Opening

Article pubs.acs.org/Macromolecules

Nonlinear Method for Determining Reactivity Ratios of Ring-Opening Copolymerizations Matthew T. Hunley and Kathryn L. Beers* Polymers Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, United States S Supporting Information *

ABSTRACT: A nonlinear errors-in-variables-model (EVM) regression method was used to determine reactivity ratios for enzyme-catalyzed ring-opening copolymerizations of ε-caprolactone (ε-CL) and δ-valerolactone (δ-VL). The cumulative copolymer composition model accurately described the experimental data, indicating that conventional models can be used to describe enzyme-catalyzed copolymerizations. Reactivity ratios were calculated from Raman spectroscopic data collected in situ and the model of monomer feed drift over the course of copolymerization. The analysis was combined for multiple experiments to improve the estimate. For the lipase-catalyzed copolymerization, the calculated reactivity ratios were rε‑CL = 0.27 and rδ‑VL = 0.39. Compared to conventional linearization techniques, the EVM method reduced the experimental work required and reduced the measurement error, as indicated by the 95% joint confidence region. In addition, the EVM method is influenced less by the apparent induction period of δ-VL. The conventional methods rely on low conversion data where the induction period is significant. The EVM method presented here determines reactivity ratios rapidly, saving both time and material waste.

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INTRODUCTION Ring-opening polymerization (ROP) of cyclic esters and carbonates offers a controlled polymerization technique for degradable and renewable polymers. Metal alkoxide catalysts for ROP remain popular due to their versatility and cost, although interest in enzyme catalysts and organocatalysts has increased dramatically due to the high efficiencies, ease of reuse, and the green chemistry appeal.1−3 However, poly(lactic acid) (PLA), the most ubiquitous renewable polyester, cannot replace commercial thermoplastics for many applications due to its lower strength and increased gas permeability compared to poly(ethylene terephthalate), as well as its low melting temperature compared to polystyrene. Recently, reports of degradable copolymers have indicated that the mechanical, thermal, and physical properties can be dramatically enhanced through copolymerization.4 For example, by varying the sequence of poly(lactic acid-co-glycolic acid) (PLGA) copolymers from blocky to alternating, the hydrolytic degradation rate can be controlled from rapid molecular weight loss to a linear degradation rate, respectively.5,6 Predictable degradation rates will improve the usefulness of these copolymers for specific applications including controlled drug release and tissue scaffolds.6 Likewise, incorporating ε-caprolactone (ε-CL) into PLA increases the hydrophobicity and increases the permeability of many therapeutic molecules for drug-delivery applications.7 For thermoplastic applications, lactone comonomer can tune the thermal transitions and reduce the brittleness of PLA.8 Similarly, the homopolymers of ε-CL and δ-valerolactone (δ-VL) are both semicrystalline with similar melting temperatures, but the copolymer exhibits a depressed melting temperature lower than both homopolymers.9 However, the copolymerization behavior of cyclic esters varies This article not subject to U.S. Copyright. Published 2013 by the American Chemical Society

dramatically and can be unpredictable. For example, the reactivity ratios of ε-CL and LA in bulk via Al(OiPr)3 are 0.58 and 17.9, respectively.10 However, statistical copolymers have been reported for catalyst systems with highly tailored ligands.11−13 As the popularity of such copolymers increases and more novel monomers and catalysts are developed, a better understanding of the copolymerization behavior and mechanisms is required. Radical and ionic chain-growth copolymerizations often conform to the terminal or penultimate reactivity models. In the case of terminal model copolymerization mechanism, reactivity ratios are often calculated using a linearization of the Mayo−Lewis equation14 (either the differential or integrated form). Two of the most common techniques are the method developed by Fineman and Ross (F−R method)15 and the subsequent refinement by Kelen and Tüdös (K−T method).16 In both methods, the copolymerization is performed over a wide range of monomer feed ratios and stopped at low conversion (≤5%) to calculate copolymer composition. The K−T method refines the F−R method to equally weight all data points and estimates reactivity ratios r1 and r2 through linear regression of the experimental data fit to the relationship η = [r1 + (r2/α)]ξ − (r2/α)

(1)

where the variables η and ξ have the following format: η=

G α+F

Received: September 24, 2012 Revised: January 16, 2013 Published: February 5, 2013 1393

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F α+F

where f1,0 and f1 represent the initial monomer feed composition and the feed composition of monomer one at overall monomer conversion X, respectively. The parameters α, β, γ, and δ are functions of the reactivity ratios with the form

⎛ [M1]0 ⎞2 ⎛ d[M 2]0 ⎞ F=⎜ ⎟ ⎟⎜ ⎝ [M 2]0 ⎠ ⎝ d[M1]0 ⎠

G= α=

[M1]0 ⎛ d[M 2]0 ⎞ ⎜1 − ⎟ [M 2]0 ⎝ d[M1]0 ⎠

α=

r2 1 − r2

β=

r1 1 − r1

γ=

1 − r1r2 (1 − r1)(1 − r2)

δ=

1 − r2 2 − r1 − r2

FminFmax

In the above equations, ([M1]0/[M2]0) is the initial monomer feed ratio and (d[M1]/d[M2]) is the instantaneous monomer consumption ratio. These methods rely on several key assumptions, particularly the low conversion assumption, which introduces large systemic errors into the calculation. Tüdös et al.17 acknowledged that this systemic error can be larger than the nominal values for reactivity ratios. In addition, large experimental data sets are required to reduce the measurement uncertainty. McFarlane et al.18 estimated that the measurement precision for K−T methods (quantified as the areas of the joint confidence regions (JCRs)) increased 66% compared to nonlinear least-squares fitting based on the Mayo−Lewis equation. Tüdös and Kelen17 later modified their technique to account for finite conversions. The so-called extended Kelen−Tüdös method (eK−T method) directly accounts for partial monomer conversions and removes the low conversion assumption. In this method, the terms F and G are replaced by

Equation 2 describes the drift of monomer feed composition as each monomer is consumed at different rates during the copolymerization. This technique can make use of all the data collected via in situ monitoring of the copolymerization, and it eliminates the low conversion assumption inherent in the linearization techniques. Van den Brink et al.22 and Kazemi et al.26 have both applied eq 2 to cumulative composition data for acrylic copolymerizations using errors-in-variables-model (EVM) nonlinear regression. The EVM analysis consistently improved the measurement precision (quantified via the joint confidence regions (JCRs)) for reactivity ratios. We recently demonstrated that in situ Raman spectroscopy can monitor the ring-opening copolymerization of ε-CL and δvalerolactone (δ-VL).21 The model system of ε-CL and δ-VL was chosen due to the similar homopolymerization behaviors of the two monomers via enzymatic catalysts. Both monomers undergo polymerization via Candida antarctica Lipase B (CALB) with pseudo-first-order kinetics, and the rate constants of homopolymerization are 1.26 and 1.37 h−1, respectively.9 As mentioned above, both homopolymers of ε-CL and δ-VL are semicrystalline, but the copolymer exhibits a melting temperature significantly lower than both homopolymers. Copolymerization has been investigated as a way to control the melting transition over a wide range. Previous work reported that the reactivity ratios for the copolymerization of ε-CL and δ-VL in the bulk using Sn(Oct)2 as the catalyst were rε‑CL = 0.25 and rδ‑VL = 0.49.27 In our recent report, the ring-stretching peaks at 696 cm−1 (ε-CL) and 745 cm−1 (δ-VL) were used to quantify monomer consumption during the reaction. Reactivity ratios determined using the K−T method agreed well with copolymer composition data. In this case, the terminal model appeared to accurately describe the enzyme-catalyzed ROP. Although the data were in agreement, the Raman spectra indicated a significant induction period for δ-VL that could have a large influence on low conversion results. In this study, ring-opening copolymerization behavior was fit to eq 2 using EVM regression to determine if these conventional techniques could adequately describe the composition of enzyme-catalyzed copolymers. Equation 2 accurately described the monomer drift during copolymerization, and this technique was used to determine reactivity ratios. As expected, the measurement uncertainty was significantly lower than the linearization techniques. The cumulative composition model provides an enhanced analytical tool to rapidly determine copolymerization reactivity ratios for ROP with low measurement uncertainty.

⎛ log(1 − χ ) ⎞2 ⎛ d[M ] ⎞ 1 0 2 ⎟⎟ ⎜ F = ⎜⎜ ⎟ ⎝ log(1 − χ1 ) ⎠ ⎝ d[M 2]0 ⎠

⎛ log(1 − χ ) ⎞⎛ d[M ] ⎞ 1 0 2 ⎟⎟⎜ G = ⎜⎜ − 1⎟ ⎠ ⎝ log(1 − χ1 ) ⎠⎝ d[M1]0

where χ1 and χ2 are the partial molar conversions of each monomer. The definitions of η and ξ remain the same, and eq 1 is again used to calculate reactivity ratios. Error analysis indicates that the eK−T method closely approximates the actual reactivity ratios up to high conversions approaching 80%. However, the other sources of systemic error of the linearization techniques remain, such as collecting only one experimental data point per reaction and the transformation of the error structure due to linearization.18,19 Recently, several in situ techniques have become popular for monitoring copolymerizations, including infrared20 and Raman spectroscopy21,22 and refractivity.23 These techniques eliminate the need to collect reaction aliquots at low conversion and improve the speed of data collection. Although these techniques can provide continuous data monitoring throughout the reaction, the F−R and K−T methods still rely on only one data point per reaction. A different model must be implemented to derive reactivity ratios from monomer conversion profiles. Skeist24 developed a relationship to describe the drift in monomer feed ratio over the course of a copolymerization, which Meyer and Lowry25 later solved into the closed form ⎛ f ⎞α ⎛ 1 − f ⎞ β ⎛ f − δ ⎞ γ M 1 ⎟ ⎜ 1,0 ⎟ = X = 1 − ⎜⎜ 1 ⎟⎟ ⎜⎜ 1− ⎟ ⎜ f −δ⎟ M0 f f − 1 ⎠ ⎝ 1,0 ⎠ ⎝ 1,0 ⎠ ⎝ 1

(2) 1394

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Scheme 1. Enzymatic Copolymerization of ε-CL and δ-VL

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EXPERIMENTAL SECTION

Materials. Toluene, ε-CL (97%), and δ-VL (technical grade) were obtained from Sigma-Aldrich,a distilled over CaH2, and stored under Ar prior to use. Novozyme 435 (N435) was obtained from Novozymes (Bagsvaerd, Denmark) and stored in a vacuum desiccator prior to use. To minimize experimental deviations caused by variations in N435 particle size, a 400 μm sieve was used to select a particle size distribution of 400 ± 50 μm. Aluminum isopropoxide and titanium isopropoxide were obtained from Sigma-Aldrich and used as received. Polymerizations. The reaction procedures, reactor design, and data processing for enzyme-catalyzed copolymerizations with in situ Raman monitoring are described in detail elsewhere.21 All reactions were carried out under inert atmosphere using standard Schlenk techniques. For a typical enzyme-catalyzed copolymerization, N435 beads (100 mg) and toluene (2 mL) were added to a 5 mL roundbottom flask under argon. The monomers, ε-CL (0.83 mL, 7.5 mmol) and δ-VL (0.70 mL, 7.5 mmol), and toluene (1.5 mL) were added to a second flask via syringe and mixed under argon. To start the reaction, the monomer mixture was transferred using a cannula to the flask containing catalyst. The exact starting monomer ratios were calculated from the monomer mixture using Raman spectroscopy. Initiation occurred from trace water present in the catalyst beads and distilled monomers; no additional initiator was added. We previously investigated the amount of water present in the beads and found the value consistently around 200 ppm.28 For a typical metal-catalyzed reaction, ε-CL (0.83 mL, 7.5 mmol), δ-VL (0.70 mL, 7.5 mmol), and toluene (1.5 mL) were added via syringe to the reaction flask under argon. In a separate flask, Al(OiPr)3 (30 mg) was dissolved in toluene (2 mL) under argon. The catalyst solution was added to the monomer mixture via syringe to initiate the polymerization. In the metalcatalyzed polymerizations, the propoxide fragments on the catalyst acted as initiator for the polymerization. For Ti(OiPr)4-catalyzed polymerizations, the starting feed ratios were fε‑CL,0 = 0.356, 0.446, 0.513, 0.645, and 0.700. For Al(OiPr)3-catalyzed polymerizations, the starting feed ratios were fε‑CL,0 = 0.534, 0.709, and 0.432. Modeling. The experimental data were modeled in MATLAB using eq 2. The MATLAB algorithm used an errors-in-variables model (EVM) of nonlinear regression to account for the measurement error in both the dependent and independent variables. A description and the pseudocode of this algorithm are provided in the Supporting Information. The fitting procedure involves the preliminary calculation of weighting factors assigned to the experimental data. The weighting factors were derived through error propagation analysis from the measurement uncertainty of the monomer concentrations, which has been discussed previously.21 The error propagation and the structure of the weighting factors are both described in detail in the Supporting Information.

Figure 1. (a) Monomer concentration and (b) monomer conversion profiles for the enzymatic copolymerization of ε-CL and δ-VL. Reaction conditions: fε‑CL,0 = 0.61, 55 °C.

polymerization, ε-CL is consumed rapidly while the concentration of δ-VL remains relatively constant. This induction period behavior has been observed previously.21 Pseudo-firstorder kinetic behavior was observed for δ-VL only after the first 5 min of polymerization, confirming the induction period (Figure S2). After the induction period, δ-VL polymerizes faster than ε-CL, as evident in the crossover of the two monomer conversion profiles. Because of this induction behavior, the reactivity ratios calculated from low conversion data likely do not represent the copolymerization behavior at moderate to high conversions. The Raman probe provided monomer consumption data over the entire reaction, so the same experiments could be analyzed using the eK−T method. Using monomer consumption data up to about 40% conversion, we calculated the reactivity ratios as rε‑CL = 0.27 ± 0.05 and rδ‑VL = 0.18 ± 0.02. The linear regression results and 95% JCRs for both K−T and eK−T methods are shown in Figure 2. The measured reactivity ratios are lower from the eK−T method compared to the K−T method. The value of rε‑CL decreased 29%, and the value of rδ‑VL decreased 38%. However, the JCRs show significant overlap, and both results are on the boundary of the other 95% confidence interval. The elliptical shape of both JCRs indicates a correlation between the two reactivity ratios.29 In addition, both JCIs are large compared to the magnitude of the reactivity ratio values.

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RESULTS AND DISCUSSION Linearization Techniques. The enzymatic ROP of ε-CL and δ-VL is illustrated in Scheme 1. We recently reported reactivity ratios of rε‑CL = 0.38 ± 0.06 and rδ‑VL = 0.29 ± 0.03 using the K−T method and in situ Raman spectroscopic data.21 The reactivity ratios from monomer consumption agreed well with the copolymer compositions determined by 1H NMR analysis, indicating that the terminal copolymerization model can accurately characterize enzymatic ROP. However, the Raman spectroscopy also revealed an induction period for δVL. Figure 1 shows the monomer concentrations and monomer conversion profiles over the course of a copolymerization with initial feed ratio fε‑CL,0 = 0.61. During the first 5 min of 1395

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Figure 2. Calculated reactivity ratios and corresponding 95% JCRs for both the K−T and eK−T methods. The K−T model results are from ref 21. The error bars indicate one standard deviation obtained from the linear regression results.

Figure 3. Monomer fraction versus total monomer conversion for the enzymatic copolymerization of ε-CL and δ-VL with initial monomer feed ratios fε‑CL,0 of (□) 0.61, (△) 0.52, (○) 0.45, and (◇) 0.40. The symbols represent experimental data, and the lines represent the best fit of eq 2 using EVM regression.

Error-in-Variables Method. The in situ Raman spectroscopy provides data over the entire polymerization at intervals as low as 20 s. However, the linearization techniques condense this spectroscopic data into one data point per reaction. On the other hand, eq 2 models the compositional drift during copolymerization and can use almost the entirety of the spectroscopic data. However, accurately modeling eq 2 is difficult for multiple reasons. Specifically, both variables (X, f1) are dependent variables with different observational errors. EVM regression explicitly accounts for the error in both regressors. Based on repeated measurements of samples with known concentration, the measurement uncertainty for both monomers is known to be 0.05.21 The error propagation analysis to determine the weighting factors and the calculation of the residuals is detailed in the Supporting Information. In addition, in the current form of eq 2 (X = g(f1)), regression becomes difficult due to the shape of the curve. For a given pair of reactivity ratios, r1 and r2, and starting composition f1,0, total conversion X is not defined over the entire range of f1 from 0 to 1. For a robust fit, we analyzed the experimental data in the form f1 = g(X) using an iterative algorithm to solve for f1 in eq 2. This iterative EVM technique was implemented in MATLAB, and the pseudocode is provided in the Supporting Information. Four copolymerizations were performed with feed compositions ranging from fε‑CL,0 = 0.40 to 0.61. Figure 3 plots the comonomer conversion profiles for the four reactions as fε‑CL,0 versus X. The starting concentrations were chosen to maximize the monomer composition drift during copolymerization and to provide two reactions on either side of the azeotropic composition, faz. The azeotropic composition was estimated as faz = 0.48 from the eK−T reactivity ratios using the relationship 1 − r1 faz = 2 − r1 − r2 (3)

Figure 4. Weighted residuals versus conversion for the experimental data fit to eq 2 using EVM regression at initial monomer feed ratios fε‑CL,0 of (□) 0.61, (△) 0.52, (○) 0.45, and (◇) 0.40. For clarity, each data set is offset around a solid line representing zero.

residuals for all four reactions. The residuals appear normally distributed, and their magnitude increases with conversion. This increase resulted from the relative increase in measurement error as the monomer concentrations approach zero. To reduce the impact of this error, any data points above 90% conversion were excluded from further analysis. The MATLAB algorithm calculated the sum of squared residuals (SSR) at each pair of rε‑CL and rδ‑VL from zero to one. The SSR profiles for all four reactions are included in the Supporting Information. Figure 5 shows the 95% JCRs calculated from the SSR profiles. The JCRs illustrate the difference in fitting results between the separate reactions. The best fit estimates for each reaction differ dramatically, and several estimates are near zero for at least one reactivity ratio. The JCRs are highly elongated and nonsymmetrical, indicating very strong rε‑CL−rδ‑VL correlations. Variations in the scatter of the data lead to smaller JCRs that appear elliptical or larger JCRs that appear boomerang-shaped. The elliptical JCRs have a strong slope that appears to decrease as the initial mole fraction of ε-CL increases. Different initial feed ratios influence the correlation between reactivity ratios, resulting in different orientations of the JCRs.22 More specifically, at high feed ratios of ε-CL, rε‑CL is undetermined; at low feed ratios of ε-CL, rδ‑VL is undetermined. The lack of overlap between JCRs can result

The azeotropic composition is evident in the conversion profiles in Figure 3. Above faz, the fraction of unreacted ε-CL remaining increases as the reaction proceeds; below faz, fε‑CL decreases as the reaction proceeds. Each individual reaction profile was fit using our MATLAB EVM algorithm described above. The fits are illustrated as the solid lines in Figure 3. The regression results appear to reasonably model the data. Figure 4 shows the weighted 1396

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size, presumably controlled by the scatter within the experimental data. Although there is only moderate overlap between the individual JCRs, the best fit estimates are relatively close and the value for rε‑CL is close to zero. In addition, all three JCRs have roughly the same slope, suggesting that the primary factor controlling shape and location of the JCR is initial monomer feed ratio. The fact that the individual JCRs overlap indicates that the systematic errors in data collection are not the dominant factor controlling the individual results. Combining multiple experiments into the analysis can reduce the impact of random error and increase the number of data points for analysis. The SSR matrices from the three individual reactions in Figure 6 were combined to calculate a composite best fit. The SSRs were combined using a relative weighting scale, with the combined SSR defined as

Figure 5. 95% JCRs for individual copolymerizations of ε-CL and δVL at initial monomer feed ratios fε‑CL,0 of (□) 0.61, (△) 0.52, (○) 0.45, and (◇) 0.40. The symbols represent the best fit estimates and the lines are the 95% JCRs.

i=1

SSR combined(rε‐CL , rδ‐VL) =

⎛ ni ⎞ SSR i(rε‐CL , rδ‐VL) ⎟ SSR i ,min ⎝ ∑ ni ⎠

∑⎜ k

(4)

from small systematic errors in data collection, including effects of incorrect baseline corrections, initial feed ratios, and normalizations on the data analysis. The influence of these systematic errors can be reduced dramatically, and the accuracy of the estimate increased by combining analysis and different feed ratios, as discussed in the following sections. Repeatability. To investigate the repeatability of the experiments and the robustness of the fitting algorithm, we ran three reactions under the same conditions and at similar starting monomer compositions. The JCRs are shown in Figure 6, and the conversion profiles are shown in the Supporting Information as Figure S4. The JCRs again vary dramatically in

where k is the number of experiments, ni is the number of data points in the ith experiment, and SSRi,min is the minimum SSR in the ith experiment (at the estimated reactivity ratio). The relative weighting accounts for differences in the number of data points in each set and prevents one data set with significantly higher errors from dominating the combined result. The combined JCR is also plotted in Figure 6. The combined analysis appears to be a composite of the three individual reactions, but the size of the JCR is not significantly reduced. However, the best fit reactivity ratios for the combined analysis are dramatically different at rε‑CL = 0.23 and rδ‑VL = 0.35. Combining experimental data sets into a single composite result reduces the impact of the systematic errors and improves the accuracy of the model fit. Combined Experiments. Combining the analysis of three copolymerizations at the same feed composition appeared to incrementally improve the determination of reactivity ratios. However, each individual reaction only covers a small range of feed compositions, which can introduce bias into the result. Combining the results of copolymerizations at different starting feed ratios leads to improved estimates.22,29 Combining the four reactions from varied feed ratios leads to estimated reactivity ratios of rε‑CL = 0.27 and rδ‑VL = 0.39. The monomer conversion profiles with overlaid best fit results are shown in Figure 7. This estimate is quite close to the combined analysis at the same feed composition presented above. Figure 8a shows the combined JCR in comparison with the individual reaction JCRs. Intuitively, the result from the composite analysis appears to occupy the extrapolated intersection of the individual JCRs. The size of the composite JCR also decreased dramatically, presumably due to the use of all data points. This result indicates the importance of choosing initial monomer feed compositions such that the combined compositional drift of all reactions spans as much of the fε‑CL axis from 0 to 1 as possible. Although the JCR from the composite EVM analysis is significantly reduced, a large discrepancy still exists between the EVM and linearization results, as shown in Figure 8b. Both linearization methods are strongly biased by the short induction period for δ-VL, which lasts until about 10% total monomer concentration. Even the eK−T technique, while analyzing the reaction up to about 40% total conversion, is heavily influenced by this induction behavior. The EVM analysis uses data from the first 90% of conversion, significantly reducing the impact of

Figure 6. Calculated 95% JCRs from EVM analysis of three reactions at similar initial compositions. The solid line is the 95% JCR for the combined reaction sets. Data points indicate the point estimate for reactions with fε‑CL,0 of (◇) 0.61, (△) 0.67, (○) 0.65, and (□) the composite result. 1397

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result of an inappropriate model, or it could be due to systematic errors in the data collection. Small measurement errors are evident in the results for the reaction at fε‑CL,0 = 0.52, which is very close to the predicted azeotropic composition of faz = 0.54. Systematic errors in the calculation of monomer ratio could explain why the reported composition is below the azeotropic composition, but the experimental data indicate that it is above the azeotrope. Recently, we modeled the kinetic pathways of enzyme-catalyzed ROP.30 The model and accompanying experimental results suggested that large numbers of cyclic oligomers formed during the initial stages of polymerization, followed by elongation and formation of long linear chains. During the late stages of polymerization, chain elongation via polycondensation becomes more prevalent. This enzyme-catalyzed mechanism may deviate from terminal model kinetics enough to account for the observed systemic errors, especially at the late stages of polymerization. Nonetheless, eq 2 still provides a reasonable fit to the experimental data. We are currently further probing the mechanism of ROP for enzyme and metal catalysts to confirm the applicability of the terminal model. This will be the subject of a future report. Other Catalyst Systems. To investigate the applicability of this reactivity ratio technique to other catalyst systems, ε-CL and δ-VL were copolymerized using the metal catalysts Al(OiPr)3 and Ti(OiPr)4. Both of these metal alkoxide catalysts function through the coordination−insertion mechanism.31,32 Unlike enzymatic catalysts, which result in cyclic oligomers and numerous propagating chains per catalytic site, the propagating chains grow in a continuous manner more consistent with the terminal model of copolymerization. Compared to the enzymatic catalysts, no strong induction period was observed in the metal-catalyzed polymerizations. The reactivity ratios for these copolymerizations via enzyme and metal catalysts are presented in Table 1. As with the enzyme-catalyzed

Figure 7. Monomer fraction versus total monomer conversion for the enzymatic copolymerization of ε-CL and δ-VL with initial monomer feed ratios fε‑CL,0 of (□) 0.61, (△) 0.52, (○) 0.45, and (◇) 0.40. The symbols represent experimental data, and the lines represent the best fit of eq 2 using composite EVM analysis.

Table 1. Reactivity Ratios for the Copolymerization of ε-CL and δ-VL Using Different Catalyst Systems catalyst/conditions N435 (enzyme) 55 °C Al(OiPr)3 30 °C Ti(OiPr)4 30 °C

Figure 8. (a) 95% JCIs for individual copolymerizations of ε-CL and δ-VL and the combined analysis. (b) 95% JCR for the composite analysis. The dashed ellipse shows the K−T method for comparison. The error bars for the K−T method in (b) indicate one standard deviation obtained from the linear regression results.

K−T method rε‑CL rδ‑VL rε‑CL rδ‑VL rε‑CL rδ‑VL

= = = = = =

0.38 0.29 0.96 0.87 0.82 1.00

EVM method rε‑CL rδ‑VL rε‑CL rδ‑VL rε‑CL rδ‑VL

= = = = = =

0.27 0.39 0.52 0.78 0.44 0.51

copolymerizations, large deviations were observed between the results of the EVM and K−T methods for metal-catalyzed copolymerizations, although these deviations were within the 95% JCRs. Compared to both metal catalysts, the enzyme catalyst exhibited reactivity ratios much further from unity, corresponding to more alternating behavior. This selectivity could result from the structure of the lipase’s active site. The metal catalysts studied are less selective between the two monomers and the reactivity ratios under those conditions are closer to unity. Further experiments are required to confirm the selectivity of the enzyme catalysts.

the low conversion behavior on the results. It is quite reasonable that the EVM analysis estimated reactivity ratios with a lower rε‑CL and a higher rδ‑VL. As observed in Figure 1, after the induction period, δ-VL is consumed significantly faster than ε-CL. It should be noted that the point estimate obtained from combined experiments introduces a bias in the residuals when applied to the individual experiments. The bias can be observed in Figure 7, where the best fit curves do not exactly model the experimental data for all reactions. The bias occurs in opposite directions for the different experiments. This bias could be the

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CONCLUSIONS We applied both conventional linearization techniques and a nonlinear model of compositional drift to estimate reactivity ratios for ring-opening copolymerizations. In situ Raman 1398

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spectroscopy monitored the simultaneous conversion of ε-CL and δ-VL to eliminate the need for offline measurements of reaction aliquots. As expected, the EVM method resulted in a significantly reduced JCR with fewer experiments. The induction period for δ-VL significantly influenced the results of linearization techniques, leading to a discrepancy between results from the different methods. The EVM method models data over most of the reaction and provides a more accurate representation of the copolymer at moderate to high conversions. However, the EVM technique must be extended to other monomer pairs to verify its applicability and consistency with results from conventional methods and to examine the model’s limitations. We can conclude that the cumulative composition model reasonably describes the enzyme- and metal-catalyzed ROP of cyclic esters. The EVM technique provides a rapid tool to determine reactivity ratios with less experimental work required, reducing both the time required and material waste.

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ASSOCIATED CONTENT

* Supporting Information S

More information relating to the error-in-variables regression analysis, residual calculation, error analysis, calculation of joint confidence intervals, and pseudocode for the determination of reactivity ratios. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Novozyme for providing the N435 beads. M.H. acknowledges the financial support of the National Research Council Fellowship Program.

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ADDITIONAL NOTE Certain commercial equipment, instruments, or materials are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose. a

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dx.doi.org/10.1021/ma302015e | Macromolecules 2013, 46, 1393−1399