Nonlinear Method for Determining Reactivity Ratios of Ring-OpeningCopolymerizationsMatthew 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 wasused to determine reactivity ratios for enzyme-catalyzed ring-opening copolymeriza-tions of ε-caprolactone (ε-CL) and δ-valerolactone (δ-VL). The cumulativecopolymer composition model accurately described the experimental data, indicatingthat conventional models can be used to describe enzyme-catalyzed copolymeriza-tions. Reactivity ratios were calculated from Raman spectroscopic data collected insitu and the model of monomer feed drift over the course of copolymerization. Theanalysis was combined for multiple experiments to improve the estimate. For thelipase-catalyzed copolymerization, the calculated reactivity ratios were rε‑CL = 0.27and rδ‑VL = 0.39. Compared to conventional linearization techniques, the EVM method reduced the experimental work requiredand reduced the measurement error, as indicated by the 95% joint confidence region. In addition, the EVM method is influencedless by the apparent induction period of δ-VL. The conventional methods rely on low conversion data where the inductionperiod is significant. The EVM method presented here determines reactivity ratios rapidly, saving both time and material waste.

■ INTRODUCTION

Ring-opening polymerization (ROP) of cyclic esters andcarbonates offers a controlled polymerization technique fordegradable and renewable polymers. Metal alkoxide catalystsfor ROP remain popular due to their versatility and cost,although interest in enzyme catalysts and organocatalysts hasincreased 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, cannotreplace commercial thermoplastics for many applications dueto its lower strength and increased gas permeability comparedto poly(ethylene terephthalate), as well as its low meltingtemperature compared to polystyrene. Recently, reports ofdegradable copolymers have indicated that the mechanical,thermal, and physical properties can be dramatically enhancedthrough copolymerization.4 For example, by varying thesequence of poly(lactic acid-co-glycolic acid) (PLGA) copoly-mers from blocky to alternating, the hydrolytic degradation ratecan be controlled from rapid molecular weight loss to a lineardegradation rate, respectively.5,6 Predictable degradation rateswill improve the usefulness of these copolymers for specificapplications including controlled drug release and tissuescaffolds.6 Likewise, incorporating ε-caprolactone (ε-CL) intoPLA increases the hydrophobicity and increases the perme-ability of many therapeutic molecules for drug-deliveryapplications.7 For thermoplastic applications, lactone como-nomer can tune the thermal transitions and reduce thebrittleness of PLA.8 Similarly, the homopolymers of ε-CL andδ-valerolactone (δ-VL) are both semicrystalline with similarmelting temperatures, but the copolymer exhibits a depressedmelting temperature lower than both homopolymers.9

However, the copolymerization behavior of cyclic esters varies

dramatically and can be unpredictable. For example, thereactivity ratios of ε-CL and LA in bulk via Al(OiPr)3 are0.58 and 17.9, respectively.10 However, statistical copolymershave been reported for catalyst systems with highly tailoredligands.11−13 As the popularity of such copolymers increasesand more novel monomers and catalysts are developed, a betterunderstanding of the copolymerization behavior and mecha-nisms is required.Radical and ionic chain-growth copolymerizations often

conform to the terminal or penultimate reactivity models. Inthe case of terminal model copolymerization mechanism,reactivity ratios are often calculated using a linearization of theMayo−Lewis equation14 (either the differential or integratedform). Two of the most common techniques are the methoddeveloped by Fineman and Ross (F−R method)15 and thesubsequent refinement by Kelen and Tudos (K−T method).16

In both methods, the copolymerization is performed over awide range of monomer feed ratios and stopped at lowconversion (≤5%) to calculate copolymer composition. TheK−T method refines the F−R method to equally weight alldata points and estimates reactivity ratios r1 and r2 throughlinear regression of the experimental data fit to the relationship

η α ξ α= + −r r r[ ( / )] ( / )1 2 2 (1)

where the variables η and ξ have the following format:

ηα

=+G

F

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

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

ξα

=+F

F

=⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟F

[M ][M ]

d[M ]d[M ]

1 0

2 0

22 0

1 0

= −⎛⎝⎜

⎞⎠⎟G

[M ][M ]

1d[M ]d[M ]

1 0

2 0

2 0

1 0

α = F Fmin max

In the above equations, ([M1]0/[M2]0) is the initial monomerfeed ratio and (d[M1]/d[M2]) is the instantaneous monomerconsumption ratio. These methods rely on several keyassumptions, particularly the low conversion assumption,which introduces large systemic errors into the calculation.Tudos et al.17 acknowledged that this systemic error can belarger than the nominal values for reactivity ratios. In addition,large experimental data sets are required to reduce themeasurement uncertainty. McFarlane et al.18 estimated thatthe measurement precision for K−T methods (quantified as theareas of the joint confidence regions (JCRs)) increased 66%compared to nonlinear least-squares fitting based on theMayo−Lewis equation.Tudos and Kelen17 later modified their technique to account

for finite conversions. The so-called extended Kelen−Tudosmethod (eK−T method) directly accounts for partial monomerconversions and removes the low conversion assumption. Inthis method, the terms F and G are replaced by

χχ

=−−

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟F

log(1 )

log(1 )d[M ]d[M ]

2

1

21 0

2 0

χχ

=−−

−⎛⎝⎜⎜

⎞⎠⎟⎟⎛⎝⎜

⎞⎠⎟G

log(1 )

log(1 )d[M ]d[M ]

12

1

1 0

1 0

where χ1 and χ2 are the partial molar conversions of eachmonomer. The definitions of η and ξ remain the same, and eq 1is again used to calculate reactivity ratios. Error analysisindicates that the eK−T method closely approximates theactual reactivity ratios up to high conversions approaching 80%.However, the other sources of systemic error of thelinearization techniques remain, such as collecting only oneexperimental data point per reaction and the transformation ofthe error structure due to linearization.18,19

Recently, several in situ techniques have become popular formonitoring copolymerizations, including infrared20 and Ramanspectroscopy21,22 and refractivity.23 These techniques eliminatethe need to collect reaction aliquots at low conversion andimprove the speed of data collection. Although thesetechniques can provide continuous data monitoring throughoutthe reaction, the F−R and K−T methods still rely on only onedata point per reaction. A different model must beimplemented to derive reactivity ratios from monomerconversion profiles. Skeist24 developed a relationship todescribe the drift in monomer feed ratio over the course of acopolymerization, which Meyer and Lowry25 later solved intothe closed form

δ

δ− = = −

−−

−

−

α β γ⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟M

MX

f

f

f

f

f

f1 1

1

10

1

1,0

1

1,0

1,0

1 (2)

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

α =−r

r12

2

β =−r

r11

1

γ =−

− −r r

r r1

(1 )(1 )1 2

1 2

δ =−

− −r

r r1

22

1 2

Equation 2 describes the drift of monomer feed composition aseach monomer is consumed at different rates during thecopolymerization. This technique can make use of all the datacollected via in situ monitoring of the copolymerization, and iteliminates the low conversion assumption inherent in thelinearization techniques. Van den Brink et al.22 and Kazemi etal.26 have both applied eq 2 to cumulative composition data foracrylic copolymerizations using errors-in-variables-model(EVM) nonlinear regression. The EVM analysis consistentlyimproved the measurement precision (quantified via the jointconfidence regions (JCRs)) for reactivity ratios.We recently demonstrated that in situ Raman spectroscopy

can monitor the ring-opening copolymerization of ε-CL and δ-valerolactone (δ-VL).21 The model system of ε-CL and δ-VLwas chosen due to the similar homopolymerization behaviors ofthe two monomers via enzymatic catalysts. Both monomersundergo polymerization via Candida antarctica Lipase B(CALB) with pseudo-first-order kinetics, and the rate constantsof homopolymerization are 1.26 and 1.37 h−1, respectively.9 Asmentioned above, both homopolymers of ε-CL and δ-VL aresemicrystalline, but the copolymer exhibits a melting temper-ature significantly lower than both homopolymers. Copoly-merization has been investigated as a way to control the meltingtransition over a wide range. Previous work reported that thereactivity ratios for the copolymerization of ε-CL and δ-VL inthe bulk using Sn(Oct)2 as the catalyst were rε‑CL = 0.25 andrδ‑VL = 0.49.27 In our recent report, the ring-stretching peaks at696 cm−1 (ε-CL) and 745 cm−1 (δ-VL) were used to quantifymonomer consumption during the reaction. Reactivity ratiosdetermined using the K−T method agreed well with copolymercomposition data. In this case, the terminal model appeared toaccurately describe the enzyme-catalyzed ROP. Although thedata were in agreement, the Raman spectra indicated asignificant induction period for δ-VL that could have a largeinfluence on low conversion results. In this study, ring-openingcopolymerization behavior was fit to eq 2 using EVM regressionto determine if these conventional techniques could adequatelydescribe the composition of enzyme-catalyzed copolymers.Equation 2 accurately described the monomer drift duringcopolymerization, and this technique was used to determinereactivity ratios. As expected, the measurement uncertainty wassignificantly lower than the linearization techniques. Thecumulative composition model provides an enhanced analyticaltool to rapidly determine copolymerization reactivity ratios forROP with low measurement uncertainty.

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■ EXPERIMENTAL SECTIONMaterials. Toluene, ε-CL (97%), and δ-VL (technical grade) were

obtained from Sigma-Aldrich,a distilled over CaH2, and stored underAr prior to use. Novozyme 435 (N435) was obtained fromNovozymes (Bagsvaerd, Denmark) and stored in a vacuum desiccatorprior to use. To minimize experimental deviations caused by variationsin N435 particle size, a 400 μm sieve was used to select a particle sizedistribution of 400 ± 50 μm. Aluminum isopropoxide and titaniumisopropoxide were obtained from Sigma-Aldrich and used as received.Polymerizations. The reaction procedures, reactor design, and

data processing for enzyme-catalyzed copolymerizations with in situRaman monitoring are described in detail elsewhere.21 All reactionswere carried out under inert atmosphere using standard Schlenktechniques. For a typical enzyme-catalyzed copolymerization, N435beads (100 mg) and toluene (2 mL) were added to a 5 mL round-bottom 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 asecond flask via syringe and mixed under argon. To start the reaction,the monomer mixture was transferred using a cannula to the flaskcontaining catalyst. The exact starting monomer ratios were calculatedfrom the monomer mixture using Raman spectroscopy. Initiationoccurred from trace water present in the catalyst beads and distilledmonomers; no additional initiator was added. We previouslyinvestigated the amount of water present in the beads and foundthe value consistently around 200 ppm.28 For a typical metal-catalyzedreaction, ε-CL (0.83 mL, 7.5 mmol), δ-VL (0.70 mL, 7.5 mmol), andtoluene (1.5 mL) were added via syringe to the reaction flask underargon. In a separate flask, Al(OiPr)3 (30 mg) was dissolved in toluene(2 mL) under argon. The catalyst solution was added to the monomermixture via syringe to initiate the polymerization. In the metal-catalyzed polymerizations, the propoxide fragments on the catalystacted as initiator for the polymerization. For Ti(OiPr)4-catalyzedpolymerizations, 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, thestarting 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 errorin both the dependent and independent variables. A description andthe pseudocode of this algorithm are provided in the SupportingInformation. The fitting procedure involves the preliminary calculationof weighting factors assigned to the experimental data. The weightingfactors were derived through error propagation analysis from themeasurement uncertainty of the monomer concentrations, which hasbeen discussed previously.21 The error propagation and the structureof the weighting factors are both described in detail in the SupportingInformation.

■ RESULTS AND DISCUSSIONLinearization Techniques. The enzymatic ROP of ε-CL

and δ-VL is illustrated in Scheme 1. We recently reportedreactivity ratios of rε‑CL = 0.38 ± 0.06 and rδ‑VL = 0.29 ± 0.03using the K−T method and in situ Raman spectroscopic data.21

The reactivity ratios from monomer consumption agreed wellwith the copolymer compositions determined by 1H NMRanalysis, indicating that the terminal copolymerization modelcan accurately characterize enzymatic ROP. However, theRaman spectroscopy also revealed an induction period for δ-VL. Figure 1 shows the monomer concentrations and monomerconversion profiles over the course of a copolymerization withinitial feed ratio fε‑CL,0 = 0.61. During the first 5 min of

polymerization, ε-CL is consumed rapidly while the concen-tration of δ-VL remains relatively constant. This inductionperiod behavior has been observed previously.21 Pseudo-first-order kinetic behavior was observed for δ-VL only after the first5 min of polymerization, confirming the induction period(Figure S2). After the induction period, δ-VL polymerizes fasterthan ε-CL, as evident in the crossover of the two monomerconversion profiles. Because of this induction behavior, thereactivity ratios calculated from low conversion data likely donot represent the copolymerization behavior at moderate tohigh conversions.The Raman probe provided monomer consumption data

over the entire reaction, so the same experiments could beanalyzed using the eK−T method. Using monomer con-sumption data up to about 40% conversion, we calculated thereactivity 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 andeK−T methods are shown in Figure 2. The measured reactivityratios are lower from the eK−T method compared to the K−Tmethod. The value of rε‑CL decreased 29%, and the value of rδ‑VLdecreased 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 indicatesa correlation between the two reactivity ratios.29 In addition,both JCIs are large compared to the magnitude of the reactivityratio values.

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

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

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Error-in-Variables Method. The in situ Raman spectros-copy provides data over the entire polymerization at intervals aslow as 20 s. However, the linearization techniques condensethis spectroscopic data into one data point per reaction. On theother hand, eq 2 models the compositional drift duringcopolymerization and can use almost the entirety of thespectroscopic data. However, accurately modeling eq 2 isdifficult for multiple reasons. Specifically, both variables (X, f1)are dependent variables with different observational errors.EVM regression explicitly accounts for the error in bothregressors. Based on repeated measurements of samples withknown concentration, the measurement uncertainty for bothmonomers is known to be 0.05.21 The error propagationanalysis to determine the weighting factors and the calculationof the residuals is detailed in the Supporting Information. Inaddition, in the current form of eq 2 (X = g( f1)), regressionbecomes difficult due to the shape of the curve. For a given pairof reactivity ratios, r1 and r2, and starting composition f1,0, totalconversion X is not defined over the entire range of f1 from 0 to1. For a robust fit, we analyzed the experimental data in theform f1 = g(X) using an iterative algorithm to solve for f1 in eq2. This iterative EVM technique was implemented inMATLAB, and the pseudocode is provided in the SupportingInformation.Four copolymerizations were performed with feed compo-

sitions ranging from fε‑CL,0 = 0.40 to 0.61. Figure 3 plots thecomonomer conversion profiles for the four reactions as fε‑CL,0versus X. The starting concentrations were chosen to maximizethe monomer composition drift during copolymerization and toprovide two reactions on either side of the azeotropiccomposition, faz. The azeotropic composition was estimatedas faz = 0.48 from the eK−T reactivity ratios using therelationship

=−

− −f

rr r

12az

1

1 2 (3)

The azeotropic composition is evident in the conversionprofiles in Figure 3. Above faz, the fraction of unreacted ε-CLremaining increases as the reaction proceeds; below faz, fε‑CLdecreases as the reaction proceeds.Each individual reaction profile was fit using our MATLAB

EVM algorithm described above. The fits are illustrated as thesolid lines in Figure 3. The regression results appear toreasonably model the data. Figure 4 shows the weighted

residuals for all four reactions. The residuals appear normallydistributed, and their magnitude increases with conversion.This increase resulted from the relative increase in measure-ment error as the monomer concentrations approach zero. Toreduce 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 theSupporting Information. Figure 5 shows the 95% JCRscalculated from the SSR profiles. The JCRs illustrate thedifference in fitting results between the separate reactions. Thebest fit estimates for each reaction differ dramatically, andseveral estimates are near zero for at least one reactivity ratio.The JCRs are highly elongated and nonsymmetrical, indicatingvery strong rε‑CL−rδ‑VL correlations. Variations in the scatter ofthe data lead to smaller JCRs that appear elliptical or largerJCRs that appear boomerang-shaped. The elliptical JCRs have astrong slope that appears to decrease as the initial mole fractionof ε-CL increases. Different initial feed ratios influence thecorrelation between reactivity ratios, resulting in differentorientations of the JCRs.22 More specifically, at high feed ratiosof ε-CL, rε‑CL is undetermined; at low feed ratios of ε-CL, rδ‑VLis undetermined. The lack of overlap between JCRs can result

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

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

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

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from small systematic errors in data collection, including effectsof incorrect baseline corrections, initial feed ratios, andnormalizations on the data analysis. The influence of thesesystematic errors can be reduced dramatically, and the accuracyof the estimate increased by combining analysis and differentfeed ratios, as discussed in the following sections.Repeatability. To investigate the repeatability of the

experiments and the robustness of the fitting algorithm, weran three reactions under the same conditions and at similarstarting monomer compositions. The JCRs are shown in Figure6, and the conversion profiles are shown in the SupportingInformation as Figure S4. The JCRs again vary dramatically in

size, presumably controlled by the scatter within theexperimental data. Although there is only moderate overlapbetween the individual JCRs, the best fit estimates are relativelyclose and the value for rε‑CL is close to zero. In addition, allthree JCRs have roughly the same slope, suggesting that theprimary factor controlling shape and location of the JCR isinitial monomer feed ratio. The fact that the individual JCRsoverlap indicates that the systematic errors in data collectionare 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 datapoints for analysis. The SSR matrices from the three individualreactions in Figure 6 were combined to calculate a compositebest fit. The SSRs were combined using a relative weightingscale, with the combined SSR defined as

∑=∑ε δ

ε δ‐ ‐

=‐ ‐

⎛⎝⎜

⎞⎠⎟SSR r r

nn

r r( , )

SSR ( , )SSRk

ii

i

i

icombined CL VL

1CL VL

,min

(4)

where k is the number of experiments, ni is the number of datapoints in the ith experiment, and SSRi,min is the minimum SSRin the ith experiment (at the estimated reactivity ratio). Therelative weighting accounts for differences in the number ofdata points in each set and prevents one data set withsignificantly higher errors from dominating the combinedresult. The combined JCR is also plotted in Figure 6. Thecombined analysis appears to be a composite of the threeindividual reactions, but the size of the JCR is not significantlyreduced. However, the best fit reactivity ratios for the combinedanalysis are dramatically different at rε‑CL = 0.23 and rδ‑VL =0.35. Combining experimental data sets into a single compositeresult reduces the impact of the systematic errors and improvesthe accuracy of the model fit.

Combined Experiments. Combining the analysis of threecopolymerizations at the same feed composition appeared toincrementally improve the determination of reactivity ratios.However, each individual reaction only covers a small range offeed compositions, which can introduce bias into the result.Combining the results of copolymerizations at different startingfeed ratios leads to improved estimates.22,29 Combining thefour reactions from varied feed ratios leads to estimatedreactivity ratios of rε‑CL = 0.27 and rδ‑VL = 0.39. The monomerconversion profiles with overlaid best fit results are shown inFigure 7. This estimate is quite close to the combined analysisat the same feed composition presented above. Figure 8a showsthe combined JCR in comparison with the individual reactionJCRs. Intuitively, the result from the composite analysis appearsto 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 resultindicates the importance of choosing initial monomer feedcompositions such that the combined compositional drift of allreactions 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 theEVM and linearization results, as shown in Figure 8b. Bothlinearization methods are strongly biased by the short inductionperiod for δ-VL, which lasts until about 10% total monomerconcentration. Even the eK−T technique, while analyzing thereaction up to about 40% total conversion, is heavily influencedby this induction behavior. The EVM analysis uses data fromthe first 90% of conversion, significantly reducing the impact of

Figure 5. 95% JCRs for individual copolymerizations of ε-CL and δ-VL 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 andthe lines are the 95% JCRs.

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

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the low conversion behavior on the results. It is quitereasonable that the EVM analysis estimated reactivity ratioswith a lower rε‑CL and a higher rδ‑VL. As observed in Figure 1,after the induction period, δ-VL is consumed significantly fasterthan ε-CL.It should be noted that the point estimate obtained from

combined experiments introduces a bias in the residuals whenapplied to the individual experiments. The bias can be observedin Figure 7, where the best fit curves do not exactly model theexperimental data for all reactions. The bias occurs in oppositedirections for the different experiments. This bias could be the

result of an inappropriate model, or it could be due tosystematic errors in the data collection. Small measurementerrors are evident in the results for the reaction at fε‑CL,0 = 0.52,which is very close to the predicted azeotropic composition offaz = 0.54. Systematic errors in the calculation of monomer ratiocould explain why the reported composition is below theazeotropic composition, but the experimental data indicate thatit is above the azeotrope. Recently, we modeled the kineticpathways of enzyme-catalyzed ROP.30 The model andaccompanying experimental results suggested that largenumbers of cyclic oligomers formed during the initial stagesof polymerization, followed by elongation and formation oflong linear chains. During the late stages of polymerization,chain elongation via polycondensation becomes more preva-lent. This enzyme-catalyzed mechanism may deviate fromterminal model kinetics enough to account for the observedsystemic errors, especially at the late stages of polymerization.Nonetheless, eq 2 still provides a reasonable fit to theexperimental data. We are currently further probing themechanism of ROP for enzyme and metal catalysts to confirmthe applicability of the terminal model. This will be the subjectof a future report.

Other Catalyst Systems. To investigate the applicability ofthis reactivity ratio technique to other catalyst systems, ε-CLand δ-VL were copolymerized using the metal catalystsAl(OiPr)3 and Ti(OiPr)4. Both of these metal alkoxide catalystsfunction through the coordination−insertion mechanism.31,32

Unlike enzymatic catalysts, which result in cyclic oligomers andnumerous propagating chains per catalytic site, the propagatingchains grow in a continuous manner more consistent with theterminal model of copolymerization. Compared to theenzymatic catalysts, no strong induction period was observedin the metal-catalyzed polymerizations. The reactivity ratios forthese copolymerizations via enzyme and metal catalysts arepresented in Table 1. As with the enzyme-catalyzed

copolymerizations, large deviations were observed betweenthe results of the EVM and K−T methods for metal-catalyzedcopolymerizations, although these deviations were within the95% JCRs. Compared to both metal catalysts, the enzymecatalyst exhibited reactivity ratios much further from unity,corresponding to more alternating behavior. This selectivitycould result from the structure of the lipase’s active site. Themetal catalysts studied are less selective between the twomonomers and the reactivity ratios under those conditions arecloser to unity. Further experiments are required to confirm theselectivity of the enzyme catalysts.

■ CONCLUSIONSWe applied both conventional linearization techniques and anonlinear model of compositional drift to estimate reactivityratios for ring-opening copolymerizations. In situ Raman

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

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

Table 1. Reactivity Ratios for the Copolymerization of ε-CLand δ-VL Using Different Catalyst Systems

catalyst/conditions K−T method EVM method

N435 (enzyme) rε‑CL = 0.38 rε‑CL = 0.2755 °C rδ‑VL = 0.29 rδ‑VL = 0.39Al(OiPr)3 rε‑CL = 0.96 rε‑CL = 0.5230 °C rδ‑VL = 0.87 rδ‑VL = 0.78Ti(OiPr)4 rε‑CL = 0.82 rε‑CL = 0.4430 °C rδ‑VL = 1.00 rδ‑VL = 0.51

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spectroscopy monitored the simultaneous conversion of ε-CLand δ-VL to eliminate the need for offline measurements ofreaction aliquots. As expected, the EVM method resulted in asignificantly reduced JCR with fewer experiments. Theinduction period for δ-VL significantly influenced the resultsof linearization techniques, leading to a discrepancy betweenresults from the different methods. The EVM method modelsdata over most of the reaction and provides a more accuraterepresentation of the copolymer at moderate to highconversions. However, the EVM technique must be extendedto other monomer pairs to verify its applicability andconsistency with results from conventional methods and toexamine the model’s limitations. We can conclude that thecumulative composition model reasonably describes theenzyme- and metal-catalyzed ROP of cyclic esters. The EVMtechnique provides a rapid tool to determine reactivity ratioswith less experimental work required, reducing both the timerequired and material waste.

■ ASSOCIATED CONTENT*S Supporting InformationMore information relating to the error-in-variables regressionanalysis, residual calculation, error analysis, calculation of jointconfidence intervals, and pseudocode for the determination ofreactivity ratios. This material is available free of charge via theInternet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*E-mail beers@nist.gov.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThe authors thank Novozyme for providing the N435 beads.M.H. acknowledges the financial support of the NationalResearch Council Fellowship Program.

■ ADDITIONAL NOTEaCertain commercial equipment, instruments, or materials areidentified in this paper in order to specify the experimentalprocedure adequately. Such identification is not intended toimply recommendation or endorsement by the NationalInstitute of Standards and Technology, nor is it intended toimply that the materials or equipment identified are necessarilythe best available for the purpose.

■ REFERENCES(1) Arbaoui, A.; Redshaw, C. Polym. Chem. 2010, 1, 801−826.(2) Kobayashi, S. Macromol. Rapid Commun. 2009, 30, 237−266.(3) Kiesewetter, M. K.; Shin, E. J.; Hedrick, J. L.; Waymouth, R. M.Macromolecules 2010, 43, 2093−2107.(4) Tripathy, A. R.; MacKnight, W. J.; Kukureka, S. N. Macro-molecules 2004, 37, 6793−6800.(5) Li, J.; Stayshich, R. M.; Meyer, T. Y. J. Am. Chem. Soc. 2011, 133,6910−6913.(6) Park, T. G. Biomaterials 1995, 16, 1123−1130.(7) McDonald, P. F.; Lyons, J. G.; Geever, L. M.; Higginbotham, C.L. J. Mater. Sci. 2010, 45, 1284−1292.(8) Wei, Z.; Liu, L.; Qu, C.; Qi, M. Polymer 2009, 50, 1423−1429.(9) Van der Mee, L.; Helmich, F.; de Bruijn, R.; Vekemans, J.A.J.M;Palmans, A. R. A.; Meijer, E. W. Macromolecules 2006, 39, 5021−5027.

(10) Vion, J.-M.; Jerome, R.; Teyssie, P.; Aubin, M.; Prud’homme, R.E. Macromolecules 1986, 19, 1828−1838.(11) Wang, Y.; Ma, H. Chem. Commun. 2012, 48, 6729−6731.(12) Darensbourg, D. J.; Karroonnirun, O. Macromolecules 2010, 43,8880−8886.(13) Nomura, N.; Akita, A.; Ishii, R.; Mizuno, M. J. Am. Chem. Soc.2010, 132, 1750−1751.(14) Mayo, F. R.; Lewis, F. M. Macromolecules 1944, 66, 1594−1601.(15) Fineman, M.; Ross, S. D. J. Polym. Sci. 1950, 5, 259−262.(16) Kelen, T.; Tudos, F. J. Macromol. Sci., Chem. 1975, A9, 1−27.(17) Tudos, F.; Kelen, T.; Foldes-Berezsnich, T.; Turcsanyi, B. J.Macromol. Sci., Chem. 1976, A10, 1513−1540.(18) McFarlane, R. C.; Reilly, P. M.; O’Driscoll, K. F. J. Polym. Sci.,Polym. Chem. Ed. 1980, 18, 251−257.(19) Behnken, D. W. J. Polym. Sci., Part A 1964, 2, 645−668.(20) Pasquale, A. J.; Long, T. E. J. Appl. Polym. Sci. 2004, 92, 3240−3246.(21) Hunley, M. T.; Bhangale, A. S.; Kundu, S.; Johnson, P. M.;Waters, M. S.; Gross, R. A.; Beers, K. L. Polym. Chem. 2012, 3, 314−318.(22) Van Den Brink, M.; Smulders, W.; Van Herk, A. M.; German, A.L. J. Polym. Sci., Part A: Polym. Chem. 1999, 37, 3804−3816.(23) Catalgil-Giz, H.; Giz, A.; Alb, A. M.; Oncul Koc, A.; Reed, W. F.Macromolecules 2002, 35, 6557−6571.(24) Skeist, I. J. Am. Chem. Soc. 1946, 68, 1781.(25) Meyer, V. E.; Lowry, G. G. J. Polym. Sci., Part A 1965, 3, 2843−2851.(26) Kazemi, N.; Duever, T. A.; Penlidis, A. Macromol. React. Eng.2011, 5, 385−403.(27) Storey, R. F.; Herring, K. R.; Hoffman, D. C. J. Polym. Sci., PartA: Polym. Chem. 1991, 29, 1759−1777.(28) Kundu, S.; Johnson, P. M.; Beers, K. L. ACS Macro Lett. 2012, 1,347−351.(29) Plaumann, H. P.; Branston, R. E. J. Polym. Sci., Polym. Chem. Ed.1989, 27, 2819−2822.(30) Johnson, P. M.; Kundu, S.; Beers, K. L. Biomacromolecules 2011,12, 3337−3343.(31) Kowalski, A.; Libiszowski, J.; Majerska, K.; Duda, A.; Renczek, S.Polymer 2007, 48, 3952−3960.(32) Ouhadi, T.; Stevens, C.; Teyssie, P. Makromol. Chem., Suppl.1975, 1, 191−201.

Macromolecules Article

dx.doi.org/10.1021/ma302015e | Macromolecules 2013, 46, 1393−13991399