Recommendation for Accurate Experimental Determination of

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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Recommendation for Accurate Experimental Determination of Reactivity Ratios in Chain Copolymerization Nathaniel A. Lynd,*,† Robert C. Ferrier, Jr.,‡ and Bryan S. Beckingham§ †

McKetta Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas 78712, United States Department of Chemical Engineering and Materials Science, Michigan State University, East Lansing, Michigan 48824, United States § Department of Chemical Engineering, Auburn University, Auburn, Alabama 36849, United States ‡

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S Supporting Information *

ABSTRACT: A set of copolymerization data at prescribed reactivity ratios was numerically generated and then fit using common methods of data analysis including the copolymer equation, Fineman−Ross, Kelen− Tüdös, and integrated methods of data analysis, such as those reported by Beckingham, Sanoja, and Lynd, and Meyer and Lowry. Significantly, the nonintegrated approaches based on the copolymer equation returned systemically inaccurate reactivity ratios, whereas the integrated methods produced consistently accurate reactivity ratios across 560 calculated data sets. Hence, to determine reactivity ratios with the greatest accuracy and efficiency, we recommend that copolymerization data be fit simultaneously to the models reported by Beckingham−Sanoja−Lynd (BSL) and Meyer−Lowry (ML). If the reactivity ratios are consistent, then a nonterminal model of copolymerization adequately describes the copolymerization with a single reactivity ratio parameter. If there is a difference in the reactivity ratios between BSL and ML, then the ML-derived values take precedence and a terminal model of copolymerization describes the kinetics of the system with two independent reactivity ratios. This prescription will ensure that the model with the least complexity will be used to interpret data, and that the reactivity ratios reported are most accurate and descriptive of the underlying copolymerization mechanism. Future use of the copolymer equation, Fineman−Ross, and Kelen−Tüdös to interpret copolymerization data is strongly discouraged due to unquantifiable inaccuracy and needlessly wasted experimental effort.

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INTRODUCTION Copolymers, which consist of more than one monomer repeat unit type in a single polymer backbone, form the basis for a wide range of polymeric materials used by society. The versatility of copolymers is due to the intimate connection between polymer structure and properties and the ease of compositional control of structure−property relationships in copolymer materials. Few concepts are more central to the ability of polymer chemists to adapt new materials to technological need than statistical control of polymer structure and composition enabled through accurately known reactivity ratios. The statistics of incorporation of two or more comonomers into a single polymeric backbone is rarely truly random. Knowledge of the deviation from randomness of a statistical copolymerization can be extracted by determining reactivity ratios that quantify the tendency of a monomer to self-propagate. Wherever polymer structure connects to polymer properties in multicomponent copolymer materials synthesized via statistical synthetic procedures,1,2 accurate reactivity ratio pairs must be considered as part of a complete understanding of polymer structure.1 The determination of reactivity ratios is generally considered to be a daunting task in both the amount of laboratory work required and in selection of the most appropriate model for © XXXX American Chemical Society

data interpretation. The literature on reactivity ratio determination is vast, with most of the foundational work occurring many decades in the past2−5 and mostly before the routine use of the computer and accurate spectroscopic tools for measuring polymer composition. Methods of determining reactivity ratios have been surveyed in the past.6−10 These studies relied on experimental data where the true reactivity ratios cannot be known ahead of time,11,12 and were focused on evaluating the precision of models and fitting routines.10−13 Compounding confusion for the experimentalist, different methods extract different reactivity ratios each with their own error from the same experimental system.6,13 It is impossible to know which value is most accurate, and true values have no independent method of verification since all methods are based on the same foundational copolymer equation and its assumptions. In this Article, we will present a renewed understanding of reactivity ratio determination and make a recommendation for the accurate experimental determination of reactivity ratios. To accomplish this, we calculate 35 numerically defined copolymerization kinetic data sets with Received: August 14, 2018 Revised: February 21, 2019

A

DOI: 10.1021/acs.macromol.8b01752 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

equation reported decades ago by Meyer and Lowry. The direct use of the copolymer equation or its various linearizations such as Fineman−Ross or Kelen−Tüdös, are strongly discouraged due to increased and unnecessary experimental effort, arbitrary implementation, and inherent inaccuracy even under the most ideal conditions. We recommend only the use of integrated methods to determine reactivity ratios accurately and with minimal experimental effort.

prescribed reactivity ratios as shown in Table 1. We then apply typical routines for determining reactivity ratios from our Table 1. Summary of Numerical Copolymerization Data Generated Using Reactivity Ratios Corresponding to Five Categories of Copolymerization Behavior run

rA × rB

pure gradients rA × rB = 1 1 1.000 2 1.000 3 1.000 4 1.000 5 1.000 6 1.000 7 1.000 8 1.000 9 1.000 10 1.000 impure gradients rA × rB ≠ 1 11 0.250 12 0.500 13 0.750 14 1.250 15 1.500 16 1.750 17 2.000 18 2.250 19 2.500 symmetrically alternating rA × rB ≈ 0 20 0.000 21 1.000 × 10−6 22 1.000 × 10−4 23 1.000 × 10−2 asymmetrically alternating rA × rB ≈ 0 24 0.000 25 0.000 26 0.000 blocky rA × rB ≫ 1 27 4.000 28 9.000 29 16.00 30 25.00 31 36.00 32 49.00 33 64.00 34 81.00 35 100.0

rA

rB

1.000 1.111 1.250 1.429 1.667 2.000 2.500 3.333 5.000 10.00

1.000 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100

0.278 0.625 1.071 2.083 3.000 4.375 6.667 11.25 25.00

0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100

0.000 0.001 0.010 0.100

0.000 0.001 0.010 0.100

0.100 2.000 10.00

0.000 0.000 0.000

2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.00

2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.00

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KINETIC MODEL AND COMPUTATIONAL METHODS To ascertain the accuracy of methods used to extract reactivity ratios from kinetic data, sets of model data with known reactivity ratios were first generated numerically. Compositional data were numerically generated by integrating a series of coupled ordinary differential equations describing the timedependent change in concentration of initiator (I) and two comonomers (A and B). Initiation rate constants for A and B were given by kA and kB. The rate-law describing initiation was given by dI = −kAI(t ) A(t ) − kBI(t ) B(t ) dt

(1)

subject to the initial conditions I(0) = I0 = 0.001 for all calculations. The initiation rate constants were set to kA = kB = 20 M−1 s−1, whereas propagation rate constants were generally much lower. The rate constants of initiation were at least 10 times as large as propagation rate constants in order to remove any artifacts associated with slow initiation. It should be noted that the chosen value (i.e., 20 M−1 s−1) for the initiation rate constant is sufficiently large as to make initiation effectively instantaneous and increasing the initiation rate constant to a much higher value has no effect other than to decrease the efficiency of the integration method. In a nonterminal model of copolymerization, one propagation rate constant is required per monomer. This simple model of copolymerization has been used recently to extract reactivity ratios for a variety of ionic and pseudoionic copolymerizations and will not be derived here.16−22 In the standard terminal model of copolymerization, two propagation rate constants are required per monomer to describe the binary reaction between two unique chain-ends and either monomer. Propagation rate constants are given by kAA and kAB for the respective reactions between an A-terminal chain-end (XA) and the A and B monomers; the reactivity ratio rA = kAA/kAB is a measure of the tendency toward self-propagation of A-chain ends to add additional A monomers. Likewise, the propagation rate constants for a B-terminal chain-end (XB) with the A and B monomers are, respectively, kBA and kBB, and the reactivity ratio for the B-chain end is given by rB = kBB/kBA. The rate laws describing the consumption of comonomers in the terminal model are given by eq 2.

model data while approximating varying levels of data quality in terms of reaction conversion and its associated compositional drift. The tolerance of each method to these experimental considerations under the best possible circumstances is presented. Our study will not provide a detailed error analysis of each method and instead will present the merit of each method based on its ability to reproduce known reactivity ratios accurately. The conclusion of this work is that the most accurate methods for determining reactivity ratios are the integrated models such as Beckingham−Sanoja−Lynd for nonterminal copolymerizations and Meyer−Lowry for terminal copolymerizations. Direct numerical integration of the copolymer equation may also be accurate,14,15 but it is not an improvement on the analytical integration of the copolymer

dA = −kAI(t ) A(t ) − kAAXA(t ) A(t ) − kBAXB(t ) A(t ) dt dB = −kBI(t ) B(t ) − kBBXB(t ) B(t ) − kABXA(t ) B(t ) dt (2)

for A and B comonomers subject to initial conditions A(0) = fA0 and B(0) = 1− fA0. We required A(0) + B(0) = 1 for all calculations, and fA0 = A(0)/(A(0)+B(0)) and fA = A(t)/(A(t) B


Article

Macromolecules + B(t)). The change in concentration of active chains possessing an A-monomer or B-monomer terminus are given by

detailed compositional and sequential characterization to distinguish penultimate from terminal model copolymerizations.25−27 However, distinguishing between all three models (i.e., nonterminal, terminal, penultimate) based on compositional data alone is not possible.28 To make sufficient measurements to distinguish between these models generally requires specific sequential characterization of heterosequences along the polymer backbone, which is not available in all copolymer systems.27 In light of these considerations, we limit our analysis to the nonterminal and terminal models. For the terminal model, the concept of reactivity ratios and the copolymer equation were introduced in 1944 by Mayo and Lewis,4 Wall,2 Alfrey and Goldfinger,3 and Walling and Briggs,29 who proposed a description for the terminal model of copolymerization making use of a steady-state assumption where the concentration of active growing chain ends is constant. While the copolymer equation (eq 4) is named for Mayo and Lewis, it was Frederick T. Wall who published the most convenient and commonly used form of the equation used today:2

dXA = kAI(t ) A(t ) + kBAXB(t ) A(t ) − kABXA(t ) B(t ) dt dXB = kBI(t ) B(t ) + kABXA(t ) B(t ) − kBAXB(t ) A(t ) dt (3)

subject to the initial conditions XA(0) = XB(0) = 0.0. All equations were integrated simultaneously using an adaptive Runge−Kutta integrator with error tolerance set to near machine precision for all calculations.23 The source code is given in the Supporting Information as well as a program to automate input file generation and a sample input script. (Source code, input files, and output data files may be obtained by contacting the authors.) Model copolymerization data were generated at different initial A-monomer feed compositions (fA0) and reactivity ratios. The reactivity ratios were related to propagation rate constants by setting rA to kAA, and rB to kBB with kAB = kBA = 1. Changing the values of cross product rate constants (kAB, kBA) at equivalent reactivity ratios did not alter the nature of a copolymerization. The behavior of a copolymerization can be characterized by the product of the reactivity ratios rA × rB = γ. Kinetic data were generated in 35 different calculation groups belonging to five classes of copolymer behavior: pure gradient (nonterminal model, γ = 1), impure gradient with alternating (γ < 1) or blocky (γ > 1) tendencies, fully alternating (γ = 0), partially alternating (γ ≈ 0), and blocky copolymerizations (γ ≫ 1). For each set of reactivity ratios, copolymerizations were conducted at initial feeds of fA0 = 0.01, 0.03, 0.05, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.85, 0.90, 0.95, 0.97, and 0.99. This resulted in a total of 560 distinct data sets that were used to evaluate the accuracy of reactivity ratios determined using the five methodologies described in the next section.

FA =

rAfA 2 + fA fB rAfA 2 + 2fA fB + rBfB2

(4)

where FA is not strictly the composition of the copolymer, but rather the relative velocity of A-monomer incorporation into the copolymer over an infinitesimally small change in conversion (eq 5), fA is the instantaneous molar feed composition of A monomers, f B = 1 − fA, and rA and rB are the reactivity ratios for the A- and B-chain ends, respectively. FA =

−dA /dt −dA /dt − dB /dt

(5)

Over a vanishingly small increment in monomer conversion, the incremental change in copolymer compositions FA would approximate the composition of the copolymer. As written, this equation can be used to extract reactivity ratios through a nonlinear least-squares fitting of eq 4 to a plot of polymer composition (FA) vs initial feed composition (fA0) over a small and undefined increment of conversion. The fact that eq 4 cannot account for the time-evolution of the copolymerization presents an inherent limitation to the accuracy of methods based on the original, nonintegrated copolymer equation. Vanishingly small conversions are required for eq 4 to be valid. However, at those low conversions a given copolymerization may not have reached a steady state concentration of growing polymer, and more importantly, the copolymerization may not yet have consumed enough monomer to present a statistically meaningful, representative, and measurable result. Mayo and Lewis presented a graphical method of determining reactivity ratios;4 however, it has been superseded by more consistent methods. At the time of the original conceptual development leading to eq 4, linearization of models was typically employed to facilitate convenient extraction of information from graphical data. The first linearization of eq 4 was put forth by Fineman and Ross in 1950.5 Their derivation was based on the version of the copolymer equation published by Mayo and Lewis, and through simple substitutions and rearrangements they generated the linear form:

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MODELS AND METHODS OF EXTRACTING REACTIVITY RATIOS Three kinetic models are currently used to model copolymerization data. The nonterminal model of copolymerization is the simplest, and is often applicable to ionic and pseudoionic copolymerizations where the compositionally determining step may be a competitive coordination event to a metal center.16 Mathematically, this system only requires a single propagation rate constant per monomer. The coupled system of ordinary differential equations describing the nonterminal model can be integrated analytically, and the compositional drift that occurs at any conversion during the copolymerization can be used to extract reactivity ratios as described by Beckingham et al.16 The nonterminal model interprets copolymerizations as pure gradients without any alternating or blocky character. The second model is the classical terminal model developed originally to describe radical copolymerization, but has since been adapted to nearly all chain copolymerization mechanisms. The terminal model requires two propagation rate constants per monomer and describes the synthesis of copolymers with random, gradient, alternating, and blocky characteristics. As a further elaboration, the penultimate model accounts for differences in chain-end reactivity depending on the last two monomers that were enchained,24 and requires four propagation rate constants per monomer. Specialists have used C


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Macromolecules f f2 = rA − rB F F f F F = A, f = A FB fB

In 1965, Meyer and Lowry published an integrated, conversion dependent version of the copolymer equation that substantially improved upon all of the inherent weaknesses of the nonintegrated versions of the copolymer equation schemes. The derivation was based on a comparison between distillation and copolymerization due originally to Skeist.30 The starting composition of the binary monomer feed in a copolymerization can be described by the mole-fraction of Amonomers fA, and the total number of A- and B-monomers, L. The total composition of the unreacted monomer and repeat units is equal to the starting composition. After the copolymerization has proceeded, the total number of moles in the monomer feed will have been reduced by those that became copolymer, dL. The composition of the monomer feed will adjust by quantity df, and the composition of the copolymer will be adjusted to the amount yA. Based on these considerations, the overall mole balance for the copolymerization is given by the following:

(F − 1)

(6)

In this formalism, a plot of (F−1)f/F vs f 2/F has a slope of rA and an intercept of −rB. Unfortunately, the same limitations of a nonintegrated copolymer equation remain, the most significant being that eq 6 is only valid if the monomer and polymer compositions have changed by infinitesimally small values. Or, in the most experimentally practical sense, that the monomer conversion is kept very low, subjectively less than 5% in practice. A clustering of data points about the origin is generally evident using the Fineman and Ross (FR) method that weighs the low composition data more heavily in the linear least-squares fitting. In 1974, Kelen and Tüdös presented a reparameterization of the FR model to spread the data points more evenly along the x-axis thus improving the error structure of the model fitting. The Kelen−Tüdös (KT) linearization resulted and is shown below: r y r i η = jjjrA + B zzzξ − B α{ α k

η=

ξ=

fA L = (fA − df )(L − dL) + yA dL

By expanding the mole balance, and ignoring the products of differentials, rearrangement leads to the integral conversioncomposition Skeist equation:

(7)

dfA dL = L yA − fA

FAFB(fA − fB ) 2

FA fB + αfA FB

2

ij L(2) yz lnjjjj (1) zzzz = kL {

(8)

FA 2fB FA 2fB + αfA FB2

∫f

(10a) fA 0

A

1 df yA − fA A

(10b)

Meyer and Lowry recognized that the term yA in eq 10b corresponded to the instantaneous change in composition of the analogous copolymer after an infinitesimally small change in conversion, which was described exactly by the copolymer equation (eq 4). Inserting eq 4 for yA, and integrating from the starting feed composition at t = 0, fA0, to the ending feed composition at time t, fA, resulted in the Meyer−Lowry equation (eq 11):

(9)

In the KT method, the reparameterized variable η is plotted versus ξ yielding an intercept equal to −rB/α, and a slope of (rA + rB/α). The value of α is discretionary, and it does not affect the accuracy of the KT method; it merely affects how spread the data will be along the ξ axis. The KT method is based on the copolymer equation (eq 4) and is subject to the same limitations and assumptions. The copolymer equation, FR, and KT represent three commonly utilized approximate models to extract reactivity ratios from low conversion experimental data. Unfortunately, the usage of these methods can be labor intensive and requires a series of copolymerizations conducted at different initial feed compositions fA0 that are terminated at low conversion. A key question for application of these approaches still remains: Which conversion produces the most accurate estimate of reactivity ratios for a given copolymerization? Vanishingly low conversion data is required for the fundamental model to be applicable. However, experimental difficulty such as low quantities of synthesized material, purification biasing the composition of the isolated materials, and sampling error due to low copolymerization conversion may affect the accuracy and precision of experimental reactivity ratios. A penetrating contribution published approximately a decade prior to KT presented a method that provided an integrated version of the copolymer equation that could in theory utilize copolymerization data at any and all conversions for homogeneous copolymerization without appreciable changes in rate constants with conversion. An integrated approach entirely circumvents the inherent contradictions plaguing the approximate linearizations based on the copolymer equation.

ij f yz B conv = 1 − jjjj A0 zzzz jf z k A{

ij 1 − f yz A A z jj zz jj j 1 − f 0 zz A k {

r /(1 − rB)

r /(1 − rA)

ji f (2 − rA − rB) − rB − 1 zyz zz × jjjj A0 j f (2 − rA − rB) − rB − 1 zz k A {

(rArB− 1) /(1 − rA)(1 − rB)

(11)

The Meyer−Lowry equation relates the overall conversion of the copolymerization given by conv = 1 −

A ( t ) + B (t ) A(0) + B(0)

(12)

to the time-dependent unreacted monomer feed composition given by fA. The first two product terms of the Meyer−Lowry equation (eq 11) account for pure gradient character, and the last term accounts for nonidealities such as alternation or blockiness in the copolymerization. The deviation of rA × rB from unity dictates the relative importance of this term. The Meyer−Lowry (ML) equation represents an exact, integrated description of copolymerization kinetics at all conversions. Eq 11 does suffer from some trivial numerical pathologies. If rA = rB = 1, then eq 11 is undefined. However, for a truly random D


É ÄÅ ÅÅ ij B(t ) yzÑÑÑÑ Å j z Å conv = 2(1 − fA )ÅÅ1 − jj zzÑÑÑ ÅÅÅ (0) B (17b) k {ÑÑÑÖ Ç Copolymerizations that are perfectly alternating follow compositional drift consistent with eq 17a, and b at all accessible conversions. In the next section, we generate numerical copolymerization kinetic data sets and fit those data with the above formalisms in order to extract reactivity ratios and compare accuracy as a function of composition, conversion, and methodology. Article

Macromolecules copolymerization, no fitting routine would be required. The composition of the copolymer would always equal that of the feed. Additional numerical pathologies exist at rA = 1, rB ≠ 1, and vice versa. However, expressions for these cases can be found in the appendix of ref 31. The ML equation provided a conceptual framework that introduced no new parameters and could naturally account for, and in fact use, the compositional drift that occurred during copolymerization to determine reactivity ratios within the context of the full terminal model of copolymerization. Finally, Beckingham et al. (BSL) reported simple expressions to calculate reactivity ratios from compositional drift of pure gradient (i.e., ideal) copolymerizations.16 These expressions were based on the following definition of the conversion for a copolymerization: i A(t ) yz i y zz − (1 − f 0 )jjj B(t ) zzz conv. = 1 − fA 0 jjjj z z A j k A(0) { k B(0) {

0

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RESULTS AND DISCUSSION An adaptive Runge−Kutta integration was applied to eqs 1−3 for 35 copolymerization types with reactivity ratios shown in Table 1. The program along with instructions for its use can be found in the Supporting Information (SI). The error tolerance of the adaptive Runge−Kutta was set close to machine precision on a 64-bit personal workstation. Representative numerically simulated data for run 15 with A(0) = 0.4 and B(0) = 0.6 are shown in Figure 1. Initiation occurs nearly

(13)

Relationships between the consumption of the comonomers were derived for pure gradient copolymers based on a nonterminal model for chain copolymerization similar to those defined originally by Wall in 1941:32 ij A(t ) yz ij B(t ) yz jj z j z j A(0) zz = jj B(0) zz k { k {

α

(14)

For the case of a pure gradient copolymer only a single reactivity ratio is required: α = kA/kB = rA = 1/rB. Inserting this into the expression for conversion resulted in eq 15a and b which relates the fractional concentration of each monomer to the overall conversion. Equation 15 is strictly correct for pure gradients, and was shown to be broadly applicable to a variety of ionic and pseudoionic catalytic copolymerizations.16 However, an excellent exposition of important cases where a coordination polymerization mechanism could result in nonideal behavior has been reported by Tsai and Register.33 i A(t ) yz i yB zz − (1 − f 0 )jjj A(t ) zzz conv = 1 − fA 0 jjjj z z A j k A(0) { k A(0) { r

i B(t ) zy A i y zz − (1 − f 0 )jjj B(t ) zzz conv = 1 − fA jj z z A j k B(0) { k B(0) { 0j j

(15a)

r

(15b)

The reactivity ratios rA and rB may be restricted to a reciprocal relationship and fit as a single parameter, or as independent parameters. An additional integrated model can be formulated for perfectly alternating copolymerizations: 1 − fA 0 1 − fA 0 ij B(t ) yz ij A(t ) yz jj zz = 1 − jj z + j A(0) z j B(0) zz 0 0 f f k { k { A A

fA 0 fA 0 ij A(t ) yz ij B(t ) yz jj zz = 1 − jj zz + j B(0) z 1 − fA 0 1 − fA 0 jk A(0) z{ k {

Figure 1. Representative copolymerization data are shown for run 15 (Table 1) with rA = 3.0, rB = 0.5, and A(0) = 0.4, B(0) = 0.6 (i.e., fA0 = 0.4). Full initiation of the copolymerization is complete before appreciable consumption of comonomers.

instantaneously so as not to interfere with comonomer consumption. The program outputs a data file formatted for each method of determining reactivity ratios described in detail in the SI. The output files were read by a Mathematica program that fit reactivity ratios at a fixed conversion goal across all starting feed compositions (fA0) for the nonintegrated methods (copolymer, FR, KT), and used all available conversion data at each starting feed composition (fA0) for integrated methods (ML and BSL). Figure 2 is a representative subset of the fits to copolymerization kinetic data. The model fit to a single starting conomoner feed composition and conversion is presented on top of a

(16a)

(16b)

When related to the overall conversion of a copolymerization the following two, new reactivity ratio independent linear expressions result: É ÅÄÅ i A(t ) zyÑÑÑÑ j 0Å Å j z Å Ñ conv = 2fA ÅÅ1 − jj zzÑÑ ÅÅÅ (17a) k A(0) {ÑÑÑÖ Ç E


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Figure 2. For a single copolymerization (run #15, rA = 3.0, rB = 0.5), the fit to each model (curve) is excellent using model numerical data (points). Reactivity ratios deviate significantly and nonsystematically from the true values (line) for approximate methods (left). Integrated methods due to Meyer−Lowry and Beckingham et al. (fit to run #8, rA = 3.33, rB = 0.30) provide a more accurate description of copolymerization kinetics (right).

determination using methods based on the copolymer equation increases as rA and rB approach unity. The deviation from the nonintegrated models is not due to the steady-state assumption employed in the derivation of eq 4. The steady-state approximation refers to a steady-state concentration of living chain ends, which is a valid assumption in many living ionic and catalytic systems.34,35 Moreover, the steady-state assumption is not a required condition for the derivation of the copolymer equation (eq 4).36,37 The steadystate assumption does not introduce any significant systematic error unless the concentration of active chain ends changes substantially during the course of the copolymerization. Significantly, there appeared to be no consensus threshold conversion below which the assumptions resulting in eq 4 were satisfied and accurate reactivity ratios could be determined on a consistent basis for all copolymerization classes and compositions that we investigated. For gradient copolymerizations, it appeared that up to ca. 1% monomer conversion produced reactivity ratios that were reasonably invariant with conversion. For alternating copolymerizations, accurate reactivity ratios could be determined at up to ca. 2% conversion. It was difficult to reproduce consistent reactivity ratios from block-type copolymerizations under any circumstances. In light of these difficulties, and the fact that more accurate methods are available that under some circumstances can require significantly less experimental effort, we recommend using only methods that account for the change in composition with conversion, and that provide a means of independently querying accuracy. Based on our survey of our numerical data, the analytically integrated methods, ML (eq 11), BSL (eq 15), and eq 17, fill this need without introducing any new parameters, and reducing the amount of experimental effort required. The integrated models consistently returned accurate reactivity ratios irrespective of copolymerization type. However, alternating copolymerizations proved the most difficult to fit. For alternating systems, comparison of experimental data to eq 17 may be diagnostic. For pure gradient copolymerizations (runs 1−10) following nonterminal copolymerization kinetics, both BSL and ML could be used to return accurate reactivity ratios as shown in Figure 3 and in SI Figures S106−115 and

comprehensive summary of reactivity ratios determined by each method. For the copolymer equation, FR, and KT, the error bars in the x-direction represent the standard deviation in conversion across fA0 used to calculate the set of reactivity ratios at the given conversion. Error bars in the y-direction represent the regression error of the fit to the respective model across all initial comonomer feed compositions (fA0). The calculated reactivity ratios for the copolymer equation, FR, and KT methods are presented as a function of conversion of the copolymerization, whereas the integrated methods (ML, and BSL) are presented as a function of the starting comonomer feed composition (fA0). The true reactivity ratio values are indicated by the horizontal lines. Additional results can be found in the SI. Figures S1−S175 present the results of determining reactivity ratios from the model copolymerization data. Figures S176−S199 show representative fits to the copolymerization data for four conversions for nonintegrated models, or all starting compositions fA0 for integrated models. In general, the nonintegrated methods (copolymer, FR, KT) produced systemically inaccurate reactivity ratios for all copolymerization types and at both low and high conversion as can be seen in Figure 2 and in SI Figures S1−S175. Reactivity ratios greater than one generally deviated more in magnitude from the known values than reactivity ratios less than one. The nonintegrated models failed to produce accurate reactivity ratios except serendipitously as the extracted values crossed the true values as conversion was increased. In general, the larger the reactivity ratio, the greater the inaccuracy. The inherent inaccuracty of these methods is compounded by that fact that all experimental uncertainty has been removed in our evaluation. Therefore, the nonintegrated methods would be at least as inherently inaccurate in an experimental context as we have found them to be here. We stress that this is not an issue of a poor fit to the data. As can be seen in Figure 2 and in SI Figures S176−S199, the fits to the data were excellent. The reason for the inherent inaccuracy is violation of the assumptions that form the basis for eq 4. The copolymer equation is most valid over an infinitesimally small increment of monomer conversion. However, the system has not developed enough to differentiate significantly from a random copolymerization. As such, the accuracy of reactivity ratio F


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Figure 3. Nonterminal model is accurate for pure gradient copolymerizations such as runs #4 and #9 shown above (left) fitted to the BSL model. The BSL model is not accurate when applied to impure gradients such as run #16 (right). Fit to the same data, the ML equation can extract accurate reactivity ratios from nearly all copolymerizations (right) and indicates that run #16 follows a terminal model of copolymerization kinetics.

lying kinetic model and accuracy of calculated reactivity ratios may be verified by conducting repeated experiments at different fA0; these experiments should produce the same reactivity ratios using the appropriate model. The accurate ML approach (eq 11) has not garnered widespread use in the literature in the decades since its derivation. It is unclear why routine direct fitting of the copolymer (eq 4), Fineman−Ross (eq 6), and Kelen−Tüdös (eq 7) continued in spite of the greater accuracy and efficiency offered by the ML approach. The integrated ML and BSL methods offer not only greater accuracy, but also greater experimental versatility; all available conversion data can be used. However, data can also be excluded from certain regimes of the polymerization to minimize experimental pathologies. For example, the first 10% of conversion data could be excluded if slow initiation were found to interfere with reactivity ratio determination. Likewise, high conversion data could be ignored if the copolymerization possessed a reversible monomer such as lactide or if the copolymerization became diffusion controlled.38 On a practical note, the ML and BSL equations require at a minimum only a single copolymerization conducted in its entirety to produce a single or pair of reactivity ratios. In contrast, the copolymer equation, FR, and KT require several experiments at different fA0 that are terminated at low conversion, and fitting produces unquantified systematic error in reactivity ratios. The nonintegrated approximate approaches (copolymer equation, FR, KT) are inherently inaccurate, and make unnecessary waste of experimental effort. We recommend that the nonintegrated methods no longer be used to describe macromolecular synthesis and architecture. Accurate determination of reactivity ratios leads not only to a greater understanding of macromolecular structure, but may also uncover details of the underlying mechanistic basis for copolymer formation.16,39 Significantly, the accurate determination of reactivity ratios is a vital prerequisite to true

S141−149. For all other copolymerizations, ML produced accurate values under nearly all circumstances, as did BSL under some circumstances. There were some numerical instabilities that we encountered at symmetric feed compositions that complicated reactivity ratio determination using ML. Because of this, we recommend that off-symmetric compositions (e.g., fA0 = 0.40 or 0.60) be used to experimentally determine reactivity ratios in order to avoid numerical pathologies. Other deviations observed for ML occurred at very low fA0, < 0.05, (as in Figure 3) and high fA0, > 0.95 compositions when the concentration of a comonomer approached that of the initiator. Based on this survey of reactivity ratio determination, we make the following recommendation for the accurate and efficient determination of reactivity ratios from experimental data: Conduct a single copolymerization at off-symmetric composition (e.g., fA0 = 0.60) and fit the appropriately formatted compositional drift with BSL and ML. The BSL model is appropriate for ionic,17 pseudoionic, and catalytic mechanisms that in some cases follow nonterminal copolymerization kinetics,35 and require a single parameter to describe the compositional drift. The terminal ML model is most appropriate for copolymerizations which exhibit significant nonidealities such as alternation, or blockiness, and two independent parameters are required to describe the compositional drift. In this sense, the appropriate kinetic model used to extract reactivity ratios can reveal mechanistic aspects of the polymerization by revealing the importance of chain-end structure on relative reactivity. The ML model poses some difficulty for highly alternating copolymerizations. Under those circumstances, we propose comparison with the reactivity ratio independent expressions given in eq 17. Data should be fit in regimes of conversion where high viscosity, slow initiation, or reversibility do not interfere with copolymerization kinetics. We then recommend that results are reported based on the model that requires the fewest number of parameters to accurately describe the copolymerization kinetics. The appropriateness of the underG


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Macromolecules

(4) Mayo, F. R.; Lewis, F. M. Copolymerization. I. A Basis for Comparing the Behavior of Monomers in Copolymerization; the Copolymerization of Styrene and Methyl Methacrylate. J. Am. Chem. Soc. 1944, 66, 1594−1601. (5) Fineman, M.; Ross, S. D. Linear Method for Determining Monomer Reactivity Ratios in Copolymerization. J. Polym. Sci. 1950, 5, 259−262. (6) Kennedy, J. P.; Kelen, T.; Tüdös, F. Analysis of the Linear Methods for Determining Copolymerization Reactivity Ratios. II. a Critical Reexamination of Cationic Monomer Reactivity Ratios. J. Polym. Sci., Polym. Chem. Ed. 1975, 13, 2277−2289. (7) O’Driscoll, K. F.; Reilly, P. M. Determination of Reactivity Ratios in Copolymerization. Makromol. Chem., Macromol. Symp. 1987, 10−11, 355−374. (8) Hautus, F. L. M.; Linssen, H. N.; German, A. L. Dependence of Computed Copolymer Reactivity Ratios on the Calculation Method. I. Effect of Experimental Setup. J. Polym. Sci., Polym. Chem. Ed. 1984, 22, 3487−3498. (9) Hautus, F. L. M.; Linssen, H. N.; German, A. L. Dependence of Computed Copolymer Reactivity Ratios on the Calculation Method. II. Effects of Experimental Design and Error Structure. J. Polym. Sci., Polym. Chem. Ed. 1984, 22, 3661−3671. (10) Kazemi, N.; Duever, T. A.; Penlidis, A. Reactivity Ratio Estimation From Cumulative Copolymer Composition Data. Macromol. React. Eng. 2011, 5, 385−403. (11) Tüdos, F.; Kelen, T.; Turcsányi, B.; Kennedy, J. P. Analysis of the Linear Methods for Determining Copolymerization Reactivity Ratios. VI. a Comprehensive Critical Reexamination of Oxonium Ion Copolymerizations. J. Polym. Sci., Polym. Chem. Ed. 1981, 19, 1119− 1132. (12) Braun, D.; Czerwinski, W.; Disselhoff, G.; Tüdös, F.; Kelen, T.; Turcsányi, B. Analysis of the Linear Methods for Determining Copolymerization Reactivity Ratios, VII. a Critical Reexamination of Radical Copolymerizations of Styrene. Angew. Makromol. Chem. 1984, 125, 161−205. (13) Riahinezhad, M.; Kazemi, N.; McManus, N.; Penlidis, A. Optimal Estimation of Reactivity Ratios for Acrylamide/Acrylic Acid Copolymerization. J. Polym. Sci., Part A: Polym. Chem. 2013, 51, 4819−4827. (14) Kazemi, N.; Duever, T. A.; Penlidis, A. Reactivity Ratio Estimation From Cumulative Copolymer Composition Data. Macromol. React. Eng. 2011, 5, 385−403. (15) Wu, Z.; Liu, P.; Liu, Y.; Wei, W.; Zhang, X.; Wang, P.; Xu, Z.; Xiong, H. Regulating Sequence Distribution of Polyethers via Ab Initio Kinetics Control in Anionic Copolymerization. Polym. Chem. 2017, 8, 5673−5678. (16) Beckingham, B. S.; Sanoja, G. E.; Lynd, N. A. Simple and Accurate Determination of Reactivity Ratios Using a Nonterminal Model of Chain Copolymerization. Macromolecules 2015, 48, 6922− 6930. (17) Herzberger, J.; Leibig, D.; Langhanki, J.; Moers, C.; Opatz, T.; Frey, H. Clickable PEG” via Anionic Copolymerization of Ethylene Oxide and Glycidyl Propargyl Ether. Polym. Chem. 2017, 8, 1882− 1887. (18) Chwatko, M.; Lynd, N. A. Statistical Copolymerization of Epoxides and Lactones to High Molecular Weight. Macromolecules 2017, 50, 2714−2723. (19) Herzberger, J.; Fischer, K.; Leibig, D.; Bros, M.; Thiermann, R.; Frey, H. Oxidation-Responsive and “Clickable” Poly(Ethylene Glycol) via Copolymerization of 2-(Methylthio)Ethyl Glycidyl Ether. J. Am. Chem. Soc. 2016, 138, 9212−9223. (20) Herzberger, J.; Leibig, D.; Liermann, J. C.; Frey, H. Conventional Oxyanionic Versus Monomer-Activated Anionic Copolymerization of Ethylene Oxide with Glycidyl Ethers: Striking Differences in Reactivity Ratios. ACS Macro Lett. 2016, 5, 1206− 1211. (21) Cho, K. Y.; Cho, A.; Kim, H.-J.; Park, S.-H.; Koo, C. M.; Kwark, Y. J.; Yoon, H. G.; Hwang, S. S.; Baek, K.-Y. Control of Hard Block

compositional control of structure−property relationships in polymeric materials.

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CONCLUSION We surveyed an array of numerically generated kinetic data for copolymerization and extracted reactivity ratios using a variety of methodologies: i.e., copolymer equation, Fineman−Ross, Kelen−Tüdös, Beckingham et al., and Meyer−Lowry. We evaluated all methods based on their accuracy, and found that the integrated methods of Meyer−Lowry and Beckingham et al. would require significantly less experimental effort than the nonintegrated traditional linearized models and offer less uncertainty in implementation and greater accuracy. Based on our evaluation of methods to determine reactivity ratios from our numerically defined data, we find no reason to continue to use the traditional approximate, nonintegrated methods to report reactivity ratios due to their arbitrary application, inaccuracy, and unnecessarily wasted experimental effort. In summary, we recommend that copolymerization data be interpreted using integrated models such as those reported by Beckingham et al. for the nonterminal model by Meyer− Lowry for the terminal model, and the results reported based on the method that requires the fewest parameters to accurately describe the experimental copolymerization system.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b01752. Source code, and fitting of models to numerical data, and data (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Nathaniel A. Lynd: 0000-0003-3010-5068 Robert C. Ferrier, Jr.: 0000-0002-5123-7433 Bryan S. Beckingham: 0000-0003-4004-0755 Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS N.A.L. acknowledges support from the Welch Foundation (Grant No.: F-1904). The authors thank Taylor A. Hatridge and Frank A. Leibfarth for critical evaluations of this article.

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REFERENCES

(1) Holmberg, A. L.; Karavolias, M. G.; Epps, T. H. RAFT Polymerization and Associated Reactivity Ratios of MethacrylateFunctionalized Mixed Bio-Oil Constituents. Polym. Chem. 2015, 6, 5728−5739. (2) Wall, F. T. The Structure of Copolymers. II 1. J. Am. Chem. Soc. 1944, 66, 2050−2057. (3) Alfrey, T., Jr.; Goldfinger, G. The Mechanism of Copolymerization. J. Chem. Phys. 1944, 12, 205−209. H


Article

Macromolecules Segments of Methacrylate-Based Triblock Copolymers for Enhanced Electromechanical Performance. Polym. Chem. 2016, 7, 7391−7399. (22) Reisman, L.; Mbarushimana, C. P.; Cassidy, S. J.; Rupar, P. A. Living Anionic Copolymerization of 1-(Alkylsulfonyl)Aziridines to Form Poly(Sulfonylaziridine) and Linear Poly(Ethylenimine). ACS Macro Lett. 2016, 5, 1137−1140. (23) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes; Cambridge University Press: New York, 2016; pp 1−1262. (24) Merz, E.; Alfrey, T.; Goldfinger, G. Intramolecular Reactions in Vinyl Polymers as a Means of Investigation of the Propagation Step. J. Polym. Sci. 1946, 1, 75−82. (25) Hill, D. J. T.; Lang, A. P.; O’Donnell, J. H.; O’Sullivan, P. W. Determination of Reactivity Ratios From Analysis of Triad FractionsAnalysis of the Copolymerization of Styrene and Acrylonitrile as Evidence for the Penultimate Model. Eur. Polym. J. 1989, 25, 911−915. (26) Kelen, T.; Tüdös, F.; Braun, D.; Czerwinski, W. K. Analysis of the Linear Methods of Determining Copolymerization Reactivity Ratios, 9. Penultimate Systems with r2 = 0. Makromol. Chem. 1990, 191, 1853−1861. (27) Hunley, M. T.; Sari, N.; Beers, K. L. Microstructure Analysis and Model Discrimination of Enzyme-Catalyzed Copolyesters. ACS Macro Lett. 2013, 2, 375−379. (28) Hunley, M. T.; Beers, K. L. Nonlinear Method for Determining Reactivity Ratios of Ring-Opening Copolymerizations. Macromolecules 2013, 46, 1393−1399. (29) Walling, C.; Briggs, E. R. Copolymerization. III. Systems Containing More Than Two Monomers. J. Am. Chem. Soc. 1945, 67, 1774−1778. (30) Skeist, I. Copolymerization: the Composition Distribution Curve. J. Am. Chem. Soc. 1946, 68, 1781−1784. (31) Meyer, V. E.; Lowry, G. G. Integral and Differential Binary Copolymerization Equations. J. Polym. Sci., Part A: Gen. Pap. 1965, 3, 2843−2851. (32) Wall, F. T. The Structure of Vinyl Copolymers. J. Am. Chem. Soc. 1941, 63, 1862−1866. (33) Tsai, S. D.; Register, R. A. Endo/Exo Reactivity Ratios in Living Vinyl Addition Polymerization of Substituted Norbornenes. Macromol. Chem. Phys. 2018, 219, 1800059. (34) Wu, Z.-C.; Liu, Y.; Wei, W.; Chen, F.-S.; Qiu, G.-X.; Xiong, H.M. Reaction Kinetics in Anionic Copolymerization: a Revisit on Mayo-Lewis Equation. Chin. J. Polym. Sci. 2016, 34, 431−438. (35) Lin, T.-P.; Chang, A. B.; Chen, H.-Y.; Liberman-Martin, A. L.; Bates, C. M.; Voegtle, M. J.; Bauer, C. A.; Grubbs, R. H. Control of Grafting Density and Distribution in Graft Polymers by Living RingOpening Metathesis Copolymerization. J. Am. Chem. Soc. 2017, 139, 3896−3903. (36) Melville, H. W.; Noble, B.; Watson, W. F. Copolymerization. I. Kinetics and Some Experimental Considerations of a General Theory. J. Polym. Sci. 1947, 2, 229−245. (37) Goldfinger, G.; Kane, T. Derivation of the Copolymerization Equation Without Steady-State Assumptions. J. Polym. Sci. 1948, 3, 462−463. (38) Marten, F. L.; Hamielec, A. E. High Conversion DiffusionControlled Polymerization. In Catalysis in Polymer Synthesis; ACS Symposium Series; American Chemical Society: Washington, D.C., 1979; Vol. 104, pp 43−70. (39) Lee, B. F.; Wolffs, M.; Delaney, K. T.; Sprafke, J. K.; Leibfarth, F. A.; Hawker, C. J.; Lynd, N. A. Reactivity Ratios and Mechanistic Insight for Anionic Ring-Opening Copolymerization of Epoxides. Macromolecules 2012, 45, 3722−3731.

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