A Pulsed-Laser Study of Penultimate Copolymerization Propagation

(1996) consider the question of why reported radical reactivities (si) are all less than unity and conclude that there is no theoretical justification...
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Ind. Eng. Chem. Res. 1997, 36, 1103-1113

1103

A Pulsed-Laser Study of Penultimate Copolymerization Propagation Kinetics for Methyl Methacrylate/n-Butyl Acrylate Robin A. Hutchinson,* John H. McMinn, Donald A. Paquet, Jr., Sabine Beuermann,† and Christian Jackson Central Research and Development, E. I. du Pont de Nemours and Company, Inc., P.O. Box 80101, Experimental Station, Wilmington, Delaware 19880-0101

The PLP/MWD technique, based upon analysis of the molecular weight distribution (MWD) of a polymer produced by pulsed-laser polymerization (PLP), has proven to be a powerful probe for the examination of free-radical homopolymerization kinetics. The same method is applicable to the study of copolymerization propagation kinetics provided that proper calibrations are established for the measurement of copolymer MWDs, as illustrated by an examination of the methyl methacrylate (MMA)/n-butyl acrylate (nBA) system. Statistical analysis indicates that although MMA/nBA copolymer composition is well described by the terminal model, propagation kinetics are not. Like many other systems recently examined, the data are well represented by the “implicit penultimate unit effect” model of Fukuda and co-workers. However, whereas previous systems studied have propagation rates that are lower than those predicted by the terminal model, it is found that the propagation rate of MMA/nBA is greater than the terminal model predictions. The implications of these results to theoretical interpretations of the penultimate model are discussed. Introduction The precise kinetic mechanisms which control rate and composition in free-radical copolymerization have been long debated. It is generally found that the copolymer composition and sequence distribution are well described by the terminal model of monomer addition, but copolymerization rate is not. Rate “abnormalities” have previously been attributed to termination reactions and represented by introducing physically unrealistic cross-termination constants (Jenkins and O’Driscoll, 1995). A growing body of evidence, however, convincingly shows that rate deviations cannot be attributed to the diffusion-controlled termination reactions but result from the failure of the terminal model to explain the propagation kinetics (Fukuda et al., 1992, and references therein). A model which accounts for the influence of the penultimate monomer unit on the chain-growth kinetics has been used to successfully represent this behavior (Fukuda et al., 1992). The penultimate model, first formulated by Merz et al. (1946), has received much attention over the last decade or so, primarily due to the convincing experimental work of Fukuda, Ma, and co-workers. Their carefully performed rotating-sector experimentation has shown that the terminal model cannot simultaneously describe the polymer composition and copolymer propagation rate coefficients (kcop p ) measured for many common systems, including styrene/methyl methacrylate (MMA) (Fukuda et al., 1985), MMA/vinyl acetate (VAc) (Ma et al., 1993), and styrene/ethyl acrylate (Ma et al., 1994). Pulsed-laser methods, which allow the direct measurement of kcop (details below), have also been p used to demonstrate the failure of the terminal model to describe the propagation kinetics for styrene/meth* To whom correspondence should be addressed. Phone: (302)695-3788. Fax: (302)695-2645. E-mail: HUTCHIRA@ esspt0.dnet.dupont.com. † Currently at Institut fu ¨ r Physikalische Chemie, Universita¨t Go¨ttingen, Tammannstrasse 6, D-37077 Go¨ttingen, Germany. S0888-5885(96)00403-4 CCC: $14.00

acrylate (Davis et al., 1989, 1990; Olaj et al., 1989) and styrene/acrylate systems (Davis et al., 1991; Schoonbrood et al., 1995). An excellent survey of most of this past work is provided by Fukuda and co-workers (1992); a condensed summary follows below. Terminal Model. For the terminal model (TM) of propagation, the polymer composition in a two-monomer system is given by

fp1 )

r1fm12 + fm1fm2 r1fm12 + 2fm1fm2 + r2fm22

(1)

where fmi is the mole fraction of monomer i, fpi is the mole fraction of segment i in the polymer, and ri are reactivity ratios defined as follows:

r1 ) kp11/kp12

r2 ) kp22/kp21

where kpij refers to the reaction of monomer j with radical i. The reactivity ratios also control the sequence length distribution; equations for the triad fractions are found in many references, including Burke et al. (1994a). In a copolymer system, the overall rate of polymerization may be written as • Rp ) kcop p [R ][M]

(2)

where [R•] and [M] denote the total concentrations of the radicals and monomers, respectively. For a twomonomer system, an expression for the overall copolymer propagation rate coefficient can be derived:

kcop p

r1fm12 + 2fm1fm2 + r2fm22 )

r1fm1/kp11 + r2fm2/kp22

(3)

Note that the polymer composition is only a function of the reactivity ratios, while kcop p is a function of all four propagation rate coefficients. Penultimate Model. The penultimate model assumes that the radical reactivity is affected by the © 1997 American Chemical Society

1104 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

preceding unit on the chain; radical Rij is differentiated from radical Rjj. Thus, for a two-monomer system, there are eight propagation reactions, with the rate coefficient kpijk representing the reaction of radical ij with monomer k. The reactivity ratios are defined as follows:

which are close to idealsMMA/BMA (Ito and O’Driscoll, 1979) and styrene/4-methoxystyrene (Piton et al., 1990)sshow no penultimate effects on the rate. Based upon stabilization energy arguments, Fukuda et al. (1987) suggest that

r11 ) kp111/kp112

r21 ) kp211/kp212

rj1rj2 ) s1s2

r22 ) kp222/kp221

r12 ) kp122/kp121

s1 ) kp211/kp111

s2 ) kp122/kp222

Since it is often difficult to obtain accurate estimates of both radical reactivities (si) from the experimental data, Fukuda et al. (1989) also suggest the simplification that s1 ) s2. Combination with eq 8 leads to

where si, the radical reactivity ratio, represents the effect of the penultimate unit on the addition rate of monomer i to radical i. It has been shown that the penultimate and terminal ratios may be equated as follows (Fukuda et al., 1985):

rj1 ) r21

rj2 ) r12

r11fm1 + fm2 r21fm1 + fm2 r22fm2 + fm1 r12fm2 + fm1

(4)

(5)

r11fm1 + fm2 kh p11 ) kp111 r11fm1 + fm2/s1

(6)

r22fm2 + fm1 k h p22 ) kp222 r22fm2 + fm1/s2

(7)

The final expressions for the copolymer properties are obtained by substituting these definitions into eqs 1 and 3. For a very few systems such as styrene/acrylonitrile (Hill et al., 1982, 1989; Jones et al., 1985) and pchlorostyrene/methyl acrylate (Ma et al., 1985), the polymer composition and triad sequences are not well described by the terminal model, with failure more evident by examining the triad distributions than the polymer composition (Hill et al., 1982, 1989; Burke et al., 1994a,b). Fukuda et al. (1991) coined the phrase “explicit penultimate unit effect” (EPUE) to describe these systems which require all eight rate coefficients in the penultimate model to describe the polymer composition and rate. It is possible that other effects, such as the “bootstrap” model proposed by Harwood (1987), may also be important. EPUE systems are rare and are not considered further in this work. For the majority of the copolymer systems studied to date, including MMA/VAc, styrene/acrylates, and styrene/ methacrylates (see references above), the polymer composition is well described by the terminal model, but kcop is not. These systems are well represented by p setting r11 ) r21 ()rj1) and r22 ) r12 ()rj2) and are said to have an “implicit penultimate unit effect” (IPUE) (Fukuda et al., 1991). For all of these systems, experimental measures of the copolymerization rate (or kcop p ) are lower than predicted by the terminal model. This observation is so universal (Fukuda et al., 1992) as to encourage speculation as to why the penultimate unit can only reduce, and not increase, the radical reactivity (Jenkins and O’Driscoll, 1995). The majority of the IPUE systems summarized in the Fukuda et al. (1992) review are “nonideal” copolymer systems, with r1r2 < 1. The two systems examined

s1 ) s2 ) (rj1rj2)1/2

(8)

(9)

It is demonstrated that this approximation does a reasonable job of correlating the available experimental data (Fukuda et al., 1992). This extensive body of work by Fukuda and coworkers has brought new and valuable insight into the mechanisms of copolymerization but has also received some criticism. One point of concern is the statistical analysis of the data. For IPUE systems, composition data are used to fit rj1 and rj2 according to the terminal model, with the resulting values then used to fit the si values from the kcop or rate data. A better underp standing of the system could be gained by fitting models to the composition and rate data simultaneously (Burke et al., 1994a,b, 1995). Indeed, Schweer (1993) showed that either the composition or the kcop styrene/MMA p data could be well represented by the terminal model but that the resulting rj1 and rj2 estimates then do a very poor job of fitting the other set of data. When the penultimate model is applied to kcop p or the composition data separately, the resulting fit is no better than that of the terminal model and is nonuniquesthe data can be represented by different combinations of the penultimate model parameters. Attempts to simultaneously fit the penultimate model to both kcop p and the composition data also illustrate that quite often the experimental results can be reasonably represented by different combinations of penultimate reactivity ratios (Schweer, 1993), bringing into question the physical significance of the model parameters. A subsequent NMR study of styrene/MMA triad sequences by Maxwell et al. (1993) provides additional insight to this issue. It is shown that both triad sequences and polymer composition are well described by the terminal model; the ri values estimated from the triad sequences are identical (within experimental error) to those estimated from the polymer composition data. This result validates that the fitting procedure of Fukuda and co-workers (estimating the rj1 and rj2 values solely from composition data), while perhaps not statistically optimal, provides meaningful estimates of the monomer reactivity ratios. The work of Maxwell and co-workers also strongly supports the assumptions behind the IPUE; there is no observable penultimate unit effect on the polymer chain structure. The second point of criticism raised against the work of Fukuda and co-workers is, in some regards, an offshoot of the first. Equation 8 (Fukuda et al., 1987, 1991) justifies the IPUE through arguments about radical stabilization energies. This derivation has been criticized for the neglect of steric (entropic) contributions (Maxwell et al., 1993; Heuts et al., 1996). No theoretical discussion on this matter will be presented in this work. However, it is interesting to note that Heuts et al. (1996) consider the question of why reported radical reactivities

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1105

(si) are all less than unity and conclude that there is no theoretical justification for this apparently general experimental result. It is also important to note that Fukuda and co-workers suggest the simplification s1 ) s2 (which leads to eq 9) only as a means to deal with the experimental uncertainty in the data analysis and the fact that s1 and s2 are highly correlated (Fukuda et al., 1989; Burke et al., 1995). It is clear from this discussion that data from additional comonomer systems could help resolve these issues, especially if these data are analyzed using sound statistical techniques. This is one goal of this work, which documents a PLP/ MWD study of copolymerization propagation kinetics for the methyl methacrylate (MMA)/n-butyl acrylate (nBA) system. PLP/MWD Technique. The rotating sector method used by Fukuda, Ma, and co-workers determines kcop p by combining rotating-sector measurements of kcop p /kt with steady-state polymerization rate measurements of 1/2 kcop p /kt , a difficult technique prone to many sources of error (Buback et al., 1992). Indeed, it is a testimonial to the experimental care of Ma and co-workers that the homopropagation kp values measured for VAc (1993) and estimated for ethyl acrylate (EA) (1994)sboth monomers have fast propagation kineticssagree reasonably well with recent determinations by the PLP/ MWD technique (Hutchinson et al., 1995; Beuermann et al., 1996a). (Although the EA kp value estimated by Ma et al. (1994) may be low by a factor of 4-5, it is shown that the exact value has no effect on the data interpretation.) Because of the experimental difficulties associated with rotating sector methodology, a technique which gives a direct measure of kcop independent of the p termination kinetics, should provide a better test of the various copolymerization models. The PLP/MWD method, pioneered by Olaj and co-workers (Olaj et al., 1987; Olaj and Schno¨ll-Bitai, 1989), does exactly that. The technique is based upon the analysis of the molecular weight distribution (MWD) of the polymer produced by pulsed-laser polymerization (PLP). In the technique, a monomer system with photoinitiator is exposed to laser flashes. Each flash generates a new population of radicals, with the radical concentration decreasing between flashes due to radical-radical termination. At the end of the period between flashes (t0), the radicals which have escaped termination have propagated to a chain length DP0, given by the simple equation

DP0 ) kp[M]t0

(10)

where [M] is the monomer concentration. When the next flash arrives, the remaining radicals are exposed to a high concentration of newly generated radicals, which leads to a greatly increased probability for their termination. Thus, the formation of dead polymer molecules with length close to DP0 is favored. With a measure of DP0, kp can be calculated from eq 10. The PLP/MWD technique has proven to be a direct and reliable method for measuring homopolymer kp values. Measurements from many laboratories show excellent agreement and have been combined to provide benchmark data sets for styrene (Buback et al., 1995) and MMA (Beuermann et al., 1996b). As mentioned above, the PLP/MWD technique has been applied to copolymer systems before. The accuracy of these previous works, however, has been questioned. It is evident from eq 10 that the accuracy of kcop is p directly related to the accuracy of DP0, usually meas-

ured by size exclusion chromatography (SEC) methods which separate polymer molecules on the basis of hydrodynamic volume (the product of chain MW and intrinsic viscosity). Past work assumes that SEC calibration for copolymers can be calculated as a weighted average of the homopolymer calibrations. It is wellknown, however, dating back to Stockmayer et al. (1955), that copolymers in dilute solution can be expanded more, with a higher intrinsic viscosity ([η]) than would be expected from the behavior of the two homopolymers. As stated in the paper of Stockmayer et al., “it is apparent that a simple linear average will not suffice”. Such deviations have been reported for styrene/ methacrylate (Stockmayer et al., 1955; Podesva et al., 1977; Kent et al., 1994) and acrylate/methacrylate (Wunderlich, 1970) systems, with the relative magnitude of the copolymer coil expansion being dependent upon the solvent quality (Wunderlich, 1970; Kent et al., 1994). For accurate SEC calibration, it is necessary to examine how the relationship between [η] and MW varies with copolymer composition on a case-by-case basis. Without this knowledge, the accuracy of kcop p values determined by the PLP/MWD technique will remain under question. In this work, we build upon experience gained in previous studies (Hutchinson et al., 1995; Beuermann et al., 1996a) which examine the potential error introduced by SEC analysis by comparing PLP homopolymer kp data analyzed using calibrations established by a triple-detector SEC (TD-SEC) instrument (differential refractometer, differential viscometer, multiangle light scattering) and the same data analyzed according to calibrations reported in the literature. It was concluded that the TD-SEC instrument gives excellent results and that with careful calibration, the absolute error in kp introduced by SEC analysis is no more than 10-15%. (It is important to note that these studies also identified that some previously published Mark-Houwink calibration parameters for nBA and methyl acrylate (Davis et al., 1991) are in significant error. These erroneous values, which have a large impact on kp estimates (Beuermann et al., 1996a), have been used in two of the previous copolymer PLP studies (Davis et al., 1991; Schoonbrood et al., 1995).) We now apply the same careful calibration techniques to a PLP/MWD study of MMA/nBA copolymerization. In addition to examining the importance of accurate SEC calibration for kcop p determined by PLP/MWD, this work is, to the best of our knowledge, the first direct study of propagation kinetics for a methacrylate/acrylate copolymerization system. As detailed below, we find not only that penultimate effects are important for this system but that the results do not follow some of the general behavior reported in previous work. Experimental Details The experimental pulsed-laser setup has been described previously (Hutchinson et al., 1995). A QuantaRay pulsed Nd:YAG GCR-190-100 laser with a harmonic generator produces light of wavelength 355 nm at pulse energies up to 70 mJ/pulse and a half-height pulse width of 6 ns. Copolymerization experiments were run at 20 and 60 °C at repetition rates of 10 and 25 Hz; repetition rates were controlled with a digital delay generator (Stanford DG-535). The bulk monomer samples containing benzoin photoinitiator (1 mmol/L) were pulsed for a total time sufficient to allow 0.5-1.5%

1106 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

conversion of the 4-mL monomer sample to polymer; conversion was measured gravimetrically. MMA, nBA, and benzoin photoinitiator were obtained from Aldrich Chemical Co. and used as received. Chemically initiated experiments were performed in bulk monomer at 50 °C with 5000 ppm of 2,2′-azodiisobutyronitrile (AIBN) as the initiator. Polymerizations were stopped after 1 h, with conversions of less than 7%; it has previously been shown for this system that the composition drift at these conversion levels is insignificant (Dube and Penlidis, 1995). An inhibitor was added to the system, and the polymer was isolated by precipitation and repeated washings with heptane. The polymer composition was determined by 1H NMR using techniques well documented in the literature (Dube and Penlidis, 1995; Aerdts et al., 1994). These samples were also used for SEC calibrations, as detailed below. For all copolymerization experiments, MMA and nBA were mixed at room temperature on a volumetric basis. The monomer molar fractions and overall monomer concentrations were calculated using MMA (F20 ) 0.9422 g/cm3; F60 ) 0.8970) and nBA (F20 ) 0.9002 g/cm3; F60 ) 0.8572) density values obtained from the DIPPR database (Daubert et al., 1993). Volume fractions (φmi) were converted to weight (wmi) and molar fractions (fmi) using the density values at 20 °C; the overall system density (Fmix) and monomer concentration were calculated using these mole fractions, the monomer molecular weights, and the densities at the reaction temperature. Volume additivity was assumed in all calculations. SEC analyses were performed using a single-detector instrument consisting of a Waters pump (Model 590), Waters autosampler (WISP 712), two SHODEX columns (KF80M), and a Waters differential refractometer (Model 410) at 30 °C. The PLP-generated samples (5-15 mg/ mL of polymer in monomer) were diluted in THF to polymer concentrations of 1-3 mg/mL for SEC injection. The pMMA and copolymer samples were analyzed based upon a calibration curve obtained from narrow MW pMMA standards; Mark-Houwink (M-H) parameters measured by a triple-detector instrument (TD-SEC) were then used to calculate the true copolymer MW according to universal calibration, a procedure described previously (Hutchinson et al., 1995). The triple-detector instrument used to measure the Mark-Houwink parameters consisted of a Waters 150C size exclusion chromatograph (Waters Associates, Milford, MA), a Viscotek 150R bridge viscometer (Viscotek Corp., Houston, TX), and a DAWN DSP-F laser photometer (Wyatt Technology, Santa Barbara, CA) equipped with a 5-mW helium neon laser operating at 633 nm. The detectors were connected in series with the photometer first after the columns and the refractometer last and were calibrated using polystyrene solutions of known molecular weight, intrinsic viscosity, and concentrations. Separation was achieved using four 300mm × 7.5-mm-i.d. columns packed with MIXED-C 5 µmPLGel packing (Polymer Laboratories, Amherst, MA). The mobile phase was HPLC-grade tetrahydrofuran (THF), and the operating temperature was 30 °C. Polymer solutions were prepared at a concentration of 4 mg/mL, and the injected volume was typically 100 µL. Data were collected and analyzed using commercial (TriSEC, Viscotek Corp.) and in-house software. The light-scattering detector required a value for dn /dc (the change in refractive index with polymer concentration), which was estimated using a weighted average of the

known homopolymer values; the estimates agree well with the values calculated by integrating the refractometer peak areas and normalizing with the known masses of injected polymer. The accuracy of this system for establishing calibrations for various methacrylate (Hutchinson et al., 1995) and acrylate (Beuermann et al., 1996a) homopolymers has been previously documented. Further details on the use of multidetector SEC systems (Garcia-Rubio, 1987), and this system in particular (Jackson et al., 1996a,b), can be found in the literature. Inflection points from the laser-generated MW peak were obtained by fitting the experimental MWD with cubic smoothing splines followed by differentiation of the splines. The SEC data (mass MWD on a logarithmic, or w(log M ), scale) were converted to mass (w(M)) and number distributions (f(M)) on a linear scale, with inflection points calculated from all three distributions; whenever possible, the data were validated by ensuring that the inflection points from all three distributions show good agreement (Hutchinson et al., 1993). For some of the copolymer samples with high nBA fraction, however, the inflection points from the w(M) and f(M) distributions were not easily discernible. Thus, the inflection point from the derivative plot of the w(log M) distribution was used for all samples to calculate kp from a rearranged form of eq 10:

kcop p )

M0 1000Fmixt0

(11)

where M0 is the polymer MW at the inflection point, kp has units of L/(mol‚s), and the comonomer mixture density Fmix has units of g/cm3. If a secondary inflection point at twice the primary was not observed, the distribution was not considered to provide a reliable measure of kp (Hutchinson et al., 1993; Buback et al., 1995; Beuermann et al., 1996a). Results and Discussion SEC Calibration. Even with sharp, well-defined PLP MWDs, the calculated kp values will not be accurate without accurate SEC calibration. This fact was recognized from the outset (Olaj et al., 1987) but has not been a major issue for most MMA and styrene studies; for these systems, SEC calibration curves are constructed directly from narrow MW polymer standards. For other polymer systems, the MWD is calculated using the principle of universal calibration, as first suggested by Benoit and co-workers (Grubisic et al., 1967), based upon the separation of molecules by hydrodynamic volume:

[ηc]Mc ) [ηp]Mp

(12)

This equation allows for the transformation of the MWD as calculated for the calibration polymer (subscript “c”) to the MWD for the polymer of interest (subscript “p”), provided that the M-H constants which relate [η] to polymer MW are known for both polymers at SEC solvent and temperature conditions:

KcMcac+1 ) KpMpap+1

(13)

The validity of universal calibration has been demonstrated for many different types of polymers and copolymers and is especially useful for the PLP/MWD technique in which kp is calculated from a single point (M0 ) from the MWD. It is thus possible to transform a

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1107 Table 1. Mark-Houwink Parameters for MMA(1)/nBA Copolymersa φ1

fm1

fp1

K, dL/g

a

[η],b dL/g

1 0.75 0.50 0.25 0.10 0

1 0.801 0.573 0.309 0.130 0

1 0.903 0.785 0.538 0.292 0

9.44 × 10-5 1.05 × 10-4 1.53 × 10-4 1.69 × 10-4 1.87 × 10-4 1.22 × 10-4

0.719 0.711 0.703 0.706 0.695 0.700

0.372 0.377 0.501 0.573 0.558 0.386

a Copolymers produced at 50 °C with AIBN; M-H parameters measured at 30 °C in THF. b [η] calculated at M ) 1 × 105.

Table 2. MMA(1)/nBA Copolymer Composition Data from the Literature ref

T, °C

Npts

r1

r2

this work (Table 1) Dube and Penlidis (1995) Aerdts et al. (1994) Grassie et al. (1965) Han and Wu (1994) Han and Wu (1994)

50 60 50 60 60 100

4 16 9 6 8 8

2.51 1.98 2.28 1.8 2.56 2.42

0.357 0.355 0.395 0.37 0.47 0.46

previously reported inflection point to an updated value calculated with new M-H constants (Hutchinson et al., 1995). Table 1 summarizes the copolymer composition as a function of monomer composition for the chemically initiated copolymerization experiments; throughout this work, subscript 1 is used to denote MMA. These samples, along with MMA and nBA homopolymer standards, were analyzed by the triple-detector SEC instrument to obtain M-H parameters, also summarized in Table 1. The pBA parameters have been published earlier (Beuermann et al., 1996a) and show excellent agreement with other literature values, as do the pMMA constants. Also tabulated are the values for [η] (dL/g), calculated at M ) 1 × 105. The values for the pMMA and pBA homopolymers almost coincide. If it were valid to assume that copolymer calibrations could be calculated from a weighted average of the two homopolymers (as done in previous PLP/MWD copolymer studies), the copolymer [η] values should be very similar. However, as seen from Table 1, the [η] values for the copolymers are as much as 50% higher than the homopolymer values. As demonstrated later, this difference has a significant effect on the measured kcop p values and penultimate model parameter estimates. Copolymer Composition. In addition to the data in Table 1, polymer composition data have been compiled from the literature and are summarized in Table 2. Although the reactivity ratios estimated from the independent data sets are scattered, the combined data lay within a relatively narrow band on a plot of polymer vs monomer composition, as illustrated by Figure 1. The 51 data points were fit according to the terminal model, eq 1, with nonlinear least-squares parameter estimation routines using a modified Newton method; the numerical routines are part of the NAG library of Fortran routines (NAG Ltd., 1993). This algorithm was used to fit all of the different models and data sets in this work. Standard deviations are calculated as the square root of the diagonal terms of the covariance matrix, also determined by numerical routines in the NAG library. The best-fit reactivity ratios are r1 ) 2.24 ( 0.09 and r2 ) 0.414 ( 0.017, with the resulting curve shown on Figure 1 as a solid line. These values agree well with the estimates obtained from the error-invariables approach outlined by Dube et al. (1991): r1 ) 2.36, r2 ) 0.437.

Figure 1. MMA mole fraction in copolymer as a function of MMA fraction in monomer mixture. Lines indicate fits by the terminal model, with parameters estimated from the fit to (see Table 4) fp1 data only (s), kcop data only (- - -), and combined data sets (‚‚‚). p Symbols indicate data from Table 1 (b), Dube and Penlidis (O), Aerdts et al. (+), Grassie et al. (*), and Han and Wu (×).

Figure 2. MWDs (top) and derivative plots (bottom) for MMA/ nBA pulsed at 60 °C and 25 Hz (experimental set 4 from Table 3). Numbers on plot indicate the volume fraction MMA in the monomer mixture, with duplicate experiments run at each condition.

kcop Values from PLP Experiments. Four sets of p PLP experiments, two each at 20 and 60 °C, were performed over a range of monomer compositions. Figure 2a shows a typical set of MWDs from PLP experiments run at 60 °C with a laser repetition rate of 25 Hz, with Figure 2b showing the corresponding derivative plots. (Experimental sets at 20 and 60 °C were performed in 1994, and again in 1996; the reproducibility is excellent.) The PLP structure of the MWDs (sharp primary peak and secondary shoulder on MWDs; clear primary and secondary inflection points from the

1108 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 Table 3. Results of the PLP MMA/nBA Experiments pMMA calibration

copolym calibration

expt

φ1

fm1

Fmix, g/cm3

M0

kp, L/mol‚s

M0

kp, L/mol‚s

set 1 20 °C 10 Hz 65 mJ/pulse [I] ) 1 mmol/L

1 1 0.80 0.80 0.50 0.50 0.20 0.20 0.10 0.10 1 1 0.75 0.75 0.50 0.50 0.25 0.25 0.10 0.10 1 1 0.80 0.80 0.50 0.50 0.20 0.20 1 1 0.75 0.75 0.50 0.50 0.25 0.25 0.10 0.10

1 1 0.843 0.843 0.573 0.573 0.251 0.251 0.130 0.130 1 1 0.801 0.801 0.573 0.573 0.309 0.309 0.130 0.130 1 1 0.843 0.843 0.573 0.573 0.251 0.251 1 1 0.801 0.801 0.573 0.573 0.310 0.310 0.130 0.130

0.9422 0.9422 0.9338 0.9338 0.9212 0.9212 0.9086 0.9086 0.9044 0.9044 0.9422 0.9422 0.9317 0.9317 0.9212 0.9212 0.9107 0.9107 0.9044 0.9044 0.8970 0.8970 0.8890 0.8890 0.8771 0.8771 0.8652 0.8652 0.8970 0.8970 0.8870 0.8870 0.8771 0.8771 0.8671 0.8671 0.8671 0.8671

2.79 × 104 2.85 × 104 3.09 × 104 3.20 × 104 3.94 × 104 3.94 × 104 6.68 × 104 6.68 × 104 1.15 × 105 1.14 × 105 2.77 × 104 2.67 × 104 2.78 × 104 3.06 × 104 3.85 × 104 3.85 × 104 5.62 × 104 5.62 × 104 9.94 × 104 8.92 × 104 7.50 × 104 7.30 × 104 8.61 × 104 8.32 × 104 1.11 × 105 1.08 × 105 1.62 × 105 1.64 × 105 3.17 × 104 3.06 × 104 3.51 × 104 3.47 × 104 4.27 × 104 4.27 × 104 6.52 × 104 6.46 × 104 1.07 × 105 1.07 × 105

296 302 331 343 428 428 735 735 1269 1255 294 283 298 328 418 418 617 617 1099 986 836 814 968 936 1264 1236 1875 1897 884 853 989 978 1217 1217 1880 1862 3100 3100

2.79 × 104 2.85 × 104 3.05 × 104 3.16 × 104 3.28 × 104 3.28 × 104 5.17 × 104 5.17 × 104 9.05 × 104 8.94 × 104 2.77 × 104 2.67 × 104 2.74 × 104 3.02 × 104 3.20 × 104 3.20 × 104 4.34 × 104 4.34 × 104 7.82 × 104 7.00 × 104 7.50 × 104 7.30 × 104 8.53 × 104 8.24 × 104 9.31 × 104 9.10 × 104 1.26 × 105 1.28 × 105 3.17 × 104 3.06 × 104 3.46 × 104 3.42 × 104 3.55 × 104 3.55 × 104 5.04 × 104 5.00 × 104 8.41 × 104 8.41 × 104

296 302 326 338 356 356 569 569 1000 989 294 283 294 324 348 348 477 477 864 774 836 814 960 927 1062 1038 1460 1477 884 853 976 965 1013 1013 1454 1440 2440 2440

set 2 20 °C 10 Hz 45 mJ/pulse [I] ) 1 mmol/L

set 3 60 °C 10 Hz 65 mJ/pulse [I] ) 1 mmol/L

set 4 60 °C 25 Hz 40 mJ/pulse [I] ) 2 mmol/L

derivative plots) becomes less pronounced as the nBA fraction in the monomer mixture increases. The MWDs of Figure 2 were measured by a singledetector SEC and calculated assuming the samples were homopolymer pMMA. Table 3 reports the complete details of all experiments, with inflection points and kp values tabulated from both the pMMA calibration and from the data transformed to the correct copolymer calibration. It is assumed that the calibrations measured for the polymers produced with 25 and 75 vol % MMA are also valid for polymers produced with 20 and 80 vol % MMA mixtures. The transformation is calculated according to Hutchinson et al. (1995):

M0cop )

(

)

KMMAM0MMAaMMA+1 Kcop

1/(acop+1)

(14)

with M-H parameters taken from Table 1. Figure 3 plots the 60 °C kcop values as a function of fm1, with p kcop calculated assuming pMMA and pBA homopolyp mer calibrations as well as the correct copolymer calibration. The results from the two homopolymer calibrations are almost identical, while the true copolymer calibration significantly reduces the kcop p values at high nBA fraction. In previous PLP copolymerization studies (Davis et al., 1989, 1990, 1991; Olaj et al., 1989; Piton et al., 1990; Schoonbrood et al., 1995), the values for kp were estimated using a weighted average of the two homopolymer calibrations. For the MMA/nBA

Figure 3. kcop p (L/(mol‚s)) at 60 °C as a function of MMA fraction in monomer mixture, with values calculated according to pMMA calibration (O), pBA calibration (+), and copolymer calibration (×). Mark-Houwink calibration parameters from Table 1.

system, this procedure would overestimate kcop by as p much as 30% for copolymers with high nBA fraction. These data illustrate the importance of SEC calibration to the accuracy of PLP results and suggest that previous studies need to be reexamined. The data of Table 3 can be reduced to a single experimental set by scaling the kcop estimates by the p corresponding MMA homopolymerization values, calculated as 295 and 847 L/(mol‚s) by averaging the four

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1109 Table 4. Terminal Model Reactivity Ratio Estimates case

Nobs

fp data only MMA kcop data only: p /kp not scaled further scaled by 5 combined data: MMA kcop scaled by 5 p /kp

51 30

aSSE

Figure 4. Terminal model fits to kcop data, with parameters p estimated from fit to (see Table 4) fp1 data only (s), kcop p data only (- - -), and combined data sets (‚‚‚). kcop p data at 20 (O) and 60 (×) °C are scaled by homopolymer MMA kp values.

data points at 20 and 60 °C, respectively. These kMMA values are in excellent agreement with previous p studies (Hutchinson et al., 1995; Beuermann et al., 1996b). In addition to collapsing the kcop data to a p single curve (see Figure 4), this treatment has the added benefit of scaling the 30 kcop data points to the same p order of magnitude as the polymer composition data. kp for nBA is much higher than that of MMA; a value of 14 200 L/(mol‚s) has been measured at 20 °C (Beuermann et al., 1996a). The estimate at 60 °C is 23 00033 000 L/(mol‚s), with the large spread being a result of extrapolating to higher temperatures with uncertain activation energy (Lyons et al., 1996; Beuermann et al., 1996a). These nBA kp values are 48 times higher than the MMA value at 20 °C and a factor of 27-39 higher at 60 °C. As shown by Fukuda et al. (1989) and demonstrated later for this system, such a large difference ensures that the reactivity ratio estimates are insensitive to the exact value of kp222/kp111; a ratio of 40 is assumed in this work. Fit with the Terminal Model. The solid line on Figure 4 shows the fit of the terminal model (TM) to the kcop p data, as calculated by eq 3 using the r1 and r2 estimates from the fit to the polymer composition. It is clear that the data are not well represented by this terminal model fit. Furthermore, the experimental kcop p values are greater than predicted by the terminal model. In all previous studies (Fukuda et al., 1992; Jenkins and O’Driscoll, 1995), experimental kcop p values are lower than predicted by the terminal model. This surprising result, first reported at a 1996 IUPACsponsored meeting on free-radical polymerization (Hutchinson et al., 1996), was also independently reported by Madruga and Fernandez-Garcia (1996) for the same system at the same conference. Rather than a direct measure of kcop p , however, these authors report a lumped rate parameter, kcop p /xkt. The same group (Arias et al., 1993) has published similar results for MMA/methyl acrylate (MA), which, as might be expected, have similar ri values to MMA/nBA. The penultimate model with an MMA s value of 10 was used to represent the observed increase in the MMA/MA copolymer rate (kcop p /xkt) over that predicted by the terminal model. The advantage of the PLP/MWD technique over rate data is that it provides a direct measure of kcop p , eliminating the need to make ad-

r1

r2

SSEa

2.24 ( 0.09 0.414 ( 0.017 0.0247 1.60 ( 0.11 0.535 ( 0.069 0.564 1.60 ( 0.11 0.535 ( 0.069 0.0226

81 1.44 ( 0.06 0.363 ( 0.026 0.146

) sum of squared errors.

ditional assumptions about the initiator decomposition rate, efficiency, and termination rates in the copolymerization system. The TM reactivity ratios can be estimated solely from kcop p data, just as they can be estimated based only on polymer composition data. As summarized in Table 4, the parameter estimates from the two fits are very different. Values of r1 ) 1.60 and r2 ) 0.535 give an excellent fit of the kcop p data (dashed line on Figure 4) but not the polymer composition data (dashed line on Figure 1). Despite the scaling of the kcop data by p kMMA , it was found that the sum of squared error (SSE) p from the fit to the rate data was significantly higher than that from the model fit to the composition data. This is an indication of the greater experimental uncertainty in the rate data, as well as the larger range of magnitudes (1-3.5 for kcop p ; 0.1-0.9 for fp1). In order to give the rate and composition data sets equal weightMMA ing for combined fits, the kcop rate data are scaled p /kp by a further factor of 5. As shown in Table 4, this scaling reduces the SSE for the kcop data to the same p level as for the composition data. This further scaling is used in the remainder of this work. An attempt to fit the TM to the combined composition and kcop data further illustrates its inadequacy to p represent the MMA/nBA system. Once again, parameter estimates are summarized in Table 4. The SSE for the combined fit is much greater than the sum of the two individual fits; the model cannot adequately represent both sets of data simultaneously, as also illustrated by the dotted lines on Figures 1 and 4. Schweer (1993) has made the same observation for the styrene/MMA system; the terminal model can adequately fit either the rate or the composition data but not both simultaneously. It must be remembered, however, that the TM reactivity ratios estimated from polymer composition data also give a good representation of the triad sequences as measured by NMR for both the styrene/MMA (Maxwell et al., 1993) and MMA/ nBA (Aerdts et al., 1994) systems. Thus, it can be concluded that polymer structure (composition and triad sequences) is well represented by the terminal model and that it is the rate (kcop p ) data that requires further interpretation. Fit with the Penultimate Model. With the failure of the TM to fit both the composition and kcop p data for MMA/nBA, we turn now to the penultimate model. It is clear that, like the majority of copolymerization systems, MMA/nBA can be well modeled by the IPUE; i.e.; composition data are well represented by the terminal model, but kcop data are not. However, folp lowing the suggestion of Burke et al. (1994a, 1995), we choose to fit all parameters of the penultimate model to the combined composition and kcop data set. This p differs from the approach typically taken, which is to fix rj1 and rj2 to the values estimated from the TM fit to polymer composition data and then use kcop p data only

1110 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 Table 5. Penultimate Model Parameter Estimatesa case

Np

rj1

rj2

s1, s2

SSE

full model

6 4

r22 ) 0.56 (0.49 r12 ) 0.60 ( 0.83 0.401 ( 0.020

IPUEb

2

2.24

0.414

s1 ) 4.3 ( 17.8 s2 ) (-82 ( 3.5) × 105 s1 ) 10.4 ( 154 s2 ) 0.026 ( 0.083 s1 ) 2.0 ( 7.8 s2 ) (-240 ( 1.1) × 103

0.0461

IPUE

r11) 2.11 ( 0.40 r21 ) 6.6 ( 24.5 2.16 ( 0.10

reduced IPUE: Ic IIb,c IIId

3 1 3

2.15 ( 0.08 2.24 2.15 ( 0.08

0.400 ( 0.017 0.414 0.400 ( 0.017

0.0488 0.0495 0.0488

0 3

2.24

0.414

1.90 ( 0.12 1.99 ( 0.05 s1 ) 1.98 ( 0.13 s2 ) 0.43 0.96

2.16 ( 0.08 2.15 ( 0.08 1.86 ( 0.08

0.401 ( 0.017 0.399 ( 0.017 0.356 ( 0.019

1.95 ( 0.12 1.87 ( 0.11 2.47 (0.19

0.0486 0.0489 0.0807

IVb-d reduced IPUE:c MMA ) 30 knBA p /kp nBA MMA ) 50 kp /kp pMMA calibc

3

0.0483 0.0495

0.466

MMA MMA All fits to combined data set, with kcop scaled by factor of 5. kBA ) 40, unless otherwise noted. b Reactivity ratios from TM p /kp p /kp fit to composition data. c s1 ) s2. d s2 ) rj1rj2/s1.

a

Figure 5. Fit of penultimate model to polymer composition and kcop data: full six-parameter model (s); four-parameter IPUE p (- - -); three-parameter IPUE with s1 ) s2 (-‚-‚); zero-parameter IPUE, with s1 ) s2 ) (rj1rj2)1/2 (‚‚‚). See Table 5 for parameter values.

to estimate s1 and s2. Table 5 contains a summary of the parameter estimates with standard deviations for the various cases discussed below. The first case examined is the full 6-parameter and knBA fixed at 1 and penultimate model, with kMMA p p 40, respectively. Although the model gives a good fit to the data (see Figure 5), standard deviations for the parameters are large, with many of the reactivity ratios encompassing zero (see Table 5). Within this uncer-

tainty, however, it can be seen that the IPUE assumption appears valid: r11 ≈ r21 and r22 ≈ r12. Thus, the next case considered is the IPUE model, for which only four parameters must be estimated. The resulting SSE is no larger than the SSE for the full sixparameter model, indicating that the model simplification was justified (also see Figure 5); this conclusion can also be statistically shown by using an F test (Hill et al., 1982; Burke et al., 1994a). Not only are the standard deviations for the monomer reactivity ratios greatly reduced, but the estimates converge to the same values obtained by the TM fit to the polymer composition data. This result is very encouraging, since it strongly indicates that the rj1 and rj2 values estimated from TM fits to the polymer composition data can be used to represent IPUE copolymerization systems. While the generality of this result must be verified by studying other systems, it supports the validity of decoupling the fitting of monomer reactivity ratios (rji) from radical reactivity ratios (si), as done by Fukuda, Ma, and co-workers. Even with the IPUE, however, the estimates for the radical reactivities (si) remain indeterminate. The same result is found if rj1 and rj2 are set to TM estimates and only s1 and s2 are estimated. As pointed out by Fukuda et al. (1989), this result is to be expected for two reasons. First of all, rj1rj2 for MMA/nBA is close to unity, greatly increasing the correlation between s1 and s2. Second, rj1/kp11 . rj2/kp22, making it very difficult to obtain a reliable estimate for s2 (see eq 3). From these results, it is apparent that a threeparameter penultimate model (rj1, rj2, and s1 ) should be examined next. Several approaches can be taken: I. Simultaneously fit all three parameters, with s2 ) s1. This approximation is suggested by Fukuda et al. (1989, 1992) to deal with data for which it is not possible to estimate both s values. II. Set rj1 and rj2 to TM values and fit the data with s2 ) s1, a one-parameter fit. III. Fit all three parameters, using the suggestion of Fukuda et al. (1987, 1991) that s2 ) (rj1rj2)/s1 (eq 8). IV. Combining cases II and III leads to eq 9, with no parameters to estimate. For MMA/nBA, this simplification leads to s1 ) s2 ) 0.96. As summarized in Table 5, the first three of these approaches leads to virtually identical estimates for the

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1111

three reactivity ratios rj1, rj2, and s1, with s1 in the range of 1.9-2.0. Furthermore, the fits cannot be distinguished from those of the higher-order penultimate models (see Figure 5). It is evident that this system is insensitive to the value of s2 and that it is not possible to discriminate between cases I, II, and III. However, it is also clear that the combination of the two assumptions which lead to eq 9 and case IV does not provide an adequate representation of the kcop p data (see Figure 5). Indeed, since MMA/nBA is close to an ideal system (rj1rj2 ) 0.93), this approximation fits the kcop p data little better than the terminal model. It is useful to examine the sensitivity of the model parameters to the estimate of knBA and to SEC calibrap tion. The three-parameter model, with s1 ) s2, is used for this purpose. As summarized in Table 5, changing MMA knBA from 40 to 30 or 50 has no effect on the fit of p /kp the penultimate model. This is not surprising for a system where r1/kp11 . r2/kp22. Thus, the uncertainty in the absolute value of knBA has no effect on the p parameter estimates or conclusions of this study. The same cannot be said about the effect of SEC calibration. Fitting the model to the kcop p values calculated assuming pMMA calibration not only yields a higher estimate for s1 (as would be expected from an examination of Figure 3) but also shifts the rj1 and rj2 estimates away from the values estimated from the TM fit to the composition data. (Since the pMMA and pBA calibrations are very close, this result is equivalent to what would be obtained assuming a composition weighted average, as done previously in PLP copolymerization studies.) Although the main conclusions of the study would not be changed, it is clear that more meaningful results and more accurate parameter estimates are obtained from a PLP/MWD study when SEC calibration issues are explicitly examined.

converged to the values estimated from the terminal model. Thus, the IPUE is particularly attractive for the modeling of copolymerization systems, since it can utilize previously determined monomer reactivity ratios from TM fits to composition data. In contrast, the parameter estimation work done by Maxwell et al. shows that the “bootstrap model” reactivity ratios must change significantly from the terminal model values in order to adequately describe the styrene/MMA composition and rate data. As for many other systems, the IPUE with a single radical reactivity adequately describes the MMA/nBA datass2 cannot be estimated for this system. If we accept the premise of eq 8, a value of 0.43 is calculated for s2 (rj1 ) 2.15, rj2 ) 0.40, s1 ) 1.98). Mechanistically, these s1 and s2 values are appealing: having nBA (rather than MMA) as a penultimate unit to an MMA radical doubles the addition rate of MMA monomer; having nBA (rather than MMA) as a penultimate unit to a nBA radical also roughly doubles the addition rate of nBA monomer. (It is also noted, coincidentally, that rj1 ≈ s1 and rj2 ≈ s2. This also seems to be the case for many styrene/acrylate and styrene/methacrylate copolymerization systems examined previously, where all s and rj values are < 1 (Fukuda et al., 1992). However, the MMA/VAc study of Ma et al. (1993) violates this observation.) Finally, we have demonstrated the importance of SEC calibration for the interpretation of kcop p PLP data. The assumption that copolymer calibration can be approximated by a weighted average of homopolymer calibrations is certainly not valid for MMA/nBA and possibly many other systems. Only when SEC calibration is specifically examined for a copolymer system should PLP kcop p data be deemed quantitative. Literature Cited

Conclusions The results from this MMA/nBA copolymerization study are consistent with previous work in many ways: although the terminal model is able to predict either composition or rate (kcop p ) data, it is unable to adequately represent both types of data simultaneously. This conclusion seems to be the norm, rather than the exception, for common monomer systems of interest. MMA/nBA, however, has proven to be a very interesting system to study for two particular reasons. First, although it is close to an ideal copolymerization system, it clearly does not obey terminal kinetics. Second, the terminal model underpredicts kcop p ; for other systems studied to date, kcop is overpredicted by terminal kinetp ics. It is clear that the radical reactivity for MMA (s1) must be greater than unity in order to fit the data with the IPUE. Also, we observe that the data cannot be well represented by eq 9, which results from combining two different suggestions published by Fukuda and coworkers. We leave the mechanistic interpretation of these observations to others. No doubt the MMA/nBA data could be equally well fit by other models, as Maxwell et al. (1993) demonstrate for the styrene/MMA system. However, we feel the statistical treatment of this work gives some support to the general form of the penultimate model. Parameters were allowed to vary freely during the fitting procedure, but the “best fit” monomer reactivity ratios

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Received for review July 12, 1996 Revised manuscript received September 9, 1996 Accepted September 10, 1996X IE9604031

X Abstract published in Advance ACS Abstracts, February 15, 1997.