Assessment of Copolymerization Models via Stochastic

Ind. Eng. Chem. Res. 2007, 46, 2455-2465

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Assessment of Copolymerization Models via Stochastic Integrodifferential Estimation Francisco Lo´ pez-Serrano,*,† Jorge E. Puig,‡ and Jesu´ s A Ä lvarez§ UniVersidad Nacional Auto´ noma de Me´ xico, Facultad de Quı´mica, Departamento de Ingenierıá Quı´mica, Ciudad UniVersitaria, D. F. 04510, Me´ xico; UniVersidad de Guadalajara, Departamento de Ingenierıá Quı´mica, CUCEI, Guadalajara, Jalisco 44430, Me´ xico; and UniVersidad Auto´ noma Metropolitana-Iztapalapa, Departamento de Ingenierıá de Procesos e Hidra´ ulica, D. F. 09340, Me´ xico

The model assessment and discrimination problems for copolymerization systems with measurements of polymer composition and average propagation rate constant versus feed composition are addressed via an integrodifferential stochastic estimation approach with error propagation analysis. The methodology is applied to perform the ultimate versus penultimate model discrimination problem for two previously studied representative systems with experimental data: styrene/methyl methacrylate (STY/MMA) and parachlorostyrene/para-chloromethoxystyrene (PCS/PMOS). It was found that, in each case, (i) the model discrimination task can be performed with more precision than with the integral method alone in the light of the experimental data uncertainty, (ii) nearly constant reactivity ratios (ri) are exhibited, (iii) a complex radical reactivity ratios (si) functionality, not reported before, was found in the STY/MMA system, (iv) approximately constant radical reactivity ratios are presented in the PCS/PMOS system, a feature that is typical of the (implicit or) penultimate model statement, and (v) the radical stabilization mechanism differs from the one proposed in previous reports (s1s2/r1r2 * 1). 1. Introduction Even though the reactivity ratios (RRs) determination problem has been addressed for over 60 years, this subject is still controversial. Basically, the model assessment, discrimination, and parameter fitting problems have been addressed via the regression method, according to the following rationale: (i) a candidate model is considered and its parameters are assumed to be constant, (ii) standard regression is applied to fit the experimental data, and (iii) the model goodness is assessed on the basis of the fitting report in conjunction with the inspection of the predicted versus actual data behavior. If the model description capability is unsatisfactory, the candidate model is either replaced by another candidate model or redesigned on the basis of the model mismatch in conjunction with physical insight. The most widely used system descriptions are the twoparameter ultimate1 (U), the four-parameter penultimate2 (PU), and the three-parameter bootstrap3 (B) models. In the case of RRs estimation, they are obtained by fitting the U model to polymer against feed composition data, and then the average propagation rate constant is predicted.4-7 The PU and B models have been proposed because the U model has, in several instances, not been able to predict adequately the average propagation rate constant behavior,5,8,9 or the triad sequence4,10 against feed composition. Moreover, an implicit penultimate5 (IPU) model has been proposed where the four reactivity ratios from the PU model are restricted to two. On the other hand, it has been claimed that any model with a sufficient number of adjustable parameters4 can describe the copolymer composition and the propagation rate constant dependences on feed composition. A drawback of the PU model is that, if only composition * To whom correspondence should be addressed. Tel.: (52) 5556225361. Fax: (52) 55-56225355. E-mail: lopezserrano@ correo.unam.mx. † Universidad Nacional Autońoma de Me´xico. ‡ Universidad de Guadalajara. § Universidad Autońoma Metropolitana-Iztapalapa.

data are used, multiple RR solutions can be obtained11 from the regression technique. In the B case, when solving for the partition parameter, two solutions exist12,13 and, unless one is negative, there are not convincing physical arguments to disregard one of them. Regarding the model discrimination task, controversial conclusions prevail. Some authors claim that the task can be performed,14-16 whereas others reported inconclusive results.4,12,17,18 Basically, the problem has been studied with regression methods: the candidate models are regarded, some of their parameters are assumed to be constant, and their values are drawn by regression against the experimental data. Recently, an instantaneous observability-based19,20 integrodifferental (ID) deterministic estimation approach13 has been presented to assess the RR dependency pair without a priori assuming the constancy or some concentration dependency of the RRs, by exploiting the additional information contained in the measurement trend derivatives. This approach is endowed with necessary solvability conditions, delimitation of the rich-in-information interval in the composition interval, and the capability of yielding an analytic RR dependency representation.13 As it stands, the application of this approach: (i) raises a key concern on the model assessment certainty, because of the employment of data trend derivatives, and (ii) needs the consideration of the two-measurement (polymer composition and average propagation versus feed composition) case. From an estimation theory perspective,21,22 the meaningfulness of the RR- homopolymerization rate (HPR) dependency assessment signifies the attainment of a suitable compromise between the information contained in the measurement derivative and the associated error propagation by data trend differentiation, depending on the number and quality of the experimental data and on the particular system data-toestimate information and error propagation mechanism. These considerations justify the development of a stochastic estimation framework to address the two-measurement copolymer system RR and HPR dependency determination in the light of a quantitative estimate certainty assessment.

10.1021/ie061479s CCC: $37.00 © 2007 American Chemical Society Published on Web 03/14/2007

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Ind. Eng. Chem. Res., Vol. 46, No. 8, 2007

In this work, the modeling assessment (i.e., RR and HPR dependency on feed composition determination) problem for copolymerization systems with polymer composition and average propagation rate constant versus feed composition is addressed. First, the two-measurement deterministic estimation problem is focused on yielding solvability conditions and a combined RR-HPR dependency construction formula. Then, the stochastic ID estimation approach version is developed, providing: (i) mean RR and HPR dependency estimates and their uncertainty bands and (ii) criteria to establish the modeling assessment certainty and/or discrimination results in the light of the errors contained in the experimental data. The approach is applied to consider the ultimate versus penultimate model discrimination problem for two previously studied representative systems with experimental data, styrene/methyl methacrylate5 (STY/MMA) and para-chlorostyrene/para-methoxystyrene8 (PCS/ PMOS), and the findings are put in perspective with the ones drawn from the application of the standard integral regression method alone. 2. Model Assessment Problem In this section, the model assessment problem for copolymerization systems with polymer composition and average propagation rate constant versus feed composition measurements is formulated. One should recall the Mayo-Lewis1 (ML) model, allow for the reactivity ratios (r1 and r2) (RRs) and homopropagation (HPR) term (k1 and k2) dependency on feed composition (x), and write the resulting algebraic equations in a format suited for the purpose at hand.

y)

z)

κ1(x, d) ) s11

r11 x + 1 - x

,

r11 x + (1 - x)/s12

κ2(x, d) ) s22

r22(1 - x) + x r22(1 - x) + (1 - x)/s21

then the GML (eq 1) model becomes the penultimate2 (PU) model, with four (or six) constant parameters23 c (or d),

c ) (r11, r21, r12, r22)′, r11 ) k111/k112, r21 ) k122/k121, r12 ) k211/k212, r22 ) k222/k221 d ) (r11, r22, s11, s22, s12, s12)′, s11 ) k111, s22 ) k222, s21 ) k211/k111, s12 ) k122/k222 where the entries of c are associated with the cross-propagation mechanism description and the entries of d contain terms (s11, s22) [or (s12, s12)] related to the homopolymerization rate constants (or radical reactivity ratios). Consider the polymer composition (δy) and the average propagation rate constant versus the discrete experimental data of feed composition (δz),

δ ) {δy, δz}, δy ) (ψ, χy), δz ) (ζ, χy)

r1 x2 + 2x(1 - x) + r2(1 - x)2

) g(x, r, k), r1 x/k1 + r2(1 - x)/k2 k ) (k1, k2)′ ) [κ1(x), κ2(x)]′ ) κ(x, r) (1b)

where x (usually denoted by f) is the feed composition, y (usually denoted by F) is the polymer composition, z (usually denoted by kp) is the average propagation proportionality term, F1(x) [or F2(x)] is the scalar function that describes the first (or second) reactivity ratio r1 (or r2) dependency on feed composition (x), and κ1(x) [or κ2(x)] is the scalar function that describes the first (or second) homopropagation term k1 (or k2) [usually denoted by k11 (or k22)] dependency on feed composition (x). Henceforth, eqs 1a and 1b will be referred to as the generalized MayoLewis model (GML). When the RRs (r) and the HPR (k) terms are constant, i.e.,

G(x) ) r ) (k11/k12, k22/k21)′, K(x, r) ) k ) (k11, k22)′ (2a,b) eq 1 reduces to the classic ML’s ultimate (U) model, with k12 and k21 being the cross-propagation constants and r1 and r2 being the homo-to-cross-propagation quotients. If the RR (F1 and F2) and HPR (κ1 and κ2) functions have the first-order polynomial quotient forms,

F1(x, c) ) r12

r11 x + 1 - x r2(1 - x) + x 2 2 (x, c) ) r , F 2 1 r12 x + 1 - x r21(1 - x) + x (3a,b)

(4)

where

ψ ) ψo + ey, ψ ) (y1, ..., yny)′, ey ) (ey1, ..., eyny)′, χy ) (x1, ..., xny)′

r1 x2 + x(1 - x)

) f(x, r), r1 x2 + 2x(1 - x) + r2(1 - x)2 r ) (r1, r2)′ ) [F1(x), F2(x)]′ ) F(x) (1a)

(3c,d)

ζ ) ζo + ez, ζ ) (y1, ..., yny)′, ez ) (ez1, ..., enz z)′, χz ) (x1, ..., xnz)′ ψ (or ψo) are the ny measured (or actual) polymer composition values at the ny feed compositions χy, ζ (or ζo) denotes the nz measured (or actual) average propagation rate constant values at the nz feed compositions χz, and ey, eyx, ez, and ezx are the corresponding experimental sampling procedure and/or instrument errors. Given the GML model (eq 1) in conjunction with the experimental data δ (eq 4), our model assessment problem consists of the following: (i) developing the means to establish conditions so that the four RR (F1 and F2) and HPR (κ1 and κ2) function dependencies (eq 3) on feed composition can be meaningfully drawn in the light of the experimental data (δ) (eq 4) and their uncertainty, (ii) applying the feasibility results to address the ultimate versus penultimate model discrimination problems for two copolymerization systems (PCS/STY, PCS/ PMOS) with experimental data,8 with standard regression techniques, and (iii) comparing the results with the previous ones drawn from regression methods. 3. RR and HPR Dependency Estimation In this section, the model assessment problem for the copolymerization systems with polymer measurements of composition and average propagation rate versus feed composition is addressed by means of an ID stochastic estimation methodology that generalizes the single-measurement deterministic one employed before13 to address the RR deterministic estimation problem on the basis of the polymer versus feed composition measurement.13 The reader more interested in the

Ind. Eng. Chem. Res., Vol. 46, No. 8, 2007 2457

application aspect may go directly to the Summary subsection and then to the Application Examples section. 3.1. Deterministic Estimation. Following the abovementioned ID deterministic estimation approach, a standard regression technique is applied to noiselessly fit the experimental data set δy (or δz) (eq 4) by means of a continuously differentiable function R (eq 5a) [or β (eq 5b)] with ny (or nz)adjustable parameters, and the analytic differentiation of these functions yields a noiseless representation (eq 5c) [or (eq 5d)] of the measurement derivative trends:

y ) R(x), z ) β(x), y’ ) R’(x), z’ ) β’(x) (5a-d)

(1 - x)[y + (1 - x)xy’] ) ι1(x, y, y’), x[2(1 - y)y - (1 - x)xy’] x{[3 - 2y]y - (1 - x)xy’ - 1} ) ι2(x, y, y’) (10a,b) r2 ) x[2(1 - y)y - (1 - x)xy’]

r1 )

for the RR vector r in terms of the polymer composition data ψ-x, where J is the Jacobian 2 × 2 matrix of the map φ with respect to the vector r. If, in addition, the nonsingularity condition

det M(x, r, k) ) [k12k22r1r2{r2 + x[2 - 2r2 + x(r1 + r2 - 2)]}]2

* 0, [xk2r1 + (1 - x)k1r2]4 M(x, r, k) ) ∂kγ(x, r, k) (11)

In compact notation, these data graphs are written as follows

ψ ) λ(x), ζ ) µ(x)

(6)

is met, equation pair 7b can also be solved analytically where

ψ ) (y, y’)′, λ(x) ) [R(x), R’(x)]′, ζ ) (z, z’)′, µ(x) ) [β(x), β’(x)]′ On the basis of the notion of instantaneous observability19-22 that underlies the design of nonlinear Luenberger observers,24 let us augment the GML (eq 1) with two (hopefully) independent equations to enable the possibility of setting an estimation scheme that exploits the information content of the measurement trend derivatives, in the understanding that: (i) this possibility depends on having sufficient experimental data, and (ii) in such a possibility resides the ID approach advantage over the standard integral regression-based approaches commonly employed for copolymerization modeling assessment and fitting studies. For the purpose at hand, the GML equations (eq 1) derivatives can be written as

y’ ) ∂xf(x, r) + [∂rf(x, r)]r’, (∂rf)r’ ) 0

(7a)

z˘ ) ∂zg(x, r, k) + [∂rg(x, r, k)]r’ + [∂kg(x, r, k)]k’, (∂rg)r’ + (∂kg)k’ ) 0 (7b) where ∂rfr’ (or ∂rgr’ + ∂kgk’) ) 0 because the function pair (or quartet) F1-F2 (or F1-F2-κ1-κ2) is a parametric representation of a curve in the r1-r2 plane25 (or r1-r2-k1-k2 space). The combination of eqs 1 and 7 yields the deterministic augmented GML model:

y ) f(x, r), y˘ ) ∂xf(x, r); z ) g(x, r, k), z˘ ) ∂zg(x, r, k) or, equivalently, in compact form

ψ ) O(x, r), ζ ) γ(x, r, k)

(8a,b)

where

ψ ) (y, y˘ )′, O(x, r) ) [f(x, r), ∂xf(x, r)]′, ζ ) (z, z˘ )′, γ(x, r, k) ) [g(x, r, k), ∂zg(x, r, k)]′ det J(x, r) )

x2(1 - x)2

* 0, {r2 + x[2(1 - r2) + x(r1 + r2 - 2)]}3 0 < x < 1, J(x, r) ) ∂kφ(x, r) (9)

is met, equation pair 7a can be analytically solved,

k1 )

xzk2r1 xk2[2 + x(r1 - 2)] + (x - 1)[(x - 1)k2 + z]r2 ) φ1(x, r1, r2, z, z˘ ) (12a)

k2 ) z2r2 xz’r2 + zr2 + x3z’(r1 + r2 - 2) - x2[2z’(r2 - 1) + z(r1 + r2 - 2)] ) φ2(x, r1, r2, z, z˘ ) (12b)

for k in terms of the propagation rate-RR data (ζ, r)-x, where M is the Jacobian 2 × 2 matrix of the map γ with respect to the vector k. In compact notation, the solution (eqs 6-8) of the augmented GML model is written as follows

r ) ι(x, ψ), k ) φ(x, r, ζ) and the combination of these expressions with the data equations (eq 6) yields the RR and HPR dependencies

r ) ι[x, λ(x)] ) G(x), x ⊂ Xr ) {x ⊂ (0, 1) | dJ(x) * 0} (13a) k ) φ[x, G(x), µ(x)] ) κ(x), x ⊂ Xk ) {x ⊂ [0,1] | dM(x) * 0} (13b) in analytic form, where

dJ(x) ) det J[x, G(x)], dM(x) ) det J[x, G(x), K(x)] (13c,d) Xr (or Xr) is the feed composition interval where, given the data ψ [or (ζ, r)] for the RR (or HPR) vector r (or k) can be uniquely determined. According to eq 13a, (i) the RR function pair cannot be estimated in the neighborhood of zero and one feed composition (nearly homopolymer behavior), and (ii) for some x-r1-r2 combination, there may exist an interior feed composition neighborhood where the RR function pair cannot be estimated. Equation 13b indicates that: (i) the HPR function can be estimated in the vicinity of zero and one feed composition, (ii) for some x-r1-r2 combination, there may exist an interior feed composition neighborhood where the HPR function pair cannot be estimated, and (iii) dM(x) ) ∞ when k12 ) k21, implying that r2/r1 ) k2/k1, and this in turn signifies that (eq 13b) is redundant because the pertinent information on k is contained in the r estimate (eq 13a).

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3.2. Stochastic Estimation. From a practical viewpoint, the preceding deterministic estimation scheme (i) identifies necessary conditions (eqs 8 and 10) for the solvability of the joint RR-HPR dependency estimation problem and (ii) provides analytic construction formulas for the derivation of the dependencies on the continuous data trend representation basis (eq 5). From an estimation theory perspective,21,26,27 the meaningfulness of the RR-HPR dependency assessment signifies the attainment of a suitable compromise between information availability and error propagation by differentiation, depending on the experimental data number and quality and on the particular system data-to-estimate information and error propagation mechanism. These comments justify the consideration of a stochastic estimation framework to address the key issue on the certainty of the RR and HPR dependency estimates in the copolymer system model assessment problem. On the basis of the mean squared estimation error qy (or qz), provided by the regression-based fitting report for the polymer composition (or average propagation) fitting function R (eq 5a) [or β (eq 5b)], let us write the stochastic data trend model,

y ) R(x) + Vy, y’ ) R’(x) + νy; z ) β(x) + Vz, z’ ) β’(x) + νz to describe the measurement trend signals (y, y’, z, and y’) random nature, in terms of their interpolated smooth representations (R, R’, β, and β’) (eq 5), now with white noise zero-mean uncorrelated deviations (Vy, νy, Vz, and νy) (E is the statistical expectation operator),

Vy ≈ N[0, qy], νy ≈ N[0, qy2]; Vy ≈ N[0, qz], νz ≈ N[0, qz2] E(Vyνy) ) E(Vzνz) ) E(VzVy) ) E(Vyνz) ) E(Vzνy) ) 0 and qy2 and qz2 are set according with random signal derivation formulas.26 In compact vector-matrix notation, the preceding stochastic data trend model is written as follows

ψ ) λ(x) + ey, ey ) (Vy, νy)′ ≈ N[0, Q], Q ) diag(qy, qy2) (14a) ζ ) µ(x) + ez, ez ) (ez, νz)′ ≈ N[0, Θ], Θ ) diag(qz, qz2) (14b) The combination of this stochastic data description with the deterministic augmented GML model (eq 7) yields the augmented stochastic GML model:

φ(x, r) ) λ(x) + ey, γ(x, r, k) ) µ(x) + ez

(15a,b)

From standard nonlinear system estimation approaches,27 the solution of this model, in the sense of a second-order statistical approximation, yields the RR and HPR stochastic estimates

r ≈ N[rˆ, Σ(x)], rˆ ) F(x), Σ(x) ) S[x, F(x)] (16a) k ≈ N[kˆ , Ω(x)], kˆ ) κ(x), Ω(x) ) O[x, F(x)]

(16b)

where rˆ (or kˆ ) is the augmented GML model (eq 12) based RR (or HPR) mean estimate, and Σ (or Ω) is the corresponding 2 × 2 estimation error covariance matrix,

Σ(x) ) E(r˜ r˜ ′) )

[

]

σ1(x) σc(x) , σc(x) σ2(x) Ω(x) ) E(k˜ k˜ ′) )

[

ω1(x) ωc(x) ωc(x) ω2(x)

]

(17a,b)

The related data-to-estimate error propagation matrix S (or O) is given by

S(x, rˆ ) ) [J-1(x, rˆ )]Q[J-1(x, rˆ )]′, L(x, rˆ , kˆ ) ) ∂rγ(x, rˆ , kˆ ) (18a) O(x, rˆ , kˆ ) ) [M-1(x, rˆ , kˆ )]{Θ + [L(x, rˆ , kˆ )][Σ(x, rˆ )] × [L(x, rˆ , kˆ )]′}[M-1(x, rˆ , kˆ )]′ (18b) J (or M) is the Jacobian 2 × 2 matrix associated with the necessary solvability condition (eq 13c) [or (eq 13d)] of the deterministic RR (or HPR) dependency estimation problem, and L is the Jacobian 2 × 2 matrix of the nonlinear map γ with respect to the vector r. Observe that if, at some point or interval along the feed composition interval, the Jacobian matrix J (or M) is nearly singular (eq 9) (or eq 11), the entries of Σ (or Ω) are very large, or equivalently, if the underlying instantaneous observability property is ill conditioned,28 the RR (rˆ) [or HPR (kˆ )] estimates become very uncertain. While eq 18a describes the ey-to-r˜ uncertainty propagation mechanism from the polymer composition data (ψ) to the RR estimate (kˆ ), eq 18b characterizes the ez-to-k˜ uncertainty propagation scheme from the propagation rate data (ζ) and the RR estimate (rˆ ) to the HPR estimate (kˆ ). Observe that the existence (eqs13c and 13d) of the deterministic estimate solution (eqs 13a and 13b) ensures the stochastic estimates (eq 16) uncertainty boundedness (eq 18), or equivalently, the deterministic estimation problem solvability is necessary but not sufficient for the stochastic estimation problem solvability, in the sense that bounded but excessively large estimate uncertainties, in comparison to the mean trend changes, signify that no meaningful RR and/or HPR function assessments can be done. According to eq 16a, (i) the uncertainty of the RR function pair estimate grows rapidly as the feed composition approaches 0 or 1 (nearly homopolymer behavior), and (ii) for some x-r1r2 combination, there may exist an interior feed composition neighborhood where the uncertainty of the RR function pair estimate is excessively large or infinite. Equation 16b says that (i) the HPR function estimate has bounded uncertainty in the vicinity of 0 and 1 feed composition, (ii) for some x-r1-r2 combination, there may exist an interior feed composition neighborhood where the uncertainty of the HPR function pair estimate is excessively large or infinite, and (iii) dM(x) ) ∞, the error covariance matrix Ω becomes undetermined because, as mentioned before, the pertinent information on k is contained in the r estimate (eq 16a). If the overall mean dependency estimate changes are sufficiently larger than their associated error bands, then conclusive results on the dependency estimates of RR and HPR (either constant or composition-varying) can be drawn. Otherwise, the error propagation mechanism analysis provides elements to perform a tailored experimental design for modeling assessment purposes, including specifications on the number of experimental data as well as their location in the feed composition interval. 3.3. Summary. The application of the ID stochastic estimation technique has two steps. In a first step, standard regression is applied to fit the experimental data sets (δy and δz) (eq 4) with the continuous function (noiseless) trends (R and β) (eq 5), and the related fitting report is employed to generate the


stochastic data trend model (eq 14). In a second step, the solution (eq 16) of the augmented stochastic GML model (eq 15) yields estimate means and their uncertainty characterization, which in a detailed application-oriented problem is given by the following: (i) estimate means

rˆ1 ) F1(x), rˆ2 ) F2(x), kˆ 1 ) κ1(x), kˆ 2 ) κ2(x)

(19a)

(ii) mean (squared root) errors

Table 1. Polymer Composition Data Trend Parameters and Error for the STY/MMA and PCS/PMOS Systems r11 STY/MMAa PCS/PMOSa STY/MMA29 set 1 STY/MMA29 set 2 a

1/2

1/2

1/2

1/2

σ1 (x), σ2 (x), ω1 (x), ω2 (x)

r12

r21

r22

R2

σ (std dev)

0.647 5 0.332 4 0.326 4 0.499 4 0.999 68 1.131 1 0.522 2 1.203 3 0.414 4 0.999 89 0.664 0 0.384 0.366 0 0.489 0

0.010 8 0.007 9 0.010 9

0.727 0 4.583

0.010 9

2.890 0 0.490 0

This work.

(19b)

and (iii) correlation coefficients

cr ) σc(x)/[σ11/2(x)σ21/2(x)], ck ) κc(x)/[κ11/2(x)κ21/2(x)] (19c) Finally, it must be pointed out that the stochastic mean and error characterization (eq 16) is in a form that is amenable to a diversity of statistical error analyses, like ellipsoidal (volume, area, and principal directions) interpretation, confidence intervals, bias measurement detection via χ2 tests, and so on.22 4. Application Examples In this section, the ID stochastic estimation approach is applied to two copolymerization systems that have been studied before with regression-based techniques,5,8 on the basis of polymer composition as well as average propagation rate with gel permeation chromatography (GPC) measurements5 for the STY/MMA system and with size exclusion chromatography (SEC) and SEC-differential viscosimetry (DV) measurements8 for the PCS/PMOS system. Then, the findings are used to perform the U versus PU model discrimination and for the identification of a functional relationship between the monomer and radical RRs (si). The results are put in perspective with those reached from previous4-6,8,29 regression approaches. 4.1. STY/MMA System. 4.1.a. Results Drawn with Integral Regression Method. For the STY/MMA system, it has been found that (i) the Mayo-Lewis or U model describes adequately the monomer composition with4 r1 ) 0.48 ( 0.03, r2 ) 0.42 ( 0.09 and5 r1 ) 0.523, r2 )0.44, HPR values4,5 of k111 ) 120, k222 ) 377 (L mol-1 s-1), radical RR4,5 of s1 ) 0.30 and s2 ) 0.53, and unprecise values extracted from kp data;6 (ii) the IPU model adequately describes both polymer composition and overall propagation rate data against feed composition; (iii) the copolymerization behavior can be described with a threeadjustable-parameter model;4 and (iv) when applying the PU model to composition data, multiple solutions exist.29 4.1.b. Continuous Data Trend Representation. Following the ID stochastic estimation approach first step (eq 14), standard regression was applied to fit the following: (i) the ny ) 32 discrete polymer versus feed composition data set5 (δz in eq 4a) with four-adjustable-parameter (c) auxiliary U model function [f is defined in eqs 1 and 3a-3d], and (ii) the nz ) 14 discrete average propagation rate versus feed composition data set5 (δz in 4b) with a four-adjustable-parameter rational equation:

R(x, cy) ) f(x, r), cy ) (r11, r21, r12, r22)′, β(x, cz) ) (1 + c1x)/(c2 + c3x + c4x2), cz ) (c1, c2, c3, c4)′ The fitting reports are presented in Tables 1 and 2, and the related data trend curves R and β are depicted in Figures 1 and

2, respectively. Observe that the polymer composition (or rate) trend errors are less than ∼6% (or 2%) of the trend change over the feed composition interval, meaning that the continuous data representation for modeling assessment purposes is to the point. 4.1.c. RR and HPR Dependency Estimation. The ID stochastic estimation second-step application, or equivalently, of formulas (eq 19), yields the mean RR and HPR dependency functions as well as the error uncertainty bands depicted in parts a and b of Figure 3, showing that (i) the RR dependencies r1 ) F1(x) and r2 ) F2(x) are nearly constant over the feed composition intervals (∼0.1 e x e 0.8) with small (≈(2-to8%) error bands; (ii) as expected from the solvability condition expression (eqs 13c and 13d), there is more uncertainty in the homopolymer-like behavior regions (close to x ) 0 and 1); (iii) the RR behavior tends to deviate from constancy when the other monomer is richer; (iv) the GPC-based HPR dependencies k1 ) k11 ) κ1(x) and k2 ) k22 ) κ2(x) are definitely not constant over the entire feed composition (0 e x e 1) interval with small (≈(2-to-6%) error bands; (v) the ID estimation for the HPR exactly coincides with the homopolymerization values at f1 ) 1 for the k11 and for f1 ) 0 for the k22 values; and (vi) the k22 ) κ2(x) function exhibits a significant decrease (from ∼377 to ∼60 L mol-1 s-1), a minimum of ∼60 at f1 ) x ≈ 0.7, and an increase from ∼60 to ∼125. Comparing with the aforestated integral regression-based results, (i) the integral regression and ID method yield similar results only for the RR values over the intermediate composition interval [0.3, 0.6]; (ii) the results are quite different for the HPR dependencies over the entire composition interval [0,1], and for the RR dependencies over the intervals [0, 0.3] and [0.6, 1]; and (iii) the regression-based RR and HPR constant values correspond to averaged values of the ID-based RR and HPR dependencies. These observations connect the integral regression and ID methods and verify the superior model assessment capability of the ID method. According to the error propagation results (parts a and b of Figure 3), the HPR dependency estimates are more certain than the ones of the RR dependencies, and this is explained by eq 18: the r-to-k error propagation term L attenuates the effect of the RR uncertainty estimate on the HPR estimates, and the measurement z-to-k error propagation matrix M-1 is smaller than the one J-1 associated with the y-to-r error propagation of the RR estimation step. In the light of the dependency function uncertainty, (i) the RR dependencies are nearly constant; (ii) the HPR dependencies are composition varying, especially for the expression k2 ) κ2(x) through the interval 0.1 e x e 1.0, and are less pronounced for k1 ) κ1(x), however, reaching a minimum at f1 ≈ 0.25 of 30% (ca. 40 L mol-1 s-1) of the HPR value (120 L mol-1 s-1). This last finding suggests that one should analyze the regions where a solvent30,31 (comonomer) can affect the propagation rate constants. 4.1.d. Modeling Implications. According to the ID approach results, the RRs are reasonably constant and the HPRs are not,

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Table 2. Rational with GPC (for STY/MMA) and Cubic Equation with SEC and SEC-DV (for PCS/PMOS) kp Measurements Parameters Fitting

STY/MMA PCS/PMOS (SEC) PCS/PMOS (SEC-DV)

c1

c2

c3

c4

R2

σ (std dev)

0.805 7 92.713 95.155

0.002 649 217.268 214.414

0.026 556 -173.110 -164.643

-0.014 125 37.307 32.678

0.996 66 0.999 58 0.998 59

14.893 3.264 6.109

or equivalently, the ultimate model does not describe the STY/ MMA experimental system5 behavior. Here, the ID approach discrimination capability allowed us to disregard the U model. The RRs constancy in conjunction with the composition-varying feature of the HPRs suggest using5,8 the IPU model consideration, and the presence of the radical stabilization mechanism, in the understanding that such a presence has been claimed5 or discarded.8 For this purpose, one should recall the RR and HPR dependencies (parts a and b of Figure 3), assume the IPU model, and draw the corresponding radical RRs dependencies on feed compositions (eqs 3c and 3d). The resulting dependencies

s1 ) s12 ) k211/k111 ) π1(x), s2 ) s21 ) k122/k222 ) π2(x) and their uncertainty bands are depicted in Figure 4, showing that π1(x) and π2(x) are not constant. This lead us to conclude that this system is not well-described by the IPU model, even though the overall propagation rate constant (kp) graphical description can be performed.4,5 Finally, to test the possibility

of a radical stabilization model, the function products and quotient

pr(x) ) F1(x)F2(x), ps(x) ) π1(x)π2(x), q(x) ) ps(x)/pr(x) are presented in Figure 5, showing that (i) the quotient varies from q ≈ 1.1 at f1 ) 0 to q ≈ 0.25 at f1 ) 1, and (ii) the products pr(x) (from f1 ≈ 0.2-0.6) and ps(x) (from f1 ≈ 0.3-0.8) are nearly constant. Thus, the ID method has enabled the characterization of the radical stabilization model, with a rather complex correlation between the radical (s1s2) and monomer (r1r2) RRs products, in the understanding that the same pursuit with integral method alone should require a more difficult trialand-error task. Although it is clear in Figure 5 that q is not constant, Figure 6 shows that several constant (erroneous) values for q (chosen at the maximum, minimum, and intermediate values presented

Figure 1. Polymer composition (y ) f) continuous data trend against feed composition (x ) f′) for the STY/MMA5 (0) and PCS/PMOS8 (b) systems, using the penultimate model.

Figure 2. Average propagation rate constant (z ) kp) continuous data trend against feed composition (x ) f), using a rational equation, for the STY/ MMA system. Experimental data (O) measurements from Fukuda et al.5

Figure 3. (a) Reactivity ratios (r1, r2) feed composition dependency and error band certainty (dotted lines) corresponding to plus and minus the standard deviation (σ1/2) for the STY/MMA system (f1 ) x). (b) Homopropagation rate constant estimates (k1 ) k11, k2 ) k22) against feed composition. The error band certainty (dotted lines) corresponds to plus and minus the standard deviation, STY/MMA system. The horizontal lines correspond to the values reported by Fukuda et al.5 (k11 ) 120 and k22 ) 377 L mol-1 s-1).


Figure 4. Radical reactivity ratios (s1, s2) behavior against feed composition using the penultimate model. Dotted lines correspond to plus/minus standard deviation (ω1/2) for the STY/MMA system.

Figure 5. Radical stabilization model5 (s1s2 ) r1r2) testing for the STY/ MMA system (q ) s1s2/r1r2).

Figure 6. Validation of kp prediction using constant monomer RRs (r1, r2) and q values obtained from Figure 5. A single parameter (s1) was fitted for the STY/MMA system.

in Figure 5) and s1 can fit the kp evolution, meaning that the integral method application to the IPU model is only a fitting device. In fact, Fukuda et al.5 reported the constants s1 ) 0.30 and s2 ) 0.53, but the difficulty of reproducing these values has been reported before.6 That the ID approach can yield information that cannot be obtained with the purely regression approach is clearly shown by Figure 4: the RRs changes are larger than their uncertainty band. In other words, the purely

Figure 7. Polymer composition (y ) F) continuous data trend against feed composition (x ) f) for the STY/MMA (b) systems using the penultimate model. Experimental data from Fukuda et al.,5 and multiple parameter sets (values shown in Table 1) taken from Kaim and Oracz.29

Figure 8. Reactivity ratios (r1) feed composition dependency and error band certainty (dotted lines) corresponding to plus and minus the standard deviation (σ1/2) for the STY/MMA system (thick line) and comparison with multiple sets presented by Kaim and Oracz.29

regression method may converge to one (local minimum) of several adjustable parameter sets that fit the experimental data equally well. Figure 6 demonstrates that (i) the data fitting is insensitive to the value of s2, and (ii) on the basis of the a priori modeling assumption of constant monomer and radical RRs, a global regression may not be able to perform the modeling discrimination task. 4.1.e. Comparison with the Regression Method. Keeping in mind that the possibility of multiple fitting parameter sets is a drawback of the integral regression method,11,29 let us test the proposed ID method behavior with respect to multiple fittings in its regression-based first step. In Table 1 are recalled two different data sets associated with the multiple PU modelbased regression solution,29 and the corresponding F1 (or r1 and r2) vs f1 plots are presented in Figure 7 (or Figure 8), showing that no parameter set discrimination is possible via standard regression because both sets describe the experimental data equally well. On the other hand, the ID method application on the basis of its step one (data trend fitting) performed with the PU model as a fitting device yields (i) the same RR and HPR dependency results when any of the fitting parameter sets is used and (ii) the dependencies (drawn with the parameter set 1) shown in Figures 8 and 9. As can be seen in these figures, the ID method enables us to say that the regression-based parameter set 1 is better than set 2, in the sense that set 1 yields

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Figure 9. Reactivity ratio (r2) feed composition dependency and error band certainty (dotted lines) corresponding to plus and minus the standard deviation (σ1/2) for the STY/MMA system (thick line) and comparison with multiple sets presented by Kaim and Oracz.29

better averages of the actual RR and HPR functions, and this corroborates that the ID method has a better modeling assessment capability than that of the integral regressor. 4.1.f. Concluding Remarks. The application of the proposed stochastic ID method to the STY/MMA system, with experimental data, enabled a meaningful assessment of RR and HPR dependencies, with results that are new in comparison with the ones previously drawn with integral regression alone. The ID and integral method functioning and results were connected and compared, establishing that (i) the model assessment ID method capability outperforms that of the integral regression alone; (ii) the regression method alone yields, possibly with multiple fitting parameter sets, averaged values of the actual RR and HPR functions over composition; and (iii) the ID technique functions well in spite of having parameter multiplicity in its regressionbased data trend continuous representation step. 4.2. PCS/PMOS System. 4.2.a. Results Drawn with Integral Regression Method. For the PCS/PMOS system, it has been reported that (i) the U model describes adequately the monomer composition with8 r1 ) 1.04, r2 ) 0.41, k111 ) 175, and k222 ) 94 (L mol-1 s-1), as well as radical RR8 s1 ) 1.08 and s2 ) 0.889; (ii) the radical stabilization model (s1s2 ) r1r2) does not hold;8 (iii) on the basis of graphical arguments (Figure 5, ref 8), the U or IPU models described the kp behaviors and (iv) there is not sufficient evidence to state that that the monomer RR (r1r2) and radical RR products (s1s2) are correlated.6 4.2.b. Continuous Data Trend Representation. Following the ID stochastic estimation approach first step (eq 14), standard regression was applied: (i) to fit the ny ) 16 discrete polymer versus feed composition data set8 (δz in eq 4a) with a fouradjustable-parameter (c) auxiliary U model function [f is defined in eqs 1 and 3a-3d], and (ii) to fit the nz ) 59 SEC, 44 SECDV discrete average propagation rate versus feed composition data set (δz in 4b) with a four-adjustable-parameter cubic polynomial:

R(x, cy) ) f(x, r), cy ) (r11, r21, r12, r22)′, β(x, cz) ) c1 + c2x + c3x2 + c4x3, cz ) (c1, c2, c3, c4)′ The fitting reports are presented in Table 1 (or 2) for composition (or kp), and the associated data trend curves R and β are depicted in Figures 1 and 10, respectively. Again, the polymer composition (or rate) trend errors are less than ∼6% (or 2%) of the trend change over the feed composition interval, meaning

Figure 10. Average propagation rate constant (z ) kp) continuous data trend against feed composition (x ) f), using a cubic equation, for the PCS/ PMOS system. Experimental data for SEC (O) and SEC-DV (b) measurements from Coote and Davis.8

the adequacy of the continuous data representation for modeling assessment aims. 4.2.c. RR and HPR Dependency Estimation. With the second-step execution (eq 19), the mean and error uncertainty bands for the RR (Figure 11a) and HPR (Figure 11b) are obtained. From these figures, (1) The RR dependencies r1 ) F1(x) and r2 ) F2(x) are nearly constant over the feed composition intervals (∼0.1 e x e 0.8) with small (≈(2-to-10%) error bands, and from the solvability condition expression (eqs 13c and 13d), the estimate uncertainty is larger in the homopolymer-like behavior regions (close to x ) 0 and 1), as in the previous case example, but here a more constant behavior is observed. (2) The SEC- and SEC-DV-based HPR dependencies are very similar. (3) k1 ) k11 ) κ1(x) is constant for the 0.2 e x e 1 interval and k2 ) k22 ) κ2(x) is constant over the feed composition (0 e x e 0.4) interval with small (≈(2-to-6%) error bands. Large deviations from the constancy on the HPR (kii) are observed when the other monomer (fj) is richer; this is more pronounced for the PCS, where the k11 drops from ∼180 to 100 (L mol-1 s-1) at x ≈ 0 and k22 increases from ca. 95 to 110 at x ≈ 0. As observed in the previous example, a possibility of a solvent effect exists.30,31 Here again, the HPR dependency estimates are more certain than the RR dependencies. (4) The HPR exactly coincides with the homopolymerization values at f1 ) 1 for the k11 values and at f1 ) 0 for the k22 values. 4.2.d. Modeling Implications. According to the ID approach results, the RRs are constant, but the HPRs are not, signifying that the ultimate model does not describe the PCS/PMOS experimental system.8 Here, the ID approach discrimination capability enabled us to disregard the ultimate model for the PCS/PMOS system, even though, before it was presented in a graphic form (Figure 4 of ref 8), no distinction could be made between the ultimate and implicit penultimate model, as seen in the kp evolution on x. The preceding RRs constancy and HPRs composition-varying features of the PCS/PMOS system suggest consideration of the IPU model and the radical stabilization mechanism. One should recall the RR and HPR dependencies (Figure 11 parts a and b), assume the IPU model, obtain the RRs dependencies (eqs 3c and 3d) plot (see parts a and b of Figure 12) of the related s1 and s2 dependencies and their uncertainty bands, observe that


Figure 11. (a) Reactivity ratios (r1, r2) feed composition dependency and error band certainty (dotted lines) corresponding to plus and minus the standard deviation (σ1/2) for the PCS/PMOS system. (b) Homopropagation rate constant estimates (k1 ) k11, k2 ) k22) against feed composition. The error band certainty (dotted lines) corresponds to plus and minus the standard deviation (ω1/2) for the SEC and SEC-DV measurements for the PCS/ PMOS system. The horizontal lines correspond to the values reported by Coote and Davis8 (k11 ) 175 L mol- and k22 ) 94 L mol-1 s-1).

their products π1(x) and π2(x) are approximately constant (Figure 13), and conclude that the system is well-described by the IPU model. In this case, the ID method, performed without any a priori RR and HPR modeling assumptions, has established that the monomer (r1, r2) and radical (s1, s2) reactivity ratios are approximately constant, or equivalently, that they achieve the same result obtained before8 with integral regression over an a priori assumed four-adjustable-parameter IPU model. To test the radical stabilization model, the function products pr(x) and ps(x) and the quotient q(x), defined above, are presented in Figure 13, showing that (i) the quotient, even different for each measurement (SEC or SEC-DV), is nearly constant in each case; and (ii) the quotient q (≈2.1, 2.3) is not equal to one, signifying that the radical stabilization model does not describe the experimental PCS/PMOS data system, and this is in agreement with previous reports.8 Keeping in mind the impossibility of predicting the average polymerization rate on the basis of a radical stabilization model8 with q ) 1, consider regression-based values of the pairs (r1, r2) and (k1, k2) reported by Coote and Davis,8 in conjunction with the quotient (q) fixed at the values provided by the ID estimation scheme. The application of a single-adjustable-parameter (s1) regression over the average polymerization discrete data set [δz in eq 4] yielded the values of s1 and the average polymerization rate versus feed

Figure 12. (a) Radical reactivity ratios (s1, s2) behavior against feed composition using the penultimate model for the SEC measurements. Dotted lines correspond to plus/minus standard deviation (ω1/2) for the PCS/PMOS system. (b) Radical reactivity ratios (s1, s2) behavior against feed composition using the penultimate model for the SEC-DV measurements. Dotted lines correspond to plus/minus standard deviation (ω1/2) for the PCS/PMOS system.

Figure 13. Radical stabilization model5 (s1s2 ) r1r2) testing for the PCS/ PMOS system (q ) s1s2/r1r2).

composition plots presented in Figure 14, showing that (i) the prediction is rather good and (ii) the values s1 ) 0.998 (or 1.076) for SEC (or SEC-DV) are similar to those of Coote and Davis.8 4.2.e. Concluding Remarks. The application of the proposed stochastic ID method to the PCS/PMOS system with experimental data yielded that (i) the monomer and radical RRs are

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method. Thus, the model assessment ID method capability outperformed that of the integral regression alone. For the PCS/ PMOS system, it was found that (i) in agreement with previous regression-based results, the IPU describes well the monomer and radical RRs behaviors and (ii) the monomer and radical RR products are correlated in a simple manner (s1s2/r1r2 ) constant). These results verify the theoretical model assessment ID approach capability over the regression method alone, provided a model assessment certainty requirement is met, depending on the number, quality, and location of the experimental data over the composition interval. In both case examples, the ith HPR constants (one in a higher degree) varied considerably at low ith monomer composition, and this suggests experimental work to find out the extent to which the HPR varies, in a solvent effect-like manner,30,31 with the monomer environment (feed composition). Figure 14. Validation of kp prediction using constant monomer RRs (r1, r2) and q values obtained from Figure 7b. A single parameter (s1) was fitted for the PCS/PMOS system.

nearly constant, in agreement with previous results obtained with the regression method; (ii) the HPR behavior analysis lead us to disregard the U model; (iii) the monomer and radical RRs products and their quotient trajectory enabled a positive assessment of the IPU model; and (iv) these RRs products are correlated in a simple fashion, and their quotient was found (not assumed) to be constant. In this particular case, the ID modeling assessment, drawn without any a priori RR and HPR modeling assumptions, coincides with the one drawn before from integral regression. The preceding results, in conjunction with the one drawn (in the last subsection) for the STY/MMA system, illustrate the modeling assessment advantage of the ID over integral regression method. The instantaneous observability property based of the ID method provides a direct modeling assessment procedure in terms of the actual RR and HPR dependencies on composition, regardless of the RR-HPR pair dependency complexity, including with constant RRs and HPRs being a particular case. The integral regression method is a more limited and indirect procedure, in the sense that (i) a trial-end-error search over a candidate models set must be executed and (ii) a successful model assessment requires, necessarily, systematic criteria (which at present are lacking) to tackle the multiple parameter fitting possibility, and that the shape of the (a priori unknown) actual RR-HPR pair dependency happens to be included among those of the candidate models (typically U, PU, IPU, and so on). 5. Conclusions An ID stochastic estimation approach for the determination of copolymerization RR and HPR dependencies on feed composition has been developed on the basis of polymer composition and average propagation rate measurements. The combination of a deterministic instantaneous observability characterization with a stochastic error propagation analysis yielded a methodology with (i) mean RR and HPR dependency estimates and their uncertainty, (ii) modeling assessment and/ or discrimination results certainty, in the light of the errors contained in the experimental data, and (iii) the possibility of discriminating between multiple solution sets obtained from the differential method. The ID method application to the STY/ MMA system yielded that neither the U nor the PU models explain the system and the IPU model works only as a fitting device to describe the average propagation constant, and this differs from results previously drawn with the integral regression

Acknowledgment Support for this work was provided by FQ-UNAM (PAPIIT IN101806 and PAIP 529030) and is gratefully acknowledged. Nomenclature AbbreViations B ) bootstrap DV ) differential viscosimetry GML ) generalized Mayo-Lewis GPC ) gel permeation chromatography HPR ) homopropagation rate constants ID ) integrodifferential IPU ) implicit penultimate ML ) Mayo-Lewis MMA ) methyl methacrylate PCS ) para-chlorostyrene PMOS ) para-methoxystyrene PU ) penultimate RR ) reactivity ratio RRR ) radical reactivity ratio SEC ) size-exclusion chromatography STY ) styrene U ) ultimate Nomenclature c ) PU model monomer RRs vector cr ) correlation coefficient {)σc(x)/[σ11/2(x)σ21/2(x)]} ck ) correlation coefficient {)κc(x)/[κ11/2(x)κ21/2(x)]} d ) PU model monomer and radical RRs vector dJ(x) ) determinant of J e ) measurement and/or instrument errors f(x, r) ) monomer composition (y) dependency on feed composition (x) and RRs (r) g(x, r, k) ) average propagation rate (z) dependency on feed composition k ) homopropagation rate constant vector pr(x) ) monomer reactivity ratios product [)F1(x)F2(x)] ps(x) ) radical reactivity ratios product [)π1(x)π2(x)] q(x) ) radical-to-monomer reactivity ratios products quotient [)ps(x)/pr(x)] r ) reactivity ratios vector r1 ) U model monomer 1 RR ()k11/k12) r2 ) U model monomer 2 RR ()k22/k21) r11 ) PU model monomer RR ()k111/k112) r21 ) PU model monomer RR ()k122/k121) r12 ) PU model monomer RR ()k211/k212)


r22 ) PU model monomer RR ()k222/k221) s1 ) PU model radical RR ()k211/k111) s2 ) PU model radical RR ()k122/k222) s11 ) PU model HPR ()k111) s22 ) PU model HPR ()k222) s21 ) PU model radical RR ()k211/k111) s12 ) PU model radical RR ()k122/k222) n ) number of data points V ) measurement zero mean uncorrected deviations x ) monomer feed composition in the copolymer (usually denoted f1) y ) monomer composition in the copolymer (usually denoted F1) z ) average propagation rate (usually denoted kp) R ) noiseless representation of y β ) noiseless representation of z δ ) discrete experimental data against feed composition φ ) k dependency function O ) augmented function representation of ψ γ ) augmented function representation of ζ ι ) estimation of r K ) homopropagation rate estimation function λ ) ψ dependency function µ ) ζ dependency function ν ) measurement derivative zero mean uncorrected deviations π ) radical reactivity ratio estimation function G(x) ) RR (r) dependency on feed composition (x) σ ) mean square root errors ω ) mean square root errors χ ) measured feed composition ζ ) measured HPR ψ ) measured polymer composition Θ ) diagonal matrix for ζ Σ ) estimation error covariance matrix for r[)E(r˜r˜T)] Ω ) estimation error covariance matrix for k[)E(k˜k˜T)] E ) statistical expectation operator E(r˜r˜T) ) estimation error covariance matrix E(k˜k˜T) ) estimation error covariance matrix J(x,r) ) Jacobian matrix of system y L ) Jacobian of γ M(x,r,k) ) Jacobian matrix of system z N ) normal matrix (zeros in diagonal) O ) error propagation matrix Q ) diagonal matrix for ψ S ) error propagation matrix X ) composition interval ∂ ) partial derivative ‘ ) denotes derivative with respect to x ′ ) denotes transpose Literature Cited (1) Mayo, F. R.; Lewis, F. M. Copolymerization. I. A basis for comparing the behavior of monomer in copolymerization; the copolymerization of styrene and methyl methacrylate J. Am. Chem. Soc. 1944, 66, 1994-1601. (2) 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. (3) Harwood, H. J. Structures and Compositions of Polymers. Makromol. Chem., Macromol. Symp. 1987, 10/11, 331-354. (4) Maxwell, I. A.; Aerdts, A. M.; German, A. L. Free Radical Copolymerization; An NMR Investigation of Current Kinetic Methods. Macromolecules 1993, 26, 1956-1964. (5) Fukuda, T.; Ma, Y. D.; Inagaki, H. Free-Radical Copolymerization. 3. Determination of Rate Constants of Propagation and Termination for

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ReceiVed for reView November 20, 2006 ReVised manuscript receiVed January 13, 2007 Accepted January 16, 2007 IE061479S

Assessment of Copolymerization Models via Stochastic

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