An Error-in-Variables Method for Determining Reactivity Ratios by On-Line Monitoring of Copolymerization Reactions

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1 162 Full Paper Summary: A new error-in-variables method was developed to estimate the reactivity ratios in copolymerization systems. It brings the power of automatic, continuous, on-line monitoring of polymerization (ACOMP) to copolymerization calculations. In ACOMP systems, monomer and polymer concentrations are measured by the monitoring of two independent properties of the system. The reactivity ratios are found by taking into account errors in the monomer concentrations determined from measurements and from calibration of the instruments. All the error sources are taken into account according to the error-in-variables method, and their effects are reflected in determining the confidence intervals of the reactivity ratios by the usual error propagation technique. Distribution of concentrations [a] and [b] for the simulated experiment I. Random errors are 1% of the initial value in both observed variables. An Error-in-Variables Method for Determining Reactivity Ratios by On-Line Monitoring of Copolymerization Reactions Didem Sünbül, 1 Huceste C atalgil-giz, 1 Wayne Reed, 2 Ahmet Giz* 1 1 I.T.Ü. Fen-Edebiyat Fakültesi, Maslak, Istanbul, Turkey Fax: þ ; giz@itu.edu.tr 2 Physics Department, Tulane University, Stern Hall, New Orleans, LA 7118, USA Received: July 21, 23; Revised: October 15, 23; Accepted: October 27, 23; DOI: 1.12/mats.238 Keywords: copolymerization; kinetics (polym.); on-line monitoring; reactivity ratio Introduction When penultimate effects are not important, copolymerization kinetics are governed by the equations: d½aš d½bš d½a*š ¼ k aa ½A*Š½aŠ k ba ½B*Š½aŠ ¼ k ab ½A*Š½bŠ k bb ½B*Š½bŠ ¼ k ba ½B*Š½aŠ k ab ½A*Š½bŠ ð1aþ ð1bþ ð1cþ in which [a] and [b] are the molar concentrations of the two monomers, [A*] and [B*] are the molar concentrations of the radical chains with the two types of monomers as their ultimate units, and the k s are the rate constants for the relevant reactions. Under steady-state conditions, the time derivatives of the two radical concentrations can be set to zero. This simplifies the above set to the well-known Mayo-Lewis copolymerization equation. [1] d½aš d½bš ¼ ½aŠ ½bŠ r a ½aŠþ½bŠ ½aŠþr b ½bŠ ð2þ d½b*š ¼ k ab ½A*Š½bŠ k ba ½B*Š½aŠ ð1dþ Here, the reactivity ratios, r a and r b, are defined as r a ¼ k aa / k ab and r b ¼ k bb /k ba. Macromol. Theory Simul. 24, 13, DOI: 1.12/mats.238 ß 24 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

2 An Error-in-Variables Method for Determining Reactivity Ratios by On-Line Monitoring of Copolymerization Reactions 163 All copolymerization reactions show a drift in the monomer ratio as the degree of conversion increases and the more active monomer is depleted. The reactivity ratios must be obtained by techniques that take this drift into account. Early and obsolete, but still commonly used linear methods, such as the Finemann-Ross and the original version of the Kelen-Tüdös, failed to compensate for this drift. [2,3] O Driscoll, German, Van Herk, and others have shown that nonlinear least square (NLLS) methods minimizing w 2 and taking error propagation and individual errors on each of the measurements into account are superior in error handling and have smaller regions for a given percent probability. [4 14] In nonlinear fitting procedures, the error in each measurement is propagated through standard error propagation techniques using partial derivatives and the chain rule. Since only first derivatives are used in error propagation, even these methods distort the error structure, but to a much lesser degree than the linear methods. Early NLLS methods could handle errors in only one measured variable, usually the monomer ratio in the polymer. The error-in-variables methods (EVM) [4,5] take into account errors in all three measured quantities, namely, the ratios of initial monomer concentrations, final monomer concentrations or the concentrations of the two monomers in the polymer and the conversion ratio. German and Heikens [6] have developed a sequential sampling method, in which they periodically withdrew samples from the reaction medium and analyzed the samples. They applied their method to gas-phase copolymerization of ethylene and vinyl acetate. In this method, the reactivity ratios are obtained by fitting the experimental data to an analytic solution of the copolymerization equation given by Skeist. [15] They described the error propagation starting from the monomer concentrations to reactivity ratios, but did not give the error propagation from the actual measurements of the observed variables to the monomer concentrations. More recently, Florenzano, Reed, and co-workers have developed an on-line technique called automatic, continuous, on-line monitoring of polymerization (ACOMP). In this technique, a steady stream of sample is withdrawn from the reaction medium and diluted with pure solvent by an automatic mixing pump. The diluted mixture is then passed through a set of detectors including a refractometer, an ultraviolet spectrophotometric detector, a viscometer, and a multi-angle light scattering cell. ACOMP has been applied to study radical chain polymerization of vinyl pyrrolidinone, [16] acrylamide, [17] and urethane formation. [18] It is clear that on-line techniques, which allow hundreds of determinations of the concentrations and take-up rates of the reactants, enable much more information to be extracted from each experiment. Here, an EVM-type calculation method was developed for obtaining the reactivity ratios by an on-line monitoring technique. As the withdrawn sample is pumped through the detectors, the monitoring of the changes in two of its independent properties gives two equations for the conversion of the two monomers. The total concentration of monomer units of each species in the monomer and polymer form is constant. Thus, two measurement results, plus the two known totals, give the concentrations of the two monomers and the two polymers as functions of time. Theoretical If we let [a], [b] and [A], [B] represent the molar concentrations of the two monomers and the molar concentrations of the corresponding monomer units in the polymer respectively, at any time during the reaction, then ½aŠþ½AŠ ¼½aŠ ð3aþ ½bŠþ½BŠ ¼½bŠ ð3bþ where [a] and [b] are the initial monomer concentrations. These equations are valid when the density of the solution is constant during the reaction. This condition is automatically satisfied for dilute solution polymerization. However, if the polymerization is conducted at a high monomer concentration, the density increase of the reaction medium can be significant. In that case, they must be replaced with ½aŠþ½AŠ ¼½aŠ r=r ð3a Þ ½bŠþ½BŠ ¼½bŠ r=r ð3b Þ where r and r are the densities of the solution at the time of the measurement and at the beginning of the reaction, respectively. Because the reaction is continuously monitored, the conversion of each monomer is known at the previous measurement time. Thus, the density correction is easily calculated from the densities of the monomers and the polymers. Alternatively, the analytical expression given in ref. [19] can be used. To find the conversion of each monomer, two independent measurements must be performed. These can be UV spectrophotometry at two different wavelengths or a spectrophotometric detection coupled with differential refractometry, fluorescence, or the measurement of some other property of the monomers, which changes upon polymerization. Both signals must change during polymerization, and the changes must be different for the conversion of the two monomers. During polymerization, the UV absorbance of many systems drops dramatically. Similarly, the differential refractive index (RI) of many polymers is greater than the value of the corresponding monomer. Thus, for many monomer pairs, UV and RI detections provide two independent measurements. For clarity, we will refer to the two independent signals as UVand RI here, but the method is general and any other pair of signals can be substituted. Macromol. Theory Simul. 24, 13, ß 24 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

3 164 D. Sünbül, H. C atalgil-giz, W. Reed, A. Giz If we let U a and U b be the UV absorption coefficients of the two monomers; U A and U B be the corresponding values for monomer units in the polymer; and U I be the coefficient of the initiator, then: U a ½aŠþU b ½bŠþU A ½AŠþU B ½BŠþU I ½IŠ ¼ UV signal UV baseline ð3cþ Here [I] is the initiator concentration. Similarly, the other monitored signal gives the equation: R a ½aŠþR b ½bŠþR A ½AŠþR B ½BŠþR I ½IŠ ¼ RI signal RI baseline ð3dþ where R a, R b, R A, R B, and R I are the differential refractive index coefficients for the two monomers, monomer units of the two species in the polymer, and the initiator, respectively. UV signal and RI signal are the signals from the detectors at the measurement time. UV baseline and RI baseline are the corresponding baseline values obtained by averaging over a time interval when pure solvent is pumped through the detectors. Defining U þ and R þ as: U þ ¼ UV signal UV baseline U A ½aŠ U B ½bŠ U I ½IŠ ð3eþ R þ ¼ RI signal RI baseline R A ½aŠ R B ½bŠ U I ½IŠ ð3fþ the Equation (3a) (3d) give the two monomer concentrations as: ½aŠ ¼ DU ð b R þ DR b U þ Þ= det ð4aþ ½bŠ ¼ðDU a R þ DR a U þ Þ= det ð4bþ where the determinant, det, is det ¼ DU a DR b DU b DR a ð4cþ and the D terms denote the differences in the values of the relevant constant between a monomer and the corresponding polymer, DU a ¼ U a U A ð4dþ The other D terms, DU b, DR a, and DR b are similarly defined. If det is zero, then Equation (3c) and (3d) are linearly dependent; if it is small (det DU a DR b þ DU b DR a ), then the equations are ill-conditioned so that the resulting errors are large. These cases must be avoided by carefully selecting the measurement variables. When the experimental data are plotted, with [a] as a function of [b], the result can be compared to a solution of Equation (2). An explicit analytical solution such as those given by Mayo, Skeist, Van der Meer can be used. It is also possible to directly fit to a numerical, rather than an analytical solution. A fit to a numerical solution is computationally more intensive, but it has the advantage that it avoids transformations, which tend to distort the error structure. Although proper error propagation correctly handles the first-order terms, under certain circumstances, higher order terms can also be significant. [2] To fit the data to a solution of Equation (2) of the form [a] ¼ X([b], [a], [b] ) which can be written as, Q ¼½aŠ Xð½bŠ; ½aŠ ; ½bŠ Þ¼ ð5þ w 2 ¼ X nðexpþ nðdataþ j j¼1 X i¼1 W ij Q 2 ij ð6aþ is minimized. Here, the index j denotes the experiment number, the index i the data point on that experiment. The sum runs over all data points in all of the experiments. The weight function W ij is, W ij ¼ðVarðQ ij ÞÞ 1 ð6bþ The derivation of this variation in terms of the expected errors in all the measured quantities and all the calibration factors is given in the appendix. Simulated Experiments The signals from the two detectors are taken as the independent variables. The two monomer concentrations [a] and [b] are obtained from these by Equation (5). The errors in the initial composition are likely to be the smallest. Calibration errors do not affect the results unless the calibrations of the instruments drift between calibration and the actual experiment. Errors in the baseline voltages can be reduced by measuring over a long time. Thus, the errors in measuring the actual signals are likely to be dominant. In the simulations, random errors were added to the RI signal and the UV signal values. The simulations are performed by selecting a set of actual reactivity ratios (r a, r b ) and three initial compositions (75, 5, and 25% of monomer a). The kinetic equations [Equation (1a) (1d)] are numerically integrated by the fourth-order Runge-Kutta method with fixed time steps to obtain the concentrations of the two monomer species and the corresponding monomer units in the copolymer. All simulated experiments were stopped at 9% conversion. Fixed time steps were preferred because during on-line monitoring, data is gathered at fixed time intervals. The experimental UVand RI signals are then determined by Equation (3c) and (3d), and random errors are added to each signal. The magnitude of the random noise is set at a fixed percentage (1%) of the initial value of the signal. Fitting Procedure To find the fit quality of the parameter set (r a, r b ), the copolymerization equation is numerically integrated by the Macromol. Theory Simul. 24, 13, ß 24 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

4 An Error-in-Variables Method for Determining Reactivity Ratios by On-Line Monitoring of Copolymerization Reactions 165 fourth-order Runge-Kutta method [21] with adjustable time steps. The integrations are started from the initial compositions used in the simulation. No attempt is made to match the timesteps of the numerical calculation to the times experimental data are obtained. The concentration values, [a], corresponding to the concentrations [b] at detection times are obtained by cubic spline interpolation. In those experiments where 95% of one monomer was consumed before 9% conversion was reached, the analysis was stopped, retaining the data up to this point and discarding the subsequent data. Finally, the theoretical solution of the copolymerization equation was compared with the experimental result obtained as described above and a w 2 value for the parameter set (r a, r b ) was obtained. The theoretical part was repeated in a sweep over the r a, r b parameter space, and a w 2 map was obtained. Such a map is more useful than just the best-fit value, because it also gives the acceptable part of the parameter space. Results and Discussion Three cases were selected as illustrative. Example I is ideal copolymerization with r a ¼ 5. and r b ¼.2. Ideal copolymerizations with widely differing reactivities give drastic composition drifts. For this reason, they pose difficulties for methods which handle composition drift inadequately. The UV coefficients for this example are selected as U a ¼.1, U b ¼.8, U A ¼ U B ¼ and the RI coefficients as R a ¼.3, R b ¼.2, R A ¼.4 and R B ¼.25. The w 2 map is given in Figure 1. The center of the 1s ellipse is r a ¼ 5.31, r b ¼.25; the error bars obtained by its projection on the axes give.41 for r a and.6 for r b. Note that the selected compositions lead to an initial take-up rate of the active monomer to be the greater take-up rate in all three cases. The results are good despite poor experimental planning. Example II is chosen with reactivities close to one, r a ¼ 1. and r b ¼.8. The UV coefficients for this example are selected as U a ¼.1, U b ¼.2, U A ¼U B ¼ and the RI coefficients as R a ¼.3, R b ¼.2, R A ¼.35 and R B ¼.25. The Skeist solution of the copolymerization equation, commonly used in EVM s: 1 P n ¼ f r b 1 r a b f a ra f b 1 ra f b f a d f a d ð1 rar b Þ ð1 raþð1 r b Þ ð7þ where P n is the degree of polymerization, f a ¼ 1 f b ¼ ð½aš=ð½ašþ½bšþþis the mole fraction of monomer a, and d ¼ð1 r b Þ=ð2 r a r b Þ, as well as the solution given by Mayo: ln x b ¼ r b ln x a þ 1 r ar b ln ð1 r bþx b ð1 r a Þx b x 1 r a ð1 r b Þx b ð1 r a Þx a x ð8þ where x a ¼½aŠ=½aŠ, x b ¼½bŠ=½bŠ and x ¼½aŠ =½bŠ, have singularities, when one of the reactivity ratios is equal to one. Methods based on them suffer from the distortion of the error structure caused by this singularity when one or both ratios are close to one. The w 2 map is given in Figure 2. The center of the 1s ellipse is r a ¼ 1.1, r b ¼.793; the error bars obtained by its projection on the axes give.1 for r a and.14 for r b. This shows that this method, based on a numerical solution, is not adversely affected in this range of reactivities. In Example III, both reactivities are less than one, with one reactivity ratio close to zero, r a ¼.5 and r b ¼.1. Under these conditions, it is difficult to distinguish the reactivity Figure 1. w 2 plot for the r a (actual) ¼ 5. and r b (actual) ¼.2 case, showing 1, 2 and 3 s contours. Figure 2. w 2 plot for the r a (actual) ¼ 1. and r b (actual) ¼.8 case, showing 1, 2 and 3 s contours. Macromol. Theory Simul. 24, 13, ß 24 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

5 166 D. Sünbül, H. C atalgil-giz, W. Reed, A. Giz Figure 3. w 2 plot for the r a (actual) ¼.5 and r b (actual) ¼.1 case, showing 1, 2 and 3 s contours. of the less active monomer from zero. Such systems also show a rapid shift in the composition in the later stages of the reaction, when the monomer less abundant than the azeotropic ratio is depleted. The UV coefficients for this example are selected as U a ¼.1, U b ¼.2, U A ¼ U B ¼ and the RI coefficients as R a ¼.3, R b ¼.2, R A ¼.35 and R B ¼.25. The w 2 map is given in Figure 3. The center of the 1s ellipse is r a ¼.5, r b ¼.99; the error bars obtained by its projection on the axes give.45 for r a, and.4 for r b. In all three cases, the method gave a very good approximation to the actual parameter values with small regions of probable error. This indicates that the method is robust and reliable. Note that the random errors given here are added to the observational variables, so that the uncertainties in the concentrations are actually much larger. Figure 4 shows the values computed for the concentrations from the experimental UV and RI values for Experiment I with 1.% random errors in the observational variables. In fact, data scatter in an actual experiment will probably be less, as was the case when C atalgil-giz et al. applied this method to styrene, methyl methacrylate copolymerization [19] shown in Figure 5. A fit by the Marquar-Levenberg method requires a few dozen w 2 evaluations. A complete w 2 map to find the statistically acceptable portion of the parameter space can be made with a few thousand such evaluations. If the data is fitted to an analytical solution of the copolymer equation, each w 2 evaluation requires that the theoretical [a] be computed as many times as the experimental points. Fits to numerical solutions involve an integration of the copolymer equation for each experiment. In the batch method, each experiment yields a single point. In ACOMP, each experiment yields several hundred to thousand points Figure 4. Distribution of concentrations [a] and [b] for the simulated experiment I. Random errors are 1% of the initial value in both observed variables. Figure 5. Scatter of an actual experimental data from ref. [19] depending on how often the data is recorded. The extra computational overhead in fitting to a numerical solution is significant in batch methods, but not in on-line monitoring. The calculation for a complete w 2 map took only a few minutes on a PIII class computer. For this reason, computational efficiency was not considered to be a significant factor. On the other hand, analytical solutions of the copolymer equation involve transformations of the variables. These transformations distort the error structure. Formal error analysis involves only the first derivatives and can compensate for the error structure distortions only up to a certain extent. Fits to numerical solutions avoid transformations and the consequent error structure distortions. Macromol. Theory Simul. 24, 13, ß 24 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

6 An Error-in-Variables Method for Determining Reactivity Ratios by On-Line Monitoring of Copolymerization Reactions 167 Conclusion This method has one constraint, namely, two independent, observational variables that change on polymerization such that the changes are different for conversion of the two monomers be found. Otherwise, Equation (4) becomes ill-conditioned. The proposed technique can also reliably determine the reactivity ratios in copolymerization with just a few reactions with preferably widely differing initial compositions. Since data is taken continuously, it is not necessary to stop the reaction at a given point. Furthermore, if the errors are correctly propagated, it is not even necessary to get rid of the useless data at the end, when one of the monomers is depleted. Such data have negligible weight and should not affect the results of the experiments. Because it can give the reactivity ratios from much fewer experiments and does not necessitate the precipitation of the resulting polymer and analysis after the experiment, the on-line method can be a very efficient and useful way of determining reactivity ratios. The method was tested in solution copolymerization of methyl methacrylate and styrene in butyl acetate solvent. [19] In that case, the UVabsorption at 282 nm was dominated by the styrene monomer, and the difference in the differential refractive index values between the monomer and the polymer was greater in methyl methacrylate; thus, the equations were well-conditioned and reliable reactivity ratios were obtained. Appendix According to the standard error propagation rules, by definition, the variation of an independent variable, V, is just its standard deviation: VarðVÞ ¼ðdVÞ 2 ða1þ The variation of a dependent variable is obtained as: VarðUÞ ¼ du 2 VarðVÞ ða2þ dv Covariation of two independent variables is zero. When two dependent variables, U and V, are functions of the same observed variable Y then their covariation is: Co varðu; VÞ ¼ du dy dv dy VarðYÞ ða3þ The error terms arise from the measurements, their baseline values, and the calibration constants. Thus the variations of the two monomer concentrations and their covariance are given by VarðQÞ ¼Varð½aŠÞ þ VarðXÞ 2Co varð½aš; XÞ; ða4aþ VarðXÞ 2 Varð½bŠÞ Þ 2 2 Þ 2 Co Þ Co Þ ða4bþ Co varð½aš; XÞ Co varð½aš; Co varð½aš; Co varð½aš; Þ ða4cþ 8 >< Varð½aŠÞ ¼ >: 8 >< Varð½bŠÞ ¼ >: ðu þ Þ 2 VarðDR b ÞþðR þ Þ 2 VarðDU b Þþ½aŠ 2 VarðdetÞþðDR b Þ 2 VarðU þ ÞþðDU b Þ 2 VarðR þ Þ 2U þ DU b Co varðdr b ; R þ Þ 2R þ DR b Co varðdu b ; U þ Þ 2DR b DU b Co varðr þ ; U þ Þ 2½aŠðU þ Co varðdr b ; detþ R þ Co varðdu b ; detþþ þ 2½aŠðDU b Co varðr þ ; detþ DR b Co varðu þ ; detþþ ðu þ Þ 2 VarðDR a ÞþðR þ Þ 2 VarðDU a Þþ½bŠ 2 VarðdetÞþðDR a Þ 2 VarðU þ ÞþðDU a Þ 2 VarðR þ Þ 2U þ DU a Co varðdr a ; R þ Þ 2R þ DR a Co varðdu a ; U þ Þ 2DR a DU a Co varðr þ ; U þ Þ þ2½bšðu þ Co varðdr a ; detþ R þ Co varðdu a ; detþþ 2½bŠðDU a Co varðr þ ; detþ DR a Co varðu þ ; detþþ 9 >= =det 2 >; 9 >= =det 2 >; 8 9 DR b DR a VarðU þ Þ DU b DU a VarðR þ Þþ½aŠ½bŠVarðdetÞþU þ ðdu a Co varðdr b ; R þ Þ >< þdu b Co varðdr a ; R þ ÞÞ þ U þ ð½ašco varðdr a ; detþ ½bŠCo varðdr b ; detþþ >= Co varð½aš; ½bŠÞ ¼ þr þ ðdr a Co varðdu b ; U þ ÞþDR b Co varðdu a ; U þ ÞÞ R þ ð½ašco varðdu a ; detþ =det 2 þ½bšco varðdu b ; detþþþðdr a DU b DR b DU a ÞCo varðu þ ; R þ Þ >: >; þð½ašdr a ½bŠDR b ÞCo varðu þ ; detþ ð½ašdu a ½bŠDU b ÞCo varðr þ ; detþ ða4dþ ða4eþ ða4fþ Macromol. Theory Simul. 24, 13, ß 24 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

7 168 D. Sünbül, H. C atalgil-giz, W. Reed, A. Giz Co varð½aš; ½aŠ Þ¼ðDU b R A DR b U A Þdð½aŠ Þ 2 = det ða4gþ Co varð½aš; ½bŠ Þ¼ðDU b R B DR b U B Þdð½bŠ Þ 2 = det ða4hþ Co varð½bš;½aš Þ¼ ðdu a R A DR a U A Þdð½aŠ Þ 2 = det ða4iþ Co varð½bš; ½bŠ Þ¼ ðdu a R B DR a U B Þdð½aŠ Þ 2 = det ða4jþ Co varðdet; DU b Þ¼ DR a VarðDU b Þ Co varðdet; DR a Þ¼ DU b VarðDR a Þ Co varðdet; DR b Þ¼DU a VarðDR b Þ: ða5nþ ða5oþ ða5pþ where VarðDU a Þ¼dðU a Þ 2 þ dðu A Þ 2 VarðDU b Þ¼dðU b Þ 2 þ dðu B Þ 2 VarðDR a Þ¼dðR a Þ 2 þ dðr A Þ 2 ða5aþ ða5bþ ða5cþ Acknowledgement: Support provided by a joint NSF- TUBITAK project [NSF Project No. INT-8624 and TUBITAK Project No. TBAG-U2 (1T141) and TBAG-2174 (12T6)] and ITU Research Fund (Project No and 1673) are acknowledged. VarðDR b Þ¼dðR b Þ 2 þ dðr B Þ 2 ða5dþ and the other variations are VarðdetÞ ¼ðDR b Þ 2 VarðDU a ÞþðDR a Þ 2 VarðDU b Þ þðdu b Þ 2 VarðDR a ÞþðDU a Þ 2 VarðDR b Þ ða5eþ VarðU þ Þ¼dðUV signal Þ 2 þ dðuv baseline Þ 2 þ½iš 2 dðu I Þ 2 þ½aš 2 dðu AÞ 2 þ½bš 2 dðu BÞ 2 þ UI 2 dð½išþ2 þ UA 2 dð½aš Þ2 þ UB 2 dð½bš Þ2 ða5fþ VarðR þ Þ¼dðRI signal Þ 2 þ dðri baseline Þ 2 þ½iš 2 dðr I Þ 2 þ½aš 2 dðr AÞ 2 þ½bš 2 dðr BÞ 2 þ R 2 I dð½išþ2 þ R 2 A dð½aš Þ2 þ R 2 B dð½bš Þ2 ða5gþ Co varðdu a ; U þ Þ¼½aŠ dðu A Þ 2 ða5hþ Co varðdu b ; U þ Þ¼½bŠ dðu B Þ 2 ða5iþ Co varðdr a ; R þ Þ¼½aŠ dðr A Þ 2 ða5jþ Co varðdr b ; R þ Þ¼½bŠ dðr B Þ 2 ða5kþ Co varðu þ ; R þ Þ¼U I R I dð½išþ 2 þ U A R A dð½aš Þ 2 þ U B R B dð½bš Þ 2 ða5lþ Co varðdet; DU a Þ¼DR b VarðDU a Þ ða5mþ [1] F.R.Mayo,F.M.Lewis,J. Am. Chem. Soc. 1944, 66, [2] M. Finemann, S. D. Ross, J. Polym. Sci. 195, 5, 269. [3] T. Kelen, F. Tudos, J. Polym. Sci., Polym. Chem. Ed. 1977, 15, 347. [4] R. Van der Meer, H. N. Linssen, A. L. German, J. Polym. Sci., Polym. Chem. Ed. 1978, 16, [5] K. K. Chee, S. C. Ng, Macromolecules 1986, 19, [6] A. L. German, D. Heikens, J. Polym. Sci., Part A: Polym. Chem. 1971, 9, [7] V. E. Meyer, G. G. Lowry, J. Polym. Sci., Part A: Polym. Chem. 1965, 3, [8] P. W. Tidwell, G. A. Mortimer, J. Polym. Sci., Part A: Polym. Chem. 1965, 3, 369. [9] D. W. Benhken, J. Polym. Sci., Part A: Polym. Chem. 1964, 2, 645. [1] R. McFarlane, P. M. Reilly, K. F. O Driscoll, J. Polym. Sci., Polym. Chem. Ed. 198, 18, 251. [11] H. Patino-Leal, P. M. Reilly, K. F. O Driscoll, J. Polym. Sci., Polym. Lett. Ed. 198, 18, 219. [12] M. Dube, R. A. Sanayei, A. Penlidis, K. F. O Driscoll, P. M. Reilly, J. Polym. Sci., Part A: Polym. Chem. 1991, 29, 73. [13] A. M. van Herk, J. Chem. Educ. 1995, 72, 138. [14] A. M. van Herk, T. Dröge, Macromol. Theory Simul. 1997, 6, [15] I. Skeist, J. Am. Chem. Soc. 1946, 68, [16] F. H. Florenzano, R. Streliktzki, W. F. Reed, Macromolecules 1998, 31, [17] A. Giz, H. C atalgil-giz, J.-L. Brousseau, A. M. Alb, W. F. Reed, Macromolecules 21, 34, 118. [18] H. C atalgil-giz, A. Giz, A. Alb, W. F. Reed, J. Appl. Polym. Sci. 21, 82, 27. [19] H. C atalgil-giz, A. Giz, A. M. Alb, A. Öncül Koç, W. F. Reed, Macromolecules 22, 35, [2] A. T. Giz, Macromol. Theory Simul. 1998, 7, 391. [21] W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Wetterling, Numerical Recipes, Cambridge University Press, New York 1986, p. 55. Macromol. Theory Simul. 24, 13, ß 24 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim