Analysis of quaternary carbon resonances of vinylidene chloride/methyl acrylate copolymers

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1 Analysis of quaternary carbon resonances of vinylidene chloride/methyl acrylate copolymers Ajaib Singh Brar *, Gurmeet Singh, Ravi Shankar Department of Chemistry, Indian Institute of Technology-Delhi, Hauz Khas, New Delhi , India Received 18 May 2004; accepted 21 July 2004 Abstract Analysis of the quaternary carbon resonance signals of vinylidene chloride in vinylidene chloride (V)/methyl acrylate (M) copolymers at pentad level of compositional sensitivity is presented in this paper. The analysis has been done by resolving overlapped and complex resonance signals using an approach based on the intensities of resonances, chemical shift prediction and spectral simulation. Intensities of the resonance signals were calculated using the reactivity ratios optimized from the dyad and triad fractions, obtained from the 13C{1H} NMR data, by applying genetic algorithm. Joint confidence interval was obtained for the optimized reactivity ratios. The chemical shift modeling of the quaternary carbon resonance signals in terms of empirical additive parameters was done. The chemical shifts of overlapping pentad resonances were predicted from the empirical additive parameters optimized using genetic algorithm. Comparison of the intensities of pentad resonances assigned by chemical shift modeling and experimental intensities of resonances has been done to ascertain the assignments made. Comparison between simulated and experimental spectra at pentad level of sensitivity has been done. Keywords: NMR; Chemical shift; Modeling; Genetic algorithm; Reactivity ratios 1. Introduction High resolution NMR spectra of polymers provide large amount of structural information [1-3]. The polymer chemists are greatly benefited by the use of this data. The interpretation of high resolution NMR spectra of polymers requires resolving overlapped and complex resonance signals. In the micro structure analysis of copolymers, it is difficult to differentiate overlapped and complex resonance signals such as AAABA, AAABB, BAABA and BAABB pentads (where A and B represents two different monomer units). To resolve this complexity, an approach based on comparison of calculated and experimental intensities of resonance signals, chemical shift modeling and spectral simulation has been applied in this report. To test the validity of the approach, quaternary carbon resonances of vinylidene chloride in vinylidene chloride/methyl acrylate were analyzed. The calculation of intensities requires determination of reactivity ratios of copolymer systems. Reactivity ratios can be determined from the dyad and triad fractions obtained from 1 H [4-6] and 13 C{ 1 H} [7-9] NMR spectra. Although, reactivity ratios can be determined from the dyad and triad fractions with high precision, this

2 methodology has not been followed and analyzed rigorously. Least square method incorporated in genetic algorithm has been used in this report for the optimization of reactivity ratios from dyad and triad fractions, van Herk et al. have discussed critically the least square methodology with different weighting schemes depending on the error structure for the reactivity ratios determination and also focused on joint confidence intervals [10,11]. In this report self-consistency approach has been used for the determination of weights. To substantiate the estimation of reactivity ratios from this methodology, reactivity ratios obtained from the dyad and triad fractions for vinylidene chloride/vinyl acetate copolymer system have been compared. For the chemical shift analysis of polymers the useful feature of additivity of 13 C{ 1 H} NMR chemical shifts i.e. chemical shift of carbon nuclei can be modeled into different additive components produced by substituents at various positions (a,b,c,d,e etc.) can be applied [12,13]. This has been used by Cheng et al. and Matlengiewicz et al. for the microstructure analysis of polymers [14 19]. Cheng et al. have proposed a general approach based on the empirical framework for the 13C{1H} NMR spectral interpretation and prediction of vinyl and vinylidene copolymers by means of proposed substituent parameters for various comonomers [14-17]. Matlengiewicz et al. have reported the successful and complete 13C{1H} NMR sequence analysis upto tetrad level based on spectral simulation and incremental calculations of chemical shifts [18,19]. The empirical chemical shift model based on additivity feature of chemical shift has been applied for the modeling of chemical shifts of assigned resonances and prediction of overlapping resonance signals. Genetic algorithm based on the Darwin's theory of survival of the fittest is highly suited for problems that are not well defined, difficult to model mathematically, are highly non-linear and consequently difficult to solve with traditional optimization techniques [20-23]. Genetic algorithm is capable of searching solutions through a large space and converges to global optima. Genetic algorithm had started to benefit chemists [24] like for automated wavelength selection [25,26], reactivity ratios optimization [27] and automated structure elucidation [28,29]. These features had prompted the application of genetic algorithm for the reactivity ratios and empirical additive parameters optimization. 2. Experimental 2.1. Copolymer synthesis Vinylidene chloride/methyl acrylate copolymers were synthesized in bulk using uranyl nitrate as the photo initiator. The conversion was kept below 5% to prevent composition drift, by precipitating the polymerization solutions in large excess of methanol. Polymers were further purified and dried in vacuum NMR studies The 13 C{ 1 H} and DEPT-135 NMR spectra were recorded on Bruker DPX-300 spectrometer in CDCl 3 at 75.5 MHz at 45 C. "Cf'H) NMR spectra were recorded with 2 s delay time. DEPT-135 spectra were recorded using dept pulse sequence from Bruker with J modulation time of 3.7ms (J CH = 135Hz) and 2s delay time Genetic algorithm In the genetic algorithm, a population of 50 uniformly distributed individuals or strings representing parameters to be optimized was randomly generated. The individuals are uniformly generated in the bounds of the search space. The strings are evaluated to find the fitness value and then operated by three operators: selection, crossover and mutation to create a new set of population as shown in Fig. 1. Based on the fitness of individuals they are selected for crossover. For the selection of individuals tournament selection operator was used. Randomly 'a' number of individuals are selected and 'V {a > b) number of better fitting individuals INITIALIZE Generate a random population of chromosomes FITNESS EVALUATION Compute fitness of each chromosome NATURAL SELECTION Choose parents based on fitness RECOMBINATION Crossover of selected parents to produce the next generation i MUTATION Mutation done to prevent stagnant gene pool Fig. 1. The flow diagram of working of genetic algorithm.

3 are carried over for crossover. The selected individuals are then allowed to produce the next generation i.e. to perform crossover to diversify the genetic makeup (the values of parameters). For the crossover, heuristic crossover operator was applied. The heuristic crossover operator uses the fitness values of the two parent chromosomes to determine the direction of the search. The offspring are created according to the following equations: X = X þ rðx - YÞ Y'=X where, X 0 and Y 0 are the progeny, X and Y are parents with X being the better fitting parent and r is a random number between 0 and 1. It is possible that X will not be feasible if r is chosen such that one or more of its genes fall outside of the allowable lower or upper bounds. For this reason, heuristic crossover has a user settable parameter (n) for the number of times to try and find an V that result in a feasible progeny. If a feasible chromosome is not produced after n tries, the lesser fitting parent is returned as X0. Thus the operator is less prone to cause changes to the better fitting individuals and so preserving them. Non-uniform mutation operator was applied for mutation to avoid the accidental trapping of individuals in the local minima. It directs the probability of the mutation towards zero as the generation number increases. This mutation operator keeps the population from stagnating in the early stages of the evolution then allows the genetic algorithm to fine tune the solution in the later stages of evolution. X 0 =X þ r 2 (Xn-X)\\ G X=X-r 2 (X-X L )(\- Gen Gen max ifr1<0 : 5 if r 1P 0 : 5 where, X and X are the individuals after and before the mutation, respectively. r 1 and r 2 are the random numbers between 0 and 1. X L and X H are the lower and upper bounds for the X parameter, respectively. Gen is the number of current generation and Gen max is the maximum number of generations. Over generations, the individuals move towards increasingly optimal solutions, making genetic algorithm highly probable for finding optimal solutions to a mathematical problem for which there may not be one correct answer. This characteristic feature and its capability to narrow down to good fitness after a small number of function evaluations prompted application of genetic algorithm for optimization of reactivity ratios and chemical shift additive parameters. The iteration was carried for 1000 generations to obtain optimized parameters. The code for genetic algorithm has been written in C++ language Calculations of resonance signals' intensities For the simulation of 13C{1H} NMR spectra, the intensities of various resonance signals were calculated following the first-order Markov statistical model [30,31]. Reactivity ratios optimized from the dyad and triad fractions were used for the calculations. A Microsoft Excel worksheet was created for calculation of the intensities of resonance signals at different levels of compositional sensitivity. The details of the calculations are given in Appendix A Simulation of NMR spectra The simulation of NMR spectra requires intensities of resonance signals, chemical shifts, line widths and line shape. Calculations for intensities of resonances have been explained above. Chemical shifts optimized from the empirical additive chemical shift modeling were used. The NMR resonance signals have been simulated for Lorentzian line shape. A Microsoft Excel workbook was created for the simulation of NMR spectrum. Line widths were attuned to obtain good match between the experimental and simulated spectra. 3. Results and discussion The steps of the approach followed for the analysis of quaternary carbon resonances were: (1) The reactivity ratios were optimized from the dyad and triad fractions and joint confidence intervals were obtained. (2) The comparison was made between the calculated and experimental dyad and triad fractions. (3) Chemical shift modeling of the assigned carbon resonance signals was done. (4) Empirical additive chemical shift parameters were optimized and chemical shifts of the overlapping resonances were calculated. (5) Intensities of the pentad compositional sequences were calculated. (6) Comparison between calculated intensities and experimental intensities of pentad resonances was done to ascertain the assignments made. (7) Spectral simulation for various copolymer compositions and comparison was done with the experimental spectra Reactivity ratios determination Dyad (AA, AB/BA and BB) fractions of a copolymer sample provide three data points in comparison to a single data point provided by the copolymer composition

4 for the determination of reactivity ratios. Reactivity ratios of both the monomers can be determined from the dyad fractions. Triad fractions provide three data points per copolymer sample for the determination of reactivity ratio of one monomer in a copolymer system. Hence the reactivity ratios for both A and B monomer units can be determined by the analysis of A and B centered triad fractions. For the rigorous analysis of this methodology, reactivity ratios determined from the dyad and triad fractions can be compared to check for the consistency. The experimental dyad and triad fractions can be used for the reactivity ratios optimization using the least square methodology. The application of least square method for the reactivity ratios optimization is based on the principle of minimizing sum of weighted residuals: where, r 1 and r 2 are reactivity ratios, w i is the weighting factor included to take account of the error involved in the measurements of triad fractions, y i is the experimental dyad or triad fraction, and x i is the infeed. (xi, r1, r 2 ) is the function of infeed and reactivity ratios giving the value of theoretical dyad or triad fractions. The weighting factor is the reciprocal variance, w i = 1/RI. When the error is unknown the assumption of constant relative error i.e. variance of the experimentally observed responses is proportional to the square of theoretical responses, r i ~/(x,,r 0 1 ; r20þ2 can be assumed. The r 0 1 and r20 corresponds to the best fit, but their values are not known in the beginning of the optimization. To solve this problem self-consistency approach has been applied. To begin with, w i = 1 is assumed then r 1 and r 2 are optimized. The values of these r1 and r 2 are then used to calculate the theoretical response i.e. r1 and r 2 are substituted as r01 and r J v respectively in the weighting factor (WJ I/a, : «\/f{x i,r' l,r J 2f) and r x and r 2 are optimized again. This iteration was continued till r 1 and r 2 converged and returned the same values of reactivity ratios as used in weighting factor i.e. optimized in the previous optimization. Genetic algorithm was then used for the determination of reactivity ratios. Advantage of the application of genetic algorithm is that it is capable of searching through very large space taking precedence over traditional search algorithms. The joint confidence interval of exact shape about the best set of reactivity ratios was obtained by applying the F-distribution, using the following inequality: F z (p,n p) is the value from the F-distribution at level z at p and n p degrees of freedom Empirical additivity parameters for 13 C{ 1 H} NMR chemical shifts of polymers Empirical additive rules for the hydrocarbons were first proposed by Grant and Paul [32]. They proposed that 1 3 C{1H} NMR chemical shifts can be broken down into linear combination of additive terms. Empirical additive parameterization has been utilized for the analysis of polymers by Cheng et al. and Matlengiewicz et al. The model begins with polymethylene backbone having chemical shift of 29.8 ppm for all the carbons, following the model of empirical additive parameters proposed by Cheng et al. [15]. Substituents are then considered at alternating carbon atoms imparting increments to the chemical shifts of neighboring carbon atoms based on their relative position. The substituent effects of main chain get incorporated in the base value of 29.8 ppm thus methylene units are not modeled as substituents. For the 1 3 C{1H} NMR chemical shifts of vinyl polymers, the chemical shifts were modeled as illustrated in Fig. 2. To obtain fine structure, d and e substituent effects were considered to be influenced by the monomer units at b and c positions, respectively. In a copolymer with A and B monomer units, e substituent additive effects are: ea(a) (i.e. e effect due to A at e position with a A monomer unit lying at c position), ea(b), eb(a) and eb(b). Similarly, due to different substituents at b position, the d substituent effects were modeled as da(a), da(b), db(a) and db(b). As illustrated in Fig. 2, the 13 C{ 1 H} NMR chemical shifts of C* carbon of A unit will have a base value of 29.8 ppm, A unit at a position will impart aa additive effect, the two substituents at c positions A and B will have ca and cb additive effect, respectively. The two B substituents at e positions will have eb(a) and e B(B) additive effects corresponding to A and B units at c positions, respectively. In the case of bisubstituted monomer units the additive effects of both the pendant units are modeled for single entity, thus in vinylidene chloride both the chlorine were modeled as a single substituent. eb(a) c c c c c c c c c eb(b) ssðr 1 ;r 2 Þ + p=ðn -pþf z ðp ; n -p) (1) where, n is the number of data points, p is the number of parameters (p = 2, for the two reactivity ratios). Fig. 2. An illustration showing modeling of the carbon-13 NMR chemical shifts of vinyl/vinylidene copolymers.

5 Table 1 Chemical shift modeling of the A centered pentads in A/B copolymer system Pentad AAAAA BAAAA BAAAB ABAAA BBAAA ABAAB BBAAB ABABA ABABB BBABB Chemical shift modeling a A + 2*c A ( A ) + EA(A) a A + 2*c A + e A(A) + e B(A) a A + 2*c A + 2*e B(A) aa + ca + cb + ea(a) + EA(B) + Q a A + c A + CB + EA(A) + e B(B) + Q a A + c A + CB + ea(b) + EB(A) + Q a A + c A + CB + e B (A) + e B (B) + Q a A + 2*c B + 2*e A(B ) AA + 2*c B + EA(B) + EB(B) K A "*" c B + 2 *e B ( B ) It was observed that when two different substituents were at c positions, the net c effect due to both the substituents was not additive. If AAB triad is considered, the net c effect will not be equal to ca + cb. To correct this effect a corrective term Q has been introduced. Thus, c effect for AAB triad was modeled as ca + cb + Q. Similar adjustment parameter q and a correction factor Q has been proposed by Cheng et al. for c additive effect [15]. The A centered pentads were modeled as described in Table 1. The chemical shifts of the assigned resonance signals were modeled into empirical additive parameters, as given in Table 1. This provide with known chemical shifts given as function of additive parameters which were optimized using genetic algorithm Vinylidene chloride (V)/methyl acrylate (M) copolymer system Analysis of carbon resonances of vinylidene chloride/ methyl acrylate copolymers was reported by Brar et al. [33]. They discussed compositional assignments and sequence distribution of quaternary, methine and methylene carbon resonances by one and two dimensional IIIppm (b) (a) VIII '7\v VI VII- VII=- 1 VIII UI J VI V(b)=- V(a) III - TIT - Fig C{ 1 H} NMR spectra of quaternary carbon of vinylidene chloride in vinylidene chloride/methyl acrylate copolymer of fv = 0.50 (a) simulated spectrum with 1 Hz line width for all the resonance signals, (b) simulated spectrum with adjusted line widths and (c) observed spectrum.

6 NMR spectroscopy. Analysis of the carbon resonances of quaternary carbon of vinylidene chloride was reported at pentad level of compositional sensitivity. 13C{ 1 H} spectrum of quaternary carbon of the copolymer fv = 0.5 is given in Fig. 3(c). Brar et al. [33] had reported the following assignments for quaternary carbon resonances. Peaks I, II and III were assigned to VVVVV, MVVVV and MVVVM pentads, respectively. Peaks IV and V were assigned to VMVVV/MMVVV and VMVVM/MMVVM pentads, respectively and peaks VI, VII and VIII to VMVMV, MMVMV and MMVMM pentads, respectively. The expanded 13C{ 1 H} spectrum showing methine carbon of methyl acrylate of fv = 0.5 is given in Fig. 4. The ranges of MMM, MMV and VMV triads have been marked. It was observed that MVM and VVV centered pentads were segregated enabling apparent assignments of VMVMV, MMVMV, MMVMM, VVVVV, MVVVV and MVVVM pentads. The analysis of change in intensities of resonance signals with composition elucidated that peaks I, II and III are due to MVVVM, MVVVV and VWW pentads, respectively. MVV triad centered pentads constituted two broad resonance signals, there by making the assignments of VMVVV, VMVVM, MMVVV and MMVVM pentads difficult. To resolve these resonances, the approach described above was applied Reactivity ratios determination For the reactivity ratios determination of vinylidene chloride/methyl acrylate copolymers five samples of different compositions,f V = 0.30, 0.40, 0.50, 0.60 and 0.70 were synthesized. Area under V centered VVV, MVV and MVM triads and M centered MMM, MMV and VMV triads were used to obtain the respective triad fractions. From the V and M centered triad fractions, both the reactivity ratios were optimized using least square residue approach as described previously. The optimized reactivity ratios were r V = 0.95 and r M = 0.75 (where, r V and r M are the reactivity ratios for vinylidene chloride and methyl acrylate, respectively). Using these reactivity ratios, theoretical triad fractions were calculated as explained in the appendix. The experimental and theoretical V and M centered triad fractions are given in Fig. 5(a) and (b), respectively (the experimental points are indicated by discreet symbols and theoretical data is plotted as continuous curve). There was good match between the experimental data points and theoretical curve. Reactivity ratios optimization from the dyad fractions given in Table 2 was done. The reactivity ratios determined were r V = 0.97 and r M = The joint confidence interval for the reactivity ratios, r V = 0.95 and r M = 0.75, optimized from triad data by genetic algorithm is given in Fig. 6. The data point (D) is well within the confidence interval of reactivity ratios optimized from triad fractions. Thus the reactivity ratios optimized ] X \ \ VMV ^*~ "^ / MMM / MMV \ / ^ ~^~^^/ fm ^ B T> (a) Fig C{ 1 H} NMR spectrum of the methine group of methyl acrylate in vinylidene chloride/methyl acrylate copolymer off V = Fig. 5. Variation of theoretical and experimental (a) V centered and (b) M centered triad fractions with change in the infeed of copolymer system.

7 Table 2 Concentration of methylene dyads in methyl acrylate copolymers Infeed W VM/MV vinylidene chloride/ MM Fig. 6. Joint confidence interval for vinylidene chloride/methyl acrylate copolymers about the best set of reactivity ratios, r V = 0.95 and r M = 0.75 optimized from triad fractions. The points labeled as (T) and (D) are the reactivity ratios optimized by triad and dyad fractions, respectively. from triad and dyad were in good agreement lending support to this methodology NMR spectral simulation 13 C{ 1 H} NMR chemical shifts of the six pentads VMVMV, MMVMV and MMVMM; VVVVV, MVVVV and MVVVM of MVM and VVV centered triads, respectively were modeled in terms of empirical additive parameters. This provided six equations and seven parameters to be optimized. The empirical additive parameters were optimized using genetic algorithm. From the comparison of experimental chemical shifts and calculated chemical shifts of MVV triad (without the correction factor Q), the correction factor Q was determined which was found to be The optimized additive parameters are given in Table 3. 13C{1H} NMR chemical shifts of VMVVV, VMVVM, MMVVV and MMVVM pentads were then calculated from the optimized parameters. Experimental and calculated chemical shift values are listed in Table 4. The calculated chemical shifts of VMVVV and VMVVM pentad resonances were in close range so it was concluded that peak IV in Fig. 3 was due these overlapping resonances. Similarly, peak V was attributed to the overlapping MMVVV and MMVVM pentads having calculated chemical shifts in close range. The pentads fractions were calculated as described in the Appendix A. Table 5 lists the experimental and theoretical V centered pentad fractions at different compositions. Fig. 3 show the experimental and the simulated spectra of the copolymerfv = Simulated spectrum Table 3 Optimized chemical shift additive parameters for the quaternary carbon resonances of vinylidene chloride in vinylidene chloride/methyl acrylate copolymers Parameter Value of parameter (ppm) av cv cm EV(V) 0.74 EV(M) 0.02 EM(M) 0.73 EM(V) 0.64 Q 0.65 Table 4 Experimental and calculated chemical shifts of the quaternary carbon resonances of vinylidene chloride in vinylidene chloride/methyl acrylate copolymers Peak Pentad Chemical shift (experimental) Chemical shift (calculated) I II III IV (a) IV (b) V(a) V(b) VVVM MWW vww VMVV VMVVM MMVV MMVVM VI VII VIII VMVMV MMVMV MMVMM

8 8 Table 5 Concentration of quaternary carbon centered pentads of vinylidene chloride/methyl acrylate copolymers vww MWW MVVVM VMVV VMVVM MMVV MMVVM VMVMV MMVMV MMVMM fv = 0.7 Expt Cal fv = 0.6 Expt Cal fv = 0.5 Expt Cal fv = 0.4 Expt Cal fv = 0.3 Expt Cal fv f; = 0.5 t; = o.6 (a) Fig. 7. C{ 1 H} NMR spectra of quaternary carbon of vinylidene chloride in vinylidene chloride/methyl acrylate copolymers at different copolymer compositions (a) observed spectra and (b) simulated spectra with adjusted line widths. (b) in Fig. 3(a) has all the line widths as 1 Hz. Simulated spectrum in Fig. 3(b) was obtained by adjusting the line widths to obtain similarity with the experimental spectrum. Peaks I, II and III are attributed to VVVVV, M W W and MVVVM pentads, respectively. Peaks IV and V constituted two broad resonance signals which were resolved as IVa and IVb corresponding to overlapping VMVVV and VMVVM pentads, where as Va and Vb were assigned to overlapping MMVVV and MMVVM pentads. VMVMV, MMVMV and MMVMM pentads are labeled as VI, VII and VIII, respectively. The simulation of NMR spectra of different compositions was done. The experimental and theoretical NMR spectra of different compositions are given in Fig. 7. Good match between the theoretical and experimental pentad fractions and between simulated spectra and experimental spectra substantiated the validity of the approach. Therefore good estimate of reactivity ratios was obtained and the application of the empirical model for prediction of chemical shifts of overlapping resonances was successful. 4. Conclusions An approach based on intensities calculation of resonances, chemical shift modeling and spectral simulation has been used for the analysis of overlapped resonances. Reactivity ratios determination of vinylidene chloride/ methyl acrylate copolymers was done from the dyad and triad fractions. The results obtained from dyad and triad fractions were in good agreement. The assigned resonances signals were modeled into empirical additive chemical shift parameters and the optimized additive parameters were used to predict the chemical shifts of the overlapping resonances. Genetic algorithm

9 gave good fitness function value for the optimization of reactivity ratios and additive parameters after a small number of function evaluations and was able to search through a very large space. Good match between the theoretical and experimental pentad fractions and between simulated spectra and experimental spectra at pentad level was obtained. The approach helped to investigate the resonance signals to higher level thus, giving deeper insight into the polymer microstructure. Acknowledgment Author (Gurmeet Singh) thanks the Council of Scientific and Industrial Research (CSIR), New Delhi, India for the financial support. Appendix A In the first-order Markov statistical model or terminal model for a growing chain the addition of monomer units is considered to be dependent on the last unit in the chain. From the reactivity ratios, reaction probabilities are calculated as: r A >A + ([M B ]/[M A ]) 1 \+r B {[M B \/[M A \) PAB = PBB = PAA PBA where, M A and M B are the infeed fractions of A and B monomers, respectively. p AA, p AB, p BA and p BB are the reaction probabilities. p AA and p AB are the probabilities of A and B monomer units, respectively getting attached to the A radical at the growing chain end. The first-order Markov model states that sum of the probabilities PAA + PAB and p BA + p BB is equal to 1. The outfeed of A and B monomers, F A and F B, respectively are given as: " r 1 f 1 2 þ 2f 1 f 2 The intensities of AA, AB/BA and BB dyad fractions are calculated from the outfeed monomer fraction of the first monomer depicted in the dyad and the probability of second monomer following it. The dyads fractions are calculated as shown: F AA FBB FBPBB F AB F BA The monomer centered triad fractions are calculated as the multiplication of the probabilities of the outer monomer units following the central monomer unit. The triad fractions are given as: FAAA = ÐPAAÞ2 F BAA = 2ÐPABÞÐPAAÞ FBAB = ÐPABÞ FBBB = F ABB = 2ðp BA Þðp BB Þ F ABA = ðp BA Þ The monomer centered pentad fractions are calculated from the fraction of central triad and the probabilities of the outer monomer units following the central triad unit. The AAA triad centered pentad fractions are calculated as: F AAAAA ~ F AAA ðp AA Þ2 F BAAAB F AAA ðp AB Þ F AAAAB F AAA ðp AA Þðp AB Þ By the similar approach, rest of the pentad fractions can be calculated. References [1] Bovey FA, Mirau PA. NMR of polymers. San Diego: Academic Press; [2] Kim Y, Harwood HJ. Polymer 2002;43:3229. [3] Brar AS, Hooda S, Kumar R. J Polym Sci Part A: Polym Chem2003;41:313. [4] Fisher T, Kinsinger JB, Wilson CW. Polym Lett 1966;4: 379. [5] O'Driscoll KF. J Polym Sci Polym Chem Ed 1980; 18:2747. [6] Rudin A, O'Driscoll KF, Rumack MS. Polymer 1981;22: 740. [7] Moritani T, Iwasaki H. Macromolecules 1978;11:1251. [8] Uebel JJ, Dinan FJ. J Polym Sci Polym Chem Ed 1980;21: 917. [9] Hill DJT, Lang AP, O'Donnell, O'Sullivan PW. Eur Polym J 1989;25:911. [10] van Herk AM. J Chem Educ 1995;72:138. [11] van Herk AM, Droge T. Macromol Theory Simul 1997;6: [12] Wehrli FW, Marchand AP, Wehrli S. Interpretation of carbon-13 NMR spectra. Singapore: John Wiley & Sons; 1988 [Chapter 2]. [13] Pihlaja K, Kleinpeter E. Carbon-13 NMR chemical shifts in structural and stereochemical analysis. New York: VCH; [14] Cheng HN, Kasehagen LJ. Anal Chim Acta 1984;285:223. [15] Cheng HN, Bennett MA. Anal Chem 1984;56:2320. [16] Cheng HN. J Chem Inf Comp Sci 1987;27:8. [17] Cheng HN. Polym News 2000;25:114. [18] Nguyen G, Nicole D, Swistek M, Matlengiewicz M, Wiegert B. Polymer 1997;38:3455. [19] Matlengiewicz M, Nguyen G, Nicole D, Itenzeu N. J Polym Sci Part A: Polym Chem 2000;38:2147. [20] Holland J. Adoption in natural and artificial system. Ann Arbor, MI: The University of Michigan Press; [21] Goldberg DE. Genetic algorithms in search, optimization, and machine learning. Reading, MA: Addison-Wesley; [22] Banzhaf W, Nordin P, Keller RE, Francone FD. Genetic programming: an introduction: on the automatic evolution of computer programs and its applications. San Francisco: Morgan Kaufmann; [23] Deb K. Optimization for engineering design: algorithms and examples. New Delhi: Prentice Hall; 2002 [Chapter 6]. [24] Shaffer RE, Small GW. Anal Chem 1997;69:236A.

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