Anal. Chem. 1982, 54, 1828-1833
1828
The fitting parameters KOand Kl are the only parameters that can be assigned any real significance. KOshould correspond to the error in the zero-order wavelength and K1to the error in the correction of the error in the dimensional constants by the sine-bar length adjustment. Additional terms when added to the fitting equation improves the fit, but only slightly compared to the additional computer time required to carry out the calculations. Thus it is possible to correct the wavelength of the monochromator to better than rt0.025 nm from an uncertainty of up to f0.12 nm by collecting wavelength information and fitting the wavelength error curve to a test equation. A monochromator control program has been written which automatically corrects the wavelength readout as the monochromator scans by using a look-up table to determine the correction factor. The lead screw runout usually determines the wavelength inaccuracy in quality monochromators. Therefore, this procedure herein described should provide an improvement similar to that shown for the test monochromator used in this study. For the method to provide reliable corrections the monochromator must have a wavelength setting reproducibility greater than the specified accuracy.
Table I. Results of Weighted Data Fit
KO K, K, K3
K, K5
eq 1 2
eq 13
1.43 E- 2 nm -5.29 E-5a -9.88 E-2' 1.39 E- 2/nm -9.16 E-la
- 1.51 E-2 nm
K6 K, a
5.44 E-6a 1.02 E- la 1.33 E- 2/nm - 3.75 E+ Oa 3.93 E- 3a 3.28 E-2/nm -3.53 E+Oa
Unitless.
(X,(zo)), accumulated error in the linear travel from zero (x,(iV)), and the functionality the composite error (f'(X)) be determined. Results of Data Fit. Inspection of the plot of the error in the wavelength as a function of wavelength as shown in Figure 2, suggests the use of a sine curve model to fit the data. This along with eq 11 suggests eq 12 and 13 as possible test equations for the wavelength error XJX) Xe(X)
=
= KO
+ Klhi + K2 sin (&hi + K4)
KO+ KIXi + K2sin (&Xi
(12)
+ K4) + K5sin (&xi + K7)
LITERATURE CITED (1) Warren, M. W.; Avery, J. P.; Lovse, D. W.; Malmstadt, H. V. J . Autom. Chem. 1981, 3, 76. (2) Nelder, J. A.; Mead, R. Comput. J . 1965, 7 , 308. (3) GCA Corporation "Scanning Monochromator EU-700 and EUE-700 Series", Acton, MA, 1968.
(13) where KO = error in zero-order wavelength, K1 = error in constant correction, K z , K6 = sine amplitude, K3,K6 = sine angular velocity, and K4,K7 = sine phase angle. The weighted sums of the residuals according to the distance between the data point and its nearest neighbors fit eq 12 poorly in the high density data region and fit eq 13 within f0.02 nm for all data points. The fit to the experimental data for eq 13 is plotted on Figure 2.
RECEIVED for review December 24,1981. Resubmitted June 8, 1982. Accepted June 8, 1982. The authors acknowledge the support provided by Grant HEW-PHS GM 21984 and computer time on the University of Illinois Cyber 175 provided by the University of Illinois Research Board.
Markovian Statistics and Simplex Algorithm for Carbon-13 Nuclear Magnetic Resonance Spectra of Ethylene-Propylene Copolymers H.
N. Cheng
Hercules Incorporated, Research Center, Wilmington, Delaware 19899
The "C NMR spectrum of ethylene-propylene copolymers contains much lnfonatlon on polymer structure and monomer reactlvlties. I n thls work a systematlc analysis Is proposed that alms to extract the maxlmum informatlon from each spectrum. The approach used Is to model the copolymerlration of ethylene and propylene by a second-order Markovlan process and to flt the lntensltles of the NMR spectrum to the model vla a slmplex algorlthm. A computer program has been wrltten that qulckly and efflclently computes all the necessary parameters. Informatlon available Includes composltlon, comonomer sequence dlstrlbutlon, Markovlan probabllltles, and reactlvlty ratlo product ( r 2r'). The role of dlffuslon In heterogeneous copolymerlsatlon Is also dellneated. The approach is valld for ethylene-propylene copolymers that contaln no Inverted propylene sequences.
13Cnuclear magnetic resonance spectroscopy has become a powerful technique for polymer characterization ( I ) . One 0003-2700/82/0354-1828$01.25/0
fruitful area of application is in copolymer analysis where such information as polymer composition, microstructure,tacticity, sequence distribution, and propagation errors in polymerization can be readily analyzed and, with care, quantified. Ethylene-propylene copolymers are no exception; indeed, in the last 6 years several studies have been made of these copolymers (2-7). In a careful study, Carman (2) examined several ethylene-propylene rubbers and used a reaction probability model to fit all the spectral lines to the model. Since the ethylene-propylene rubbers contain inverted propylene sequences, Carman was only able to fit the data to first-order Markovian statistics. In the alternative approaches of Ray (3) and Randall ( 4 , 5 ) ,selected lines in the spectrum were taken, from which copolymer sequence distributions were calculated. The polymers studied were made with heterogeneous catalysts and were free of inverted propylene structures. For some of these polymers, it may actually be more appropriate to use a second-order Markovian model (i.e., the addition of each monomer to the polymer chain depends on both the preceding 0 1982 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 54, NO. 11, SEPTEMBER 1982
1829
-
Table I. Calculated Markovian Intensities for shift ranges, carbon line no. PPm type 1-3 45.0-46.5 sa a 4-5 3'7.7 -38.0 say 6-7 37.0-37.6 8
9 10 11
12 13 14-15 16-1 8 19-21 22-23 24 a
3 3.1
30.7 30.6
NMR Lines of Ethylene-Propylene Copolymers
sequence
first-order Markov
PP 2PEP 2P(E)rl>2P EPE PPE
P33P23'K
2F32p23'
30.1 29.8 Z!8.7 27.0-28.0
24.0-24.9 '2 1.O-2 1.8 20.2-20.9 3.9.8
For the meaning of a , p ,
y,
and
6,
see text.
2 denotes ethylene; 3 denotes propylene.
and penultimate units in the chain). The objectives of this work are to model the copolymerization of ethylene-propylene by both first-order and second-order Markovian processes, to derive the theoreticall Markovian probabilities for the NMR line intensities, and to fit the observed NMR spectrum to the models by a simplex algorithm. The calculations of sequence distribution as described by Ray and Randall are also incorporated and compared with the predictions of the Markovian models. The analytical scheme delineated above has been written into a computer program. The use of the Markovian models and the automated computation ensures that for each 13C spectrum, maximum iinformation can be extracted. Furthermore, since the Markovian statistical models use all of the spectral lines, experimental errors in measurements are reduced, giving more confidence to the fitted results. The errors are further diminished by the use of superconducting magnetic fields.
EXPERIMENTAL SECTION The four copolymers used here as illustrations are all experimental samples made with Ziegler-Natta stereoregular type catalysts. The samples were dissolved in 1,2,4-trichlorobenzene at a concentration of 20 w t % with benzene-d6added as the lock material. The 13CNMR spectra were recorded at 90.55 MHz and 120 "C on a Nicolet NT360WB spectrometer equipped with an 80K computer and NTCFT software system. Instrumental conditions used were as follows: pulse angle, 90"; pulse delay, 12 s; sweep width, 6000 Hz; and gated proton decoupling. The long delay time and the' gated decoupling ensure that any differential Ti's and nuclear Overhauser enhancements will be removed. The spectral areas were obtained by spectral integration using the software feature of the NTCFT program. In the process of fitting the data, it became apparent that the accuracy of the results depended strongly on the quality and the resolution of the NMR spectra. The use of tho higher magnetic field increased both sensitivity and chemical shift dispersion and generally produced data with greater precision. The computer program EPCO was written in BASIC language and run on the General Electric time-sharing terminal. RESULTlS AND DISCUSSION The 13C NMR spectrum of an ethylene-propylene copolymer is given in Figure 1. From each spectrum, 24 lines can be identified and separately assigned. The terminology used here follows that of Carman (2),where S, T, and P refer, respectively, to the methylene, methine, and methyl carbons, and the Greek subscripts refer to the next nearest methyl substitution (Figure 2). The complete assignments are summarized in Table 1.
I Sa*
Flgure 1. 13C NMR
spectrum of ethylene-propylene copolymer at 90
MHz. 0
0
I
0-
0
- 0-I
PP
0
0
P
0-x-7.ay 88
0-
I
PEP
0- 0
ay
0
0
I
0-0-x-Lx-x-0-0 a6 86 yy
I
PEEP
86 a6 0
0
I
0-0-x-x-x-
a6
Flgure 2.
86 y6
66
x-x-xy6 86
I
0-0
PEEEP
a6
Nomenclature used to designate the carbons.
Copolymer Composition. The calculation of copolymer composition is most conveniently made on methylene carbons in the NMR spectrum (3). The propylene content P'and ethylene content E'are given by
s,, + '/Z(S,, + Sea) E' = 72[S&?+ s,, + s,, + S,s+ Ssa + '/,cs,, + S,,)I P'=
(1)
(2)
The mole fractions of ethylene and propylene are then calculated from the relationships E = kE'and P = kP', where k = (E' P?-1. Sequence Distribution. Ethylene and propylene units when polymerized may arrange themselves in different sequences, depending on feed ratios, monomer reactivities,
+
1830
ANALYTICAL CHEMISTRY, VOL. 54, NO. 11, SEPTEMBER 1982
catalyst systems, and reaction conditions. Ray (3) has shown that for a 13CNMR spectrum, complete dyad and triad distributions and partial tetrad and pentad distributions can be calculated. The expressions for the sequence distributions used in this work are summarized in Appendix I. Markovian Model of Binary Copolymerization. In the second-orderMarkovian model the addition of each monomer depends on the two last units in the chain (8). This is also known as the "penultimate effect".
--
+ P -PPP. -PP. + E -PPE. -PE. + P -PEP-PE* + E -PEE. -EP. + P -EPP. -EP. + E -EPE* -EE* + P -EEP. -EE. + E -.+ -EEE* -PP.
+
-+
+
+
-+
k333
(3)
k332
(4)
k323
(5)
k322
(6)
12233
(7)
k232
(8)
k223
(9)
k222
(10)
Eight probabilities will describe the system. For example
where P322 is the probability of a copolymer chain ending in PE reacting with monomer E. Seven other equations can be written in a similar manner. In the derivation, 2 refers to ethylene, 3 refers to propylene, and Pijkis the probability parameter. The following abbreviations will be used: p333 = a P332 = p322 = p p323 = p p232 = 7 p233 = 7 p223 = p222 = 6 From the definition of P322, it can be seen that cy is equal to 1 - a, p is equal to 1 - p, and so on. There are actually only four independent probabilities for the second-order Markovian process. In order to tailor this model to the 13CNMR spectrum of ethylene-propylene copolymers,one must associate all of the NMR peaks in Table I with the appropriate intensities as predicted by the second-order Markov model. The first step is to calculate all the sequence probabilities. For dyads through tetrads, this is given as follows (9): PPPP = a2ys PP = y6 PPPE = 2 a ~ y S PE = 2n6 PPEP = 2n&6 EE = a$ PPEE = 2npy6 PEPE = 2@pS PPP = ays PEEP = nps2 PPE = 2ny6 PEEE = 2ap& PEP = a@ EPPE = n2y6 PEE = 2@6 EPEE = 2~2276 EPE = nqtj
EEE = apF
EEEE =
Four sets of peaks require special consideration: Sa*,So,, Sy6,and Sa,. The first two, Sa*and S,,, consist of all P(E),P sequences where n 2 2 and Sy6consists of sequences where n 2 3, whereas Sd6consists of all long E sequences. To calculate these peak intensities, one needs to know the probabilities of all P(E),P sequences. The calculations are rather tedious; only the answers are given below:
PEP PEEP PEEEP PEEEEP P(E),P For Sa6or S,, (sequence =
npsz
npszs npm
np628"-2(by induction) P(E),,,P), the probability is
Pi = 2@62(1
+ F + 8 2 +F3 + ...)
m
= 2np62
c F"
n=O
since 6 is equal to 1 - 8, as we have seen. For Sy6(sequence = P(E),,,P), the probability is Pi = 2nps2am = 2npss For SSs,the addition of every E unit actually adds two Saa carbon carbons. Consequently, although there is only one in the PEEEP sequence, there are three in the PE4Psequence and five in the PE6psequence. The probability is as follows: Pi =nps2q1 36 522 783 ...) = a p s 2 E ( 1 2n)F" = ap62q26-2 - 6-11 = zap8 - npss
+ +
+
+
+
The calculated probability and the corresponding peak position are summarized in the last column of Table I. In this table K is a normalization constant K = (5~+ 6 376 2@)-'
+
Hitherto the results have been those of the second-order Markovian model. To reduce the second-order to first-order Markov, one simply drops the first subscript; thus p = 6 = P23, a = y = P33, = 7 = P32, p = 6 = P22 and since from the conservation relationships a + a = 1 and p /3 = 1, one actually has only two independent parameters a and p for the first-order Markovian statistics. Direct substitution of these equations in Table I gives the intensities for the first-order Markovian models. These are also given in Table I. In this case the total number of carbons is 3PZ3+ 2P32;thus K = (3P23 2P3J1. It can readily be shown (in Appendix 11) that Carman's probabilities ( 2 ) can be shown to be equivalent (minus all inversion terms) to our first-order Markov intensities. The EPCO Program. The EPCO program was written in BASIC language so that it can be easily adapted for laboratory computers. It accepts as input the observed spectral intensities and performs the following functions: (a) It calculates the copolymer composition via eq 1 and 2. (b) It computes the dyad and triad distributions and the monomer sequence distribution (eq A1 through -418). (c) It takes the observed spectral intensities and fits them with the theoretical Markovian probabilities. Either the firsborder or the second-order Markovian case can be selected. (d) From the fitted model, it calculates the optimized sequence distribution, and the comonomer reactivity ratio product (r2r3).A schematic diagram of the program is given in Figure 3. The first two tasks (composition and sequence) are fairly straightforward. The third task requires an efficient fitting algorithm with reasonably fast convergence. For this purpose, the modified simplex algorithm is used (IO). In the first-order Markovian process, since there are two probabilities (P32 and PZ3),one needs three vertices (Le., three sets of Pi,)to establish the initial simplex, The EPCO program automatically works out the optimal P32 and P23for the data. In the second-order Markovian process, there are four probabilities ( a , P, y, F), and five vertices are needed. The response function in both cases is defined to be the standard deviation between the calculated and the experimental spectral line intensities.
+
+
[li(obsd) - 1i(calcd)l2
'={'
i
N(N-1)
ANALYTICAL CHEMISTRY, VOL. 54, NO. 11, SEPTEMBER 1982
1831
Table 11. 13CNMR Spectral Data of Six Ethylene-Propylene Copolymers normalized peak areas no.
6,
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
ppm
carbon type
46.3 45.8 45.6 37.9 37.8 37.8 37.4 33.11 30.7 30.6 30.11 29.8 28.7 27.6 27.6 24.:' 24.6 24.4 21.6 21.4 21.2 20.;' 20.5 19.13
8a!,
sa!, sa, saya
sa!, a
sa!6
sa 6 T6 6
TPY SYY
SY6 s6 6
TOP SP 6 SP6
SPP SPS SPP PPP PPP PPP
PPY PPY'
+
PY +Y
+
1
2
3
4
24.03 4.38 0.27 0 5.23 0 0.94 0.56 5.15 0.23 0.16 0.38 24.19 0.19 0.44 0 0.19 2.60 18.99 5.21 0.96 4.25 1.09 0.55
26.66 3.04 0.32 0 1.84 0.13 0.16 0.85 2.40 0.05 0.77 0.56 29.91 0.05 0.08 0 0.08 0.80 25.09 2.98 0.7 5 2.13 0.53 0.80
16.83 6.19 1.38 3.28 5.08 1.51 1.04 1.35 7.03 0.77 1.26 1.67 19.54 0.93 1.36 0.39 1.36 2.69 12.23 4.53 1.11 5.12 1.95 1.38
3.73 3.77 2.22 6.33 2.7 2 6.16 2.31 3.86 7.05 1.42 5.45 22.66 4.88 5.02 3.14 1.20 2.10 1.27 1.89 1.81 0.70 2.78 3.05 4.47
Ray A
Ray C
23.75 4.53 0.49 0.0 5.51 0.0 0.61 0.37 5.63 0.0 0.24 0.12 25.34 0.0 0.0 0.0 0.61 1.83 19.09 4.90 0.49 4.29 0.98 1.22
4.46 4.40 1.47 0.0 9.86 0.0 7.20 4.13 7.66 1.40 4.06 19.19 6.00 5.13 3.06 1.oo 2.13 1.47 2.73 2.53 0.80 3.93 3.53 3.86
Not resolved; arbitrarily set to zero,
ENTER OBSERVED
L ($----e
Table 111. Mole Fraction of n-ad Distributions 1 2 3 4 Ray A Ray C
1
E P PP PE EE PPP PPE EPE PEP PEE EEE E' E2 En>3 PI
CALCULATE
1 1 COMPOSITION
2 SEQUENCE DISTRIBUTION
MARK0
;YES
ZndORDER
{T VERTICES
VERTICES
P,l
qlkl
1
I
SPECTRAL INTENSITIES FOR E A C H VERTEX
MINIMIZES R
p*
Pn>3
0.10 0.90 0.81 0.19 0.02 0.73 0.15 0.02 0.08 0.02 0.01 0.72 0.13 0.15 0.02 0.02 0.97
0.12 0.88 0.79 0.19 0.03 0.71 0.16 0.01 0.08 0.03 0.01 0.68 0.15 0.17 0.01 0.00 0.99
0.21 0.79 0.65 0.29 0.06 0.54 0.21 0.04 0.11 0.06 0.03 0.53 0.13 0.34 0.05 0.09 0.86
0.58 0.42 0.22 0.40 0.38 0.13 0.17 0.13 0.10 0.19 0.29
0.18 0.11 0.72 0.30 0.24 0.46
0.08 0.92 0.83 0.18 0.00 0.72 0.16 0.04 0.07 0.01 0.00 0.94 0.00 0.06 0.04 0.03 0.93
0.55 0.45 0.25 0.41 0.35 0.16 0.19 0.10 0.11 0.18 0.25 0.21 0.18 0.60 0.22 0.16 0.62
Table IV. Markovian Probabilities Calculated for the Six EP Copolymers COMPlJTES r2
Figure 3. Schematic diagram of the
EPCO
5,
sample
@>
1 2 3 4
program.
Ray A Ray C
Since the simplex algorithm (or most fitting routines) can only search for local minima., it is advisable to feed in as the initial vertices values of Marklovian probabilities that are fairly close to the optimal values. Alternatively, different guess values should be tried and if different minima are obtained, the minimum with the smallest R value and which is still consistent with the chemical system in question is taken. Illustrative Examples. The computational scheme delineated in the previous sections has been applied to four experimental polymers. For comparison, two copolymers reported by Ray et al. ((3)were also included in the analysis.
p32
0.10 0.10 0.19 0.46 0.10 0.44
PZ3 0.86 0.83 0.76 0.28 0.91 0.35
r23
P3,3
'323
PZ3,
'223
1.5 1.8 1.3 3.1 0.9 2.4
0.91 0.97 0.84 0.63 0.90 0.60
0.86 0.95 0.78 0.51 0.91 0.56
0.83 0.97 0.72 0.36 0.92 0.48
0.96 0.95 0.76 0.24 0.86 0.25
The data for the six copolymers are reproduced in Table 11. Analysis of these copolymers by the EPCO program is straightforward. The sequence information has been consolidated in Table 111. It is readily seen that as the ethylene/propylene ratio increases, the percentages of the E containing sequences (dyads and triads) likewise increase, but not randomly. In fact in samples 4 and C, there is a slightly disproportionate amount of EE and EEE sequences. This reflects the greater reactivity of ethylene, a fact brought out more clearly by fitting the data to Markovian models.
1832
ANALYTICAL CHEMISTRY, VOL. 54, NO. 11, SEPTEMBER 1982
The Markovian probabilities are summarized in Table IV. The first-order Markovian model gives a good idea concerning the nature of the copolymer; this depends on the magnitude of P32 and P23: (1) if P32 < 0.5 and P23 < 0.5, the copolymer tends to be blocked, (2) if P32 > 0.5 and P23 > 0.5, the copolymer tends to alternate, (3) if P32 < 0.5 and P23 > 0.5, the polymer is polypropylene-like,and (4) if Plz< 0.5 and Pzl> 0.5, the polymer is polyethylene-like. In cases 3 and 4, if in addition, P32 P23 = 1, then one has the Bernoullian (random) case. It appears that sample A is very close to Bernoullian, whereas the other samples have varying tendencies toward blockiness in comonomer sequence distribution. This can be indicated in an alternative way by using reactivity ratios. It can be easily shown that
Table V. Number of Each Carbon Species in Representative Sample" (from Ref 2) with Inverted Propylene Terms Omitted
"
code
species
number
+
The values of r2r3 are also given in Table IV. Thus sample A appears to result from almost ideal copolymerization, whereas the other samples exhibit block copolymerization behavior (i.e., large r2r3). This agrees with the literature values for similar systems where r2r3 of 1.7 was obtained by conventional methods (11). The second-order Markovian model provides additional information on comonomer reactivities. In the case of polymers 1, 2, and A, the second-order Markov offers no substantial improvement over the first order. In fact =1 - P32 and p323 = P23. Samples 4 and C, however, do show noticeable deviations from the first-order process. Particularly significant is the probability P223 which tends to be much smaller than expected. Since Pzzz= 1- P223, this implies that eq 10 in the copolymerizationscheme is more important than expected. Yet interestingly P323,correspondingto eq 5, is just normal. It seems that once a block of two ethylenes is formed, it tends to propagate to form longer ethylene blocks. It is of interest that samples 4 and C from very different sources are so similar, in composition as well as in Markovian probabilities. Role of Diffusion. The copolymerization equations as depicted by eq 3-10 are usually applied to homogeneous reactions. In heterogeneous catalysis, as is the case here, additional complication may arise. A chief problem is the role of diffusion. This can be examined by a reasonable model
A
k kC + B '-k L( A B ) -AB ,
(14)
Thus species A and B (e.g., monomer and polymer/catalyst complex) diffuse to the proximity of each other and can either diffuse away or react to form AB (12). From steady-state considerations, the rate can be shown to be as follows:
thus if kc >> kD, the reaction is diffusion controlled. For the copolymerization case, one can write four equations.
kz-D. After some algebraic manipulation, one obtains an apparent r2r3
If kij > kLD, then the polymerization is diffusion controlled, and ( ~ 2 ~ =3 1.) ~ ~ ~ It may be concluded, therefore, that if the copolymerization is entirely diffusion controlled, the resulting copolymer will have random placement of comonomers, thus approximating ideal copolymerization. This will be valid even if the intrinsic rzr3 departs far away from unity. If the diffusion is only partially operative, the observed r2r3 will then lie somewhere in between the intrinsic value and unity. As a corollary to this fact, one may not extrapolate the results obtained from diffusion dominated systems to reaction controlled systems unless the relative importance of diffusion can be estimated. ACKNOWLEDGMENT Thanks are due to W. J. Freeman for helpful discussions and to G. A. Lock for supplying the samples used in this work. APPENDIX I Sequence Distributions of Ethylene-Propylene Copolymers. The distribution of dyad sequences is related to methylene peak intensities by the following equations (3): PP = kIaa ('41) (A2) PE = k(Ia7 + Ia8)
EE = k/2(Ipa + 1 8 8 ) + (k/4)178
(A3)
The accuracy of these dyad sequences can be checked by these relationships P = PP 1/,PE (A4)
+
E = EE
+ 1/,PE
(-45) The triad distributions are taken from methylene and methyl intensities: Let Pt = Ppp+ Ppy+ PYy The equations assume that the rate of diffusion is primarily determined by the monomer and independent of the last unit in the polymer chain, e.g., eq 16 and 18 have the same kzDand
PPP = P(Ppp/Pt)
(-46)
PPE = P(PB,/Pt)
(A7)
ANALYTICAL CHEMISTRY, VOL. 54, NO. 11, SEPTEMBER 1982
EPE = P(P,,/Pt)
(A81
PEF’ = kI6p = (k/2)Ia,
(A91
PlEE = kIa8 = kIp6
(A101
E E E := (k/2)16,3
+ (k/4)1,6
E , = (k/2)Ia,/E
(-412)
Ea = k(I68 - I y a ) / E
(AJ-3)
1- El - E2
0414)
P1 = f’y,/(P~(9
+ Poy + P,,)
6415)
~ L - E P P E / ~ ’(A161
Pz =
1 - P1 - P2 (AX7) Finally, the number of average sequence length of ethylene is simply given as ;YE = 1 2 E / P (AM) PiZ23=
+
APPENDIX I1 To show that the first-order Markov probabilities given in Table I are equivalent to the probability expressions derived by Carman et al. (2), one can simply drop the terms involving inverted propylene (index = 3) in the conditionalprobabilities (eq 3 and 4 in ref 2) (A19) Pi1
+ Pi2 = 1
Pi3
=0
(A201
Equation A20 can be used to simplify all the expressions derived in ref 2. For example, eq 6 in ref 2 takes on these simplified forms:
N1 = Kp21
Nz = Kp12
s 3=
0
NzPzz
NzPZlPlZ
(-422)
85 = NzPllPZlPlZ CS = Nz
(All)
The monomer sequence distribution is defined as the number of moles of one monomer in a given sequence length divided by the total moles of that monomer in the copolymer. If E,, and P, are the monomer distribution (mol %) of ethylene and propylene, respectively, in sequences n units long, then
En23
s 1=
1833
All even terms of Si become zero. The probability expressions given in Table I11 of ref 2 can be recast in Table V, valid for the case here where inverted propylene is absent or negligible. Using eq A20-A22, one can readily show that the expressions given in Table V are equivalent to the first-order Markov probabilities shown in the fifth column of Table I. As an example, the probability expression for Sa,(code E) is derived as follows: probability
(Sa&) = 2(N2 - S1- S,) = 2N2 - 2NzP22 - 2NzPzlP12 = 2Nz(l - Pzz) - 2NzPZlP12 = 2NZpz1- 2NzPzlP12 = 2NzPZAl - PlZ) = 2NzPZlPll = 2KPlzPZlPll
Note that the numbering scheme used in this work is different from that of ref 2.
LITERATURE CITED (1) Randall, J. C. “Polymer Sequence Determination, Carbon-I3 NMR Method”; Academic Press: New York, 1977. (2) Carman, C. J.; Harrington, R. A,; Wlikes, C. E. Macromolecules 1977, 70, 536-544. (3) Ray, G. J.; Johnson, P. E.; Knox, J. R. Macromolecules 1977, 70, 773-776. (4) Randall, J. C. Macromolecules 1976, 7 7 , 33-36. (5) Paxson, J. R. R.; Randall, J. C. Anal. Chem. 1976, 50, 1777-1786. (6) Brame, E. G., Jr.; Hoimquist, H. E. Rubber Chem. Techno/. 1979, 52, 1-8. (7) Sanders, J. M.; Komoroski, R. A. Macromolecules 1977, 70, 1214-1 2 16. (8) Ham, G. E.. Ed. “Copolymerization”; Wiiey-Interscience: New York, 1964. (9) Bovey, F. A. ”High Resoiutlon NMR of Macromolecules”; Academic Press: New York, 1972. (10) Morgan, S. L.; Deming, S. N. AnalChem. 1974, 4 6 , 1170-1181. (11) Natta, G., et al. J . Polym. Sci. 1961, 57, 429-454. (12) Alien, P. E. M.; Patrick, C. R. “Kinetics and Mechanisms of Polymerization Reactions”; Wiiey: New York, 1974; p 84.
(A211
RECEIVED for review October 26, 1981. Resubmitted June 11,
The expressions correisponding to the numbers of methylene sequences of different lengths (Table 11, ref 2) are also reduced:
1982. Accepted June 17,1982. This is Contribution number 1743 from Hercules Research Center.
N3=
