Statistical Models and NMR Analysis of Polymer ... - ACS Publications

Chapter 22

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Statistical Models and NMR Analysis of Polymer Microstructure H. N. Cheng*,1 and Massoud J. Miri2 1Southern

Regional Research Center, USDA Agricultural Research Service, 1100 Robert E. Lee Blvd., New Orleans, LA 70124, U.S.A. 2Department of Chemistry, Rochester Institute of Technology, Rochester, NY 14623, U.S.A. *E-mail: [email protected]

Statistical models can be used in conjunction with NMR spectroscopy to study polymer microstructure and polymerization mechanisms. Thus, Bernoullian, Markovian, and enantiomorphic-site models are well known. Many additional models have been formulated over the years for additional situations. Typically spectral interpretation and data treatment can be done through either “analytical” or “simulation” approaches. These can be combined into “integrated” approaches for specific situations. An alternative (and more general) approach considers the kinetics of the polymerization process and carries out predictions of polymer microstructures and NMR spectra. These various methodologies are briefly reviewed here. Also reviewed is a recent effort in the simulation category involving a user-friendly Excel program (“Polytact”) that can simulate the tacticities of a large number of statistical models, particularly those that pertain to polyolefins made with single-site catalysts.

Introduction High-resolution solution NMR is now a routine technique for polymer analysis (1–5). It is fairly straightforward to dissolve a polymer in a solvent and obtain a 1H or 13C spectrum. Analyses of a polymer NMR spectrum can be carried out at different levels (Figure 1). At the simplest level, one can use the gross © 2011 American Chemical Society In NMR Spectroscopy of Polymers: Innovative Strategies for Complex Macromolecules; Cheng, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.


NMR spectral features to search the spectral libraries for routine identification or pattern recognition. At a more involved level, one can interpret the spectrum (in whole or in part), using a variety of techniques, and assigning the observed peaks to specific structures. From the assignments, it may then be possible to derive detailed information on polymer microstructure, such as homopolymer tacticity, copolymer composition and sequence, branching, defect structures, and chain ends. With additional work, it may be possible to infer information on monomer reactivity and polymerization mechanisms. In general, the amount and the quality of information that can be extracted from the NMR data depend on the methodologies used and the efforts expended.

Figure 1. Information content available in the NMR data together with methods and techniques used to extract the information. Statistical models (also known as reaction probability models) have been used extensively in polymer studies (3–7). They provide fundamental understanding of polymerization processes and serve as a theoretical framework whereby NMR, molecular weight, fractionation, and polymerization kinetics data can be analyzed in a rational manner. In many cases the models can assist in NMR spectral assignments and help interpret the spectral intensities. Sometimes they provide information on the mechanisms of initiation and propagation. Suitably applied, they may permit maximum amount of information to be obtained from each spectrum. In this work a brief review is made of common statistical models being used and selected methodologies being applied in the NMR analysis of polymers, with a particular emphasis placed on the work done in the authors’ laboratories. 372 In NMR Spectroscopy of Polymers: Innovative Strategies for Complex Macromolecules; Cheng, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.

Statistical Models Over the years, many statistical models have been used to depict polymerization. New models have been devised for specific situations. The interrelationships of some of these models are shown in Figure 2.


One-State Models The simple one-state models, such as Bernoullian (B), first-order Markovian (M1), second-order Markovian (M2), and enantiomorphic-site models (E) are well-known and have been previously described (6, 8, 9). It is important to note that with B, M1 and M2 models, the polymerization are chain-end controlled, whereas with E model, the polymerization is catalytic-site controlled (8, 10–12). Models that provide dual control (both chain-end and catalytic-site) are also known and used (7, 11–13). Other useful models include Markovian with reversible propagation (14–16), complex participation (17, 18), and bootstrap models (19). Two-State Models These models are used when two separate states are involved in polymerization. The two states can be consecutive or concurrent (20). The consecutive two-state model can be applied to a polymerization where the catalytic site switches back and forth between two states as it polymerizes; the result may be a block copolymer consisting of blocks that conform to different statistics. The consecutive two-state model can also be used for other block copolymers or for a polymerization where the reaction condition changes during polymerization (e.g., changes in monomer feed, pressure, or temperature). The concurrent two-state model is basically a mixture model. It can be used for a polymerization where two separate active sites are present, each site making its own polymer according to its propagation statistics; in effect, a blend of two polymers is formed. The concurrent two-state model can also be used if a catalytic site switches back and forth between two states but the rate of switching is low relative to chain propagation and chain termination, and separate chains are formed that can be attributed to each of the two states (20). Each of the two states can be Bernoullian (B) or enantiomorphic-site (E), giving rise to two-state B/B, B/E, or E/E models. The first two-state B/B model has been formulated by Coleman and Fox (21) and included both consecutive and concurrent features. The concurrent B/B model has been used to analyze NMR data of olefin copolymers (22–25). Similarly, the concurrent B/E model has been used to analyze the tacticity of polypropylene (26, 27). Yet another two-state model has been reported by Ewen, et al (13, 28) and used to analyze the NMR data of syndiotactic polypropylene made with metallocene catalysts; the model includes dual catalytic-site and chain-end control. In addition, a general treatment of the two-state consecutive models has been formulated (20); this is a structural approach that is different from the Coleman-Fox treatment and is applicable to two-state B/B, B/E, and E/E models These models have been used for the analysis 373 In NMR Spectroscopy of Polymers: Innovative Strategies for Complex Macromolecules; Cheng, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.


of NMR tacticity data of elastomeric polypropylene (29, 30). Variant models have also been proposed (31).

Figure 2. A general scheme for selected statistical models commonly used for NMR characterization. Multistate Models Many polymers are found to have multiple polymeric components due to polymerization made at different catalytic sites, at different phases, or (in programmed monomer addition) at different times. The multistate models are useful for the analysis of these polymers. A general methodology involving multistate models has been reported (25, 32). The analysis is facilitated when NMR data of polymer fractions (from fractionation or chromatographic 374 In NMR Spectroscopy of Polymers: Innovative Strategies for Complex Macromolecules; Cheng, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.

separation) are available. The methodology has been applied to a large number of synthetic polymers, particularly those made from Ziegler-Natta catalysts which may contain more than two active sites, e.g., for the analysis of copolymer sequences (25, 32–39) and homopolymer tacticity (25, 33, 38). In addition, it has been used on natural polymers, e.g., pectin (40, 41) and alginate (41–43).


Perturbed Models Many industrial (and some lab-made) polymers exhibit varying degrees of compositional heterogeneity. Compositional heterogeneity may also influence polymer microstructure, such as tacticity, composition, and sequence distribution. Perturbed Markovian (44, 45) and enantiomorphic-site (46) models have been designed as convenient tools for the interpretation of compositional and tacticity data. In general, compositional heterogeneity can arise from many sources (47–50). Depending on the source(s) of heterogeneity, the chemical composition distribution (CCD) curve may take on a symmetric or non-symmetric appearance (Table 1) (48). For the analysis of CCD or NMR data, multistate heterogeneity can usually be treated with multistate models. Perturbed models can be used for the treatment of the other three types of heterogeneities. Computations can be done via either analytical (44–48) or simulation approaches (47–50).

Table 1. Different types of compositional heterogeneity. (adapted from (45)) type

possible source(s) of heterogeneity

CCD curve

statistical

statistical fluctuations in copolymer composition

symmetric

conversion

different comonomer reactivities

skewed or tent-shaped

multistate

1. different polymers made at different initiator sites or phases 2. programmed comonomer feeds 3. polymer blending 4. polymers from biological sources

variable (skewed to multimodal)

process

fluctuations in polymerization process conditions

symmetric or slightly skewed

Kinetic Models Instead of reaction probabilities, an alternative treatment (51) uses the actual reactions taking place during polymerization. The resulting kinetic scheme can then be used to generate all requisite information corresponding to the observed NMR data, e.g., composition, sequence, branching, defect structure, and chain ends. These kinetic models are especially useful for the treatment of complex NMR data. For example, the kinetic model of a simple binary copolymerization 375 In NMR Spectroscopy of Polymers: Innovative Strategies for Complex Macromolecules; Cheng, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.

has been reported (51). A detailed study of short- and long-chain branching in low-density polyethylene using kinetic modeling has also appeared (52).


Other Models Most of the preceding discussion deals with homopolymer tacticity or sequence distribution of binary copolymers. The first-order Markovian model for ternary copolymerization (terpolymerization) is also known (53); an excellent example of its utility has been shown by Carman et al (54) for the analysis of NMR spectra of ethylene-propylene rubbers. The first-order Markovian model for quaternary copolymerization (tetrapolymerization) has also been formulated (55) with mathematical expressions for all relevant sequences. It has been used successfully for styrene-butadiene copolymers (56), where 1,2-, 1,4-trans, and 1,4-cis additions of butadiene to the propagating chain are regarded as three separate comonomers. The same model has been used for stereoirregular and regioirregular polypropylene (57), where primary and secondary insertions of propylene in two configurations are considered the four comonomers. The mathematical expressions for a pentapolymerization (involving 5 comonomers) would be very complex (55). A simulation approach for a pentapolymerization has been published for ethylene-propylene rubber (48, 58); in that case, the five “comonomers” are ethylene, propylene in the primary insertion (up and down configuration), and propylene in the secondary insertion (up and down configuration). The versatility of metallocene catalysts gives rise to new statistical models. Some of these have been recently reviewed (59). A new model (“E-B gen”) will be described in the last section of this article.

Application Methodologies Analytical Approaches Whereas the models are useful for an understanding of polymerization mechanism and statistics, an appropriate methodology is needed to apply these models to the treatment of NMR data. Many methods and techniques have been developed for this purpose (1–9). Most of the methods can be grouped under the category of “analytical” approaches (5, 6, 60):

Thus, the NMR data of a polymer are first acquired and the spectral features assigned to pertinent polymer microstructures. Spectral intensities are then obtained, and calculations made to give copolymer composition and sequence distribution. A more refined treatment takes all assignable intensities in a spectrum and fits them to an appropriate statistical model, from which reaction probabilities can be derived (54, 60–62). This approach has also been extensively 376 In NMR Spectroscopy of Polymers: Innovative Strategies for Complex Macromolecules; Cheng, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.

used for many statistical models (5–7, 11, 12, 25, 32, 48), often with some variations in methodologies (63, 64).


Simulation Approaches The second category of approaches can be called “simulation” (or "synthetic") approaches. The starting point here is an appropriate statistical model, from which composition and sequence distribution for a set of model parameters can be calculated through theory or computer simulation. The observed and the calculated sequence distribution can be compared and adjustments in the model parameters made until a satisfactory fit is obtained. A number of people have utilized this approach over the years (59, 60, 62, 65–68). The advantage of this approach is that it is often easier to calculate the sequence distribution or to simulate the polymerization than to engage in a direct data fitting procedure for many polymeric systems.

In one variation of this approach, one can take the reaction probabilities of a model and use Monte Carlo method to generate an ensemble of polymer chains, from which composition and sequence (and their heterogeneities) can be gathered (49, 50, 66–68). When good correlations between microstructures and chemical shifts are known, one can produce a predicted NMR spectrum. As an example, a general computer program has been written that can be used to calculate the copolymer sequences and to simulate the 13C NMR spectra of vinyl and vinylidene copolymers and terpolymers (67, 68). For polymers exhibiting stereoisomerism and regioisomerism, detailed 13C shift rules incorporating tacticity, head-to-tail, head-to-head, and tail-to-tail structures are needed to produce simulated shifts or spectra. For example, such 13C rules for polypropylene and ethylene/propylene copolymer have been obtained (69). For these two polymer systems, it is possible to simulate a range of polymer microstructures, including tacticity, regiosequences, and chain ends (58, 70).

Integrated Approaches A third category of methods incorporates elements of analytical and simulation approaches. The use of both approaches in suitable cases provides more detailed understanding of polymer microstructure and NMR shift/structure correlations. For example, for copolymers obtained at high conversion, the direct use of a statistical model can lead to wrong results; an integrated approach permits more correct results to be obtained (71). Other examples of integrated approaches have also been published (48, 72).

377 In NMR Spectroscopy of Polymers: Innovative Strategies for Complex Macromolecules; Cheng, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.

Kinetic Simulation Approaches


These approaches can be used with kinetic models (above). A schematic of this type of approaches is shown below.

If we can formulate a realistic kinetic scheme of a polymerization under consideration, the scheme can then be used for simulation of polymer chain growth and NMR spectral prediction. Of course, rate constants need to be known either a priori or through model-fitting. Detailed descriptions of this approach have been reported earlier (51, 52).

Program Polytact Recently a convenient and pragmatic program, called Polytact, was reported by Miri, et al (59). It was written on the Excel software with the objective to calculate all relevant characteristics of the polymer tacticity and to present them in graphical form in a user-friendly manner using macros. Six types of tacticities can be produced with the program: isotactic, syndiotactic, atactic, hemiisotactic, stereoblock, and heterotactic (the latter meaning that ideally two adjacent substituents point in one direction followed by two substituents pointing in the other direction). Seven one-state statistical models were built into the program: B, M1, M2, E-B iso, E-B syn, E-B gen, and the E-M1 combination. The “E-B gen” model is introduced for the first time in this program (59). It enables a user to model and simulate four different types of enantiomorphically controlled tacticities with one single model: isotactic, syndiotactic, atactic and hemiisotactic polymer. It is based on a simplified scheme, in which it is only relevant what the probabilities are for each of the two types of enantiofaces of the monomer to coordinate with each of the catalyst’s two lateral, enantiotopic sites. Specific types of catalyst symmetry groups need not be explicitly considered in this model. Also, the equations for the pentads for the E-M1 model have been published for the first time in the article cited (59). The “E-B iso” and “E-B syn” models are to be used for isotactic and syndiotactic polyolefins, respectively. The model simulations have been applied to twenty polymers with different tacticities found in the literature (59). A screen shot of the summary sheet of the program is shown in Figure 3. The program’s user can select each of these models on an input window, and enter the corresponding numerical probabilities, e.g., Pmm and Prr in case of the M1 model. The program then produces the following output: (1) the numerical dyad, triad, tetrad and pentad values for the specific input probability(ies), (2) bar diagrams for the triad and pentad values, (3) 2D or 3D graphs for the triad and pentad distribution over the entire probability range(s), depending on the model, (4) the average sequence lengths for the meso and racemic dyads, (5) a sequence length distribution graph, and (6) a graph for the simulation of 50 units in the polymer. 378 In NMR Spectroscopy of Polymers: Innovative Strategies for Complex Macromolecules; Cheng, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.


In addition, a tab can be selected on the input window to use short-cuts for the models (with stored ideal probabilities) to produce the six types of tacticities mentioned above. These are intended to help the beginning user to get familiar with the concepts more easily. Furthermore, a third tab can be selected to choose from different metallocene symmetries that cause certain tacticities; for example, by clicking the icon with the figure representing a metallocene with a C1 chirality, the program will produce data for a hemiisotactic polymer as output.

Figure 3. Screenview of the summary sheet of program Polytact with the movable input window on the right (applying the “E-B gen” model). More detailed views of the 2D and 3D plots of the triad and pentad distributions are available on a separate spreadsheet in Polytact. These can be used to determine quickly which types of triads or pentads cannot be produced with certain models or what their maximum or minimum value is. For example, the [mmmm] pentad reaches a maximum 6.25% for a syndiotactic polymer (with a = 0.5), which is also the minimum value of [mmmm] for an isotactic polymer (with b = 0.5) using the E-B syn and E-B iso models, respectively. However, the [mmmm] pentad can become zero using the E-B gen model (when either both of its probabilities c and d approach zero or both probabilities approach 1). A major use of the program Polytact is to compare experimental n-ad sequences, which can be obtained from NMR spectroscopy, with those calculated by the program. To test if the experimental data are in agreement with one of the seven models, one needs to take a set of n-ads and decide on the particular probabilities associated with a model. These probabilities then need to be entered into the program, and the resulting n-ad sequences can be compared to the corresponding experimental sequences. If the differences between the experimental n-ads and those calculated by the program lie within a reasonable 379 In NMR Spectroscopy of Polymers: Innovative Strategies for Complex Macromolecules; Cheng, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2011.


margin of error, the actual polymerization is considered compatible with the particular model within the limit of experimental error. The sequence distributions obtained can also be used to make predictions about the degree of order and crystallinity of a polymer or its other physical properties. If a polymer with a certain tacticity is to be grafted or modified during post-polymerization, predictions can be made about the polymer’s reactivity as well. The limitation of the program is that it currently includes only the seven given statistical models. Thus, it does not include tactictities which are based on two-state or multi-state models.. It may also be noted that the program represents a simulation approach; thus, optimization of reaction parameters can be done manually, but there is currently no provision for automated iteration. The program will be made available by the authors upon request (e-mail: [email protected], or [email protected]).

Acknowledgments Thanks are due to the many collaborators over the years. These include (in alphabetical order) G. N. Babu, Mark A. Bennett, J. C. W. Chien, John A. Ewen, Leo J. Kasehagen, R. A. Newmark, Michael T. Roland, and Stanley B. Tam. G. H. Lee, Thomas G. Neiss, and David A. Smith contributed to the applications of some of the models to several polymer systems. Moreover, B. P. Pritchard at RIT contributed to the writing of Program Polytact. Mention of trade names or commercial products in this publication is solely for the purpose of providing specific information and does not imply recommendation or endorsement by the U.S. Department of Agriculture. USDA is an equal opportunity provider and employer.

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