Configurational Statistics and the Propagation Mechanism in Chain-Growth Polymerization

Analyses of polymers to determine stereosequence distributions and understand the propagation mechanism can be carried out with NMR spectroscopy aided by statistical propagation models [222, 402, 403]. A detailed discussion of the subject is beyond this book. The following is a brief explanation of the concepts. The Bernoulli, Markov, Colman-Fox, models describe propagation reactions with chain end control over monomer placement. The Bernoulli model assumes that the last monomer unit in the propagating chain end determines the stereochemistry of the polymer. No consideration is given to the penultimate unit or other units further back. In such an event, two modes of propagation are possible, meso and racemic:

The statistical probabilities, Pm and Pr , which are called conditional probabilities or transitions ,of forming meso or racemic dyads are defined by the following equations:

where Rm and Rr are the rates of meso and racemic dyad placement. These dyads can be isotactic or syndiotactic to one another. Such dyads are more frequently called meso and racemic:

The probabilities represent the dyad tactic fractions m and r. The triad tacticity represent isotactic, syndiotactic, and heterotactic (or atactic) arrangements. They are designated as mm, rr, and mr respectively. One way to understand these definitions is by examining a representation of a section of a polymeric chain which has a total of 9 repeating units but only 8 dyads and 7 triads, as shown below:

In the above segment there are 2 racemic dyads and 6 meso dyads. In addition, there are also 4 isotactic, 1 syndiotactic, and 2 atactic dyads. The triads are defined as follows:

Note that the atactic triad can be produced in two ways, as mr and rm. The tetrad probabilities can be defined as:

In the Bernoulli model there is no influence of the stereochemistry of the growing polymeric chain on the incoming monomer. The first order Markov model, on the other hand, is based on the concept that the adding monomer is influenced by the stereochemistry for the growing end. This end may by m or r. This means that there are four probabilities for characterizing the addition process. The monomer can add in an m fashion to an m chain end, or m to an r chain end and so on. These can be designated as P_m/m, P_r/m, P_m/r, and P_r/r. These modes of propagation can be illustrated as follows:

There are four probabilities for propagation, Pm/m, Pm/r, Pr/m, and Pr/r. With conservation of the relationship we can write:

Pm/m +Pm/r = 1

Pr/m + Pr/r = 1

We can recast the dyads, triads, tetrads, and pentads in terms of Pm/r and Pr/m. The dyads functions are:

The second order Markov model requires the specification of eight conditional probabilities. This is due to the influence of the last three pseudo asymmetric centers of the growing chain. The details of the second Markov were described by Bovey [403]. For details the reader is advised to consult the reference. For convenience the eight conditional probabilities are designated by Greek letters [403]:

In the above designation, P(mm/m) is the probability of a monomer adding to the chain end in the m manner. P(mm/r) is the probability of it adding to the chain in the r manner, and so on. Because,

there are four independent probabilities. When α = Y and B = σ the model reduces to the first order Markov model. The Coleman-Fox model attempts to explain “block like” configurations that are exhibited in varying degrees by most propagating species that deviate from Bernoullian statistics. They proposed that “block” configurations are generated because the propagating chain ends might exist in two or more states. These states would correspond to chelation by the counterion and to interruption of chelation by solvation. Here too, for further details the reader is advised to go to the original literature [403]. The enantiomorphic site control is based on the probability of the monomer adding either to an R or to an S site of the catalyst. The propagation occurs through both faces of R and S monomers. The model is described in terms of a single parameter that is commonly designated as s [404]. It is the probability of an R monomer adding at the R site and S monomer adding at an S site. The dyads are described as,

NMR spectroscopy allows testing whether in a particular polymerization the propagation follows the Bernoulli, Markov, or enantiomorphic statistical form. Attempts are usually made to fit data for dyads, triads, tetrads, and higher sequence fractions to the equation for the different models. Spectral intensities can be associated with theoretical expression involving reaction probability parameters. Theoretical intensities are compared with the observed ones. This is optimized to obtain the best-fit values of reaction probability parameters and fully characterize the structure of the macromolecule. The fitting of data can be carried out with the aid of computers.

By determining which statistical model is followed in a polymerization, such as Bernoullian, or Markov, or others, it should be possible to understand better the mechanism of steric control. Thus the Bernoulli model describes those reactions in which the chain ends determine the steric arrangement. These are polymerizations that are carried out under conditions that yield mostly atactic polymers. The high isotactic sequences follow the enantiomorphic site model and the high syndiotactic ones usually follow the Markov models. The dyads are described as,

NMRspectroscopy allows testing whether in a particular polymerization the propagation follows the Bernoulli, Markov, or enantiomorphic statistical form. Attempts are usually made to fit data for dyads, triads, tetrads, and higher sequence fractions to the equation for the different models. Spectral intensities can be associated with theoretical expression involving reaction probability parameters. Theoretical intensities are compared with the observed ones. This is optimized to obtain the best-fit values of reaction probability parameters and fully characterize the structure of the macromolecule. The fitting of data can be carried out with the aid of computers. Cheng published a very useful program in Basic language that applies the model-fitting approach to various polymeric systems [405]. Many other programs in various computer languages are available today. By determining which statistical model is followed in a polymerization, such as Bernoullian, or Markov, or others, it should be possible to understand better the mechanism of steric control. Thus the Bernoulli model describes those reactions in which the chain ends determine the steric arrangement. These are polymerizations that are carried out under conditions that yield mostly atactic polymers. The high isotactic sequences follow the enantiomorphic site model and the high syndiotactic ones usually follow the Markov models.

اخر الاخبار

اخبار العتبة العباسية المقدسة

العتبة العباسية المقدسة تختتم فعاليات يوم الخدمة بذكرى ولادة أبي الفضل العباس (عليه السلام)

العتبة العباسية المقدسة تكرم منتسبيها المتميزين خلال عام 2025

العتبة العباسية المقدسة تكرم أقسامها المتميزة في تقديم أفضل تقرير سنوي ضمن فعاليات يوم الخدمة

ضمن فعاليات يوم الخدمة.. العتبة العباسية المقدسة تكرِّم عوائل المنتسبين المتوفين خلال عام 2025

المزيد

الآخبار الصحية

دراسة: محاولة متعمدة لنشر الإنفلونزا تنتهي بـ"مفاجأة"

من خبير في هارفارد.. 6 نصائح يومية "فعالة" لإبطاء الشيخوخة

7 مشروبات طبيعية لتقوية الذاكرة وزيادة التركيز

عرض شائع على الأصابع.. قد يكشف سرطان الرئة مبكرا

المزيد

الآخبار التكنلوجية

وسائل التواصل بين الإفراط والامتناع.. ما هو الحد الآمن للمراهقين؟

أعمق خندق في الكوكب الارض !

روساتوم: روسيا تقترب من تطوير ساعات ذرية فائقة الدقة

مفاجأة علمية عن الإنجاب.. كم طفلا يطيل عمر المرأة؟

المزيد

مواضيع ذات صلة

Steric Control in Polymerizations of Oxiranes

Polymerization by Coordination Mechanism

Cationic Polymerization

Kinetics of Ring-Opening Polymerization

Chemistry of Ring-Opening Polymerizations

Group Transfer Polymerization

Copolymerizations by Ionic Mechanism

Polymerization of Ketones and Isocyanates

Polymerizations of Di Aldehydes

Polymerization of Unsaturated Aldehydes

Anionic Polymerization of Aldehydes

Polymerization of Aldehydes