Unit dyad

Another important application of NMR to polymer systems is the elucidation of the stereochemical configurations of Polymer chains. Poly (methyl methacrylate) was first studied by Bovey in 1960. It is now possible to analyse for the statistical frequency of occurrence of all possible combinations of up to four successive pairs of units (dyads) capable of occurring with either the same (meso) or opposite (racemic) configurations. 

The extra terms in the bottom row are a result of nonvanishing unit-vector derivatives. The tensor products of unit vectors (e.g., ezer) are called unit dyads. In matrix form, where the unit vectors (unit dyads) are implied but usually not shown, the velocity-gradient tensor is written as... 

After associating unit dyads (e.g., exey) with each element in the tensor, and assuming that n is a row vector, carry out the operation... 

Here, m and rai are the mass and position vector of beads, respectively. is the friction tensor, which is assumed to be isotropic for simplicity in our simulation, that is, = Fl, where I is the unit dyad and r = 0.5t 1 (t = cr(m/ )° 5j (Grest, 1996). Further, f aj is the Brownian random force, which obeys the Gaussian white noise, and is generated according to the fluctuation—dissipation theorem ... 

The stereochemistry resulting from ROMP is of interest because this can influence the properties of the material produced. For polymerization of nor-bomene and related bicyclic alkene, there are four possible stereochemical results (other than random stereochemistry). These are presented in Figure 11-1, which shows stereochemical relationships for two adjacent monomeric units (dyads). 

The unit dyads may be multiplied with each other and with the unit vectors ... 

Formation of OlqfiM AMdt.—By uniting dyad radical with... 

E/NB copolymers can be obtained as random amorphous materials with high TgS or as alternating, partially crystalline materials with high TmS. Amorphous copolymers have short blocks of norbornene units (dyads or triads), which account for their high TgS and excellent optical properties. All norbornene homo- and copolymers made by single site catalysts are characterized by narrow molecular weight distributions, which make technical processing easier. The first commercial norbornene copolymer products are already available. 

ROMP of 4-methylcyclopentene leads to the formation of a polymer that can adopt different stereochemical configurations owing to the relative positioning of the methyl substituents in neighboring repeating units (dyads). " The four possible dyad stmctures are displayed in Figure 20.2. 

Here we have left out the unit dyads that belong with each scalar component, so the = sign does not really signify equals but rather should be interpreted as scalar components are. Usually the unit dyads are understood. Matrix notation is convenient because the dot operations correspond to standard matrix multiplication. In matrix notation eq. 1.2.10 becomes a row matrix times a 3 x 3 matrix. 

Remember again that we have left out the unit dyads (xx, etc). In matrix notation the vector scalar product of eq. 1.2.4 becomes the multiplication of a row with a colutim matrix. 

This numbering of components leads to a convenient index notation. As indicated the nine scalar components of the stress tensor can be represented by 7,7, where i and j can take the values from 1 to 3 and the unit vectors Xi, X2. Xa become x,. Thus, we can write the stress tensor with its unit dyads as... 

When index notation is used, usually the summation signs are dropped and the unit vectors and unit dyads are understood. Here is how it works ... 

If an index is not repeated, multiplication of each component by a unit vector is implied (e.g., t,- or n, T,y). If two indices are not repeated, we will have two unit vectors or a unit dyad (e.g., T/y). If an index is repeated, summation before multiplication by a unit vector, if any, is implied. Since the indices all go from 1 to 3, the choice of which index letters is arbitrary, as indicated in eq. 1.2.23. 

The scattered wave reaching location r is constructed from the Green s tensor G(r- /), which is the solution to Equation [34] with the right-hand side, or source term, replaced by Id r-/) where I is the unit dyad. The delta function represents a point scatterer situated at /. For a field point far from the scatterer, i.e. r /, the form of the Green s tensor is... 

Sequence of 2 successive units (dyad) of cis-1,4-polypentadiene, presenting opposite chiralities and... 

Figure C2.1.1. (a) Constitutional isomerism of poly (propylene). The upper chain has a regular constitution. The lower one contains a constitutional defect, (b) Configurational isomerism of poly(propylene). Depending on tire relative configurations of tire asymmetric carbons of two successive monomer units, tire corresponding dyad is eitlier meso or racemo.

These observations suggest how the terminal mechanism can be proved to apply to a copolymerization reaction if experiments exist which permit the number of sequences of a particular length to be determined. If this is possible, we should count the number of Mi s (this is given by the copolymer composition) and the number of Mi Mi and Mi Mi Mi sequences. Specified sequences, of any definite composition, of two units are called dyads those of three units, triads those of four units, tetrads those of five units, pentads and so on. Next we examine the ratio NmjMi/Nmi nd NmjMiMi/NmiMi If these are the same, then the mechanism is shown to have terminal control if not, it may be penultimate control. To prove the penultimate model it would also be necessary to count the number of Mi tetrads. If the tetrad/triad ratio were the same as the triad/dyad ratio, the penultimate model is proved. 

Once these features have been identified, the spectra can be interpreted in terms of the numbers of dyads, triads, tetrads, and pentads ( ) of the butadiene units and compared with predicted sequences of various lengths. Further con sideration of this system is left for Problems 4-6 at the end of the chapter... 

The probabilities of the various dyad, triad, and other sequences that we have examined have all been described by a single probability parameter p. When we used the same kind of statistics for copolymers, we called the situation one of terminal control. We are considering similar statistics here, but the idea that the stereochemistry is controlled by the terminal unit is inappropriate. The active center of the chain end governs the chemistry of the addition, but not the stereochemistry. Neither the terminal unit nor any other repeat unit considered alone has any stereochemistry. Equations (7.62) and (7.63) merely state that an addition must be of one kind or another, but that the rates are not necessarily identical. 

The number of racemic dyads in a sequence is the same as the number of syndiotactic units n. The number of meso dyads in a sequence is the same as the number of iso units nj. These can also be verified from structure [XVIII] above. 

Propylene oxide and other epoxides polymerize by ring opening to form polyether stmctures. Either the methine, CH—O, or the methylene, CH2—O, bonds ate broken in this reaction. If the epoxide is unsymmetrical (as is PO) then three regioisomers are possible head-to-tad (H—T), head-to-head (H—H), and tad-to-tad (T—T) dyads, ie, two monomer units shown as a sequence. The anionic and... 

It should be stressed that this treatment of polymer stereochemistry only deals with relative configurations whether a substituent is "up or down" with respect to that on a neighboring unit. Therefore, the smallest structural unit which contains stereochemical information is the dyad. There are two types of dyad meso (m), where the two chiral centers have like configuration, and racemic /-), where the centers have opposite configuration (Figure 4.1). 

The configuration of a center in radical polymerization is established in the transition state for addition of the next monomer unit when it is converted to a tetrahedral sp1 center. If the stereochemistry of this center is established at random (Scheme 4.1 km = k,) then a pure atactic chain is formed and the probability of finding a meso dyad, P(m), is 0.5. 

The mechanism of B polymerization is summarized in Scheme 4,9. 1,2-, and cis- and trews-1,4-butadiene units may be discriminated by IR, Raman, or H or nC MMR speclroseopy.1 92 94 PB comprises predominantly 1,4-rra//.v-units. A typical composition formed by radical polymerization is 57.3 23.7 19.0 for trans-1,4- c7a -1,4- 1,2-. While the ratio of 1,2- to 1,4-units shows only a small temperature dependence, the effect on the cis-trans ratio appears substantial. Sato et al9J have determined dyad sequences by solution, 3C NMR and found that the distribution of isomeric structures and tacticity is adequately described by Bernoullian statistics. Kawahara et al.94 determined the microslructure (ratio // measurements directly on PB latexes and obtained similar data to that obtained by solution I3C NMR. They94 also characterized crosslinked PB. 

The probability that a chain with a terminal MAA dyad will add a MA unit is eq. 29 ... 

Statistical characteristics of the second type define the microstructure of copolymer chains. The best known characteristics in this category are the fractions P [/k) (probabilities) of sequences Uk involving k monomeric units. The simplest among them are the dyads U2, the complete set of which, for example, for a binary copolymer is composed of four pairs of monomeric units M2M, M2M2. The number of the types of k-ad in chains of m-component copolymers grows exponentially as mk so that with practical purposes in mind it is generally enough to restrict the consideration to sequences Uk] with moderate values of k. Their calculation turns out to be rather useful... 

As the result of theoretical consideration of polycondensation of an arbitrary mixture of such monomers it was proved [55,56] that the alternation of monomeric units along polymer molecules obey the Markovian statistics. If all initial monomers are symmetric, i.e. they resemble AaScrAa, units Sa(a=l,...,m) will correspond to the transient states of the Markov chain. The probability vap of transition from state Sa to is the ratio Q /v of two quantities Qa/9 and va which represent, respectively, the number of dyads (SaSp) and monads (Sa) per one monomeric unit. Clearly, Qa(S is merely a ratio of the concentration of chemical bonds of the u/i-ih type, formed as a result of the reaction between group Aa and Ap, to the overall concentration of monomeric units. The probability va0 of a transition from the transient state Sa to an absorbing state S0 equals l-pa where pa represents the conversion of groups Aa. 

A preferable system is poly(p-fluorostyrene) doped into poly(styrene). Since rotations about the 1,4 phenyl axis do not alter the position of the fluorine, the F spin may be regarded as being at the end of a long "bond" to the backbone carbon. In standard RIS theory, this polymer would be treated with dyad statistical weights to automatically take into account conformations of the vinyl monomer unit which are excluded on steric grounds. We have found it more convenient to retain the monad statistical weight structure employed for the poly(methylene) calculations. The calculations reproduce the experimental unperturbed dimensions quite well when a reasonable set of hard sphere exclusion distances is employed. 

Any or all of these forces may result in local stresses within the fluid. Stress can be thought of as a (local) concentration of force, or the force per unit area that bounds an infinitesimal volume of the fluid. Now both force and area are vectors, the direction of the area being defined by the normal vector that points outward relative to the volume bounded by the surface. Thus, each stress component has a magnitude and two directions associated with it, which are the characteristics of a second-order tensor or dyad. If the direction in which the local force acts is designated by subscript j (e.g., j = x, y, or z in Cartesian coordinates) and the orientation (normal) of the local area element upon which it acts is designated by subscript i, then the corresponding stress component (ay) is given by... 

Fig. 11 (a) A view of the unit cell of [Cp2Mo(dddt)][TCNQ]. (b) (c) Side view and top view of the dyad association with the short intermolecular S S distance (dotted line) involving the S atoms of the metallacycle... 

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