Chain composition instantaneous copolymer

In order to obtain the expression for the components of the vector of instantaneous copolymer composition it is necessary, according to general algorithm, to firstly determine the stationary vector ji of the extended Markov chain with the matrix of transitions (13) which describes the stochastic process of conventional movement along macromolecules with labeled units and then to erase the labels. In this particular case such a procedure reduces to the summation ... 

Upon expressing from the equilibrium condition the complex concentration M12 through the concentrations of monomers, and substituting the expression found into relationship (21) we obtain, invoking the formalism of the Markov chains, final formulas enabling us to calculate instantaneous statistical characteristics of the ensemble of macromolecules with colored units. A subsequent color erasing procedure is carried out in the manner described above. For example, when calculating instantaneous copolymer composition, this procedure corresponds to the summation of the appropriate components of the stationary vector jt of the extended Markov chain ... 

An alternative rationale for the unusual RLi (hydrocarbon) copolymerization of butadiene and styrene has been presented by O Driscoll and Kuntz (71). Rather than invoking selective solvation, these workers stated that classical copolymerization kinetics is sufficient to explain this copolymerization. They adapted the copolymer-composition equation, originally derived from steady-state assumptions for free-radical copolymerizations, to the anionic copolymerization of butadiene and styrene. Equation (20) describes the relationship between the instantaneous copolymer composition c/[M,]/rf[M2] with the concentrations of the two monomers in the feed, M, and M2, and the reactivity ratios, rt, r2, of the monomers. The rx and r2 values are measures of the preference of the growing chain ends for like or unlike monomers. 

It should be noted that the copolymer equation (7.11) describes the instantaneous copolymer composition on a macroscopic scale, that is, composition in terms of the overall mole ratio or mole fraction of monomer units in the copolymer sample produced, but it does not reveal its microstructure, that is, the manner in which the monomer units are distributed along the copolymer chain. Thus for two monomers Mi and M2, the ratio Fi/(I — F ) gives the overall mole ratio of Mi and M2 units in the copolymer but no information about the average lengths (i.e., number of monomer units) of Mi and M2 sequences, as illustrated (Allcock and Lampe, 1990) for a typical copolymer by... 

Various factors are important in determining the composition and molecular weight distribution of multicomponent copolymers e.g. monomer reactivity ratios, reaction conditions). Stockmaycr was one of the first to report on the problem and presented formulae for calculating the instantaneous copolymer composition as a function of chain length. Othershave examined the variation in copolymer composition with chain length by computer simulation. One method of ensuring a functionality of at least one is to use a functional initiator or transfer agent. 

In the FRC of vinyl and multivinyl monomers, a drift in the instantaneous copolymer composition throughout the reaction will be undergone due to the different reactivities of the vinyl groups. This compositional drift is caused by the fact that the more reactive monomer will be consumed faster than the less reactive ones. In the simplest instance, assuming equal reactivity of the vinyl groups in mono-and divinyl monomers present in the reaction system, the reactivity of the crosslinker would be twice that of the monovinyl monomer, and therefore, the polymer chain... 

Tip 13 (related to Tip 12) Copolymerization, copolymer composition, composition drift, azeotropy, semibatch reactor, and copolymer composition control. Most batch copolymerizations exhibit considerable drift in monomer composition because of different reactivities (reactivity ratios) of the two monomers (same ideas apply to ter-polymerizations and multicomponent cases). This leads to copolymers with broad chemical composition distribution. The magnirnde of the composition drift can be appreciated by the vertical distance between two items on the plot of the instantaneous copolymer composition (ICC) or Mayo-Lewis (model) equation item 1, the ICC curve (ICC or mole fraction of Mj incorporated in the copolymer chains, F, vs mole fraction of unreacted Mi,/j) and item 2, the 45° line in the plot of versus/j. 

Fig. 2.32 Schematic representation of monomer addition method (upper) and relationships between instantaneous copolymer composition (Cinst. move) and chain length ratio (a) of gradient copolymers obtained by IBEA/Eti sAlCh s in the presence of ethyl acetate in toluene at 0 °C [EOVE]o=0.8 M, [MOVE]totai=0.8 M, [IBEA]o=4.0 mM, [Et,.5AlCl,.5]o=20 mM, [ethyl acetate]=1.0 M...

The terms A and B represent the number of moles of the two comonomers in the feed at any given instant, and and are the comonomer reactivity ratios defined earlier. The differential dA/dB is sometimes termed the instantaneous copolymer composition since it represents the composition of polymer chains forming at any instant. A variety of methods have been developed to determine the reactivity ratios. Most of these methods are based on the assumption that for conversions up to approximately 5%, the ratio of the two monomers in the feed does not change appreciably. Thus, equation (2.24) can be rewritten as... 

If r = r, the structure is strictly alternating since each monomer wants only to add the other. If r r =l, the structure is random, since P and Q chains have an equal probability of adding either monomer. If and are both greater than unity, a block copolymer results, since cross polymerization is unlikely. Equation 16.151 describes the instantaneous copolymer composition. If the monomers are not consumed at the same rate, there will be significant compositional drift over the course of the polymerization. 

The quantity Fi, the instantaneous copolymer composition, is analogous to x , the instantaneous number-average chain length in free-radical addition polymerization. Like x , it depends on the conditions in the reactor at a particular instant. It, too, is really an average, since not all the copolymer formed at a particular instant has exactly the same composition. However, the instantaneous distribution of compositions is normally much narrower than the instantaneous distribution of chain lengths, and because the fact that it is an average is not normally of great practical importance and cannot be controlled anyhow, the overbar is left off of Fj. 

In addition to the instantaneous copolymer composition another quantity of interest is (F/), the cumulative composition of the copolymer that has been formed up to a particular conversion. Fj) is exactly analogous to (x ), the cumulative number-average chain length from Chapter 9. For a batch reactor, it is obtained through a material balance ... 

The instantaneous composition of a copolymer X formed at a monomer mixture composition x coincides, provided the ideal model is applicable, with stationary vector ji of matrix Q with the elements (8). The mathematical apparatus of the theory of Markov chains permits immediately one to wright out of the expression for the probability of any sequence P Uk in macromolecules formed at given x. This provides an exhaustive solution to the problem of sequence distribution for copolymers synthesized at initial conversions p l when the monomer mixture composition x has had no time to deviate noticeably from its initial value x°. As for the high-conversion copolymerization products they evidently represent a mixture of Markovian copolymers prepared at different times, i.e. under different concentrations of monomers in the reaction system. Consequently, in order to calculate the probability of a certain sequence Uk, it is necessary to average its instantaneous value P Uk over all conversions p preceding the conversion p up to which the synthesis was conducted. 

In addition to the above mentioned dynamic problems of copolymerization theory this review naturally dwells on more traditional statistical problems of calculation of instantaneous composition, parameters of copolymer molecular structure and composition distribution. The manner of presentation of the material based on the formalism of Markov s chains theory allows one to calculate in the uniform way all the above mentioned copolymer characteristics for the different kinetic models by means of elementary arithmetical operations. In Sect. 3 which is devoted to these problems, one can also find a number of original results concerning the statistical description of the copolymers produced through the complex radical mechanism. 

The parameters a = l/rij5 the number of which equals m(m — IX are reciprocal reactivity ratios (2.8) of binary copolymers. Markov chain theory allows one, without any trouble, to calculate at any m, all the necessary statistical characteristics of the copolymers, which are formed at given composition x of the monomer feed mixture. For instance, the instantaneous composition of the multicomponent copolymer is still determined by means of formulae (3.7) and (3.8), the sums which now contain m items. In the general case the problems of the calculation of the instantaneous values of sequence distribution and composition distribution of the Markov multicomponent copolymers were also solved [53, 6]. The availability of the simple algebraic expressions puts in question the expediency of the application of the Monte-Carlo method, which was used in the case of terpolymerization [85,99-103], for the calculations of the above statistical characteristics. Actually, the probability of any sequence MjMjWk. .. Mrl 4s of consecutive monomer units, selected randomly from a polymer chain is calculated by means of the elementary formula ... 

The composition of instantaneously generated chains is not sufficiently described by the copolymerization equation which only informs us about the mean populations of monomers in a statistical copolymer. This problem was studied by Goldfinger and Kane [190] based on the following considerations. 

Once the distribution of propagating chain types is known, the instantaneous composition of the copolymer is given by... 

The copolymer equation provides a means of calculating the amount of each monomer incorporated in the chain from a givrai reaction mixture or feed when the reactivity ratios are known. It shows that if monomer Mj is more reactive than Mj, then Mi will raiter the copolymer more rapidly consequraitly, the feed becomes progressively poorCT in M, and composition drift occurs. The equation is that an instantaneous expression, which relates only to the feed composition at any given time. 

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