The recursive approach

The method is demonstrated below for the case of a linear AB homopolymerization. This serves to establish the validity and usefulness of the approach, especially in calculating the average values of the distributions in linear copolymerizations. 

Simple AB step-growth homopolymerization AB homopolymerization is shown below  

Picking an A group at random, we define the in direction from the chosen A towards the B group from the same original monomer unit. Out is the opposite direction, from the chosen A towards the remainder of the chain on the A side of the monomer. Similar definitions apply to the in and out directions associated with B groups. 

First the expected weight of the polymer chain attached to a randomly chosen A group looking out is determined, This expected weight can 

The second term of equation (3.98) is clearly, zero, thus  

---

The recursive approach uses an elementary law of conditional expectation. Let A be an event and A its complement. Let Y be a random variable, E Y) its expectation (or average value) and E Y A) is conditional expectation, given that the event A has occurred. P A) is the probability that event A occurs. Then the law of total probability for expectation is [31] ... 

In addition to being a simpler method for obtaining the average properties such as My, and Mn compared to the Flory and similar approaches [31], the recursive approach also more easily allows an evaluation of the effect of unequal reactivity and unequal structural unit molecular weights on the average properties [27,28,33]. 

Problem 5.34 Calculate the gel-point conversions for the systems cited in Problem 5.27 using the recursive approach for comparison with the corresponding values calculated according to Flory-Stockmayer theory. 

Although the relations for probability of a finite chain and cross-link probability derived by the recursive approach remain unchanged on taking into account the formation of condensation by-products, the latter do affect the calculated value of sol fraction. Modify Eq. (5.231) for sol fraction w in homopolymerization of Af (f = 3), taking into account the formation of condensation by-product which is removed from the reaction mixture. 

Problem 5.32 Use the recursive approach to derive expressions for Mw and M,i for a polymerization system composed of equimolar amounts of A—A and B—B. To keep the system as simple as possible take the molecular weights of the two structural units as equal (denoted by Mq). 

In spite of its apparent simplicity, the direct method has several major shortcomings. First, the computational effort involved in solving the matrix eigenproblem in Eq. (53) scales as 0(N3). This feature, along with the storage requirements, which scale as N2, for H, C, and the required workspace, limit N in practice to no more than several thousand. The recursive approach described below is designed to overcome these unfavorable characteristics. 

Keeping the recursive approach of K-RAMP in mind, the final situation is a stmcrnred sequence of device-specific process segments including handling... 

The recursive approach is suitable for rigorous evaluation of the expression for resonance energies within the model of conjugated circuits for structurally related benzenoid hydrocarbons of increasing... 

---