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If free interchange occurs, the molecular weight distribution will be the Flory distribution described by Eqs. 2-86 and 2-89. Free interchange coresponds to all interunit linkages in all polymer molecules having equal probabilities of interchange. This is analogous to the concept of functional group reactivity independent of molecular size as applied to the interchange reaction. It is apparent that the presence of interchange during a polymerization will not affect the size distribution from that expected for the random polymerization. The Flory or most probable distribution is also that expected for the random scission of the interunit linkages of polymer chains, for example, in the hydrolysis of cellulose. 2-7d Alternate Approaches for Molecular-Weight Distribution
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A number of treatments other than that by Flory have been given for the molecular weight distributions in linear step polymerizations [Burchard, 1979; Durand and Bruneau, 1979a,b]. However, a knowledge of the average properties (M n , M w , and PDI) is often suf cient for many practical purposes. Macosko and Miller [1976] developed a useful statistical approach for obtaining the average properties without the need to calculate the molecular weight distributions. This approach will be described for a polymerization system composed of A A, B and B0 0 where A groups can react with B groups and with B0 groups [Ozizmir and B, B Odian, 1980]. To keep the system as simple as possible, the molecular weights of the three structural units are taken as equal (and denoted by Mo ) and the initial system contains equimolar amounts of B and B0 0 monomers where the total moles of A is equal to the B B A sum of B and B0 0 . B B
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The expected masses of polymers which contain a randomly selected A, B, or B0 group in the system are denoted by wA, wB , and wB0 . In order to obtain these quantities it is convenient to introduce the quantities wiA , wiB , wiB0 , and wo , wo , wo 0 . Here wiA represents the expected A B B mass attached to a randomly selected A group in the system looking inward toward the other A group of the A structural unit of which it is a part and wo represents the expected mass A A attached looking outward from the randomly selected A group as shown in XII. Similarly, wiB , wo , wiB0 , and wo 0 represent the expected inward and outward masses attached to B and B0 B B groups.
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The following relationships hold for this system:
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wA wiA wo A wB wiB wo B wo A wB0 wiB0 wo 0 B Mo & ' pB wiB pB0 wiB0 pB wiB pB0 wiB0 wo pA A 2 pB pB0 pB pB0 wiA wiB Mo wo B wo B wo 0 B pB wiA pB0 wiA wiB0 Mo wo 0 B pB pB0 pA 2 2-99 2-100 2-101 2-102 2-103 2-104 2-105 2-106 2-107 2-108
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Most of these relationships are simple material balance statements. Equations 2-99 through 2-101 state that the total mass of polymer attached to an A, B, or B0 group is the sum of the inward and outward masses attached to that group. Equations 2-102, 2-104, and 2-106 state that the difference between the inward and outward masses attached to a group is Mo . Equation 2-107 indicates that the polymer mass attached to a B0 group looking outward from the B0 group equals the probability of that group having reacted with an A group multiplied by the mass of polymer attached to the A group looking inward from that A group. The corresponding descriptions for wo and wo are Eqs. 2-103 and 2-105. A B Equations 2-102 through 2-108 can be solved and their results combined with Eqs. 2-99 through 2-101 to yield
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