Chemical Egieerig 60/60 Polymer Sciece ad Egieerig Lecture 7 - Statistics of Chai Copolymerizatio Jauary 3, 00
Objectives! To determie the compositioal relatioships betwee lower ad higher order sequeces of uits i a two- compoet copolymer.! To evaluate the coditioal probability i terms of the simple probabilities of occurrece of sequeces of differet orders.! To demostrate how probability theory may be used to develop iformatio about local structure i copolymers,, e.g, average sequece legth.
Outlie! Relatioships betwee simple ad coditioal probability! Evaluatio of microstructure " Markovia statistics Beroullia or radom statistics Termial model Peultimate model " Departure from radom statistics " Number fractio of sequeces of uits " Number average legth of or B rus
Defiitios of Probability Simple probability: P {} N N The simple probability of evet occurrig is just the relative fractio of evets compared to the total umber of evets. Coditioal probability: PB { / } PB { / } N N B PB { } P {} The coditioal probability of evet B occurrig give that has preceded it is just the ratio of the umber of compoud B evets to the umber of simple evets.
Relatioships Betwee Lower ad Higher Order Sequeces: Moad/Diad P{} + P{} B Moad mole fractios sum to. Cosider successor/predecessor relatios: P{} P{ } + P{ B} P{ } + P{ B} P{} B P{ BB} + P{ B} P{ BB} + P{ B} P{ B} P{ B] This is the Priciple of Microscopic Reversibility. Cosider diad mole fractios: P { } + P { B} + P { BB}
Relatioship Betwee Lower ad Higher Order Sequeces: Diad/Triad P { } P{ } + P{ B} P{ } + P{ B} 3 3 3 3 P { B} P{ B} + P{ BB} P{ B} + P{ BB} 3 3 3 3 P { B} P{ B} + P{ BB} P{ B} + P{ BB} 3 3 3 3 P { BB} P{ BB} + P{ BBB} P{ BB} + P{ BBB} 3 3 3 3 P{ B} P{ B} 3 3 P{ BB} P{ BB} 3 3 microscopic reversibility (equal to mirror image) Cosider triad mole fractios: P{ } + P{ B} + P{ B} + P{ BB} + P{ BB} + P{ BBB} 3 3 3 3 3 3
Coditioal ad Simple Probabilities Coditioal probability: Simple probability: PX { / XX X} L + P{ } P{ } P{ / } P{ B} P{ B} P{ / B} P{ } P{ } P{ / } P{ / } 3 P{ B} P{ B} P{ / B} P{ / B} 3
Coditioal Probability i Terms of Ratios of Simple Probabilities P { / } P { / B} PB { / } PB { / B} P { } P{} P { B} P{} B P { B} P{} P { BB} P{} B Moad fractios may be determied from elemetal aalysis. Diad, triad, tetrad, ad petad fractios may be determied from NMR.
Coditioal Probability i Terms of Ratios of Simple Probabilities P { / } P { / B} P { / B} P { / BB} { } P{ } { B} P{ B} { B} P{ B} { BB} P{ BB} PB { / } PB { / B} PB { / B} PB { / BB} { B} P{ } { BB} P{ B} { BB} P{ B} { BBB} P{ BB} Use NMR to determie all triad ad diad fractios.
Probability Sums P { / } + PB { / } P { / B} + BBB { / } P{ / } + P{ B / } P{ / B} + P{ B / B} P{ / B} + P{ B / B} P{ / BB} + P{ B / BB}
Coditioal Probability of Differet Orders If the probability of fidig a particular sequece of uits depeded upo every oe of the precedig (-) uits, the umber fractio of the -uit sequece would be: P{ } P{ } P{ / } P{ / } LP{ / } If the local chemical iteractios oly ifluece reactivity over a sequece legth of k uits, the umber fractio of the -uit sequece would be: P{ } P{ } P{ / } k k k
Coditioal Probability of Differet Orders The umber fractio of a sequece of uits for a kth order Markovia process is give by: P{ } P{ } P{ / } k k k Zero-order Markovia statistics: k 0 P{ } ( P{ }) First-order Markovia statistics: (Beroulia or radom statistics) (Termial model) k P P P { } { }( { / }) Secod-order Markovia statistics: (Peultimate model) k P P P { } { }( { / })
Example: the sequece BB Beroullia statistics (radom model): P { BB} P{} P{} B P{} P{} B P{} 5 3 P { BB} ( P{ }) ( P{ B}) 5 First-order Markovia statistics (termial model): P { BB} P{ } P{ B / } P{ / B} P{ B / } P{ / B} 5 P { BB} P{ }( P{ B / }) ( P{ / B}) 5 Secod-order Markovia statistics (peultimate model): P { BB} P { B} P{ / B} P{ B / B} P{ / B} 5 P { BB} P { B}( P{ / B}) P{ B / B} 5
Measure of the Departure from Radom Statistics χ P B { } P{} P{} B χ χ > χ χ < χ 0 Completely radom copolymer Copolymer with a alteratig tedecy Completely alteratig copolymer Copolymer with a blockig tedecy Completely block copolymer
Number Fractio of Sequeces of Uits N ( ) P + P+ { B B} { B B} P+ { BB} { BB} + P4{ BB} + P5 { BB} + L P+ { BB} P { B} N ( ) Sequeces of uits must be preceded ad succeeded by at least oe B uit. P + ll possible cotributios from B diads will be icluded i the sum. { BB} P{ B} For example, N( ) 3 5 P { BB} P{ B}
Number verage Legth of or B Rus l N N ( ) ( ) Use the umber distributio fuctio for the sequece legth to evaluate the average legth. N ( ) P + { BB} P{ B} Number distributio fuctio N ( ) l P+ { BB} P{ B}
Number verage Legth of or B Rus l P+ { BB} P{ B} P+ { BB} { BB} + P { BB} + 3 P { BB} + L 4 5 P+ { BB} P { } l P{} P{ B} l P {} B B P{ B} ll uits will be icluded i the sum. Recall that P{ B} P{ B}