Programs for Calculating Structural Features of Terpolymers

8

Downloaded by NANYANG TECHNOLOGICAL UNIV on October 15, 2017 | http://pubs.acs.org Publication Date: September 24, 1982 | doi: 10.1021/bk-1982-0197.ch008

Programs for Calculating Structural Features of Terpolymers YASUTO KODAIRA and H. JAMES HARWOOD The University of Akron, Institute of Polymer Science, Akron, OH 44325

Two computer programs are described for calculating structural aspects (composition, dyad-, triad-, tetrad-, and pentad-distributions, number and weight monomer sequence distributions, monomer centered triad- and pentad-fractions) of terpolymers prepared in either low or high conversion. The programs are applicable for either terminal (1st order Markoffian) or penultimate (2nd order Markoffian) terpolymerization system. Calculation methods employed in preparing the programs are discussed. The chemical and physical properties of terpolymers are influenced by their compositions and by the way the monomer units are arranged along their chains. It is important to be able to calculate the relative amounts of structural features present in terpolymers that arise from differences in monomer unit arrangements. For terpolymers prepared in conversions low enough that the composition of the monomer mixture does not change during the course of their preparation, structural features can be calculated easily from the proportions of monomers present in the polymerization mixture and from kinetic constants (reactivity ratios) appropriate for the terpolymerization system. When the composition of the monomer mixture changes during the reaction, due to some monomers being consumed faster than others, numerical integration is necessary to calculate average values for the relative amounts of structural features present. The computer is practically indispensable for such calculations. Because of the large number of structural features that are present in terpolymers, it is advisable to use a computer for such calculations even when the composition of the monomer mixture does not change during the process. Shown below is a representative portion of a terpolymer chain derived from monomers A, B and C. The structural features of interest are: the relative amounts A-, B- and C-monomer units; the relative amounts of AA-, (BA+AB)-(CA+AC)-, BB-, (BC+CB)- and CC-pairs (dyad distributions); the relative amounts of groups of 0097-6156/82/0197-0137$06.00/0 © 1982 American Chemical Society Provder; Computer Applications in Applied Polymer Science ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

138

COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE

three ( t r i a d d i s t r i b u t i o n s ) , four (tetrad d i s t r i b u t i o n s ) , f i v e ( p e n t a d d i s t r i b u t i o n s ) o r more ( n - a d d d i s t r i b u t i o n s ) monomer u n i t s ; t h e p e r c e n t a g e o f monomer u n i t s o f a g i v e n t y p e i n p a r t i c u l a r e n v i r o n m e n t s ( t r i a d , p e n t a d o r n-add f r a c t i o n s ) ; and t h e num b e r and w e i g h t d i s t r i b u t i o n s o f monomer s e q u e n c e s o f v a r i o u s lengths. These v a r i o u s f e a t u r e s a r e a l s o i l l u s t r a t e d below.


^ A - B - C - A - A - B - B - A - C - B - C - C - A ^ Dyads

AB

CA BC

Triads

AB AA

ABC

AAB BCA

B-Sequences C-Sequences

B

CA

BCC CCA CBC

ABBA

CBCC BBAC

BCCA BACB

AABB

ACBC

AA

A

A ^

BB C

CC BC

ACB BBA

ABCA BCAA CAAB

A - S e q u e n c e s Â

CB AC

BAC ABB

CAA Tetrads

BA BB

B C

CC

We h a v e p r e v i o u s l y r e v i e w e d ( 1 , 2 ) t h e methods u s e d t o calculate s t r u c t u r a l f e a t u r e s o f copolymers and t e r p o l y m e r s f r o m monomer r e a c t i v i t y r a t i o s , monomer f e e d compositions and c o n v e r s i o n s , and h a v e w r i t t e n a p r o g r a m f o r c a l c u l a t i n g s t r u c t u r a l f e a t u r e s o f c o p o l y m e r s from e i t h e r t e r m i n a l model o r p e n u l t i mate model r e a c t i v i t y r a t i o s ( 3 ) . T h i s p r o g r a m h a s b e e n d i s t r i buted w i d e l y and i s i n g e n e r a l u s e . A l i s t i n g o f an i n s t r u c t i v e program f o r c a l c u l a t i n g s t r u c t u r a l f e a t u r e s o f i n s t a n t a n e o u s t e r p o l y m e r s f r o m monomer f e e d c o m p o s i t i o n s and t e r m i n a l m o d e l r e a c t i v i t y r a t i o s was appended t o one o f o u r e a r l i e r r e v i e w s C I ) . Our r e v i e w s ( 1 , 2 ) c i t e o t h e r e x a m p l e s o f p r o g r a m m i n g t e r p o l y m e r c o m p o s i t i o n and/or s t r u c t u r e c a l c u l a t i o n s , e i t h e r by s t o c h a s t i c o r M o n t e - C a r l o m e t h o d s . I n t h e p r e s e n t p a p e r , we d e s c r i b e two g e n e r a l programs f o r c a l c u l a t i n g s t r u c t u r a l f e a t u r e s o f t e r p o l y m e r s from monomer f e e d c o m p o s i t i o n s , c o n v e r s i o n s , and e i t h e r p e n u l t i m a t e o r t e r m i n a l model r e a c t i v i t y r a t i o s . Program A i s c o m p l e t e l y s e l f c o n t a i n e d and w i l l accommodate a p e n u l t i m a t e e f f e c t w i t h one mono mer. P r o g r a m B c a l l s a m a t r i x m u l t i p l i c a t i o n s u b r o u t i n e (GMPRD f r o m t h e IBM S c i e n t i f i c S u b r o u t i n e P a c k a g e , o r VMULFF f r o m t h e IMSL L i b r a r y ) ( 4 ) , b u t i s c o m p l e t e l y g e n e r a l ; p e n u l t i m a t e e f f e c t s c a n b e a s s o c i a t e d w i t h a l l t h r e e monomers i n t h i s c a s e . The p r o grams p r o v i d e monomer c o n c e n t r a t i o n s , d y a d t h r o u g h p e n t a d d i s t r i b u t i o n s , t r i a d a n d p e n t a d f r a c t i o n s , a s w e l l a s t h e number and w e i g h t d i s t r i b u t i o n o f A-, B- a n d C - s e q u e n c e s o f monomer u n i t s .

Provder; Computer Applications in Applied Polymer Science ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

8.

General


Structural Features of

KODAIRA AND HARWOOD

Terpolymers

139

Method

The d e v e l o p m e n t o f t h e s e p r o g r a m s f o l l o w e d t h e g e n e r a l p r o cedure o u t l i n e d p r e v i o u s l y ( 1 ) . T h i s i n v o l v e d t h e f o l l o w i n g s t e p s : 1. P r o p a g a t i o n r e a c t i o n s i n v o l v e d i n t h e p o l y m e r i z a t i o n s y s tem u n d e r s t u d y a r e w r i t t e n , a l o n g w i t h a p p r o p r i a t e r a t e expressions. Appropriate r e a c t i v i t y r a t i o s are defined. 2. C o n d i t i o n a l p r o b a b i l i t i e s f o r monomer o r s e q u e n c e p l a c e m e n t s a r e c a l c u l a t e d f r o m monomer f e e d c o n c e n t r a t i o n s a n d r e a c t i v i t y r a t i o s , u s i n g expressions d e r i v e d from t h e r a t e expressions w r i t t e n i n the f i r s t step. The c o n d i t i o n s e m p l o y e d i n d e f i n i n g these p r o b a b i l i t i e s correspond t o t h e types o f propagating species ( e . g . , Â«, ^BAO i d e n t i f i e d i n t h e f i r s t s t e p . 3. U n c o n d i t i o n a l p r o b a b i l i t i e s f o r monomer o r s e q u e n c e placements a r e c a l c u l a t e d from t h e c o n d i t i o n a l p r o b a b i l i t i e s . T h i s may b e a c c o m p l i s h e d b y w r i t i n g s t a t i o n a r y r e l a t i o n s h i p s ( n e c e s s a r y n-add r e l a t i o n s h i p s ) i n v o l v i n g c o n d i t i o n a l a n d u n c o n d i t i o n a l p r o b a b i l i t i e s , f o l l o w e d by s o l v i n g such r e l a t i o n s h i p s simultaneously f o r the c o n d i t i o n a l p r o b a b i l i t i e s . While t h i s can be done a l g e b r a i c a l l y , i t i s s i m p l e r t o do t h i s b y c o n d u c t i n g o p e r a t i o n s on a m a t r i x c o n s t r u c t e d from t h e c o n d i t i o n a l p r o b a b i l i ties. The p r o c e d u r e d e s c r i b e d b y P r i c e ( 5 ) c a n p r o v i d e a n a l g e b r a i c s o l u t i o n i f d e s i r e d , b u t i t can be t h e b a s i s o f a program that provides numerical r e s u l t s . The m a t r i x m u l t i p l i c a t i o n method (1), provides numerical r e s u l t s o n l y , b u t i t seems t o b e t h e p r e f e r r e d a p p r o a c h when t h e p o l y m e r i z a t i o n s y s t e m i s c o m p l e x . 4. U n c o n d i t i o n a l p r o b a b i l i t i e s o f s e q u e n c e s s m a l l e r t h a n those e v a l u a t e d i n step t h r e e , i f any, a r e c a l c u l a t e d by a d d i t i o n of u n c o n d i t i o n a l p r o b a b i l i t i e s e v a l u a t e d i n step t h r e e . Uncondi t i o n a l p r o b a b i l i t i e s of sequences l a r g e r than those e v a l u a t e d i n step t h r e e a r e c a l c u l a t e d by m u l t i p l i c a t i o n o f a p p r o p r i a t e c o n d i t i o n a l and u n c o n d i t i o n a l p r o b a b i l i t i e s . Program A T h i s p r o g r a m was w r i t t e n t o c a l c u l a t e s t r u c t u r a l f e a t u r e s of terpolymers prepared v i a t h e f o l l o w i n g p r o p a g a t i o n r e a c t i o n s f r o m r a d i c a l s AA-, BA-, CA-, B- a n d C- a n d f r o m monomers A, B a n d C. R e a c t i v i t y r a t i o s u t i l i z e d b y t h i s p r o g r a m a r e a l s o d e f i n e d i n t h e f o l l o w i n g scheme: k WAA+ A > ÂA. a

a

a

r

AAT> AAB =

, OA.AA-

+

B

k

,b a a -/k a aa-

-h A

A

> OA,Br.._ = k /k AAC aa-a aa-c



140


+

C

^BA-

+

A

^BA-

+

B

^BA-

4-

C

^\AC;A-

+

A

r

aa-c

ba-a

ba-b

^ba-c

^ba-a^^ba-b

BAC

^ba-a^^ba-c

OA.AACAB

ca-b

v\CA-

BAB

CAC

+

A

k

k

ca-a

/k , ca-b

ca-a

/k ca-c

ba

BA

BC

"be

^B •

+

A

k

bb

B

CA

cb

CB +

C

b

G

^ A ' k

+

/ k

k

cc

/k ca

cc

/k , cb

'VAC*

On t h e b a s i s o f t h e a b o v e , i t i s c l e a r t h a t u n c o n d i t i o n a l p r o b a b i l i t i e s o f AA, BA, CA, B a n d C n e e d t o b e e v a l u a t e d t o p r o v i d e a b a s i s f o r o t h e r c a l c u l a t i o n s . These p r o b a b i l i t i e s w i l l be d e s i g n a t e d P ( A A ) , P ( B A ) , e t c . The c o n d i t i o n a l p r o b a b i l i t i e s n e c e s s a r y f o r t h i s e v a l u a t i o n a r e d e s i g n a t e d P ( J / I ) , where I and J a r e t h e i n i t i a l and f i n a l s p e c i e s i n v o l v e d i n a p r o p a g a t i o n step. These a r e c a l c u l a t e d by c o n s i d e r i n g t h e k i n e t i c e x p r e s s i o n s w r i t t e n above. F o r e x a m p l e , i n c a l c u l a t i n g P ( A A / A A ) , one must compare t h e r a t e o f AfbAA* r a d i c a l s a d d i n g monomer A t o t h e r a t e o f 'V\,AAr a d i c a l s a d d i n g monomers A, B o r C ( A f , B f a n d C f r e p r e s e n t t h e c o n c e n t r a t i o n s o f monomers A, B a n d C i n t h e p o l y m e r i z a t i o n m i x t u r e )


8.

k

aa-a

1

/

(

1

CvÂAOA

aa-b

1 +

=

141

Terpolymers

A

P(AA/AA) -



KODAIRA AND HARWOOD

k

+

f

/ r

AAB f A

aa D

(wAA* ) B + k Ov\,AA-)C I aa—C i f

f

k

f A, r

aa-a B

I + k

i

+

k

aa-c aa-a

V

W

f A* f V

Similarly, P(B/AA) = ( B / f

r A A B

A )ya + B y f

P(C/AA) - 1 - P (AA/AA)

f

r A A B

A

f

+ C /r f

M C

A ) f

P(B/AA)

etc. The t r a n s i t i o n m a t r i x i n v o l v i n g t h e s e p r o b a b i l i t i e s i s a s follows: i (Initial AA BA CA B C^- S t a t e ) / T

AA

P (AA/AA)

P(AA/BA)

BA

0

0

CA

0

0

0

0

0

P(BA/B)

0

0

0

P(CA/C)

P(AA/CA)

B

P(B/AA)

P(B/BA)

P(B/CA)

P(B/B)

P(B/C)

C

P(C/AA)

P(C/BA)

P(C/CA)

P(C/B)

P(C/C)

(Final State) E v a l u a t i o n o f P ( A A ) , P ( B A ) , P ( C A ) , P(B) and P(C) from t h i s m a t r i x b y t h e method o f P r i c e ( 5 ) y i e l d e d t h e f o l l o w i n g r e s u l t s , where X i s t h e n o r m a l i z a t i o n f a c t o r r e q u i r e d t o make t h e sum o f t h e s e q u a n t i t i e s t o t a l 1.0. ( A amounts t o t h e r e c i p r o c a l o f t h e sum of t h e e x p r e s s i o n s t h a t a r e m u l t i p l i e d by X i n t h e e q u a t i o n s g i v e n below) P(AA) = X * [ (1-P (B/B) )*P (CA/C)*P (AA/CA)-P ( B A / B ) * P ( C A / C ) * ( P ( B / B A ) * P (AA/CA) -P (B/CA) *P (AA/BA) )+P (BA/B) *P (B/C) *P (AA/BA) ] P(B) =

X*[(1-P(AA/AA))*(P(B/C)+P(CA/C)*P(B/CA))+P(AA/CA)* P(CA/C)*P(B/AA)]



142

P(C) = A*[(P(AA/AA)-l)*(P(B/B)-l+P(BAyB)*P(B7BA))-P(AA7BA)* P(B/AA)*P(BA/B)] P(BA) = A * P(B)*P(BA/B)


P(CA)= A*P(C)*P(CA/C) The r e m a i n i n g monomer and d y a d c o n c e n t r a t i o n s c a n be c a l c u l a t e d f r o m t h e above q u a n t i t i e s and a p p r o p r i a t e p r o b a b i l i t i e s . P ( A ) = P(AA)+P(BA)+P(CA) P(AB) = P ( A A ) * P ( B / A A ) + P ( B A ) * P ( B / B A ) + P ( C A ) * P ( B / C A ) P(AC)= P(AA)*P(C/AA)+P(BA)*P(C/BA)+P(CA)*P(C/CA) P(BB)= P(B)*P(B/B) P(BC)= P(B)*P(C/B) P(CB)= P(C)*P(B/C) P(CC)= P(C)*P(C/C) By f o l l o w i n g t h i s g e n e r a l a p p r o a c h , n-add d i s t r i b u t i o n s ( p r o b a b i l i t i e s ) up t o p e n t a d s ( B - c e n t e r e d p e n t a d s o n l y a t t h e present time) are c a l c u l a t e d , v i z . , P(AAB)=

P(AA)*P(AB/AA)

P(BBA)= P(B)*P(B/B)*P(BA/B) P(CBBAC)= P ( C ) * P ( B / C ) * P ( B / B ) * P ( B A / B ) * P ( C / B A ) S i n c e a l a r g e number o f n-add d i s t r i b u t i o n s a r e i n v o l v e d i n the c a l c u l a t i o n s , the program r e p o r t s complete d i s t r i b u t i o n s o n l y f o r d y a d s and t r i a d s . C o l l e c t e d d i s t r i b u t i o n s a r e r e p o r t e d f o r d y a d s , and t e t r a d s . These q u a n t i t i e s a r e o f g r e a t e r i n t e r e s t t h a n t h e i n d i v i d u a l n-add d i s t r i b u t i o n s , s i n c e t h e p r o p e r t i e s of the c e n t r a l p o r t i o n s of u n s y m m e t r i c a l sequences a r e u s u a l l y t h e same a s t h o s e o f t h e r e v e r s e d s e q u e n c e s ( e . g . , t h e c e n t r a l u n i t s i n ABSCB and BCCBA s e q u e n c e s u s u a l l y h a v e i d e n t i c a l chem i c a l and p h y s i c a l p r o p e r t i e s . ) C o l l e c t e d n-add d i s t r i b u t i o n s a r e t h u s c a l c u l a t e d by a d d i n g a p p r o p r i a t e n-add d i s t r i b u t i o n s . E x a m p l e s o f s u c h c a l c u l a t i o n s a r e shown b e l o w , where P (n-add) d e f i n e s a c o l l e c t e d n-add d i s t r i b u t i o n . 1


8.

KODAIRA AND HARWOOD


T

= P(AA)

T

= P(AB)+P(BA)

P (AA) P (AB)

Terpolymers

143

!


P ( C A A B ) = P(CAAB)+P(BAAC) T r i a d f r a c t i o n s ( i n d i v i d u a l and c o l l e c t e d ) and B-centered pentad f r a c t i o n s ( c o l l e c t e d ) a r e a l s o p r o v i d e d by t h e program. T h e s e c o r r e s p o n d t o t h e f r a c t i o n s o f monomer u n i t s r e s i d i n g i n p a r t i c u l a r e n v i r o n m e n t s i n t h e polymer. F o r example, fcAB d e s i g n a t e s t h e f r a c t i o n o f A - u n i t s c e n t e r e d i n CAB t r i a d s . Such q u a n t i t i e s a r e c a l c u l a t e d b y d i v i d i n g a p p r o p r i a t e n-add d i s t r i b u t i o n s by monomer p r o b a b i l i t i e s , v i z . , f

CAB

=

p

(

C A B

p

A

>/ ( )

O b v i o u s l y , c o l l e c t e d t r i a d and pentad f r a c t i o n s a r e o f g r e a t e r general i n t e r e s t than i n d i v i d u a l f r a c t i o n s . Number (N.D.) a n d w e i g h t (W.D.) d i s t r i b u t i o n s o f A-, B- a n d C- monomer s e q u e n c e s a r e a l s o c a l c u l a t e d . N.D.(A)

1

= 1 - (P(BA)*P(AA/BA)+P(CA)*P(AA/CA))/(P(BA)*P(CA))

N.D.(A) ^- = ( P ( B A ) * P ( A A / B A ) + P ( C A ) * P ( A A / C A ) ) * ( 1 - P ( A A / A A ) * n>l P ( A A / A A ) ~ / (P (BA)+P (CA) ) n

N.D.(B)

n

n

2

1

= P(B/B) " *(l-P(B/B)) n

1

N.D.(C) = P ( C / C ) " " * ( l - P ( C / C ) ) n W.D. (A)

n

= (n*N.D.(A) ) / ( 1 + ( P ( B A ) * P ( A A / B A ) + P ( C A ) * P ( A A / C A ) ) ] n ((P(BA)+P(CA))*(1-P(AA/AA))) 1 1

1

W.D.(B) = n * P ( B / B ) " * ( 1 - P ( B / B ) ) n n

1

W.D.(C) = n * P ( C / C ) " " * ( l - P ( C / C ) ) n

2

2

C a l c u l a t i o n s a t h i g h c o n v e r s i o n - The q u a n t i t i e s d i s c u s s e d above a r e c a l c u l a t e d a t r e g u l a r c o n v e r s i o n i n c r e m e n t s a n d a r e i n t e g r a t e d u s i n g t h e t r a p e z o i d a l method a s i s d e s c r i b e d i n s e v e r a l o f o u r e a r l i e r p a p e r s (1»3). A v e r a g e v a l u e s o f n-add d i s t r i b u t i o n s , e t c . , a r e then r e p o r t e d f o r c o n v e r s i o n s s p e c i f i e d by the u s e r . A s an o p t i o n t h e program w i l l a l s o p r o v i d e i n f o r m a t i o n a b o u t t h e c o n c e n t r a t i o n s o f u n r e a c t e d monomers a t v a r i o u s c o n v e r sions. T y p i c a l o u t p u t f r o m t h i s p r o g r a m i s shown i n F i g u r e 1. T h i s o u t p u t i s f o r a c a l c u l a t i o n i n v o l v i n g o n l y t e r m i n a l model r e a c t i v i t y r a t i o s , w h i c h i s an o p t i o n o f t h e program.


144

COMPUTER. APPLICATIONS IN APPLIED POLYMER SCIENCE

OO — N O O O O J O CO X> (M in in O O *l • • • oc»c

II

a «r :r

i/i < z —•

II T:

il o

X N- O ON o ^ — o o o • • • O o o II

n

II

z < < < J - t i u —

0 N '«J OJ — — .-> n — to j") C> ^ o - " rr rt c (/)••• — • • • — o o o ^ Oo o n -J II .i 'i G t < < < < < < < < a o — < < < cc < c a o o

-

^ N O OJ

•> OJ

*

O

—

^

_0 o f- — in vP T o n o i o c < r o x ) — *; o oj— n j ? o o n — ui oj r\j — c > OJ t\ o o o o o — — — o • • • • • • • •• o ooo

— o c nr »- n ii ii n II ii ^ < o r. r o J 1

0 : a z) 0 a CD < < -rx - 2 0

«r ') x — o *, r i o cj :*J c- oj — —' v. i r o — — — oj t >I.I r . N. ^ OJ ^ Oj o ^ O "j — „0 ^ ( V — T >C O ~! r- c r r r o «r o o o o o rj r- — if • • • • • • 'i • • • • • • • • • ^ o o o o o o u »*. ° o O O O O O O « r n

co o O O 0 CP o — o o

< O CO CO O o O O »- O X • O o — o

O O CO om n n o o o n ao o o o

o

X o

o o o o o o o o o o o o in OJ c\j • • • o o o

K. a> — o O S> O N O oj n >. 0 . 7 0 9 9 3 0 . 96 7 05 c . 8 2 9 5 6 0 . 2 0 5 7 9 0 . 3 3 13 7 0 • 14202 0 . 00 1 05 0 . C243 7 0 . 0 5 9 7 6 0 . 0 1 7 3 8 0. 0 0 0 0 3 0 • 004 1 9 0 . 0 0 5 0 7 0 . o o o o o 0 . 0 0 0 7 2 0 . 0 0 1 4 8 0. 000 0 0 0 . 0 0 01 2 0 . 0 0 0 4 3 0. o o o o o 0 • 0 0 0 0 2 0. o o o o o 0. 000 1 3 0 000 0 . 0 0 0 0 4 0 . o o o o o 0. o o o o o 0. 0000 1 0. o o o o o 0 . o o o o o - 0 .00 —0 . 0 0 0 0 1 301 - 3 . 3 0 3 0 1

AABBAAAHUil: A ABBC = BAbBABABJLU B A B B C =• CABQA= C A R U L U CABHC=

0 0 0 2 3 3 304 1 ^ 1 7 J 0 3 1 r. 2 0 3 ?. t 7

0 112 4 50 3 0. J]>21 0 . 0 J«>3H

0 . 0. 0 . 3 . 3 .

C U L L F C T c.> AABA A = 0 0 AARAo0 A A BA C GADA M= « 0 UABAC= 0 CABAC=

B C C C

A A A B H


2

s

5? •a

I

«•>*.

?

2

3

Co

§

o

ss

a

o

146



Program B T h i s p r o g r a m f u n c t i o n s i n e s s e n t i a l l y t h e same manner a s P r o g r a m A, e x c e p t i t c a n a c c e p t a s i n p u t 18 r e a c t i v i t y r a t i o s , c o r r e s p o n d i n g t o a s i t u a t i o n where a l l monomers e x h i b i t p e n u l t i mate e f f e c t s . As i s a l s o t h e c a s e i n P r o g r a m A, t e r m i n a l m o d e l r e a c t i v i t y r a t i o s c a n be u s e d , h o w e v e r . The programming i n v o l v e s the c a l c u l a t i o n o f 27 c o n d i t i o n a l p r o b a b i l i t i e s . These p r o b a b i l i t i e s a r e c a l c u l a t e d i n t h e same way t h a t P ( A A / A A ) , P(AAyBA) a n d P(AA/CA) a r e c a l c u l a t e d i n t h e c a s e o f P r o g r a m A. F i g u r e 2 shows t h e 9x9 t r a n s i t i o n m a t r i x t h a t i s c o n s t r u c t e d from these c o n d i t i o n a l p r o b a b i l i t i e s . I t s h o u l d be n o t e d t h a t a l l i n i t i a l and f i n a l s t a t e s a r e d y a d s i n t h i s c a s e . Since i t would be d i f f i c u l t t o u s e P r i c e ' s c o f a c t o r method (5) w i t h s u c h a l a r g e m a t r i x , t h e m a t r i x m u l t i p l i c a t i o n method we h a v e d e s c r i b e d p r e v i o u s l y (1) was u s e d t o e v a l u a t e u n c o n d i t i o n a l dyad p r o b a b i l i t i e s (e.g., P(AB), P(CC), e t c . ) . Thus, r e p e a t e d m u l t i p l i c a t i o n of the t r a n s i t i o n m a t r i x by i t s e l f c a u s e s i t t o c o n v e r g e t o t h e m a t r i x shown i n F i g u r e 3, w h i c h c o n t a i n s u n c o n d i t i o n a l d y a d p r o b a b i l i t i e s as i t s e l e m e n t s . These a r e t h e n added t o o b t a i n monomer p r o b a b i l i ties , viz., P(A)

= P(AA) + P(AB) + P(AC)

P(B)

= P(AB) + P(BB) + P(CB)

P(C)

= P(AC) + P(BC) + P(CC)

T r i a d and h i g h e r n-add p r o b a b i l i t i e s a r e c a l c u l a t e d by m u l t i p l y i n g a p p r o p r i a t e u n c o n d i t i o n a l and c o n d i t i o n a l p r o b a b i l i t i e s . A f e w e x a m p l e s a r e shown b e l o w . P(AAA) = P ( A A ) * P ( A A / A A ) P(AAB) = P ( A A ) * P ( A B / A A ) P(BAA) = P ( B A ) * P ( A A / B A ) P(AABC)= P ( A A ) * P ( A B / A A ) P ( B C / A B ) P(BABC)= P ( B A ) * P ( A B / B A ) * P ( B C / A B ) = P ( B A B ) * P ( B C / A B ) P(BACBC)=P(BA)*P(AC/BA)*P(CB/AC)*P(BC/CB) T r i a d and p e n t a d f r a c t i o n s a r e c a l c u l a t e d i n t h e u s u a l way, f *

= P(AAB)/P(B)

^BACBC

=

P(BACBC)/P(C)


viz.,

8.


KODAIRA AND HARWOOD

Terpolymers

o


Q_

PQ

o

o o

I o

\

PQ O

o

S

PQ

PQ Q_

PQ

o

Programs for Calculating Structural Features of Terpolymers

Recommend Documents