studies on chemical heterogeneity of multi-component ... filestudies on chemical heterogeneity of...

Studies on chemical heterogeneity ofimulti-component polymers:

Sequence Length Distributionsq g

Woosung Jung, T. A. Duever, A. PenlidisWoosung Jung, T. A. Duever, A. Penlidis

Department of Chemical Engineeringp g gIPR annual symposium, University of Waterloo

May 13, 2008

IPR 20

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Outline

• Introduction

• Copolymerization characteristics

• Macroscopic approach

• Microscopic approach

• Summary

1/31

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Introduction- Multicomponent polymerization

M1

O

O

R

Propagation-M1M2M1M3-……-M2M3M1M1-

M2

O

R

2 3 1 1

Polymer chain

M3

O

R

Composition & Arrangement

2/31

3

Monomer speciesIP

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Introduction- Multicomponent polymerization

• Competitive reactions between same or• Competitive reactions between same or

different radical/monomer speciesp

• Governed by probabilistic nature of reactions

• Numerous combinations of monomer species

• Macroscopic approach (Composition)• Macroscopic approach (Composition)

• Microscopic approach (SLD & Triad fraction)

3/31

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Copolymerization characteristics

Model• Random (Bernoullian)• Statistical (1st/2nd order Markov)

Structures• Alternating (-M1M2M1M2M1M2M1M2-)• Block ( M M M M M M M M M M M )• Block (-M1M1M1M1M1M2M2M2M2M1M1-)• Graft (-M1M1M1M1M1M1M1M1M1M1-)

4/31M2M2M2M2M2-

IPR 20

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Copolymerization characteristicst- 1st order Markov (terminal) model

Mi+ Mj

Mjkpij

Reactivity of the propagating chain depends only on the monomer unit at the growing endy g gand independent of chain composition.

5/31

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Copolymerization characteristicst- 1st order Markov (terminal) model

kM1 + M1kp11 M1

k 12

111

p

p

kk

r =

M1 + M2kp12 M2

k

12p

M2 + M1kp21 M1

k 22

222

p

kk

r =M2 + M2

kp22 M2 21pk

6/31

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Copolymerization characteristicsd- 2nd order Markov (penultimate) model

MiMj+ Mk

MjMkkpijk

Reactivity of the propagating chain is affected by the last and the next-to-last monomer unitsyand independent of chain composition.

7/31

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Copolymerization characteristics- Reactivity ratios

• Relative rate of homo- to cross-propagation• Relative rate of homo- to cross-propagation• Estimated from experimental (NMR) data• r > 1 : homo-propagation favored• r < 1 : cross-propagation favoredr 1 : cross propagation favored• r = 0 : No homo-propagation (alternating)• Q-e scheme, Hammet and Taft equation for

starting values

8/31• Determines polymerization tendencyIP

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Macroscopic approach- Instantaneous composition

∑ •N

i MRkMd ]][[][Material balances∑

=

•=−j

ijpjii MRk

dtMd

1]][[][

]][[]][[ MRkMRk ••Steady-state hypothesis]][[]][[ ijpjijipij MRkMRk •• =

iMf ][Monomer feed composition

∑=

= N

ii

ii

Mf

1][

Instantaneous compositionof multi-component polymerN = 2: Mayo-LewisN 3 W lli B i∑

= N

i

ii

Md

MdF][

][

9/31

N = 3: Walling-Briggs∑=i

i1

][IP

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Macroscopic approach- Instantaneous composition

0.8

0.9

1Instantaneous Copolymer Composition of Sty and AN

rSty = 0.36

0.5

0.6

0.7

F Sty

rAN = 0.078FSty > fSty

FSty < fSty Direction

0.2

0.3

0.4

Experimental data

Sty fSty Directionof Drift

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

fSty

Experimental dataModel predictionAzeotropic point

10/31Figure 1. Simulation of Styrene-Acrylonitrile bulk co-polymerization

T = 60 (Experimental data from Hill et al., 1982)

IPR 20

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Macroscopic approach - Instantaneous composition

0.8

0.9

1Instantaneous Copolymer Composition of MMA and MA

rMMA = 2.6FMMA > fMMA

0.5

0.6

0.7

F MM

A

rMA = 0.27

Direction

0.2

0.3

0.4

F Directionof Drift

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

fMMA

Experimental dataModel prediction

11/31Figure 2. Simulation of Methyl methacrylate-Methyl acrylate bulk co-polymerization

T = 50 (Experimental data from Kim and Harwood, 2002)

IPR 20

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Macroscopic approach- Cumulative (average) composition

∫if df

Skeist, Meyer-Lowry∫ −=−

if ii

i

fFdfX

0

)1ln(

][][][ PMMConversion][][

][][

][][

0

0

PMP

MMMX

+=

−=

Cumulative compositionof multi-component polymer∑

= N

i

ii

P

PF__

][

][

Composition changes during polymerization;

∑=i 1

12/31‘Composition drift’

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Macroscopic approach - Cumulative composition drift

0.7

0.8Cumulative polymer composition vs conversion

rSty = 0.717

0.5

0.6

sitio

n(ac

c.) rEA = 0.128

0.3

0.4

Com

po

f10 = 0.152

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

Conversion

f10 = 0.453

f10 = 0.762

13/31Figure 3. Simulation of Styrene-Ethyl acrylate bulk co-polymerization

T = 50 (Experimental data from McManus and Penlidis, 1996)

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Macroscopic approach - Limitation

• Does not give information on monomerDoes not give information on monomer arrangements

M1M1M1M1M1M2M2M2M2M2 Block

Diff i i i

M1M2M1M2M1M2M1M2M1M2 Alternating

• Different properties, same composition

F1 Bl k = F1 Al i = 0 514/31

F1, Block F1, Alternating 0.5IP

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Microscopic approach - Sequence Length Distribution (SLD)

• Shows intramolecular heterogeneityShows intramolecular heterogeneity

M1M1M1M1M1M2M2M2M2M2 BlockM1M1M1M1M1M2M2M2M2M2

M M M M M M M M M M

Block

AlternatingM1M2M1M2M1M2M1M2M1M2 Alternating

Sequence length of M = 5Sequence length of M1, Block = 5

Sequence length of M1 Al = 115/31

Sequence length of M1, Alter. 1IP

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M MkMi+ Mj

Mjkpij

∑∑∑ •

•

=== N

jpij

jpijN

jpij

jpijN

jipij

jipijij

fk

fk

Mk

Mk

MRk

MRkP

][

][

]][[

]][[

∑∑∑=== j

jp jj

jp jj

jp j111

)(1 ikPPPN

ikii

N

ij ≠=+= ∑∑

Probability of reaction between radical species id i j

11 kj ==

16/31

and monomer species jIP

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Probability of having1 PPNN

n= − ∑ Probability of havingn consecutive units

of monomer species i( ) )(111

ikPP

PPN

iin

ii

kikiiin

≠−=

=

−

=∑

-M1M1M1M1M1M2- N15 = (P11)4P12

( ) )(iiii

1 1 1 1 1 2 15 ( 11) 12

Long Chain Approximation1PPN nN

k ⎟⎞

⎜⎛

= ∑∑∑∞

−∞

Long Chain Approximation

)(111111

ikPP

PPN

iiN

ik

nii

kik

nin

≠=−

=≈

⎟⎠

⎜⎝

=

∑

∑∑∑===

17/31

)(111 PP iiiik

ik −−∑=

IPR 20

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Sequence length distribution of Sty monomerq g y

0.7

0.8

0.9

n)

0 3

0.4

0.5

0.6

babili

ty (

N1n

0.0

0.1

0.2

0.3

Pro

b

1 2 3 4 5 6 7 8

Chain length (n)

f10 = 0.4 f10 = 0.5 f10 = 0.6 f10 = 0.7 f10 = 0.8 f10 = 0.9

18/31Figure 4. Sequence Length Distribution of Sty monomer

in Styrene-Acrylonitrile co-polymer, T = 60

IPR 20

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Microscopic approach - Average Sequence Length

∑∞

( ){ }∑∑∑

∑ ∞

=

−∞

=∞= −=+++===

nii

nii

niiiin

nin

i PnPNNNnNN

nNn

1

1

1321

1__

132 L

( ) ( )∑ ∑

∑∞ ∞

−

=

==−≈−= Niin

iin

ii

nin

PnPnP

N

221

1

11

11( ) ( ) ∑

∑ ∑=

= = −−− N

kik

iiiiiin niiii

PPPP1

221 1 111

Instantaneous number average sequence length

19/31

IPR 20

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Microscopic approach - Average Sequence Length

∑∑∞∞

22

nii

n

N

kik

nin

N

kik

nin

nin

i PnPNnPn

Nn

nN

Nnw ⎟

⎠

⎞⎜⎝

⎛==== ∑∑∑∑

∑

∑

∑−

∞

==

∞

==

=∞= 1

1

22

11

2

1__

1

2

1

2__

( ) ( )( )

iiiiii

niiii

in

in

PPPPnP

nnN

+=

+−≈−= ∑

∑∞

−

=

1111 32122

1

( ) ( )( ) iiii

iin

iiii PP −−∑= 11 3

1

Instantaneous weight average sequence length

20/31

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Microscopic approach - Sequence Length Distribution (Ray)

∫∫ ==X

iin

X

iinRayin dXFNdXNNi

00

________

,

( ){ }∫∫ ∑ −=⎟⎠

⎞⎜⎝

⎛= −

=

−X

iiin

ii

X

i

N

kik

nii dXFPPdXFPP

0

1

0 1

1

00

1⎠⎝ 00

Probability of having n consecutive unitsof monomer species i during polymerization

21/31

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Microscopic approach - Sequence Length Distribution (HMP)

2⎪⎫⎪⎧ ⎞⎛⎟

⎞⎜⎛

⎟⎞

⎜⎛

∫∫X N

XX NNi

0 1

1

0__

0__

_________

,

⎟⎞

⎜⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠

⎞⎜⎝

⎛

=⎟⎟⎟

⎠⎜⎜⎜

⎝=⎟⎟⎟

⎠⎜⎜⎜

⎝=

∫ ∑

∫ ∑

∫

∫

∫

∫=

−

X N

i

N

kik

nii

Xi

i

i

in

X

i

i

in

HMPin

dXFPP

dXF

dXFn

N

dX

dXn

N

Ni

( ){ } ( )1 221

0 10__

0__

⎫⎧

⎟⎠

⎞⎜⎝

⎛

∫∫

∫ ∑∫∫=

XX

ik

ik

i

i

i

i dXFPn

dXF

n

dX

( ){ }

( )

( )

( )1

1

11

1

,1

10

2

1

_________

,0

21

=−

⎭⎬⎫

⎩⎨⎧

−−

=−

−=

∫

∫∑

∫

∫ ∞

=

−

X

iiiii

nHMPinX

iiin

ii

dXFP

dXFPP

NdXFP

dXFPP

( ) ( )1100∫∫ iiiiii dXFPdXFP

Probability of having n consecutive units

22/31

Probability of having n consecutive unitsof monomer species i during polymerization

IPR 20

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Microscopic approach - Differences between Ray & HMP

• Acc Number average Sequence LengthAcc. Number average Sequence Length

( )∫ −

X

idXFP1

1∫X

idXF

Ray HMP( )

∫X

i

ii

dXF

P

0

0 1

( )∫ −X

iii dXFP0

0

1

• Acc. Weight average Sequence Length

HMPRay( )

∫

∫ −+

X

X

iii

ii

dXF

dXFPP

02

111

∫

∫ −+

X

X

iii

ii

dXF

dXFPP

0 11

23/31

∫ − iii

dXFP0 1 ∫ idXF

0

IPR 20

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Cumulative average sequence length of Sty vs conversion Cumulative average sequence length of Sty vs conversion

2

2.1

2.2

2.3

h Number-average (Ray)3

3.2

3.4

3.6

h

Number-average (Ray)Number-average (HMP)Weight-average (Ray)Weight-average (HMP)Experimental data

1.6

1.7

1.8

1.9

Seq

uenc

e le

ngth Number average (Ray)

Number-average (HMP)Weight-average (Ray)Weight-average (HMP)Experimental data

2 2

2.4

2.6

2.8

Seq

uenc

e le

ngth

Figure 5 Figure 6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.3

1.4

1.5

Conversion

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1.8

2

2.2

Conversion

Cumulative average sequence lengths of Sty in Styrene-Acrylonitrile co-polymerT = 60 , [AIBN]0 = 0.05 M, fSty0 = 0.6 (Fig. 5), and fSty0 = 0.7 (Fig. 6)

(N b l th i t l d t f G i R bi t l 1985)

Figure 5. Figure 6.

24/31

(Number avg. sequence length experimental data from Garcia-Rubio et al., 1985)IP

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Cumulative average sequence length of Sty vs conversion Cumulative average sequence length of Sty vs conversion

8

9

10

11

h


40

50

60

h


4

5

6

7

Seq

uenc

e le

ngth

20

30

Seq

uenc

e le

ngth

Figure 7 Figure 8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

3

4

Conversion

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

10

Conversion

Cumulative average sequence lengths of Sty in Styrene-Acrylonitrile co-polymerT = 60 , [AIBN]0 = 0.05 M, fSty0 = 0.8 (Fig. 7), and fSty0 = 0.9 (Fig. 8)

(N b l th i t l d t f G i R bi t l 1985)

Figure 7. Figure 8.

25/31

(Number avg. sequence length experimental data from Garcia-Rubio et al., 1985)IP

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Microscopic approach - Dyad/triad fractions

• Dyad fractionsDyad fractions

-M1M1- -M2M1-

T i d f ti

-M1M2- -M2M2-

• Triad fractions

-M1M1M1- -M1M1M2-1 1 1 1 1 2

-M1M2M1--M2M1M2--M2M1M1--M1M2M2-

26/31

1 2 1 1 2 2-M2M2M1- -M2M2M2-

IPR 20

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Microscopic approach - Triad fraction calculation

2

2 ⎟⎞

⎜⎛ iij fr

PA

Triad fractions( )

2

1

⎟⎟⎠

⎜⎜⎝ +

==

jiij

iijj

iijiiiii

ffrPPPPAA

frff

PA

centered onmonomer species i

( )

2

2)(1

⎟⎞

⎜⎛

+=−===

iijj

jiijiiiiijiijiiiij

f

frfff

PPPPAA

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

+==

iijj

jijjij frf

fPA

( )

( )+++=

∑jijjiiiijiii AAAA

imonomeroncenteredfractions

27/31( ) 12 222 =+=++= ijiiijijiiii PPPPPP

IPR 20

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Simulation of triad fraction data for Sty/AN Simulation of Triad fraction for Sty/ANSimulation of triad fraction data for Sty/AN

0.7

0.8

0.9

1

n

Simulation of Triad fraction for Sty/AN

0.7

0.8

0.9

1

n

0.2

0.3

0.4

0.5

0.6

Tri

ad f

ract

ion

0.2

0.3

0.4

0.5

0.6

Tri

ad f

ract

ion

0

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

f1 (Sty)

A111 A211+A112 A212 A111 (exp.) A112+A211 (exp.) A212 (exp.)

0

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

f1 (Sty)


Triad fraction calculation of Styrene-Acrylonitrile co-polymerT = 60 Sty-centered (Fig 9) and AN-centered (Fig 10)

Figure 9. Figure 10.

28/31

T = 60 , Sty-centered (Fig. 9), and AN-centered (Fig. 10)(Experimental data from Hill et al., 1982)

IPR 20

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Simulation of triad fraction data for MMA/MA

0 70.80.9

1

0.30.40.50.60.7

Tria

d fr

actio

n

00.10.20 3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 11 Triad fraction calculation of Methyl methacrylate-Methyl acrylate co-polymer

f1 (MMA)


29/31

Figure 11. Triad fraction calculation of Methyl methacrylate-Methyl acrylate co-polymerT = 50 , MA-centered (Experimental data from Kim and Harwood, 2002)

IPR 20

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Summary

• Even with a limited number of monomers,Even with a limited number of monomers,

a very large number of combinations

• Complicated multi-component system

• Lack of experimental data

30/31

IPR 20

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Many Thanks to…

• Prof. T. A. DueverProf. T. A. Duever

• Prof. A. Penlidis

• BASF SE

Questions?

31/31

IPR 20

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Supplementaries- Simulation work

Kinetic studyM l i

Literature searchModel testing

Multi-componentFree-radical

P l i ti

Modeling study

PolymerizationSimulation

Modeling studyCodingTrouble shootingParameter estimation

32/31

IPR 20

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Copolymerization characteristicsd- 2nd order Markov (penultimate) model

M M + M kp111 M MM1M1 + M1kp111 M1M1

M1M1 + M2kp112 M1M2

112

1111

p

p

kk

r =212

2111

p

p

kk

r =′

M1M2 + M1 M2M1

M1M2 + M2 M2M2

kp121

kp122

p p

2222

pkr = 122

2pk

r =′1 2 2 2 2

M2M1 + M1 M1M1

M M + M M M

kp211

kp212

2212

pkr

1212

pkr

k kM2M1 + M2 M1M2

M2M2 + M1 M2M1kp221

k

111

2111

p

p

kk

s =222

1222

p

p

kk

s =

33/31M2M2 + M2 M2M2

kp222IP

R 2008

Microscopic approach - Average Sequence Length (Ray)

( )∫ ∑∑ ⎟⎞

⎜⎛ ∞∞ X

1 ( )

( )∫ ∑

∫ ∑

∑

∑

⎟⎞

⎜⎛

−⎟⎠

⎞⎜⎝

⎛

==∞

−

=

−

∞=

Xn

iiin

nii

nRayin

Rayi

dXFPP

dXFPnP

N

NnN

1

0 1

1

_______1

_______

,_______

,

1

1

( )

( ) ∫∫

∫ ∑∑ −⎟⎠

⎜⎝ ==

XX

iiin

nii

nRayin

dXFdXFP

dXFPPN0 1

1

1,

111

1

( )( )

( ) ( )

( )

∫

∫

∫

∫ −=

−

−−

≈ X

iii

X

iiiii

dXF

dXFP

dXFP

dXFPP 00

2 1

11

11

Cumulative number average sequence length

( ) ( ) ∫∫ − iiiiii

dXFdXFPP 00

11

34/31

Cumulative number average sequence lengthIP

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Microscopic approach - Average Sequence Length (Ray)

( )∫ ∑∑ ⎟⎞

⎜⎛ ∞∞ X

12 ( )

( )∫ ∑

∫ ∑

∑

∑

⎟⎞

⎜⎛

−⎟⎠

⎞⎜⎝

⎛

==∞

−

=

−

∞=

Xn

iiin

niiRayin

nRayi

dXFPP

dXFPPn

Nn

NnW

1

0 1

12

________

________

,1

2_______

,

1

1

( )

( ) ∫∫

∫ ∑∑

++

−⎟⎠

⎜⎝ ==

Xii

Xii

iiin

nii

nRayin

dXFPdXFPP

dXFPnPNn0 1

1

1,

111

1

( )( )

( )

( )

∫

∫

∫

∫ −=

−

−−

≈ X

iii

ii

X

iiiii

ii

dXF

dXFP

dXFP

dXFPP 0

20

3

11

11

11

Cumulative weight average sequence length

( )( ) ∫∫ −− i

iiiii

ii

dXFP

dXFPP 00

2 11

1

35/31

Cumulative weight average sequence lengthIP

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Microscopic approach - Average Sequence Length (HMP)

∫⎞⎛ ∞∞ X

( )

( )∫

∫ ∑

∑

∑ −⎟⎠

⎞⎜⎝

⎛

== =

−

∞

∞

=X

iiin

nii

nHMPin

HMPi

d

dXFPnP

N

NnN 0

2

1

1

_________1

_________

,________

,

1

( )

( ) ∫∫

∫∑ −=

XX

iiin

HMPin

dXFdXFP

dXFPN

2

01,

11

1

( )( )

( ) ( )∫

∫

∫

∫=

−−

≈ X

i

X

iiiii

dXFP

dXF

dXFP

dXFPP 00

22

11

11

1

( ) ( )∫∫ −− iiiiii dXFPdXFP00

11

C l ti b l th36/31

Cumulative number average sequence lengthIP

R 2008

Microscopic approach - Average Sequence Length (HMP)

( )∫ ∑∑∑ −⎟⎠

⎞⎜⎝

⎛ −∞∞∞ X

iiin

iiHMPiHMPi dXFPPnNnNn 212_________2

_________2 1( )

( )∫

∫ ∑∑

∑

∑

−

⎟⎠

⎜⎝=== ==

∞

=

=X

iiiHMPi

iiiiin

HMPi

HMPinn

nHMPin

HMPinn

HMPi

dXFPNN

Nn

Nn

NnW

0

________

,

0 1________

,

,1

1

_________

,

,1

________

,

1

( ) ( ) ∫∫

∫

∫+

−+

+ XX

X

iii

ii

X P

dXFPP

P0

2 111

1( )

( )

( )

( )

∫

∫

∫

∫

∫

∫ −+

=−

=−

−−+

≈ X

i

iii

ii

X

i

iii

X

iiiHMPi

iiiii

ii

dXF

dXFPP

dXF

dXFP

dXFPN

dXFPPP

00________

,

0

23 1

11

1

111

( )

( )

∫

∫

∫∫

−

i

X

iii

iiiiHMPi

dXFP

0

0

00,

1

37/31Cumulative weight average sequence length

IPR 20

08

Microscopic approach - Triad fractions

No ofNo. ofMonomerSpecies

Distinguishabletriads

Total possibletriads

1 (homo-) 1 1( )2 (co-) 6 83 (ter-) 18 274 (t t ) 40 644 (tetra-) 40 645 (penta-) 75 1256 (hexa-) 126 2167 (hepta-) 196 343… … …19 3610 6859

38/3120 4200 8000

IPR 20

08

studies on chemical heterogeneity of multi-component ... filestudies on chemical heterogeneity of...

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