UNESCO/IUPAC Postgraduate Course in

Polymer Science

Molecular weight and dimensions of polymers and their assemblies

Jiří Horský

Institute of Macromolecular Chemistry ASCR, Heyrovsky sq. 2, Prague -162 06

http://www.imc.cas.cz/unesco/index.html

unesco.course@imc.cas.cz

Lecture:

Polymers

M gives independent information

M affects

properties such as density, degree of crystallization, Young modulus, melt viscosity, glass transition temperature etc

Consequently, M affects application use and processing methods

M

cRTc =Π → 0

[ ] aKM=η

+

+=

→

...2

sin3

161

1 2222

0

θλ

πθ

g

c

Rn

MR

Kc

Osmotic pressure (absolute, 5 103-106)

Static Light scattering (absolute, 104-107)

Intrinsic viscosity (relative, >103)

And many more - analytical ultracentrifugation, ebullioscopy, end-group analysis, vapor pressure osmometry, ….

Classical methods for determination of molecular weight of polymers(see Polymer solutions in a nutshell )

(Mark-Houwink Eq.)

Zimm plot (2 D plot for 2 independent variables) => M, A2, Rg

Polymer sample (usually) contains molecules of various polymerization degree

Experimental methods give us only some average of the molecular weight

a

c

KMc

Y =

→0

∑=i

iYY ∑=i

aii

a KMcMcK a

i

aiiMwM

1

= ∑

rigid 0.8 flexible8.05.011ηLS >−==−=Π aaa

∑

∑∑ −

−− =

=

iii

iii

iii

Mw

MwMwM 1

01

1n ∑

∑∑ ==

iii

iii

iii

Mw

MwMwM 0

1

w

∑

∑=

iii

iii

Mw

MwM 1

2

z

number-average weight-average z-average mol. weight

∑

∑∑ =

=−

iii

iii

iii

Mx

MxMxM 0

11

n∑

∑=

iii

iii

Mx

MxM 1

2

w∑

∑=

iii

iii

Mx

MxM 2

3

z

Averages with integer a can be defined as the ratio of moments around the origin

w weight fraction, x molar fraction

Combined averages give only partial information on the sample molecular weight non-uniformity : polydispersity index Mw/MnFull information:relative amount of each polymerization degree – a distribution function

number DF - xi, weight (mass) DF – wi

cumulative discrete vs continous DF

cumulative vs differential DF

∑∑==

i

ji

i

jj wx

11

dM

dcumDFdifDF =

Mathematically defined DF

Discrete DF Poisson most-probable)!1)(1(

1

−+=

−−

Pa

aPew

Pa

P12 )1( −−= P

P aPaw

0 25 50 75 1000.00

0.02

0.04

0.06

0.08

0.10

x

DP

Poisson PI=1.01Most probable PI=1.95

0 25 50 75 1000.00

0.02

0.04

0.06

0.08

0.10

w

DP

Poisson PI=1.01Most probable PI=1.95

Continuous DF

Schulz-Zimm

MaMMb

aw b

b

d)exp()1(

d1

−+Γ

=+

Mb

M

aMaw d)ln

1exp(

1d 2

2−=π

0 50000 100000 1500000.00000

0.00002

0.00004

dx/d

M

M

Mn=40000

PI = 1.1,1.3, 1.75, 2.0

0 50000 100000 1500000.00000

0.00002

0.00004

dw/d

M

M

Mn=40000

PI = 1.1,1.3, 1.75, 2.0

logarithmic normal

0 50000 100000 1500000.0

0.2

0.4

0.6

0.8

1.0

cummulative DFs

x, w

M

Mn=40000 M

w=60000

number DF, weight DF

0 50000 100000 1500000.00000

0.00001

0.00002

dw/d

M

M

Mn=40000 M

w=60000

log-normal, Schulz-Zimm

Principle of Size Exclusion Chromatography

solvent flow

Partition between the flowing solvent and the solvent in the pores

Elution time (volume) is a function of the particle (hydrodynamic) size

Above certain size all particles are equal, flow only through the interstitial volume

log M

V0 V0+ Vi

log M=a +b V +….

Detectors: (i) mass sensitive (refractometer, evaporative light scattering detection)(ii) molecularly sensitive (end group UV absorption)(iii) molecular-mass sensitive (light scattering, viscometric)

Standard configuration (i); up-to-date = multidetector (i)&(iii)

Me

5 6 7 8 9

V [mL]

I [a

rbitr

ary

units

] Baseline substraction

Normalization

∫=

R

L

d)(

)()( V

V

VVI

VIVw

5 6 7 8 90.0

0.5

1.0

1.5

2.0

2.5

V [mL]

d w

/d V

[1/

mL

]

Point to point correspondence of cumulative V and M or log M distributions

∫ ∫∫ ==L

LL

log

log

log

log

dlogdlog

d)(d)(dlog)(log

V

V

M

M

M

M

MM

VVwVVwMMw

V

MVw

Mw

dlogd

)()(log

−=M

MwMw

303.2)(log

)( =

5 6 7 8 90.0

0.5

1.0

1.5

2.0

2.5

V [mL]

d w

/d V

[1/

mL

]

3 40

1

2

3

M

d w

/d lo

gM

log M

∫ ∫∞ ∞

==0 0

1ddd

dloglogdd

MM

wM

M

w Mn = 3100 Mw = 3780 PI = 1.22

0 5000 10000 150000.0

0.1

0.2

0.3

0.4

M

104 d

w/d

M

M(V) – valid for the column, polymer, solvent etc

SEC with a mass detector only

• calibration by polymer standards

• universal calibration – hydrodynamic volume [η]M (V) Mark-Houwink Eq. log [η]M =log K+(1+a)log M

• polystyrene calibration

SEC with mass & molecular-mass detectors

viscosity detector – gives M through the universal calibration without Mark-Houwink parameters

static light-scattering detector

Averages of M: directly according discreet definitions

Distribution: in two steps (i) LS data are used for the calibration equation (ii) RI data are treated in standard way

Sedimentation Analysis – Analytical Ultracentrifuge(macromolecular archelogy? – Swedberg 1923, Nobel prize 1926)

Declared clinically dead in 1980’s but then resurrected (natural macromolecules)

centrifugal force=friction resistance (f u) => sedimentation velocity u

sedimentation coefficient sω

sedimentation velocity

T

vMD

f

vM

r

us

R

)1(

N

)1(

A2

ρρω

−=−==

ω

sedimentation equilibriumsmall ω sedimentation creates temporal concentration gradient

which is opposed by diffusion, equilibrium gradient is achieved

c

Mr

c

r~

d

d1

Principle of Field –Flow Fractionation(J. C. Giddings)

solvent flow

Elution time (volume) is a function of the particle (hydrodynamic) sizebigger particles (up to 0.1 mm vs. SEC <100 nm) smaller adsorption

change of mechanism (bigger particles first), absolute detection LS

Force perpendicular to the flow

fieldthermal, electrical..assymetrical flow

Why MS if we have SEC? ResolutionMS: M/w1/2=10000 SEC: Ntp=5.54(V/w1/2)2=10000 V/w1/2=42

The Nobel Prize in Chemistry 2002

“ for the development of soft desorptionionisation methods for mass spectrometricanalyses of biological macromolecules”

MALDI-TOF mass spectrometry

Mass Spectrometry

Mass spectrometry

zeUmv =2

21

Principle: a charge particle is accelerated in electromagnetic field

For measurement of molecular weight, M,

(i) the sample must be vaporized and charged (ion source)(ii) particles must be separated according to M/z (mass analyzer)(iii) particles with the same M/z must be counted (detector)

In electric field

Low-molecular-weight compounds:

M is used for identification

vaporization puts an upper limit on measured M

the most widely used electron impact ionizations fragments the sample molecules

Fragmentation gives information on the structure of the compound

For a long time, polymer researches used MS only for analyzing

At the same time, they were fascinated by the MS precision

polymer additives pyrolysis products

etc…

M atrixA ssitedL aserD esorptionI onization

T imeO fF light

M assS pectrometry

Matrix:2,5-dihydroxybenzoic acid1,8,9-trihydroxyantracene

Laser:337 nm, pulse <4 ns, 40 kW/4mW

Sample preparation

1- 2 µµµµL per spot (2-10x)

0.2% sample

1-2% matrix

& ionizitation agent (NaCl, AgTFA)

+

++

PST Mw/Mn(kDa)

MALDI 62.3/61.9

SEC 71.3/71.1

For samples with M>20 000 multidetector SEC is the first choice

3000 4000 50000

2000

4000

a.i.

m/z

40000

2000

4000

a.i.

m/z

3940 3960 39800

2000

4000

a.i.

m/z

MALDI-TOF MS can distinguish individual oligomers

and even isotopic resolution(in reflector mode)

Monoisotopic peak (C12 only)

The strength of MALDI-TOF is with samples M<20000 (For samples with M>20 000, the first choice is the multi-detector SEC)

M = Meg1 + n Mm + Meg2+Mia

HO [CH2CH2O] [CH2CH2O].....[CH2CH2O] H Na+

1200 1400 1600 1800 2000 2200 2400 2600 28000

1000

2000a.

i.

M/z

17+ 39 * 44 + 1 + 23= 1758

PEG:MPEGPEG:MPEG1 : 1

18000

1000

a.i.

M/z

1714

1758

17721816

1200 1400 1600 1800 2000 2200 2400 2600 28000

1000

2000

3000

4000

5000

6000

7000

a.i.

M/z

Mixture of two homopolymers

PEG:PPG

1700 1720 1740 1760 1780 1800 1820 1840 18600

1000

2000

3000

4000

5000

6000

7000

a.i.

M/z

PEG : PPG2 : 1

1758

1782

18021724

1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 35000

1000

2000

a.i.

M/z

Binary copolymers (PEG/PPG -Pluronic)

M = Meg1+ n Ma + m Mb + Meg2 + Mia

17500

1000

2000

a.i.

M/z

1750 1760 1770 1780 1790 18000

1000

2000

a.i.

M/z

1782

1782=17+30*58+1+23 1782=17+29*44+8*58+1+2322*58 = 29*44

d e n d r i m e r

Si

Si

Si

Si

Si

Si

Si

Si

Si

Si

Si

Si

Si

10 00 1 500 2 000 25 000

2 00

4 00

6 00

8 00

10 00

12 00a.

i.

M /z

∆ 126

1812

Si

SiSi Si

Si

Si

Si

Si

Si

SiSi

Si

Si

Ag

MALDI-TOF mass spektrumof 2nd generatiom carbosilane dendrimer

(defect detection)

Montaudo G et al. Prog. Polym. Sci 31, 277-357 (2006 )

www.iupac.org/publications/books/pbook/PurpleBook-C3.pdf

Kostanski LK et al. J. Biochem. Biophys. Methods 58, 159–186 (2004)

Literature:

UNESCO/IUPAC Postgraduate Course in

Polymer Science

Molecular weight and dimensions of polymers and their assemblies

Jiří Horský

Institute of Macromolecular Chemistry ASCR, Heyrovsky sq. 2, Prague -162 06

http://www.imc.cas.cz/unesco/index.html

unesco.course@imc.cas.cz

Lecture: