Polymer Characterisation

7/30/2019 Polymer Characterisation

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Polymer Characterization

1. Polymer Properties and Characterization Methods

2. The Single Macromolecule, Polymer Chain

2.1 Configuration2.2 Conformation

- models of polymer chains, end-to-end-distance, radius of gyration,

concentration range, entanglement, rubber elasticity

2.3 Constitution

- Linear, Branched and Cross-linked Polymers

2.4 The Statistical Character of Polymer Properties

- molecular weight, chemical heterogeneity, tacticity

2.5 The Polymer Solution

- solubility of polymers, dilute polymer solution, the FLORY-HUGGINS-eq., solubility parameter,

- phase separation, theta-temperature

3 Determination of Molar Masses

3.0 Overview

3.1 Colligative Properties

- thermodynamic relations

3.2 Vapor Pressure Osmosis

3.3 Membrane Osmosis

(3.4 End group analysis)

4 Scattering Methods

4.1 RAYLEIGH-Scattering

- determination of weight-average molecular weight4.2 DEBYE-Scattering

- determination of shape and size of macromolecules, the scattering

function4.3 Dynamic Light Scattering

- diffusion coefficient, hydrodynamic radius

5. Ultracentrifuge

6. Viscometry of Dilute Polymer Solution

- KUHN-MARK-HOUWINK-eq., FLORY-FOX-eq.


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7. Molecular Weight Distribution

7.1 General Characterization of Distribution Curves

- mathematical formalism, types of m.w.d.

7.2 Separation effects and methods for polymer fractionation

7.3 Polymer-Chromatography

- Fundamental, HPLC, SEC, equipment, application, detectors

(7.4 Field-Flow-Fractionation)

8. Nonlinear Polymers

8.1 Branched Polymers

- short chain branching

- long chain branching

8.2 Cross-linked Polymers- mechanical properties

- swelling

Copyright 2008 by Karl-Fr. ArndtPhysical Chemistry of Polymers

TU Dresden, D-01062 Dresden, Germany

No part of this booklet (esp. figures!) may be reproduced or copied in any

form or by any means without permission!

Only for personal use!


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1. Polymer Properties and Characterization Methods

Polymer solution: Polymer chains in dilute solutions are isolated and

interact with each other only during brief times of encounter

determination of structural parameters of single chains by

measurements of the properties of a polymer solution:

molecular weightosmometry (membrane, vapour)

light scattering (static, dynamic)

viscometry of dilute polymer solution

ultracentrifugtion

molecular weight distributionfractionationgel chromatography

chemical heterogeneityspecial techniques of chromatography and fractionation

(combined methods)

chain lengthlight scattering

viscometry (hydrodynamic properties)

diffusion coefficientdynamic light scattering

shape of polymer chainscattering methods

branchingdetermination of typical dimensions of macromolecules LCB

spectroscopy (number of end-groups) SCB

swellingchange of macroscopic dimension of a cross-linked polymer determination of cross-linking density

further aims of investigations of polymer solutions:

solubility of polymers (thermodynamic of polymer solution,

interaction polymer solvent)

concentration range (dilute regime semi-dilute regime)

polymer solution at high concentration


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2. The Single Macromolecule, Polymer Chain2.1 Configuration

Position and arrangement of atoms in a polymeric chain can

be divided into two categories:

- Position fixed by chemical bonds (cis- or trans-isomers).This kind of order which can be changed only by disrupting chemical bondsand

is known as configuration.

- Structural differences originating from rotations around single bonds (various

forms of polymer chains in solution); conformation

2.2 Conformation

Freely Rotating Chain

C lCC: 0.154 nm (1.54 )CCC (): 109.5

C C

0.252 nm

=

+

=h n l n l M o f CC CC CC CC cos

cos

~,.2 2 1 01

1

2

Restricted Rotation

=+

+

=

h n l

h

o r b b

o f

cos

cos

cos

cos

,

,

2 1

1

1

1

: steric factor, hindrance parameter

=

cosexp( ( ) / ) cos

exp( ( ) / )

U kT d

U kT d

0

0

Freely Jointed Chain

KUNH segment chain ns = 7

ns = number of segments lsls = length of one segment; ls = f (chain stiffness)


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h: end-to-end distance

Cartesian coordinates

w(h): density function of distribution of h:

spherically symmetrical

Mean-Square Value: = =

h dh w h h n l M o s s ( ) ~ .1 00

*)

unperturbed (undisturbed) dimension

(sum of all interactions = 0 ! see theta-condition)

Contour length: maximum chain length

hmax = ns ls hmax = nsls ~ M , for vinyl-polymer:hmax = nCC lcc(eff) = 2 (M/Mo) lcc(eff) = ns ls ns = 2 (M/Mo) (lcc(eff)/ls)

*) x e dxn

a

n ax

n n

2

0

1 1 2

1 3 2 1

2

+ + =

/

...( )

w h dhh

h

hh dh

o o

( )

4

/

=

3

2

3

2

3 2

exp

w dxdydzh

x y z

hdxdydz

o o

(x,y,z)

/

=

+ +

3

2

3

2

3 2

exp

( )


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Radius of Gyration

= =+ s n r n so ij i

iji

1

2

1

1

2 2

Characteristic Ratio C:

= = = h n l n l C C C o s s CC CC n n ; lim2 2


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Radius of gyration and characteristic chain dimension of

different types of macromolecules:

Shape of

macromolecule

Characteristic dimension

radius of gyration

Characteristic dimension

molecular weightunperturbed coil o = 6 o o ~ M

random coil,

good solvent = (2 +)(3 + ) ~ M 1 +

thin rod,

length L

= 12 L ~ M

hard sphere,radius R *)

= (5/3) R~ M1/3

disc, radius R = 2 R ~ M1/2

ellipsoid (a + b + c) = 5

*) Example:< > = ==

=

=

=

r

r r dr

r dr

Rr

r R

r

r R

4

4

3

5

0

0

2.3 Constitution

Branching

long chain branching (LCB): < 10 C-atoms

broader m.w.d.

influences melt and solution propertiesshort chain branching (SCB): 2...6 C-atoms

influences properties in solid state

(crystallinity, density)

e.g. polyethylene:

PE-HD (high-density): 1..10 SCB/1000 C-atoms

PE-LD (low density): SCB + LCB

PE-LLD (linear-low-density): copolymerisation with 1-olefins,10...40 SCB/1000 C-atoms


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other branched structures: hyperbranched polymers

dendritic polymers

Crosslinking

elastomers (rubber)

high-crosslinked polymers (e.g. epoxy-resin)

interpenetrated networks

2.4 The Statistical Character of Polymer Properties

Polydispersity and Mechanism of Polymerization

Formation of polymers from monomers is divided into two reaction

classes:

-

step-growth (step polymerization, step polyaddition)polymerization

- chain-growth (or addition) polymerizationDistinction between these processes can be made on mechanistic

grounds (dissimilar reaction kinetics):

different distributions of species as a function of the extent of

reaction

different molecular weight distribution (m.w.d.)

step growth reaction: 1

monomer + monomer, monomer + oligomer, monomer/oligomer

+ macromolecules, macromolecules + macromolecules

(polycondensation : PA, PC ; polyaddition : PUR, EP)

chain-growth reaction: 2

monomer + macromolecule

(polymerization: PE, PVC, PP, other vinyl-polymers)


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P 2

n t

2

T2 > T1 1

1

M

reaction time t1 < t2 < t3 < t4

n = Zahl der Molekle

Distributions and Statistical Weights

g = fraction distribution of a property Q

moment of a distribution (general definition): Q

g Q

gg

i ia

i

i

i

a=

1

frequency- cumulative-distribution

g g

discontinuous

Q Q

g

continuous


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Average Molecular WeightsAverage Statistical Weight Definition Alternative Form

Number, Mn niM

n M

nn

i i

i

i

i

=

M

w

w Mn

i

i

i i

i

=

/

Weight, Mw wiM

w M

ww

i i

i

i

i

=

M

n M

n Mw

i i

i

i i

i

=

2

z-averaged, Mz ziM

z M

zz

i i

i

i

i

=

M

w M

w Mw

i i

i

i i

i

=

2

Viscosity-average,

M0

wiM

w M

w

i ia

i

i

i

a =

1

Polydispersity and Mechanism of Polymerization

The width of a molecular mass distribution curve is characterized by

the ratios of averages of molecular mass (polydispersity index, Q =

Mw/Mn), combined with one absolute value of an average mass (Mw).

Also (Q 1) = U is used, U = non-uniformity index (SCHULZscheUneinheitlichkeit);

U = (n/Mn)

( ) ( ) ; ( )= =

+

+

x f x dx x f x dx

n n nM M h M dM M M h M dM

h M d M

2

0 0

0

1

= =

=

+ +

+

( ) ( ) ; ( )

( )

or: gw = (w/Mw) =Mz/Mw = 1 as a measure of polydispersity


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The numerical value of Q is determined by the reaction mechanism of

the polymerization and by the condtions (p, T..) under which it was

carried out.

Living Polymers Anionic

Group-transfer

Mw/Mn

= 1.01.05

Condensation

polymers

Step reaction of

bifunctional monomers ~ 2

Addition

polymers

Radical addition

Kationic addition

Coordination-polym.

(metal-organic complexes)

2 10

2 30

Branched polymers Radical addition 2 50

Network polymers Step-reaction of tri-, tetra-

functional monomers

at the gel-point

2.5 Thermodynamics of Polymer SolutionsNotation: A = solvent; B = solute (polymer)

in case of copolymers or multi-component systems:

1 = solvent; 2, 3polymer

Entropy of mixing: The Flory-Huggins theory (1)

Deviation of polymer solutions from ideal behavior is mainly due to low mixing

entropy. This is the consequence of the range of difference in moleculardimensions between polymer and solvent.

Flory (1942) and Huggins (1942)

Calculation ofGm = G(A,B) - {G (A) + G (B)}

H = 0Gm = -T Sm lattice model transfer of the polymer chains from a pure, perfectly ordered state

to a state of disorder


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mixing process of the flexible chains with solvent moleculesSm = S(NA, NB) - {S(NA) + S(NB)}

Sm = -R (nA ln A + nB ln B)

A = volume fraction solvent = NA/K = nAVA/( nAVA + nBVB)B = volume fraction polymer = NB/K = nBVB/( nAVA + nBVB)

Flory-Huggins theory (3); chemical potential

A = RT ln aA = RT (ln A + (1 - VA/VB) B)

B = RT ln aB = RT (ln B + (1 VB/VA) A)

A = f (M) !

VA/VB = [BMA/AMO] 1/P VA/VB ~ 1/P ~1/MB

Enthalpy change of mixing

H = RT nAB= Huggins interaction parameter

Gibbs enthalpy based on Flory-Huggins theory:

Gm = RT (nA ln A + nB ln B + nAB)


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Theta-temperature and Phase separation (1)

critical point:

B cB A

cA

B

A

BV V

V

V

V

V,

/;=

+= + +

1

1

1

2 2


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Solubility Parameter

The strength of the intermolecular forces between the polymer

molecules is equal to the cohesive energy density (CED), which is the

molar energy of vaporization per unit volume. Since intermolecular

interactions of solvent and solute must be overcome when a solute

dissolves, CED values may be used to predict solubility.

1926, Hildebrand showed a relationship between solubility and theinternal pressure of the solvent;

1931, Scatchard incorporated the CED concept into Hildebrandseq.

= HV2 (nonpolar solvent; H= heat of vaporization)heat of mixing: Hm = VAB(A - B) = nAVAB (A - B)

Like dissolves like is not a quantitative expression!

Problems: polymers with high crystallinity;

polar polymers hydrogen-bonded solvents or polymers

additional terms


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The square root of cohesive energy density is called

solubility parameter. It is widely used for correlating

polymer solvent interactions. For the solubility of polymer P

in solvent S (P - S) has to be small!


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3. Determination of Molar Masses3.0 Overview

Method Molar mass

average

Range (g/mol)

**)absoluteOsmometry Mn 10

4 < M < 106

Isothermal

distillation

Mn M < 104

Ebullioscopy Mn M < 104

Cryoscopy Mn M < 104

Vapor pressureosmometry*) Mn M < 10

4

Scattering

methods

Mw M > 5 102

Ultracentrifugation Mw, Mz M > 102

relative

Viscosity M

M > 103

Size exclusion

chromatography

Mn, Mw, Mz M < 106

equivalentEnd group

determination

Mn M < 5 104

*) relative method due to experimental condition**) depends on experimental condition (solvent, polymer, equipment..)

***) parameter of Kuhn-Mark-Houwink equation were determined only in asmall range of molar mass, KMH is not valid in low molar mass range.


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3.1 Colligative properties:

cryoscopy:

T T TRT x

Hxm m m

m B

mB= = ,

,0

02

ebulliometry:

T T TRT x

Hx x c

M

Mb b b

b B

bB B

A

B

= = ,,

;00

2

Relative Comparison of the Sensitivity of Various Colligative Properties*)

Property Mn = 104

g/mol Mn = 105

g/mol Mn = 106

g/mol

Osmotic pressure (cm

solvent)

30 3 0.3

Vapor pressurelowering (cm Hg)

8 x 10-3

8 x 10-4

8 x 10-5

Freezing point

depression (C)

5 x 10-3 5 x 10-4 5 x 10-5

Boiling pointevaluation (C)

2.5 x 10-3

2.5 x 10-4

2.5 x 10-5

*) 1% solution of polystyrene in benzene, 25C

only vapor pressure osmometry is used in polymer (oligomer)

characterization

3.2 Vapor Pressure Osmometry

Measurement Data evaluation

TRT

Ha with FH equo

bA=

2

ln .

Tc

KM

A cen

v A= +

12,

solv. solution T/c

solvent

Ke = calibration constant;

c = g/g; Mn > 103

g/mol: A2,v = A2,0 c


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3.3 Osmotic PressureThe measurement of osmotic pressure is the most important method

for the determination of the number average molar mass Mn.

Schematic representation of the osmotic pressure between solvent ()

and solution ()

Osmotic equation:

c

RT

MB c RT

MA c

B nB

nB= + = +

12

Data evaluation:


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4. Scattering Methods4.1 Rayleigh-ScatteringLord RAYLEIGH (1842-1919):

Intensity of scattered light ~ (wave length)-4

We measure the reduced scattered intensity (RAYLEIGH

ratio) R :R =

I( ) r

IR solution) - R solvent)

2

o

( (

angle of observation, r distance sample-detector, Io intensity of theprimary radiation (incident beam)

Concentration fluctuations determine the intensity of the

scattered light:Kc

R R T cR T

c

MA c

cR T

1

MA c

B

B

B2 B

2

B2 B

=

= +

= +

1

2

T

T

;

with K = optical constant: ( )

F

c

nn4=K

2

B

4

o

2o

2

LN

F(): polarization of the primary light and dependence of the

effective scattering volume on the scattering angle (1/sin):

vertical pol.: ( ) ( )F =

sin1

non-polarized: ( )F

=

+12

2cos

sin

horizontal pol.: ( )F

=

cos

sin

2

Light-scattering equation

monodisperse:

Kc

R MA cB 2 B

= + +1

2 ......


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polydisperse:Kc

R MA cB

w2 B

= + +1

2 ......

4.2 DEBYE-Scattering (M > 105 g/mol)

interference between the light scattered from various positions on the

particle (intramolecular interference):

K c

R P MA cB

w2 B

= + +

12

( )......

particle scattering function P(): P() = R()/R( = 0)

P( )= 1N

sin(qr )

qr

P( )=1

M

R

KcB

2ij

ij

wC 0

ji

q scattering vector: q =( 4 / )sin / 2 ; = / no

For q rij


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Zimm-Plot

4.3 Dynamic Light Scattering Fundamental

Due to the Brownian motion of the scattering particles, a shift of the

frequency of the scattered light occurs in analogy to the well-known

Doppler effect of acoustic waves

Light ~ 1015

Hz; : 105....107 Hz).

In principle, the diffusion coefficient D of the particles may be

obtained from an analysis of the line width of the spectral density

profile of the scattered light intensity.

Hydrodynamic radius:

Rk T

Dh

B

A

=6


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5. Ultracentrifugation

- absolute method- large scale of molar masses between 10 and 109 g/mol- besides the molar mass and molar mass distribution the AUC

makes it possible to determine sedimenation coefficients, diffusion

coefficients, and virial coefficients.

Sedimentation velocity

large angular velocities, Svedberg equationfriction force = centrifugal force

buoyancy forceK K

fdf

dt

M

N x

f s

B

L

=

= ( )1 A Bv

with f = kBT/Df = friction c. ; D = diffusion c.

and sdx / dt

x=

s = sedimentation c.; s ~ Ma

Ms

D

R T

1B = A Bv

Svedberg-equ.

Sedimentation equilibriumcross-section q net-mass transport:

dm/dt = dms/dt + dmD/dt

Diffusion Sedimentation

dmD/dt dms/dt

equilibrium: dm/dt = 0 !dm

dtq s c D

cB

B=

=

xx 0

sc Dc

BB

xx =

c (x) c (x x )M

R TB B b

B= =

exp ( ) ( )1

2 A B bv x x

xb = bottom

( ) ( )[ ]M R Tlnc

c x xB

B,2

B,122

12

=

1 A Bv ; concentration at two x-values


23/36

Experiments

Powerful tool, especially for complicated systems such as

polyelectrolytes, polymers with microgels, heterogeneous polymers,

and copolymers (sedimentation in a density gradient).

....60,000 rpm; equilibrium hours...2 days; multicell-rotors

different detectors

6. Viscometry of Dilute Polymer Solution

Viscometry is the most widely utilized method for the characterization

of polymer molecular mass since it provides the easiest and most rapid

means of obtaining molecular massrelated data and requires a

minimum amount of instrumentation.A most obvious characteristic of polymer solutions is their high

viscosity, even when amount of added polymer is small.

left: velocity profile, flow through a tube

right: velocity gradient in a macromolecule relatively to the center of mass:- - - - freely-draining; partially-draining

As a result of the velocity distribution the macromolecule rotates, and

energy is dissipated (internal friction). The higher the internal friction

in the fluid, the higher is the stress to maintain the velocity gradient.

increase of viscosity


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Measurement of viscosity:

HAGEN-POISEUILLE`s law:

= = r gh t

l VA t t

4

8~

Hagenbach-correction, correction for contribution to kinetic energy:

tA

B

tfor t s no correction= >

; 100 ;

experimental determination of B: /t vs. 1/t

The limiting viscosity number (intrinsic viscosity):

Nomenclature of solution viscosity (IUPAC):

Viscosity ratio (relative viscosity): rel t t= =0 0/

Specific viscosity: sp relt t

t= =

1 0

0

Viscosity number (reduced viscosity):

redsp

Bc=

Logarithmic viscosity number (inherent viscosity): inh relBc

= ln

Limiting viscosity number (STAUDINGER index, intrinsic viscosity): []o = viscosity of solvent

Determination of the limiting viscosity:

SCHULZ-BLASCHKE: red= kSB []sp + []

HUGGINS: red= kH [] cB + []

KRAEMER: inh= kK[] cB + []

Limiting viscosity number and molecular weight

KUHN-MARK-HOUWINK: [ ] = K Ma

rodlike molecule: a = 2

random coil a = 0.6....0.9

freely-draining limit : a = 1

non-draining limit (theta conditions): a = 0.5

sphere: a = 0

disk: a = 0.5

[ ]

=

= lim lim

c

sp

B

10 00

0c

c G laminar

B

B; ; ,


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Experimental:Common solution

viscometers:

a)OSTWALD-viscometerb)UBBELOHDE-viscometer

PHILIPOFFs rule (empirical): [ ][ ]

=

w

w

i i

i

Mw M

w

i i

a

i

a

=

1/

FLORY-FOX eq. (r= /o)

[ ]

=

632

32

0

3

,

h

r

r

M


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7. Molecular Weight Distribution

7.1 General Characterization of Distribution Curvesh(M) density function of number frequency distribution

H(M) density function of mass frequency distribution

H(M) = M h(M)

I(M) distribution function, frequently termed integral mmd

(M = P M0; P = polymerization degree

I(M) = H(M)dM ; H(M)dI(M)

dM

0

M

=

molecular inhomogeneity, polydispersityU P P P 1; g P P P 1n

2n2

w n w w2

w2

z w= = = =

Distribution Functions:SCHULZ-ZIMM (FLORY): 1-parameter (Pn)

h(P)1

Pexp

P

P; h(P)dP 1

n n 0

=

=

SCHULZ-ZIMM (FLORY): 2-parameter (k,Pn)

( )H(P)

k!P exp

k P

P; H(P)dP 1

kP

k+1

k

n 0

n=

=

Evaluation of mean values:

Pk

kP P

k

kP

dH P

dPfor P P

w n z n

n

= = + = = +

= =

P H(P)dP

H(P)dP

P H(P) dP

P H(P) dP

P(max):

21 2

0

;

( )

U = Pw/Pn 1 = 1/k

x e dx n n! if nn ax

0

+ =+

+ = =

(n 1)

a; (n 1) (n) ( > 0; whole number)

n 1


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POISSON-Distribution

h(P)P exp( P )

P!P

nP

n

n

=

Pw = 1+ Pn; U = 1/Pn

living polymerization

WESSLAU-Distribution

h(P)=1 1

P

l n P / Po

exp

2

I(lnPo) = 0.5; = (dI(P)/dP)P=Po

Pn = Poexp(-/4); Pw = Poexp(/4);

SCHULZ-ZIMM (FLORY):

k = coupling constant, numerical value depends on terminationreaction (radical polymerization)

k = 1 : disproportionation ( hydrogen abstraction from one growing chain

by the radical end of another)

two chains - unsaturated terminal unit

- saturated chain end

k = 2 : combination (formation of a -bond, when free radical sites

collide, formation of one chain

k = 1.5 : chain transfer to solvent molecules, monomers, polymerization

regulators (no branches) or other polymer chain backbone branching


28/36

7.2 Separation Effects and Methods for Polymer Fractionation

Separation effects Separation methods

Thermodynamic properties

precipitation fractionation

fractionated extractionsolubility solubility fractionation with or

without supporting materials

with solvents/precipitation agents

and/or temperature gradients

in separation columns

adsorbability thin-layer chromatography

(separating copolymers acc. to

chemical heterogeneity)

Kinetic effectsdiffusion and sedimentation ultrazentrifugation

field-flow-fractionation

migration in an electrical electrophoresisfield

Particle size effectssize exclusion chromatography

field-flow-fractionation

Combination of different separation effects!

Experimental

- Fractional PrecipitationAddition of nonsolvent (or precipitant)

(continuously column fractionation, discontinuously fractionationflask)

Elimination of solvent by evaporation (not common)

Lowering the temperature of the system (ucst, or vice versa)

(experiment is carried out in one solvent, volume of system is constant,

control of fraction size can be adjusted more precisely)

- Fractional SolutionFractionation by direct extraction

(successive extraction of polymer with a series of solvents)Fractionation by film extraction


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(FUCHS, 5-10 m film of the polymer is deposited onto an aluminum foil

by dipping the foil into the polymer solution)

Fractionation by coacervate extraction

(coacervate = polymer-rich liquid phase; addition of a nonsolvent until

essentially all of the polymer is in the coacervate, polymer-poor phase is

removed and polymer in it is isolated, then extraction of the coacervate...)

- Fractionation by column elution(polymer on a support, e.g. glass beads, gradient elution increasing

content of solvent in the nonsolvent, additional temperature gradient)

- Turbidimetric Titrationfractional precipitation, a nonsolvent is slowly added to a dilute polymer

solution, the polymer is precipitated selectively, the concentration of

precipitated polymer is measured by optical methods (turbidity of theliquid phase)

Applying the method to copolymers (block and graft copolymers):

information is desired concerning the extent of copolymer formation and

the amount of homopolymer present

cloud point titration: cloud point depends on the chemical constitution


30/36

7.3 Separation effects in high performance liquid chromato-

graphy (HPLC) and size exclusion chromatography (SEC):

HPLC: energetic effects (enthalpic);interaction between the stationary phase and the molecules. HPLC utilizes

columns that are intended to encourage adsorption and partition mechanism.Reversed phase columns: the packings are less polar than the mobile phaseBinary, ternary...miscible mixtures of organic solvents and water are employed

(isocratic elution: constant ratio; gradient elution: ratio vary with time)

HPLC is used to analyze small molecules (end-group), and

copolymers

retention time ~ A + B lg M

SEC: entropic effects, conformation of the

macromolecules

retention time ~ A - B lg M

Detector combinations and their application (SEC):

concentration RI; UV; copolymersFTIR

concentration RI and molecular weigth

and size viscosity distribution, long-chain-

branching,

constants of KMH

RI and molecular masslight-scattering (absolute!), radius of

gyration, microgels,

association, separation

process


31/36

Calibration of SEC

Calibration with narrow standards: Ve = f (lg Mstandard)

Ve = A B lg M

Problem: A, B = f (polymer, solvent, exp. condition)In the general case the lgM Ve dependence is not linear and can be

approximated by polynomials of the form:V ae i

i

n

== (lgM) i

0

(Calibration by using broad (polydisperse) standards with known molecular-

weight distribution is possible, but complicate. )

MWD of standards logarithmic normal distribution: maxima in theirchromatograms can be related to M = (Mw Mn)

1/2.

Universal Calibration:hydrodynamic volume: Vh = [] M/2.5Vh = molar volume of impenetrable spheres which would have the same

frictional properties or enhance the viscosity to the same degree as the actualpolymer in solution

1966 Benoit, Grubisic, Rempp:[]x Mx = []PS MPS at constant Ve

x indicate data of unknown polymer, PS indicate data of calibration standard

With Kuhn-Mark-Houwink follows:

lg lg lg,

,

,

,

,

Ma

K

K

a

aMx

x

PS

x

PS

x

PS= ++

+

+1

1

1

1

MW of unknown sample can be calculated if K0,x and a0,x are known!

If K0,x and a0,x are not available, Mx can be derived from measurements

performed with an SEC equipped with an online viscosity detector.

[]x,i Mx,i vs Ve,i is measured , equivalent to []PS,i MPS,i vs Ve,iMx,i

or: direct monitoring of MW by light-scattering detector!

Molecular Weight Distribution: w(M) = F(Ve) [dVe/dM]


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7.4 Field-Flow-Fractionation

The separation takes place in a thin, open channel through which

samples are transported by a laminar flow of liquid (1966 Giddings).

The action of a field, applied perpendicular to the channel flow, causes

a partitioning of the various components into regions of different flowvelocities differential migration and separation.

centrifugal fields for the characterization of colloidal materials; electrical fields for protein separation and purification; thermal gradient thermal FFF; hydraulic gradient flow FFF;In contrast to SEC: analysis in an open channel with defined

dimension, formulation of rigorous theories for both retention and

column dispersion!

Retention Mechanism:

Side view of the

FFF channel

field perpendicular to

separation axis (z)

parabolic velocity profile v(x): v(x) 6 vx

w

x

w=

Sample is exposed to the field minimization of potential energy

steady state velocity U (proportional to the strength of interaction with

the field). The migration is hindered by the channel wall formationof a layer with thickness l = D/U;

D = diffusion coefficient (back diffusion)

concentration distribution: c(x) = c(x=0) exp(-x/l);l = thickness of the distribution of particles (fractionation, if lA lB)

with: U = F/f; D = kT/f l = kT/FF = force, f = friction coefficient

x x=w field

flow (v(x))

x=0 z


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8. Nonlinear Polymers

8.1 Branched Polymers: short chain branching (SCB)

long chain branching (LCB)

comb-model

star-model

tree-model

Structural parameters for describing branching structures:- type of branching, e.g. star, comb- average number of branching-points per molecule,

branching density or branching frequency

- functionality of branching points-

number of branching end points- average length of side branch and no-branched chain- distribution of branch lengths, non-branched chain

segments,

- average molecular weight and distribution of backbone chainInfluence of chain branching on polymer properties:

SCB LCB

Crystallinity Limiting viscosityDensity Elution volume in SEC

Melting point Sedimentation

Dielectric constant Virial coefficient

Creep and fracture Angular distribution of scattered

light intensity

Radius of gyration


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Determination of total number of branches:

determination of the concentration of end-groups comparedwith C-atoms in the backbone (CH3/ 1000 CH2);

IR - Method (LDPE: absorption at 1378 cm-1

)1-H nmr (LDPE: CH3 at 9.11 ppm; CH2 at 8.72 ppm)

13-C nmr (side chains up to C

6)

Specific methods for determination of SCB:13

-C nmr spectrum; branch length and their concentration (e.g.LDPE: n-butyl:

67 %; n-pentyl: 12 %; n-ethyl: 8; n-hexyl and longer: 5 %)

Degradation, scission of branches at branch points

(PVAc: saponification of ester branch units and calculation of initial and

final molecular weight)

Radiolysis gas chromatography: analysis of the decomposition gases

Determination of LCB:Branched molecules have a smaller coil dimension than a linear molecule of the samemolecular weight. shrinking factor g

radius of gyration gr

ro

o br

o lin M Mbr lin

=< >

< >

=

,

,

viscosity g g h hR

Ro

o br

o lin M M

oh br

h linbr lin

=< >

< >

= = =

=

,

,

/ ,

,

3 2 3;

experiments: g g bb = ; 0 5 15. .

Theoretical equations relating the shrinking factor (branching degree) and the number of

branch points, were derived - by ZIMM and STOCKMAYER for randomly branched mono-

disperse and polydisperse polymers; - by OROFINO and BERRY for star-branched polymers.

Experimental background:

Viscosity Light scattering

lg[]

lg M lg M

(--- branched polymer;____

linear polymer)


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8.2 Cross-linked Polymers

chemically cross-linked polymer interpenetrating network

physically cross-linked

f = functionality;

= cross-linking densityMc = Molecular weight of the cross-linked chain

= B/Mc


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Rubber-Elasticity-Theory (RET):A rubber represents an ensemble of polymer chains, each one running

between two cross-links (junction points). It is necessary two assume

correlation between the macroscopic dimension of the sample and the

chain dimension (deformation model).

no hindrance at deformation phantom-networkGaussian distribution of chain length;

only change of entropy (U = 0);

no volume change at deformation (Vdef= Vnondef; xyz = 1)

affine deformation: the cross-link points are fixed within the sample;

their displacement is linear in the macroscopic strain

(other limit: free-fluctuating phantom network)

2= TRA

= stress (force/area non-deformed sample)= deformation ratio (x = xdef/xo; xo = non-deformed dimension)

= memory-term (depends on polymer concentration at cross-linking,(dry state: = 1)

A = microstructure factor; A = 1 affine deformation, A = (1 2/f) free-

fluctuating

Experimental Background:a) compression of swollen sample

b) stress-strainSwelling, the FLORY-REHNER-eq.:

Two competing processes take place:

- increase of entropy as a result of the introduction of solvent molecules- decrease of entropy of the network chain as a result of the dilationSwelling degree:

Q = Vswollen/Vnon-swollen = 1/(volume fraction polymer)

or: Qm = (mpolymer+ msolvent)/mpolymer(independent of T !)

FR: 1943, statistical theory, calculation of change of free-

energy or chemical potential of solvent due to swelling

measurement of QMc;

Polymer Characterisation

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