Polymer Physics: Finding Molecular Stats

8/12/2019 Polymer Physics: Finding Molecular Stats

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weights Oli omers Telomers

Pleistomers Telechelic polymers Macromers

Measurement of MW are usually ondissolved polymers


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Wetting of surface Then diffusion of solvent into polymer (case I Fickian

diffusion, or case II swelling)

Finally, polymer diffuses and dissolves

Solubility parameter determines which solventwill dissolve which polymer, from:

21/ 2 1/ 2

M M MG H T S

E E

1 2

1 2

anM M

V V


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Solubility parameter is the cohesive energy

density, EV


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experiments: For cross-linked polymers, from swelling coefficient,

0 1m m

Q m

For non-cross-linked polymers, from intrinsic

viscosity of the polymer in different solutions

Or theoretically, from group molar attraction constant

G

M


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. , . ,104g/mol3 1/21.05 .

104


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,

Ideal solutions is when the heat of mixing is zero, andRaoults law is obeyed

For incompressible mixtures,

1 1 2 2 1 1 2 2( ln ln )MG N G N G kT N n N n

1 1 2 2( ln ln )MS k N n N n

,relationlnMS k

0 1 2where !/ ! !N N N


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For ol mer ofx-mers then

1 1 2 2( ln ln )MS k N v N v

ori iiM

i

i

kV V

1

1

1 2

where vN xN

2

1 2

and v N xN

solutions also exist


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For dilute solutions we use Flor -Hu ins theor with

1 to represent heat of mixingMH

1 2

1 1 2 2 1 1 2

so ( ln ln )M

v

G kT N v N v N v

2

1 2 2 1 2

1ln(1 ) 1G RT v v v

and the osmotic pressure is:

2 2 1 2

1 1

n(1 ) 1v v vv V x


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By separating out the second virial coefficient, we can

n e ory - empera ure:2

22 1

1vA

1

1 2

1

At T when , 02

A

A To relate 1 to solubility parameter,

21V 1 1 2

and is related to the -temperature by

RT

5 3 1/ 2

12 1mC M

T


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Example:


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-

divided by total number of molecules

End-group analysis Specific functionalities at the ends ofmolecules measured from titration etc,

Usually for low MW because of poorsensitivity at high MW.


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2 1T RT

Properties thatdepends on number of

0

2

mc v nc H M

molecules and nottheir chemistry

0lim

f

c

f n

c H M

,

depression, vaporpressure lowering,

1 12

1

o

o

P PX

P

osmo c pressure0

limc

n

RT

c M


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Osmotic ressure exists because of the difference

between chemical potentials of the pure solvent andthe solvent in a solutionc

2

rom ,

1

nM

At Flor -tem erature, = 0, so osmotic ressure is

2 3 ...n

c M

independent of concentration, one concentration cangiveMn


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-

light-scattering (or X-ray and small-angle neutronscattering) Scattering occurs when radiation hits a particle with

size close to the radiation wavelength ue o uc ua ons n re rac ve n ex, g sscattered, and measurements of the intensities atdifferent angles give the size

2 2 2 2

1 0

4

2 ( / )1where ( ) and

I w n dn dcHcR H

R RT c I V N


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The basic equation is

For very small particles, P() = 1, is the scattering form2

2( ) ( ) ( )w

A cR R solvent M P

factor

0

3

( ) where turbidity fromx

R I I e

For particles > 0.05 times wavelength, P() 1, is thesingle chain form factor. At very small angles,

2 2 2 24 42( ) 1 exp( ) 1 ...3g

g g

g

R KP R K R K R K

14where sin2

K


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Limiting to zero angle and zero concentration

2

0

1

2 ...( ) w

c

H A cR M

2 21

0

41 11 sin ...

( ) 3 ' 2g

wc

cH R

R M

Summing both equations gives the Zimm plot

PlottingH[c/R()] vs sin2 (/2), we get

22

2

1

3( ') (initial slope)

16 (intercept)g

R


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The radius of gyration must be determined when

So the range of angles are limited

1gK R

g ,usually between 45 to 135

The radius of gyration is related to the end-to-enddistance, r, by

22

g

rR

For random coils, rdepends on MW, r2 = CM, and

r n l


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D namic li ht scatterin measures fluctuation of

scattering intensity with time at very small volumes forvery short times

g ves mo ecu ar mo on an us on coe c en

Used to study aggregation, adsorption, structural

chan es chain d namics

2 2

0( , ) (1 )(1 )app z g DcD K c D CR K k


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expensive Fast and chea methods measures relative MW

such as by intrinsic viscosity, and using Gel

Permeation Chromatography Intrinsic viscosity

Calculates the viscosity-average MW

1/1

, where a between 0.5 and 0.8aa

i iiv

i i

N MM

N M


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Dissolvin ol mers in solution increases solution

viscosity Define:

0solvent viscosity =

relative viscosity,

0

specific viscosity, 1sp rel

0 0

nintrinsic viscosity, sp rel

c cc c

ninherent viscosity =

rel

c


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For a dilute dis ersion of uniform ri id non-interactin

spheres, 0 2(1 2.5 )v

Assuming the polymer molecule is an equivalentsphere of radiusRe, then

0 2

0

2.5sp en

VV

2 1/ 2 301

0

42.53

sp eA

c

RN Mc M


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An em irical correlation for MW and intrinsic viscosit

the Mark-Houwink-Sakurada equation:K a


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Theoreticall the constant a can be used to determine

the shape of the molecule (Table 3.11) The MHS equation can also be expressed as

3/ 2

21/ 2 3 1/ 2 30K

rM M

In a Flory -solvent or a good solvent,

3/


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Ex erimentall a series of dilute ol mer solutions is

prepared and their viscosity measured (usingUbbelhode viscometer usually),

en ca cu a e rom nown an a

2

'sp

k cc

2ln''rel k c

c


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Chromatography(GPC) Separation based on

size-exclusion

so ca e zeExclusionChromatography

(SEC) Related to HPLC


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Stationary phase usually made of cross-linkedpolymers with pores

Mobile phase usually solvent that can dissolve thepolymers of interest

,time is the length of time a particular fraction remainsin the column

Distribution of analyte between mobile and stationaryphase


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The distribution coefficient is related to Gibbs freeenergy:R i

d

V VK

V

exp

d

d

S HK

GPC HPLCbut usually dK K K


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Smaller particles pass through the column slower Larger particles pass through the column faster

Detection usually by concentration (refractive index or

UV absorption) or by molar mass (light-scattering,

Calibration is required, usually with polymer standardsof PS, PMMA, PEO etc

But there is error when using non-similar polymers(Rg2/Mnot similar)


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So use a universal calibration From MHS equation, a polymer hydrodynamic volume

is proportional to MW:

3/ 22 3

0M r o ns ea o p o ng vs e u on vo ume, we p oVHvs elution volume

,

but what will it not work for?


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Polymer Physics: Finding Molecular Stats

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