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093-

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CONTENTS

MOLECULAR WEIGHTS OF POLYMERS

INTRINSIC VISCOSITY

GEL PERMEATION CHROMATOGRAPHY

MASS SPECTROMETRY

INSTRUMENTATION FOR MOLECULARWEIGHT DETERMINATION

SOLUTION THERMODYNAMICS ANDMOLECULAR WEIGHTS

3.7

3.8

3.9

3.10

3.11

3.12

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3.7 MOLECULAR WEIGHTS OF POLYMERS

3.7.1 Molecular Weight of CommercialPolymers

Commercial use30,000-1,000,000 g/mol

High enoughmolecular weights toobtain good physical

properties

Low enough molecularweights to permit

reasonable processing

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3.7.2 Thermodynamics and

Kinetics of Polymerization

Thermodynamics

Kinetics

The molecularweights and

the

polydispersityindex

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AFFECT

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3.7.2.1 Thermodynamics of Chain

PolymerizationKRTG ln0

Consider a chain polymerization of monomerM:

1nn MMM

The rate constant of theforward reaction, polymerization, is kp

The rate constant of the reverse reaction, depolymerization, is given bykdp

][

1

]][[

][ 1

MMM

M

k

kK

n

n

dp

p

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Ceiling temperaturestemperatures above which themonomer cannot be polymerized,but the polymer willspontaneously depolymerize back to the monomer.

Commercially this fact leads to an important method

of polymer recycling whereby scrapped polymer isheated under anaerobicconditions to allow distillingoff the resultant monomers.

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)()( mpmp SSTHHSTHG

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3.7.2.2 Kinetics of Chain Polymerization

RMMR

RI

k

ki

2

2Initiation

Propagation:

1nn

2

RMMRM

RMMRM

p

p

k

k

Termination bycombination RRMRMRMk

mnmntc

Termination bydisproportionation mnmn

td RMRMRMRMk

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Rp represents the rate of polymerization

2/1

2/1

]][[ IMk

kkR

t

ipp

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Kinetic chain length: Radical chain polymerization is the average

number of monomer molecules consumed for

each radical initiating a chain.Thus, at steadystate:

2/1])[(2

][

Ikfk

Mk

R

R

R

R

vti

p

t

p

i

p

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3.7.2.3 Thermodynamics of Step Polymerization

Polyesterification where carboxyl groups andhydroxyl groups react to form a polyester and water;

The equilibrium constant, K:

OHCOOOHCOOH K 2

]][[

]][[ 2

OHCOOH

OHCOOK

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p:fractional conversion

number-average degreeof polymerization

weight-average degreeof polymerization

Polydipersity Index

pDPn

1

1

p

pDPw

1

1

pDP

DPPDI

n

w 1

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3.7.2.4 Kinetics of Step

Polymerizations

Step

polymerizations

Self-catalyzed

External-catalyzed

Catalyzed bysome externallyadded chemical

such as an acid

StepPolymerizations

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Self-catalyzed polymerization

][][][ 2 OHCOOHk

dt

COOHd

OHCOOOHCOOH COOH 2

])[][(3

OHCOOHckc

dt

dc

pDPpcc n

11),1(0

12

)1(

1 202

2

ktc

p

DPn

tc

ckdt

c

dc

030

ktcc

211

22

0

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3.7.3 Molecular Weight Distributions

Chain polymerization Termination by disproportionation:PDI=2

Termination by combination:PDI=1.5

Stepwise polymerizations, such as polyesterformation,PDI=1+p=2

Anionic polymerizations :narrow distribution, sometimesPDI

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The various molecular weight distributions have beenmodeled. Two of the most important are the Schultzdistribution and the Poisson distribution.

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3.7.4 Gelation and Network

Formation

Linear:

functionality of the

monomer is 2

Branched or cross-linked:

trifunctional, tetrafunctional,or higher

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3.7.4 Gelation and Network

Formation gelation point:

2/1)1(

1

fPc

f :the functionality of the branch

units

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3.8 INTRINSIC VISCOSITY Both the colligative and the scattering methods result

in absolute molecular weights; that is, the molecularweight can be calculated directly from first principlesbased on theory.

Defects : SlowExpensive;

In order to handle large numbers of samples,

especially on a routine basis, rapid, inexpensivemethods are required. This need is fulfilled byintrinsic viscosity and by gel permeationchromatography.

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Flow rate & Shear rate

Frictional drag androtational forces

Viscosity increase bythe polymer in the

solution.

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Mechanism ofviscosityincrease:

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3.8.1 Definition of Terms Solvent viscosity:

Polymer solution viscosity:

Relative viscosity:

Specific viscosity:

0

0

rel

1

-

rel0

0

sp

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Intrinsic viscosity:0

][

c

sp

c

][ln

2

ln

2)1ln(ln

0

2

2

2

c

rel

spsprel

sp

spsprel

c

c

c

cc

For dilute solutions:

Inherent viscosity:

c

relln

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Two sets of units are in use for [].

The American units are 100 cm3/g,

The European units are cm3/g.

Of course, this results in a factor of 100 difference inthe numerical result. Lately, the European units arebecoming preferred.

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3.8.2 The Equivalent Sphere

Model In assuming a dilute dispersion of uniform, rigid,

noninteracting spheres,

Einstein derived an

equation expressing

the increase in viscosity

of the dispersion

the quantityv2 represents

the volume fraction of spheres

20

v5.21

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In shear flow, it exhibits a frictional

coefficient off0. Then according to Stokeslaw

Re:a hydrodynamic sphere of equivalent radius

The Einstein viscosity relationship for

spheres may be written:

eRf 010 6

esp VV

n

20 5.2

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M

cN

V

n A2

13

1

31

e

0

5.2 gcmmolg

cmmol

M

VN

c

A

c

sp

2/1

2/32

e1

3

e1e

3

4

3

4 MM

R

M

R

M

V

Note

that:

32/12/3

0A

1

3

45.2 M

M

RN e

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is the expansion of the coil in a good

solvent over that of a Flory -solvent

0ee

RR

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3.8.3 The MarkHouwink

Sakurada Relationship In the late 1930s and 1940s Mark, Houwink, and

Sakurada arrived at an empirical relationship betweenthe molecular weight and the intrinsic viscosity:

a

vKM][

K and a are constants for a particular polymersolvent pair at a particular temperature.

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More generally, it should be pointed out that avaries from0 to 2; see Table 3.11.

0.5(Flory -solvent )

0.8(thermodynamicall

y good solvent)

a

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3.8.4 Intrinsic Viscosity

Experiments In most experiments, dilute solutions of about 1%

polymer are made up.

The quantityrel should be about 1.6 for the highestconcentration used.

The most frequently used instrument is theUbbelhode viscometer, which equalizes the pressure

above and below the capillary.

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Huggins equation

Kraemer equation

Algebraically:

If either of these requirements is not met, molecularaggregation, ionic effects,or other problems may beindicated.

ckc

sp 2]['][

ckcrel 2][''][

ln

5.0''' kk

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While use of the viscosity-average molecular weight ofa polymer in calibrating K and a in equation,MVvalues usually are not initially known.

The calibration problem may be alleviated by one or

more of the following methods: 1. Use of fractionated polymers.

2. Use of polymers prepared with narrow molecularweight distributions,such as via anionic polymerization.

3. Use of weight-average molecular weight of thepolymer, since it is closer than the number-averagemolecular weight.

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3.8.5 Example Calculation

Involving Intrinsic ViscosityWe dissolve 0.10 g of the polymer in 100 ml of

butanone and measure the flow times at 25in anUbbelhode capillary viscometer. The results are

sample time

Pure butanone 110 s

0.10% Polystyrene solution 140 s

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3.9 GEL PERMEATION

CHROMATOGRAPHY Gel permeation chromatography(GPC), sometimes

called size exclusion chromatography(SEC), makesuse of the size exclusion principle.

The size of the molecule, defined by its hydrodynamicradius, can or cannot enter small pores in a bed ofcross-linked polymer particles, the most commonform of the stationary phase.

The smaller molecules diffuse in and out of the poresvia Brownian motion (see Figure 3.16) and are delayed.

The larger molecules pass by and continue in themobile phase.

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The stationary phase consists of small, porous particles.While the mobilephase f lows at a specified rate controlled by the solvent delivery system, thesample is injected into the mobile phase and enters the columns.

The length of time that a particular fraction remains in the columns is calledthe retention time .

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As the mobile phasepasses the porousparticles, theseparation betweenthe smaller and thelarger moleculesbecomes greater(see Figure 3.18) .

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3.9.1 Theory of Gel Permeation

Chromatography

molecularweight

distribution(MWD)

Mw/Mnchemical

composition

Statistical

Alternating

Block Graft

functionality End groupsshape ofthe chain

Random coils

Rod shaped Rings

Most polymersexhibit some

form(s) ofheterogeneity

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3.9.2 Utilization of Distribution

Coefficients in GPC and HPLC

phasestationarytheofvolume

columntheofvolumeialinterestitsolutetheofvolumeretention

dK

V

VVK iRd

phasemobilein theionconcentratanalyte

phasestationarythetoattachedorinionconcentratanalytedK

Kd : the distribution coefficient

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the quantityKd is related to the Gibbs free energy, G,

After rearranging, we have

S: The limited dimensions of the pores relative to the sizeof the polymer chains causes S of the polymer chains to

decrease. Interactions between the pore walls and the polymer

chains are expressed in changes in H, and are negative ifthe polymer and the wall are attracted to each other.

GSTHKRT d ln

RT

H

R

S

Kd exp

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In the general case, Kdmay be expressed

the subscripts GPC and HPLC indicate quantities

involving only entropic or enthalpic interactions,respectively.

In the ideal GPC case, KHPLC equals unity, and Kd=KGPC.

HPLCGPCd KKK

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3.9.3 Types of Chromatography

GPC, which uses porous particles toseparate molecules of different sizes.

HPLC, by contrast, utilizes interactionsbetween the polymers and the surfaceof the particles composing thestationary phase.

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Instruments today are sometimes called universalHPLCs, containing both GPC and HPLC columns and

measurement capability; see Table 3.12 forspecifications.

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3.9.4 GPC Instrumentation The most important parts of the instruments are :

the columntypes for themolecular

weight rangeof analysis

the detectorsystem

the pumps formaintainingconstant,

pulseless ratesof flow

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3.9.5 Calibration

Noting GPC is a relative molecular weight method,such instrumentation needs to be calibrated. Narrowmolecular weight distribution, anionically synthesized

polystyrenes are used most often for the purpose.

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Figure 3.20 The molecular weight and molecular weightdistribution are determined with standards precalibrated

via an absolute method such as light-scattering.2013/9/20 46

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3.9.6 Selected Current Research

Problems In the case of copolymers, any single detection method

will have variable sensitivity for each type of mer.

If the copolymer composition is itself a variable, thenthe use of dual or even multiple detectors will berequired for accurate results.

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The molecular weight of each component in a polymerblend may also be determined.

In a model experiment, poly(methyl methacrylate),PMMA, molecular weights were estimated in the

presence of polystyrene, PS.

Anionically polymerized polystyrene and free radicallypolymerized poly(methyl methacrylate) weredissolved in tetrahydrofuran in a 50/50 w/w mix.

A dual detector GPC was used, equipped withrefractive index (RI) and ultraviolet (UV) detectors.

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Figure 3.22 Analysis of the molecular weights of the polymers in apolymer blend of polystyrene and poly(methyl methacrylate).This method requires a dual detector GPC system.

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Current research problems include chain geometryand solution aggregation , ABC triblock copolymers astopological isomers , and hyperbranched polymers

(85); see Section 14.5.

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3.9.7 The Universal Calibration

32/320][ rM

Figure 3.23 The universal calibration curves for polystyrene and poly(vinylacetate) (94). The number 5 in thex-axis units means that the scale is insiphon counts of 5 cm3, so that the x-ordinate 30 corresponds to anelution volume of 150 cm3. (R. Dietz, private communication,November1984.)Mr is the peak GPC molecular weight, usually the unknown, Mrvalues are close to the geometric mean ofMn and Mw.2013/9/20 51

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The right-hand side is proportional to the polymershydrodynamic volume.

This is the universal calibration, which calls for a plotof []M versus elutionvolume.

It cannot be used for highly branchedmaterials or

polyelectrolytes, which have different or varyinghydrodynamic volume relationships.

3

2/3

20][ rM

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3.9.8 Properties of Cyclic Polymers

The properties of the linear and cyclic polymers arecompared in Figure 3.24 .

1. The more compact cyclic polymers possessed smaller

hydrodynamic volumes (i.e., they eluted later via GPCin Figure3.24A).

2. They had lower intrinsic viscosities than their linearanalogs, with []cyclic/[]linear = 0.4 (Figure 3.24B).

3. The root-mean-square radius of gyration, Rg, wasmeasured using GPCcoupled to a multiangle light-scattering detector. Figure 3.24C.

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3.10 MASS SPECTROMETRY

Mass spectrometry is the study of the mass, ormolecular weight, of ions created via ionization orfragmentation and determined electrically in the gas

phase.

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In the study of polymers, mass spectrometry has twobroad applications:

1. To characterize functionality. Unknownpolymers, residual volatile chemicals, andadditives can be identified.

2. To provide a new basis for the determination

of absolute molecular weights.

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3 10 1 High Molecular Weight

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3.10.1 High Molecular Weight

Studies

The spectrum shows three different charge statescentered at about m/z of 17,000, 32,000, and 65,000g/mol.

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3.10.2 Advances Using MALDI

Techniques

Matrix-assisted laser desorption ionization (MALDI)

Typical MALDI matrices are aromatic organic acids. Asmall aliquot of the mixture is applied to the MALDItarget and solidifies as the solvent evaporates.

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3.10.2.1 Small Sample Size

Very tiny amounts of polymer are utilized.

For example, an analyte solution contains 5 mg/ml.Approximately 0.30.5 mLof this solution are placed on

the target.With these very small samples,an electrospray

technique was found to give more reproducible results

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3.10.2.2 Oligomer and Telomer-

Type Studies

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3.10.2.3 Calibration of Results The molecular weights reported by the laboratories gave

values slightly lower than the measurements made bythe classical methods.

However, there was evidence that the MALDI methodwas probably the more accurate.

For example, traces of dust raise the apparent molecularweight obtained via lightscattering.

Overall, however, good agreement was obtained.

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3.11 INSTRUMENTATION FOR MOLECULAR

WEIGHT DETERMINATION

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3.12 SOLUTION THERMODYNAMICS

AND MOLECULAR WEIGHTS

Below the Flory -temperature, polymer solutions mayphase-separate.The higher the molecular weight is, thehigher the upper critical solution temperature.

At infinite molecular weight, the Flory -temperatureis reached. Thus the Flory -temperature is defined by

several different criteria:

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1. It is the temperature whereA2 is zero for dilutesolutions, and 1 = 1/2.

2. It is the temperature where the radius of gyration

approximates that of the bulk polymer (see Chapter 5).

3. It is the temperature at which an infinite molecularweight fraction would just precipitate (see Chapter 4).

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The methods are divided into absolute methods, whichdetermine the molecular weight from first principles,and

relative methods, which depend on prior calibration.The latterare usually selected because they are fast and inexpensive.Values obtained from the several methods are summarized inTable 3.15.

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