Molecular Weight and Branching "Drop the idea of large molecules.Organic molecules with a molecularweight higher than 5000 do not exist."—Advice given to Hermann Staudinger

Mn = ∑ Nx Mx

∑ Nx

Mw = ∑ Nx Mx

2

∑ Nx Mx

30000020000010000000

2

4

6

8

10

12

Mol

es

Molecular Weight

Mw ≅ 118, 200Mn ≅ 108, 500

Definitions

Number Average Weight Average

End group analysisLowering of vapor pressure

Ebulliometry (elevation of boiling point)Cryoscopy (depression of freezing point)

Osmometry (osmotic pressure)

Methods for the Determination ofMolecular Weight

Number Average—Absolute methods

Weight Average—Absolute methods

Light scatteringNeutron scatteringUltracentrifugation

Relative methods

Solution ViscometrySize Exclusion Chromatography

Osmotic Pressure

h

PureSolvent

PolymerSolution

Cap

Piston

Pressure = π

Membrane

[A] [B] [C]

Schematic diagram of the osmotic pressure experiment.

Osmotic pressure—belongs to a family of techniques that come under the heading ofcolligative property measurements.

Osmotic Pressure Analogy to Ideal Gases and Virial Equations

1086420

1.01

1.00

0.95

0.97

0.98

0.96

0.99

P (atm)

PVNRT

Ideal gas

N2

CH4

C2H4

Schematic plots of PV/NRT versus P.

The Ideal Gas Law

PVNRT = 1or

The Idea of Virial Equations

PVNRT = 1 + B'P + C'P

2 + D'P

3 + --------

The coefficients B', C', D', etc., are the second,third, fourth, etc., virial coefficients.

Relationship to Molecular Weight

P VN = RT NV = # moles

volume = wM 1

V = cMand Hence: P

c = RTM

Osmotic Pressure

πc = RT

Mn + Bc + Cc

2 + Dc

3 + ------

πc = RT

MIdeal Solution

Not So IdealSolution

c x 103 g cm

- 3

πc x 10

- 5

cm2 sec

- 2

10864200.0

0.5

1.0

1.5

2.0

0.75

1210864200.0

0.4

0.8

1.2

1.6

0.21

32100.2

0.3

0.4

0.224πc x 10

- 3

cm

c x 102 g cm

- 3

Graph of π/c versus c for polyisobutylene in chlorobenzene.

Graph of π/c versus c for polystyrene in toluene.

Osmotic Pressure

Derivation of a Virial Equation from the Flory–Huggins Equation

µs - µs0

RT = ln Φs + (1 - 1Mn

)Φp + Φp

2 χ

π = - RTVs

ln Φs + Φ

p (1 - 1

Mn) + Φ

p

2 χ

ln Φs = ln (1 - Φp) = - Φp - Φp

2

2 - Φp

3

3 - -------

π = RTVs

Φp

Mn + Φp

2 (1

2 - χ) + Φp

3

3 + ------

Osmotic Pressure can be related to the chemical potential via the Flory—Huggins equation:

and:

Expanding the Ln term:

Leads to:

This has the same form as the virial equation, but uses the concentration variable Φ instead of c. However, we must be careful because the Flory-Huggins theory does not strictly apply to dilute solutions.

p

Light Scattering Looks fiendishly difficult because of all the equations,but the Crucial Point is that we end up with aVirial Equation similar to that used Osmometry

1.21.00.80.60.40.20.01

2

3

4

5

100 c + sin2( θ

2 )

θ = 0 line

c = 0 line

K(1+cos2θ)cRθ

K (1 + cos2θ) cR

θ = 1

Mw (1 + 2Γ

2c + ---)(1+ S sin

2(θ2 ))

Zimm PlotExperimentallymeasured parameters

Note:Weight AverageMolecularWeight

Dependence uponangle of observation

VirialExpansion

Double extrapolation

c → 0

θ → 0

Measuring the Viscosity of Polymer Solutions

Most common method used to determine the viscosity of a polymer solution is to measure the time taken to flowbetween fixed marks in a capillary tube under the draining effect of gravity. The (volume) rate of flow, υ, is thenrelated to the viscosity by Poiseuille's equation:

υ = π P r4

8 η l

where P is the pressure difference maintaining the flow, r and l are the radius and length of the capillaryand η is the viscosity of the liquid.

Relative Viscosity

Defined as the viscosity of a polymer solution divided by that of the pure solvent and for dilute solutions:

where t is the time taken for a volume V of solution (no subscript) or solvent (subscript 0) to flow between the marks.

ηrel

= ηη

0 ≈ t

t0

543210

1.0

1.2

1.4

1.6

ηrel

Concentration g/100 cc

Plot of versus c for PMMA in chloroform.Plotted from the data of G. V. Schultz and F. Blaschke.

ηrel

Relative Viscosity as aFunction of Concentration

A power series, similar to that used in the treatment ofosmotic pressure and light scattering data,is commonly used to fit relative viscosity data:

ηrel

= ηη

0 = 1 + [η] c + k c 2 + .. .. ..

Both [η] and k are constants.

[η] is called the intrinsic viscosity

(ηrel

- 1c ) = 1

c (η - η0

η0

) = [η] + k c

If viscosity measurements are confined to dilute solution, so that we can neglect terms in c and higher:

Note also that as c goes to zero (infinite dilution), then the intercept on the y-axis of a plot of ( ) against cis the intrinsic viscosity, [η]:

3

[η] = (ηsp

c )c → 0

ηsp/c

The Specific V iscosity is defined as: ηsp = ηrel

- 1

Measuring the Intrinsic Viscosity

∇

◊

∇

∇∇

∇

◊◊

◊◊

0 Concentration, c (g/dl)

ln ηrel

c

ηsp

c

0.25

[η]η sp c o

r ln

ηr e

l

c

Schematic diagram illustrating thegraphical determination of the intrinsic viscosity.

∇ ∇

∇

∇

◊◊ ◊◊

∇

◊

0 Concentration, c (g/dl) 0.25

η sp c o

r ln

ηr e

l

c

ln ηrel

c

ηsp

c

[η]

Schematic diagram illustrating the effectof strong intermolecular interactions.

Most "extrapolation to zero concentration" procedures have a serious limitation.Where one would like to perform measurements is at the lowest concentrations possible,

but this is generally where the greatest error in measurement occurs.

In practice, we use two semi-empiricalequations suggested by Huggins and Kraemer

ηsp

c = [η] + k' [η]2 c

ln ηrel

c = [η] + k"[η]2c

The Mark-Houwink-Sakurada Equation

76540

1

2

3

Log Molecular Weight

0

1

2

3

7654

PS PMMA

Log [η]

Plots of the log [η] versus log M for PS and PMMA.Replotted from the data of Z. Grubisic, P. Rempp and H. Benoit

log M

∇∇

∇∇

Slope = a

Intercept = log K

log [η]Solvent

Temp

∇∇ Monodisperse

standards

Schematic diagram of the Determinationof the Mark-Houwink-Sakurada constants

K and "a".

The Relationship BetweenIntrinsic Viscosity and Molecular Weight

If the log of the intrinsic viscosities of a range of samples is plotted against the log of their molecular weights,

then linear plots are obtained that obey equation:

[η] = KMa

Note that K and "a" are not universal constants,but vary with the nature of the polymer,

the solvent and the temperature.

The Viscosity Average Molecular Weight

For Osmotic Pressure and Light Scattering we saw that there is a clear relationship betweenexperimental measurement and the number and weight molecular weight average, respectively.

Viscosity measurements are related to molecular weight by a semi-empirical relationship anda new average, the Viscosity Average for polydisperse polymer samples is defined.

Mv = ∑i

Ni M

i

(a + 1 )

∑i

Ni M

i

1a

In very dilute solutions

Now: Hence: ηsp = K ∑i

Mi

a c

i

(ηsp)i

ci

= K Mi

a

ηsp = ∑i

(ηsp)i

[η] = ηspc =

K ∑i

Mi

a c

i

cAnd:

By substitution and rearranging we obtain:

Note that the Viscosity Molecular Weight is Not an Absolute Measure as it is

a function of the solvent through the Mark-Houwink parameter "a".

Size Exclusion (or Gel Permeation) Chromatography

Pump

MixingValve

Solvent Reservoir

Detectors

SEC Columns

Injection Port

Schematic diagram of an SEC instrument.

BulkMovementofSolvent

LargeExcludedMolecules

SmallPermeatingMolecules

VoidVolume

Pores

Schematic diagram depicting the separation ofmolecules of different size by SEC.

For a given volume of solvent flow, molecules of different size travel different pathlengths within the column.The smaller ones travel greater distances than the larger molecules due to permeation into the molecular maze.Hence, the large molecules are eluted first from the column, followed by smaller and smaller molecules.

The Calculation of Molecular Weight by SEC

Polydisperse Sample

Monodisperse Standards

LogMolecularWeight

Mi

Elution Volume

Vi

SelectivePermeation

∇∇

∇∇

TotalPermeation

Exclusion

Intensity

Schematic diagram depicting the calibration of an SEC instrument.

Elution Volume

Con

cent

rati

on

Area normalized

Vi

wi

Σ w i = 1

The Simplest Case where Monodispersed Standards of the Polymer are Available

Mn = 1

∑ wi

Mi

Mw = ∑ wiM

i

[η] = ∑ wi [η

i] = K ∑ w

i M

i

a

wi =

hi

∑ hi

How Does SEC Separate Molecules ?

Benoit and his coworkersrecognized that SEC separates noton the basis of molecular weight

but rather on the basisof hydrodynamic volume of the polymer molecule in solution.

Elution Volume

LogMolecularWeight

Linear PolyA

Calibration Curves for:Linear PolyB

Star-shaped PolyA

M

Same solventSame temperature

VLA VS

A VLB

If the molecular weight of monodisperse polystyrenes of different molecular architecture(e.g., linear, star-shaped,comb-like, etc.) are plotted against elution volume they do not fall on a single calibration curve.

In other words, if we had three monodisperse polystyrenes, one linear, one star -shaped and one comb-like,all with the same molecular weight, they would not come off the column at the same time (elution volume).

Similarly, different monodisperse polymers of the same molecular weight generally elute at different times.Thus, for example, monodisperse samples of polystyrene and PMMA having the same molecular weightmight come off the column at different times.

In effect, this means we would require different calibration curves for different polymers and even the sametype of polymer if the architecture is different.

302826242220185

6

7

8

9

10

linear polystyrenelinear PMMA loglinear PVCpolystyrene combs

polystyrene starsPS/PMMA graft copolymers (comb)PS/PMMA heterograft copolymers

Poly(phenyl siloxane) ladder polymersPS/PMMA statistical copolymers

Elution Volume

Log [η] M

A universal calibration plot of log [η]M vs elution volume for various polymers.Redrawn from the data of Z. Grubisic, P. Rempp and H. Benoit.

The Universal Calibration CurveIf we model the properties of the polymer coilin terms of an equivalent hydrodynamic sphere,then the intrinsic viscosity, [η], is related to thehydrodynamic volume V via the equation:h

[η] = 2. 5 A V

hM

A is Avogadro's number and M is the molecular weight.

Benoit and his coworkers recognized that the productof intrinsic viscosity and molecular weight was directly

proportional to hydrodynamic volume.

Mi =

Ji

KPS

1 / ( 1 + aPS

)

log J = log [η] M

Elution Volume

Ji

Linear PolystyreneStandards

Vi

Schematic of a universal calibration plotprepared from linear PS standards.

The Calculation of Molecular Weight by SECThe Universal Calibration Method

Ji = [η]

i M

iDefine:

A universal calibration curve is preparedusing e.g. monodisperse polystyrene standards

Ji = [η]

i M

i = K

PS M

i

( 1 + aPS

)

Note: we can simply calculate the intrinsic viscosityif we can obtain the Mark-Houwink-Sakurada constants,K and a , from the literature for polystyrene in thesame solvent at the same temperature as the SEC experiment.

PS PS

Rearranging:

Mn = 1

∑ wi

Mi

Mw = ∑ wiM

i

Important result because it relates the molecular weightof the ith species to the hydrodynamic volume of that species

Let us assume that the SEC data was obtained from apolydisperse sample of PMMA on an SEC instrument usingthe same solvent and temperature that was used to preparethe universal calibration curve from PS standards.

If we have K and "a" for PMMA in the same solvent andtemperature then the "true" molecular weights for thepolydisperse PMMA may be calculated from:

Mi =

Ji

KPMMA

1 / ( 1 + aPMMA

)

And:

[η] = ∑ wi [η

i] = ∑ w

i ( J

i

Mi)

Long Chain Branching

Comb-type Random Star-shaped

Schematic representation of different long chain branches.

Long chain branching can have a major effect upon the rheological and solution properties of polymers.

Difficult to quantitatively determine the amount of long chain branching using conventional analyticaltechniques, such as NMR or vibrational spectroscopy.

Very low concentration of any species that can be attributed to the presence of a long chain branch.

BBBBB

----

-

----AAAAAAAAAAAAAAAAA----

Long Chain Branching and Mean Square Dimensions

The introduction of only one or two long chain branch points leads to a significant decrease in themean-square dimensions of macromolecules compared to linear molecules of the same molecular weight.

This statement may be expressed in terms of the ratio of the respective radii of gyration, g.

B. H. Zimm and W. H. Stockmayer, J. Chem. Phys., 17, 1301 (1949).B. H. Zimm and R. W. Kilb, J. Polym. Sci., 37, 19 (1959).

g = < S 2 >

b

< S 2 >l

( for the same molecular weight)

Subscripts b and l denote branched and linear moleculesg is a function of the number and type of long chain branch points in the molecule.

g = 3f - 2

f2

For Star Shaped Polymers

(Functionality f and Equal Arm Length)

For Randomly Branched Monodisperse Polymers

(Tetrafunctional)

g4 = (1 +

mn6

)1 /2

+ 4 mn3 π

- 1/ 2

mn is the number average number of branch points per molecule

Long Chain Branching—Relation to Intrinsic Viscosity

g, the ratio of the radii of gyration of the branched to linear polymer chain of the same molecular weight, is related to the intrinsic viscosity by a branching function, g' :

g' = [η]

b[η]

l (f or the same molecular weight)

g' = g 0 .5

For Randomly Branched Monodisperse Polymers

(Empirical Relationship-Kurata et al.)

For Star Shaped Polymers

(Theoretical Relationship)

g' = g 0 .6

The experimentally determined intrinsic viscosity of a branched polymer will be less than that calculated from theSEC data using the universal calibration curve (which assumes that the polymer chains are perfectly linear).

An appropriate branching function, g'(λ, M) that contains a branching parameter, λ, is defined such that:

[η]b

= g'(λ, M) [η]l = g'(λ, M) KM

a

Elution Volume

Con

cent

ratio

n

Area normalized

Vi

wi

Σ w i = 1

Lets say this SEC is from a randomly tetrafunctionallybranched polydisperse polychloroprene (PC)

Using the universal calibration curve calculate the [η]assuming the polychloroprene is linear

log J = log [η] M

Elution Volume

Ji

Linear PolystyreneStandards

Vi

[η] = ∑ wi [η

i] = ∑ w

i ( J

i

Mi)

Mi =

Ji

KPC

1 / ( 1 + aPC

)

If K and "a" are known for linear PC, then:

And the theoretical [η] for the polydispersePC assuming it is linear is given by:

The experimentally measured [η] will be less thanthat calculated for a distribution of linear PC chainsThis is the key to a measure of Long Chain Branching

SEC and the Determinationof Long Chain Branching – I

∇

◊

∇

∇∇

∇

◊◊

◊◊

0 Concentration, c (g/dl)

ln ηrel

c

ηsp

c

0.25

[η]η sp c o

r ln

ηr e

l

c

Experimentally determine [η] for the poydisperse PC

SEC and the Determinationof Long Chain Branching – II

Elution Volume

Con

cent

ratio

n

Area normalized

Vi

wi

Σ w i = 1

The same SEC is from a randomly tetrafunctionallybranched polydisperse polychloroprene (PC)

Now use the universal calibration curve calculate [η]assuming a value of the branching parameter , λ

log J = log [η] M

Elution Volume

Ji

Linear PolystyreneStandards

Vi

Ji = K M

i

(1 + a) (1 +

λMi

6 )0 .5

+ 4 λM

i

3 π

- 0. 3

M now has to be determined iteratively from:i

And the theoretical [η] for the polydisperse PCassuming it is randomly branched

with a given value of λ is given by:

[η]b = K ∑

i

wi

Mi

a (1 +

λΜ ι

6 )0 .5

+ 4 λΜ ι

3 π

- 0. 3

Mn = 1

∑ wi

Mi

Mw = ∑ wiM

i

This theoretical value of [η] is compared to theexperimental [η]. The whole procedure is repeated

with different values of λ until:

[η]calculated

= [η]observed

Then:

CH2 C

CH

CH2

Cl

CH2

CH2 C

CH

CH2

Cl

CH2

chloroprene

Tetrafunctional Branch Point

Long Chain Branching in Polychloroprene–Experimental

1008060402000.0

0.1

0.2

0.3

0.4

0.5

0.6

% Conversion

Bra

nchi

ng P

a ram

e te r

, λ x

105

The branching parameter, λ, for polychloroprene samplesisolated as a function of conversion.

1008060402000

1

2

3

4

5

6

7

8

% Conversion

Mw

Mn

Mol

e cul

a r W

eigh

t , M

x 1

0-5

Calculated molecular weight averages for polychloroprenesamples isolated as a function of conversion.